Revised5 Report on the Algorithmic Language
Scheme
RICHARD
H. ABELSON
N. I. ADAMS IV
D. H. BARTLEY
G. BROOKS

KELSEY, WILLIAM CLINGER, AND JONATHAN REES (Editors)
R. K. DYBVIG
C. T. HAYNES
G. J. ROZAS
D. P. FRIEDMAN
E. KOHLBECKER
G. L. STEELE JR.
R. HALSTEAD
D. OXLEY
G. J. SUSSMAN
C. HANSON
K. M. PITMAN
M. WAND
Dedicated to the Memory of Robert Hieb

20 February 1998

SUMMARY
The report gives a defining description of the programming language Scheme. Scheme is a statically scoped and
properly tail-recursive dialect of the Lisp programming
language invented by Guy Lewis Steele Jr. and Gerald
Jay Sussman. It was designed to have an exceptionally
clear and simple semantics and few different ways to form
expressions. A wide variety of programming paradigms, including imperative, functional, and message passing styles,
find convenient expression in Scheme.
The introduction offers a brief history of the language and
of the report.
The first three chapters present the fundamental ideas of
the language and describe the notational conventions used
for describing the language and for writing programs in the
language.
Chapters 4 and 5 describe the syntax and semantics of
expressions, programs, and definitions.
Chapter 6 describes Scheme’s built-in procedures, which
include all of the language’s data manipulation and input/output primitives.
Chapter 7 provides a formal syntax for Scheme written in
extended BNF, along with a formal denotational semantics.
An example of the use of the language follows the formal
syntax and semantics.
The report concludes with a list of references and an alphabetic index.

CONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . .
1 Overview of Scheme . . . . . . . . . . . . . . . .
1.1 Semantics . . . . . . . . . . . . . . . . . .
1.2 Syntax . . . . . . . . . . . . . . . . . . . .
1.3 Notation and terminology . . . . . . . . .
2 Lexical conventions . . . . . . . . . . . . . . . .
2.1 Identifiers . . . . . . . . . . . . . . . . . .
2.2 Whitespace and comments . . . . . . . . .
2.3 Other notations . . . . . . . . . . . . . . .
3 Basic concepts . . . . . . . . . . . . . . . . . . .
3.1 Variables, syntactic keywords, and regions
3.2 Disjointness of types . . . . . . . . . . . .
3.3 External representations . . . . . . . . . .
3.4 Storage model . . . . . . . . . . . . . . . .
3.5 Proper tail recursion . . . . . . . . . . . .
4 Expressions . . . . . . . . . . . . . . . . . . . .
4.1 Primitive expression types . . . . . . . . .
4.2 Derived expression types . . . . . . . . . .
4.3 Macros . . . . . . . . . . . . . . . . . . . .
5 Program structure . . . . . . . . . . . . . . . . .
5.1 Programs . . . . . . . . . . . . . . . . . .
5.2 Definitions . . . . . . . . . . . . . . . . . .
5.3 Syntax definitions . . . . . . . . . . . . .
6 Standard procedures . . . . . . . . . . . . . . .
6.1 Equivalence predicates . . . . . . . . . . .
6.2 Numbers . . . . . . . . . . . . . . . . . . .
6.3 Other data types . . . . . . . . . . . . . .
6.4 Control features . . . . . . . . . . . . . . .
6.5 Eval . . . . . . . . . . . . . . . . . . . . .
6.6 Input and output . . . . . . . . . . . . . .
7 Formal syntax and semantics . . . . . . . . . . .
7.1 Formal syntax . . . . . . . . . . . . . . . .
7.2 Formal semantics . . . . . . . . . . . . . .
7.3 Derived expression types . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . .
Additional material . . . . . . . . . . . . . . . . .
Example . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . .
Alphabetic index of definitions of concepts,
keywords, and procedures . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

2
3
3
3
3
5
5
5
5
6
6
6
6
7
7
8
8
10
13
16
16
16
17
17
17
19
25
31
35
35
38
38
40
43
45
45
45
46

. . . . . . .

48

2 Revised5 Scheme

INTRODUCTION
Programming languages should be designed not by piling
feature on top of feature, but by removing the weaknesses
and restrictions that make additional features appear necessary. Scheme demonstrates that a very small number of
rules for forming expressions, with no restrictions on how
they are composed, suffice to form a practical and efficient
programming language that is flexible enough to support
most of the major programming paradigms in use today.
Scheme was one of the first programming languages to incorporate first class procedures as in the lambda calculus,
thereby proving the usefulness of static scope rules and
block structure in a dynamically typed language. Scheme
was the first major dialect of Lisp to distinguish procedures
from lambda expressions and symbols, to use a single lexical environment for all variables, and to evaluate the operator position of a procedure call in the same way as an
operand position. By relying entirely on procedure calls
to express iteration, Scheme emphasized the fact that tailrecursive procedure calls are essentially goto’s that pass
arguments. Scheme was the first widely used programming language to embrace first class escape procedures,
from which all previously known sequential control structures can be synthesized. A subsequent version of Scheme
introduced the concept of exact and inexact numbers, an
extension of Common Lisp’s generic arithmetic. More recently, Scheme became the first programming language to
support hygienic macros, which permit the syntax of a
block-structured language to be extended in a consistent
and reliable manner.
Background
The first description of Scheme was written in 1975 [28]. A
revised report [25] appeared in 1978, which described the
evolution of the language as its MIT implementation was
upgraded to support an innovative compiler [26]. Three
distinct projects began in 1981 and 1982 to use variants
of Scheme for courses at MIT, Yale, and Indiana University [21, 17, 10]. An introductory computer science textbook using Scheme was published in 1984 [1].
As Scheme became more widespread, local dialects began to diverge until students and researchers occasionally found it difficult to understand code written at other
sites. Fifteen representatives of the major implementations
of Scheme therefore met in October 1984 to work toward
a better and more widely accepted standard for Scheme.
Their report [4] was published at MIT and Indiana University in the summer of 1985. Further revision took place
in the spring of 1986 [23], and in the spring of 1988 [6].
The present report reflects further revisions agreed upon
in a meeting at Xerox PARC in June 1992.
We intend this report to belong to the entire Scheme com-

munity, and so we grant permission to copy it in whole or in
part without fee. In particular, we encourage implementors
of Scheme to use this report as a starting point for manuals
and other documentation, modifying it as necessary.
Acknowledgements
We would like to thank the following people for their
help: Alan Bawden, Michael Blair, George Carrette, Andy
Cromarty, Pavel Curtis, Jeff Dalton, Olivier Danvy, Ken
Dickey, Bruce Duba, Marc Feeley, Andy Freeman, Richard
Gabriel, Yekta Gürsel, Ken Haase, Robert Hieb, Paul
Hudak, Morry Katz, Chris Lindblad, Mark Meyer, Jim
Miller, Jim Philbin, John Ramsdell, Mike Shaff, Jonathan
Shapiro, Julie Sussman, Perry Wagle, Daniel Weise, Henry
Wu, and Ozan Yigit. We thank Carol Fessenden, Daniel
Friedman, and Christopher Haynes for permission to use
text from the Scheme 311 version 4 reference manual.
We thank Texas Instruments, Inc. for permission to use
text from the TI Scheme Language Reference Manual [30].
We gladly acknowledge the influence of manuals for MIT
Scheme [17], T [22], Scheme 84 [11], Common Lisp [27],
and Algol 60 [18].
We also thank Betty Dexter for the extreme effort she put
into setting this report in TEX, and Donald Knuth for designing the program that caused her troubles.
The Artificial Intelligence Laboratory of the Massachusetts
Institute of Technology, the Computer Science Department
of Indiana University, the Computer and Information Sciences Department of the University of Oregon, and the
NEC Research Institute supported the preparation of this
report. Support for the MIT work was provided in part by
the Advanced Research Projects Agency of the Department
of Defense under Office of Naval Research contract N0001480-C-0505. Support for the Indiana University work was
provided by NSF grants NCS 83-04567 and NCS 83-03325.

1. Overview of Scheme 3

DESCRIPTION OF THE LANGUAGE
1.

Overview of Scheme

1.1. Semantics
This section gives an overview of Scheme’s semantics. A
detailed informal semantics is the subject of chapters 3
through 6. For reference purposes, section 7.2 provides a
formal semantics of Scheme.
Following Algol, Scheme is a statically scoped programming language. Each use of a variable is associated with a
lexically apparent binding of that variable.
Scheme has latent as opposed to manifest types. Types
are associated with values (also called objects) rather than
with variables. (Some authors refer to languages with
latent types as weakly typed or dynamically typed languages.) Other languages with latent types are APL,
Snobol, and other dialects of Lisp. Languages with manifest types (sometimes referred to as strongly typed or statically typed languages) include Algol 60, Pascal, and C.
All objects created in the course of a Scheme computation,
including procedures and continuations, have unlimited extent. No Scheme object is ever destroyed. The reason that
implementations of Scheme do not (usually!) run out of
storage is that they are permitted to reclaim the storage
occupied by an object if they can prove that the object
cannot possibly matter to any future computation. Other
languages in which most objects have unlimited extent include APL and other Lisp dialects.
Implementations of Scheme are required to be properly
tail-recursive. This allows the execution of an iterative
computation in constant space, even if the iterative computation is described by a syntactically recursive procedure.
Thus with a properly tail-recursive implementation, iteration can be expressed using the ordinary procedure-call
mechanics, so that special iteration constructs are useful
only as syntactic sugar. See section 3.5.
Scheme procedures are objects in their own right. Procedures can be created dynamically, stored in data structures, returned as results of procedures, and so on. Other
languages with these properties include Common Lisp and
ML.
One distinguishing feature of Scheme is that continuations,
which in most other languages only operate behind the
scenes, also have “first-class” status. Continuations are
useful for implementing a wide variety of advanced control
constructs, including non-local exits, backtracking, and
coroutines. See section 6.4.
Arguments to Scheme procedures are always passed by
value, which means that the actual argument expressions
are evaluated before the procedure gains control, whether
the procedure needs the result of the evaluation or not.

ML, C, and APL are three other languages that always
pass arguments by value. This is distinct from the lazyevaluation semantics of Haskell, or the call-by-name semantics of Algol 60, where an argument expression is not
evaluated unless its value is needed by the procedure.
Scheme’s model of arithmetic is designed to remain as independent as possible of the particular ways in which numbers are represented within a computer. In Scheme, every
integer is a rational number, every rational is a real, and
every real is a complex number. Thus the distinction between integer and real arithmetic, so important to many
programming languages, does not appear in Scheme. In its
place is a distinction between exact arithmetic, which corresponds to the mathematical ideal, and inexact arithmetic
on approximations. As in Common Lisp, exact arithmetic
is not limited to integers.

1.2. Syntax
Scheme, like most dialects of Lisp, employs a fully parenthesized prefix notation for programs and (other) data; the
grammar of Scheme generates a sublanguage of the language used for data. An important consequence of this simple, uniform representation is the susceptibility of Scheme
programs and data to uniform treatment by other Scheme
programs. For example, the eval procedure evaluates a
Scheme program expressed as data.
The read procedure performs syntactic as well as lexical
decomposition of the data it reads. The read procedure
parses its input as data (section 7.1.2), not as program.
The formal syntax of Scheme is described in section 7.1.

1.3. Notation and terminology
1.3.1. Primitive, library, and optional features
It is required that every implementation of Scheme support
all features that are not marked as being optional. Implementations are free to omit optional features of Scheme
or to add extensions, provided the extensions are not in
conflict with the language reported here. In particular,
implementations must support portable code by providing
a syntactic mode that preempts no lexical conventions of
this report.
To aid in understanding and implementing Scheme, some
features are marked as library. These can be easily implemented in terms of the other, primitive, features. They are
redundant in the strict sense of the word, but they capture
common patterns of usage, and are therefore provided as
convenient abbreviations.

4 Revised5 Scheme
1.3.2. Error situations and unspecified behavior
When speaking of an error situation, this report uses the
phrase “an error is signalled” to indicate that implementations must detect and report the error. If such wording
does not appear in the discussion of an error, then implementations are not required to detect or report the error,
though they are encouraged to do so. An error situation
that implementations are not required to detect is usually
referred to simply as “an error.”
For example, it is an error for a procedure to be passed an
argument that the procedure is not explicitly specified to
handle, even though such domain errors are seldom mentioned in this report. Implementations may extend a procedure’s domain of definition to include such arguments.
This report uses the phrase “may report a violation of an
implementation restriction” to indicate circumstances under which an implementation is permitted to report that
it is unable to continue execution of a correct program because of some restriction imposed by the implementation.
Implementation restrictions are of course discouraged, but
implementations are encouraged to report violations of implementation restrictions.
For example, an implementation may report a violation of
an implementation restriction if it does not have enough
storage to run a program.
If the value of an expression is said to be “unspecified,”
then the expression must evaluate to some object without
signalling an error, but the value depends on the implementation; this report explicitly does not say what value
should be returned.
1.3.3. Entry format
Chapters 4 and 6 are organized into entries. Each entry describes one language feature or a group of related features,
where a feature is either a syntactic construct or a built-in
procedure. An entry begins with one or more header lines
of the form
template

category

indicates zero or more occurrences of a hthingi, and
hthing1 i hthing2 i . . .
indicates one or more occurrences of a hthingi.
If category is “procedure”, then the entry describes a procedure, and the header line gives a template for a call to the
procedure. Argument names in the template are italicized .
Thus the header line
(vector-ref vector k )

procedure

indicates that the built-in procedure vector-ref takes two
arguments, a vector vector and an exact non-negative integer k (see below). The header lines
(make-vector k )
(make-vector k fill )

procedure
procedure

indicate that the make-vector procedure must be defined
to take either one or two arguments.
It is an error for an operation to be presented with an argument that it is not specified to handle. For succinctness,
we follow the convention that if an argument name is also
the name of a type listed in section 3.2, then that argument must be of the named type. For example, the header
line for vector-ref given above dictates that the first argument to vector-ref must be a vector. The following
naming conventions also imply type restrictions:
obj
list, list1 , . . . listj , . . .
z, z1 , . . . zj , . . .
x, x1 , . . . xj , . . .
y, y1 , . . . yj , . . .
q, q1 , . . . qj , . . .
n, n1 , . . . nj , . . .
k, k1 , . . . kj , . . .

any object
list (see section 6.3.2)
complex number
real number
real number
rational number
integer
exact non-negative integer

1.3.4. Evaluation examples
The symbol “=⇒” used in program examples should be
read “evaluates to.” For example,
(* 5 8)

for required, primitive features, or
template

hthing1 i . . .

qualifier category

where qualifier is either “library” or “optional” as defined
in section 1.3.1.
If category is “syntax”, the entry describes an expression
type, and the template gives the syntax of the expression
type. Components of expressions are designated by syntactic variables, which are written using angle brackets,
for example, hexpressioni, hvariablei. Syntactic variables
should be understood to denote segments of program text;
for example, hexpressioni stands for any string of characters which is a syntactically valid expression. The notation

=⇒

40

means that the expression (* 5 8) evaluates to the object 40. Or, more precisely: the expression given by the
sequence of characters “(* 5 8)” evaluates, in the initial
environment, to an object that may be represented externally by the sequence of characters “40”. See section 3.3
for a discussion of external representations of objects.
1.3.5. Naming conventions
By convention, the names of procedures that always return
a boolean value usually end in “?”. Such procedures are
called predicates.

2. Lexical conventions 5
By convention, the names of procedures that store values
into previously allocated locations (see section 3.4) usually
end in “!”. Such procedures are called mutation procedures. By convention, the value returned by a mutation
procedure is unspecified.
By convention, “->” appears within the names of procedures that take an object of one type and return an analogous object of another type. For example, list->vector
takes a list and returns a vector whose elements are the
same as those of the list.

2.

Lexical conventions

This section gives an informal account of some of the lexical
conventions used in writing Scheme programs. For a formal
syntax of Scheme, see section 7.1.
Upper and lower case forms of a letter are never distinguished except within character and string constants. For
example, Foo is the same identifier as FOO, and #x1AB is
the same number as #X1ab.

2.1. Identifiers
Most identifiers allowed by other programming languages
are also acceptable to Scheme. The precise rules for forming identifiers vary among implementations of Scheme, but
in all implementations a sequence of letters, digits, and “extended alphabetic characters” that begins with a character
that cannot begin a number is an identifier. In addition,
+, -, and ... are identifiers. Here are some examples of
identifiers:
lambda
q
list->vector
soup
+
V17a
<=?
a34kTMNs
the-word-recursion-has-many-meanings

Extended alphabetic characters may be used within identifiers as if they were letters. The following are extended
alphabetic characters:
! $ % & * + - . / : < = > ? @ ^ _ ~

See section 7.1.1 for a formal syntax of identifiers.
Identifiers have two uses within Scheme programs:
• Any identifier may be used as a variable or as a syntactic keyword (see sections 3.1 and 4.3).
• When an identifier appears as a literal or within a
literal (see section 4.1.2), it is being used to denote a
symbol (see section 6.3.3).

2.2. Whitespace and comments
Whitespace characters are spaces and newlines. (Implementations typically provide additional whitespace characters such as tab or page break.) Whitespace is used for
improved readability and as necessary to separate tokens
from each other, a token being an indivisible lexical unit
such as an identifier or number, but is otherwise insignificant. Whitespace may occur between any two tokens, but
not within a token. Whitespace may also occur inside a
string, where it is significant.
A semicolon (;) indicates the start of a comment. The
comment continues to the end of the line on which the
semicolon appears. Comments are invisible to Scheme, but
the end of the line is visible as whitespace. This prevents a
comment from appearing in the middle of an identifier or
number.
;;; The FACT procedure computes the factorial
;;; of a non-negative integer.
(define fact
(lambda (n)
(if (= n 0)
1
;Base case: return 1
(* n (fact (- n 1))))))

2.3. Other notations
For a description of the notations used for numbers, see
section 6.2.
. + - These are used in numbers, and may also occur
anywhere in an identifier except as the first character. A delimited plus or minus sign by itself is also an
identifier. A delimited period (not occurring within a
number or identifier) is used in the notation for pairs
(section 6.3.2), and to indicate a rest-parameter in a
formal parameter list (section 4.1.4). A delimited sequence of three successive periods is also an identifier.
( ) Parentheses are used for grouping and to notate lists
(section 6.3.2).
’ The single quote character is used to indicate literal data
(section 4.1.2).
` The backquote character is used to indicate almostconstant data (section 4.2.6).
, ,@ The character comma and the sequence comma atsign are used in conjunction with backquote (section 4.2.6).
" The double quote character is used to delimit strings
(section 6.3.5).

6 Revised5 Scheme
\ Backslash is used in the syntax for character constants
(section 6.3.4) and as an escape character within
string constants (section 6.3.5).
[ ] { } | Left and right square brackets and curly braces
and vertical bar are reserved for possible future extensions to the language.
# Sharp sign is used for a variety of purposes depending
on the character that immediately follows it:

The region is determined by the particular binding construct that establishes the binding; if the binding is established by a lambda expression, for example, then its region
is the entire lambda expression. Every mention of an identifier refers to the binding of the identifier that established
the innermost of the regions containing the use. If there is
no binding of the identifier whose region contains the use,
then the use refers to the binding for the variable in the
top level environment, if any (chapters 4 and 6); if there
is no binding for the identifier, it is said to be unbound.

#t #f These are the boolean constants (section 6.3.1).
#\ This introduces a character constant (section 6.3.4).
#( This introduces a vector constant (section 6.3.6). Vector constants are terminated by ) .
#e #i #b #o #d #x These are used in the notation for
numbers (section 6.2.4).

3.

Basic concepts

3.1. Variables, syntactic keywords, and regions
An identifier may name a type of syntax, or it may name
a location where a value can be stored. An identifier that
names a type of syntax is called a syntactic keyword and is
said to be bound to that syntax. An identifier that names
a location is called a variable and is said to be bound to
that location. The set of all visible bindings in effect at
some point in a program is known as the environment in
effect at that point. The value stored in the location to
which a variable is bound is called the variable’s value.
By abuse of terminology, the variable is sometimes said
to name the value or to be bound to the value. This is
not quite accurate, but confusion rarely results from this
practice.
Certain expression types are used to create new kinds of
syntax and bind syntactic keywords to those new syntaxes,
while other expression types create new locations and bind
variables to those locations. These expression types are
called binding constructs. Those that bind syntactic keywords are listed in section 4.3. The most fundamental of
the variable binding constructs is the lambda expression,
because all other variable binding constructs can be explained in terms of lambda expressions. The other variable
binding constructs are let, let*, letrec, and do expressions (see sections 4.1.4, 4.2.2, and 4.2.4).
Like Algol and Pascal, and unlike most other dialects of
Lisp except for Common Lisp, Scheme is a statically scoped
language with block structure. To each place where an
identifier is bound in a program there corresponds a region
of the program text within which the binding is visible.

3.2. Disjointness of types
No object satisfies more than one of the following predicates:
boolean?
symbol?
char?
vector?
procedure?

pair?
number?
string?
port?

These predicates define the types boolean, pair, symbol,
number, char (or character), string, vector, port, and procedure. The empty list is a special object of its own type;
it satisfies none of the above predicates.
Although there is a separate boolean type, any Scheme
value can be used as a boolean value for the purpose of a
conditional test. As explained in section 6.3.1, all values
count as true in such a test except for #f. This report uses
the word “true” to refer to any Scheme value except #f,
and the word “false” to refer to #f.

3.3. External representations
An important concept in Scheme (and Lisp) is that of the
external representation of an object as a sequence of characters. For example, an external representation of the integer 28 is the sequence of characters “28”, and an external
representation of a list consisting of the integers 8 and 13
is the sequence of characters “(8 13)”.
The external representation of an object is not necessarily unique. The integer 28 also has representations
“#e28.000” and “#x1c”, and the list in the previous paragraph also has the representations “( 08 13 )” and “(8
. (13 . ()))” (see section 6.3.2).
Many objects have standard external representations, but
some, such as procedures, do not have standard representations (although particular implementations may define
representations for them).
An external representation may be written in a program to
obtain the corresponding object (see quote, section 4.1.2).

3. Basic concepts 7
External representations can also be used for input and
output. The procedure read (section 6.6.2) parses external representations, and the procedure write (section 6.6.3) generates them. Together, they provide an
elegant and powerful input/output facility.
Note that the sequence of characters “(+ 2 6)” is not an
external representation of the integer 8, even though it is an
expression evaluating to the integer 8; rather, it is an external representation of a three-element list, the elements of
which are the symbol + and the integers 2 and 6. Scheme’s
syntax has the property that any sequence of characters
that is an expression is also the external representation of
some object. This can lead to confusion, since it may not
be obvious out of context whether a given sequence of characters is intended to denote data or program, but it is also
a source of power, since it facilitates writing programs such
as interpreters and compilers that treat programs as data
(or vice versa).
The syntax of external representations of various kinds of
objects accompanies the description of the primitives for
manipulating the objects in the appropriate sections of
chapter 6.

3.4. Storage model
Variables and objects such as pairs, vectors, and strings
implicitly denote locations or sequences of locations. A
string, for example, denotes as many locations as there
are characters in the string. (These locations need not
correspond to a full machine word.) A new value may be
stored into one of these locations using the string-set!
procedure, but the string continues to denote the same
locations as before.
An object fetched from a location, by a variable reference or
by a procedure such as car, vector-ref, or string-ref,
is equivalent in the sense of eqv? (section 6.1) to the object
last stored in the location before the fetch.
Every location is marked to show whether it is in use. No
variable or object ever refers to a location that is not in use.
Whenever this report speaks of storage being allocated for
a variable or object, what is meant is that an appropriate
number of locations are chosen from the set of locations
that are not in use, and the chosen locations are marked
to indicate that they are now in use before the variable or
object is made to denote them.
In many systems it is desirable for constants (i.e. the values of literal expressions) to reside in read-only-memory.
To express this, it is convenient to imagine that every
object that denotes locations is associated with a flag
telling whether that object is mutable or immutable. In
such systems literal constants and the strings returned by
symbol->string are immutable objects, while all objects

created by the other procedures listed in this report are
mutable. It is an error to attempt to store a new value
into a location that is denoted by an immutable object.

3.5. Proper tail recursion
Implementations of Scheme are required to be properly tailrecursive. Procedure calls that occur in certain syntactic
contexts defined below are ‘tail calls’. A Scheme implementation is properly tail-recursive if it supports an unbounded number of active tail calls. A call is active if
the called procedure may still return. Note that this includes calls that may be returned from either by the current continuation or by continuations captured earlier by
call-with-current-continuation that are later invoked.
In the absence of captured continuations, calls could return
at most once and the active calls would be those that had
not yet returned. A formal definition of proper tail recursion can be found in [8].
Rationale:
Intuitively, no space is needed for an active tail call because the
continuation that is used in the tail call has the same semantics
as the continuation passed to the procedure containing the call.
Although an improper implementation might use a new continuation in the call, a return to this new continuation would
be followed immediately by a return to the continuation passed
to the procedure. A properly tail-recursive implementation returns to that continuation directly.
Proper tail recursion was one of the central ideas in Steele and
Sussman’s original version of Scheme. Their first Scheme interpreter implemented both functions and actors. Control flow
was expressed using actors, which differed from functions in
that they passed their results on to another actor instead of
returning to a caller. In the terminology of this section, each
actor finished with a tail call to another actor.
Steele and Sussman later observed that in their interpreter the
code for dealing with actors was identical to that for functions
and thus there was no need to include both in the language.

A tail call is a procedure call that occurs in a tail context. Tail contexts are defined inductively. Note that a tail
context is always determined with respect to a particular
lambda expression.
• The last expression within the body of a lambda expression, shown as htail expressioni below, occurs in a
tail context.
(lambda hformalsi
hdefinitioni* hexpressioni* htail expressioni)
• If one of the following expressions is in a tail context,
then the subexpressions shown as htail expressioni are
in a tail context. These were derived from rules in

8 Revised5 Scheme
the grammar given in chapter 7 by replacing some occurrences of hexpressioni with htail expressioni. Only
those rules that contain tail contexts are shown here.

In the following example the only tail call is the call to f.
None of the calls to g or h are tail calls. The reference to
x is in a tail context, but it is not a call and thus is not a
tail call.
(lambda ()
(if (g)
(let ((x (h)))
x)
(and (g) (f))))

(if hexpressioni htail expressioni htail expressioni)
(if hexpressioni htail expressioni)
(cond hcond clausei+ )
(cond hcond clausei* (else htail sequencei))
(case hexpressioni
hcase clausei+ )
(case hexpressioni
hcase clausei*
(else htail sequencei))
(and hexpressioni* htail expressioni)
(or hexpressioni* htail expressioni)
(let (hbinding speci*) htail bodyi)
(let hvariablei (hbinding speci*) htail bodyi)
(let* (hbinding speci*) htail bodyi)
(letrec (hbinding speci*) htail bodyi)
(let-syntax (hsyntax speci*) htail bodyi)
(letrec-syntax (hsyntax speci*) htail bodyi)

Note: Implementations are allowed, but not required, to recognize that some non-tail calls, such as the call to h above, can be
evaluated as though they were tail calls. In the example above,
the let expression could be compiled as a tail call to h. (The
possibility of h returning an unexpected number of values can
be ignored, because in that case the effect of the let is explicitly
unspecified and implementation-dependent.)

4.

Expressions

Expression types are categorized as primitive or derived.
Primitive expression types include variables and procedure
calls. Derived expression types are not semantically primitive, but can instead be defined as macros. With the exception of quasiquote, whose macro definition is complex,
the derived expressions are classified as library features.
Suitable definitions are given in section 7.3.

(begin htail sequencei)
(do (hiteration speci*)
(htesti htail sequencei)
hexpressioni*)

4.1. Primitive expression types
4.1.1. Variable references

where

hvariablei

hcond clausei −→ (htesti htail sequencei)
hcase clausei −→ ((hdatumi*) htail sequencei)

An expression consisting of a variable (section 3.1) is a
variable reference. The value of the variable reference is
the value stored in the location to which the variable is
bound. It is an error to reference an unbound variable.

htail bodyi −→ hdefinitioni* htail sequencei
htail sequencei −→ hexpressioni* htail expressioni

• If a cond expression is in a tail context, and has a
clause of the form (hexpression1 i => hexpression2 i)
then the (implied) call to the procedure that results
from the evaluation of hexpression2 i is in a tail context.
hexpression2 i itself is not in a tail context.
Certain built-in procedures are also required to perform
tail calls. The first argument passed to apply and to
call-with-current-continuation, and the second argument passed to call-with-values, must be called via a
tail call. Similarly, eval must evaluate its argument as if
it were in tail position within the eval procedure.

(define x 28)
x

syntax

=⇒

28

4.1.2. Literal expressions
(quote hdatumi)
’hdatumi
hconstanti

syntax
syntax
syntax

(quote hdatumi) evaluates to hdatumi. hDatumi may be
any external representation of a Scheme object (see section 3.3). This notation is used to include literal constants
in Scheme code.
(quote a)
(quote #(a b c))
(quote (+ 1 2))

=⇒
=⇒
=⇒

a
#(a b c)
(+ 1 2)

4. Expressions 9
(quote hdatumi) may be abbreviated as ’hdatumi. The
two notations are equivalent in all respects.
’a
’#(a b c)
’()
’(+ 1 2)
’(quote a)
’’a

=⇒
=⇒
=⇒
=⇒
=⇒
=⇒

a
#(a b c)
()
(+ 1 2)
(quote a)
(quote a)

4.1.4. Procedures
(lambda hformalsi hbodyi)

Numerical constants, string constants, character constants,
and boolean constants evaluate “to themselves”; they need
not be quoted.
’"abc"
"abc"
’145932
145932
’#t
#t

=⇒
=⇒
=⇒
=⇒
=⇒
=⇒

"abc"
"abc"
145932
145932
#t
#t

As noted in section 3.4, it is an error to alter a constant
(i.e. the value of a literal expression) using a mutation procedure like set-car! or string-set!.
4.1.3. Procedure calls
(hoperatori hoperand1 i . . . )

syntax

A procedure call is written by simply enclosing in parentheses expressions for the procedure to be called and the
arguments to be passed to it. The operator and operand
expressions are evaluated (in an unspecified order) and the
resulting procedure is passed the resulting arguments.
(+ 3 4)
((if #f + *) 3 4)

=⇒
=⇒

Note: In many dialects of Lisp, the empty combination, (),
is a legitimate expression. In Scheme, combinations must have
at least one subexpression, so () is not a syntactically valid
expression.

7
12

A number of procedures are available as the values of variables in the initial environment; for example, the addition
and multiplication procedures in the above examples are
the values of the variables + and *. New procedures are created by evaluating lambda expressions (see section 4.1.4).
Procedure calls may return any number of values (see
values in section 6.4). With the exception of values the
procedures available in the initial environment return one
value or, for procedures such as apply, pass on the values
returned by a call to one of their arguments.
Procedure calls are also called combinations.
Note:
In contrast to other dialects of Lisp, the order of
evaluation is unspecified, and the operator expression and the
operand expressions are always evaluated with the same evaluation rules.
Note: Although the order of evaluation is otherwise unspecified, the effect of any concurrent evaluation of the operator and
operand expressions is constrained to be consistent with some
sequential order of evaluation. The order of evaluation may be
chosen differently for each procedure call.

syntax

Syntax: hFormalsi should be a formal arguments list as
described below, and hbodyi should be a sequence of one
or more expressions.
Semantics: A lambda expression evaluates to a procedure.
The environment in effect when the lambda expression was
evaluated is remembered as part of the procedure. When
the procedure is later called with some actual arguments,
the environment in which the lambda expression was evaluated will be extended by binding the variables in the formal
argument list to fresh locations, the corresponding actual
argument values will be stored in those locations, and the
expressions in the body of the lambda expression will be
evaluated sequentially in the extended environment. The
result(s) of the last expression in the body will be returned
as the result(s) of the procedure call.
(lambda (x) (+ x x))
((lambda (x) (+ x x)) 4)

=⇒
=⇒

a procedure
8

(define reverse-subtract
(lambda (x y) (- y x)))
(reverse-subtract 7 10)

=⇒

3

(define add4
(let ((x 4))
(lambda (y) (+ x y))))
(add4 6)

=⇒

10

hFormalsi should have one of the following forms:
• (hvariable1 i . . . ): The procedure takes a fixed number of arguments; when the procedure is called, the
arguments will be stored in the bindings of the corresponding variables.
• hvariablei: The procedure takes any number of arguments; when the procedure is called, the sequence
of actual arguments is converted into a newly allocated list, and the list is stored in the binding of the
hvariablei.
• (hvariable1 i . . . hvariablen i . hvariablen+1 i): If a
space-delimited period precedes the last variable, then
the procedure takes n or more arguments, where n
is the number of formal arguments before the period
(there must be at least one). The value stored in the
binding of the last variable will be a newly allocated
list of the actual arguments left over after all the other
actual arguments have been matched up against the
other formal arguments.

10

Revised5 Scheme

It is an error for a hvariablei to appear more than once in
hformalsi.
((lambda x x) 3 4 5 6)
((lambda (x y . z) z)
3 4 5 6)

=⇒

(3 4 5 6)

=⇒

(5 6)

(cond hclause1 i hclause2 i . . . )
(htesti hexpression1 i . . . )

4.1.5. Conditionals

where htesti is any expression. Alternatively, a hclausei
may be of the form
(htesti => hexpressioni)

The last hclausei may be an “else clause,” which has the
form
(else hexpression1 i hexpression2 i . . . ).

(if htesti hconsequenti halternatei)
(if htesti hconsequenti)

syntax
syntax

Syntax: hTesti, hconsequenti, and halternatei may be arbitrary expressions.
Semantics: An if expression is evaluated as follows: first,
htesti is evaluated. If it yields a true value (see section 6.3.1), then hconsequenti is evaluated and its value(s)
is(are) returned. Otherwise halternatei is evaluated and its
value(s) is(are) returned. If htesti yields a false value and
no halternatei is specified, then the result of the expression
is unspecified.
=⇒
=⇒

yes
no

=⇒

1

4.1.6. Assignments
(set! hvariablei hexpressioni)

syntax

hExpressioni is evaluated, and the resulting value is stored
in the location to which hvariablei is bound. hVariablei
must be bound either in some region enclosing the set!
expression or at top level. The result of the set! expression
is unspecified.
(define x 2)
(+ x 1)
(set! x 4)
(+ x 1)

library syntax

Syntax: Each hclausei should be of the form

Each procedure created as the result of evaluating a lambda
expression is (conceptually) tagged with a storage location,
in order to make eqv? and eq? work on procedures (see
section 6.1).

(if (> 3 2) ’yes ’no)
(if (> 2 3) ’yes ’no)
(if (> 3 2)
(- 3 2)
(+ 3 2))

4.2.1. Conditionals

=⇒
=⇒
=⇒

3
unspecified
5

4.2. Derived expression types
The constructs in this section are hygienic, as discussed
in section 4.3. For reference purposes, section 7.3 gives
macro definitions that will convert most of the constructs
described in this section into the primitive constructs described in the previous section.

Semantics: A cond expression is evaluated by evaluating
the htesti expressions of successive hclauseis in order until
one of them evaluates to a true value (see section 6.3.1).
When a htesti evaluates to a true value, then the remaining hexpressionis in its hclausei are evaluated in order,
and the result(s) of the last hexpressioni in the hclausei
is(are) returned as the result(s) of the entire cond expression. If the selected hclausei contains only the htesti and no
hexpressionis, then the value of the htesti is returned as the
result. If the selected hclausei uses the => alternate form,
then the hexpressioni is evaluated. Its value must be a procedure that accepts one argument; this procedure is then
called on the value of the htesti and the value(s) returned
by this procedure is(are) returned by the cond expression.
If all htestis evaluate to false values, and there is no else
clause, then the result of the conditional expression is unspecified; if there is an else clause, then its hexpressionis are
evaluated, and the value(s) of the last one is(are) returned.
(cond ((> 3 2) ’greater)
((< 3 2) ’less))
=⇒ greater
(cond ((> 3 3) ’greater)
((< 3 3) ’less)
(else ’equal))
=⇒ equal
(cond ((assv ’b ’((a 1) (b 2))) => cadr)
(else #f))
=⇒ 2

(case hkeyi hclause1 i hclause2 i . . . )

library syntax

Syntax: hKeyi may be any expression.
should have the form

Each hclausei

((hdatum1 i . . . ) hexpression1 i hexpression2 i . . . ),

where each hdatumi is an external representation of some
object. All the hdatumis must be distinct. The last
hclausei may be an “else clause,” which has the form
(else hexpression1 i hexpression2 i . . . ).

Semantics: A case expression is evaluated as follows.
hKeyi is evaluated and its result is compared against each
hdatumi. If the result of evaluating hkeyi is equivalent
(in the sense of eqv?; see section 6.1) to a hdatumi, then
the expressions in the corresponding hclausei are evaluated
from left to right and the result(s) of the last expression in

4. Expressions 11
the hclausei is(are) returned as the result(s) of the case expression. If the result of evaluating hkeyi is different from
every hdatumi, then if there is an else clause its expressions are evaluated and the result(s) of the last is(are) the
result(s) of the case expression; otherwise the result of the
case expression is unspecified.
(case (* 2 3)
((2 3 5 7) ’prime)
((1 4 6 8 9) ’composite)) =⇒
(case (car ’(c d))
((a) ’a)
((b) ’b))
=⇒
(case (car ’(c d))
((a e i o u) ’vowel)
((w y) ’semivowel)
(else ’consonant))
=⇒

composite

unspecified

consonant

library syntax

The htesti expressions are evaluated from left to right, and
the value of the first expression that evaluates to a false
value (see section 6.3.1) is returned. Any remaining expressions are not evaluated. If all the expressions evaluate
to true values, the value of the last expression is returned.
If there are no expressions then #t is returned.
=⇒
=⇒
=⇒
=⇒

(let hbindingsi hbodyi)

library syntax

Syntax: hBindingsi should have the form
((hvariable1 i hinit1 i) . . . ),

(and htest1 i . . . )

(and (= 2 2) (> 2 1))
(and (= 2 2) (< 2 1))
(and 1 2 ’c ’(f g))
(and)

initial values are being computed, thus allowing mutually
recursive definitions.

#t
#f
(f g)
#t

where each hiniti is an expression, and hbodyi should be a
sequence of one or more expressions. It is an error for a
hvariablei to appear more than once in the list of variables
being bound.
Semantics: The hinitis are evaluated in the current environment (in some unspecified order), the hvariableis are
bound to fresh locations holding the results, the hbodyi is
evaluated in the extended environment, and the value(s) of
the last expression of hbodyi is(are) returned. Each binding of a hvariablei has hbodyi as its region.
(let ((x 2) (y 3))
(* x y))

=⇒

6

(let ((x 2) (y 3))
(let ((x 7)
(z (+ x y)))
(* z x)))

=⇒

35

See also named let, section 4.2.4.
(let* hbindingsi hbodyi)

library syntax

Syntax: hBindingsi should have the form
(or htest1 i . . . )

library syntax

The htesti expressions are evaluated from left to right, and
the value of the first expression that evaluates to a true
value (see section 6.3.1) is returned. Any remaining expressions are not evaluated. If all expressions evaluate to
false values, the value of the last expression is returned. If
there are no expressions then #f is returned.
(or
(or
(or
(or

(= 2 2) (> 2 1))
(= 2 2) (< 2 1))
#f #f #f)
(memq ’b ’(a b c))
(/ 3 0))

=⇒
=⇒
=⇒

#t
#t
#f

=⇒

(b c)

((hvariable1 i hinit1 i) . . . ),

and hbodyi should be a sequence of one or more expressions.
Semantics: Let* is similar to let, but the bindings are
performed sequentially from left to right, and the region of
a binding indicated by (hvariablei hiniti) is that part of
the let* expression to the right of the binding. Thus the
second binding is done in an environment in which the first
binding is visible, and so on.
(let ((x 2) (y 3))
(let* ((x 7)
(z (+ x y)))
(* z x)))

=⇒

70

4.2.2. Binding constructs

(letrec hbindingsi hbodyi)

The three binding constructs let, let*, and letrec give
Scheme a block structure, like Algol 60. The syntax of the
three constructs is identical, but they differ in the regions
they establish for their variable bindings. In a let expression, the initial values are computed before any of the
variables become bound; in a let* expression, the bindings and evaluations are performed sequentially; while in a
letrec expression, all the bindings are in effect while their

Syntax: hBindingsi should have the form

library syntax

((hvariable1 i hinit1 i) . . . ),

and hbodyi should be a sequence of one or more expressions. It is an error for a hvariablei to appear more than
once in the list of variables being bound.
Semantics: The hvariableis are bound to fresh locations
holding undefined values, the hinitis are evaluated in the

12

Revised5 Scheme

resulting environment (in some unspecified order), each
hvariablei is assigned to the result of the corresponding
hiniti, the hbodyi is evaluated in the resulting environment,
and the value(s) of the last expression in hbodyi is(are) returned. Each binding of a hvariablei has the entire letrec
expression as its region, making it possible to define mutually recursive procedures.
(letrec ((even?
(lambda (n)
(if (zero? n)
#t
(odd? (- n 1)))))
(odd?
(lambda (n)
(if (zero? n)
#f
(even? (- n 1))))))
(even? 88))
=⇒ #t

termination condition is met, the loop exits after evaluating
the hexpressionis.
Do expressions are evaluated as follows: The hiniti expressions are evaluated (in some unspecified order), the
hvariableis are bound to fresh locations, the results of
the hiniti expressions are stored in the bindings of the
hvariableis, and then the iteration phase begins.
Each iteration begins by evaluating htesti; if the result is
false (see section 6.3.1), then the hcommandi expressions
are evaluated in order for effect, the hstepi expressions
are evaluated in some unspecified order, the hvariableis
are bound to fresh locations, the results of the hstepis are
stored in the bindings of the hvariableis, and the next iteration begins.
If htesti evaluates to a true value, then the hexpressionis
are evaluated from left to right and the value(s) of the
last hexpressioni is(are) returned. If no hexpressionis are
present, then the value of the do expression is unspecified.

One restriction on letrec is very important: it must be
possible to evaluate each hiniti without assigning or referring to the value of any hvariablei. If this restriction is
violated, then it is an error. The restriction is necessary
because Scheme passes arguments by value rather than by
name. In the most common uses of letrec, all the hinitis
are lambda expressions and the restriction is satisfied automatically.

The region of the binding of a hvariablei consists of the
entire do expression except for the hinitis. It is an error
for a hvariablei to appear more than once in the list of do
variables.
A hstepi may be omitted, in which case the effect is the
same as if (hvariablei hiniti hvariablei) had been written
instead of (hvariablei hiniti).
(do ((vec (make-vector 5))
(i 0 (+ i 1)))
((= i 5) vec)
(vector-set! vec i i))

4.2.3. Sequencing
(begin hexpression1 i hexpression2 i . . . )

=⇒

#(0 1 2 3 4)

library syntax

The hexpressionis are evaluated sequentially from left to
right, and the value(s) of the last hexpressioni is(are) returned. This expression type is used to sequence side effects such as input and output.

(let ((x ’(1 3 5
(do ((x x (cdr
(sum 0 (+
((null? x)

7 9)))
x))
sum (car x))))
sum)))
=⇒

25

(define x 0)

(let hvariablei hbindingsi hbodyi)

(begin (set! x 5)
(+ x 1))

“Named let” is a variant on the syntax of let which provides a more general looping construct than do and may
also be used to express recursions. It has the same syntax and semantics as ordinary let except that hvariablei
is bound within hbodyi to a procedure whose formal arguments are the bound variables and whose body is hbodyi.
Thus the execution of hbodyi may be repeated by invoking
the procedure named by hvariablei.

=⇒

6

(begin (display "4 plus 1 equals ")
(display (+ 4 1)))
=⇒ unspecified
and prints 4 plus 1 equals 5

4.2.4. Iteration
(do ((hvariable1 i hinit1 i hstep1 i)
...)
(htesti hexpressioni . . . )
hcommandi . . . )

library syntax

Do is an iteration construct. It specifies a set of variables
to be bound, how they are to be initialized at the start,
and how they are to be updated on each iteration. When a

library syntax

(let loop ((numbers ’(3 -2 1 6 -5))
(nonneg ’())
(neg ’()))
(cond ((null? numbers) (list nonneg neg))
((>= (car numbers) 0)
(loop (cdr numbers)
(cons (car numbers) nonneg)
neg))
((< (car numbers) 0)

4. Expressions 13
`(a `(b ,(+ 1 2) ,(foo ,(+ 1 3) d) e) f)
=⇒ (a `(b ,(+ 1 2) ,(foo 4 d) e) f)
(let ((name1 ’x)
(name2 ’y))
`(a `(b ,,name1 ,’,name2 d) e))
=⇒ (a `(b ,x ,’y d) e)

(loop (cdr numbers)
nonneg
(cons (car numbers) neg)))))
=⇒ ((6 1 3) (-5 -2))

4.2.5. Delayed evaluation
(delay hexpressioni)

library syntax

The delay construct is used together with the procedure force to implement lazy evaluation or call by need.
(delay hexpressioni) returns an object called a promise
which at some point in the future may be asked (by the
force procedure) to evaluate hexpressioni, and deliver the
resulting value. The effect of hexpressioni returning multiple values is unspecified.
See the description of force (section 6.4) for a more complete description of delay.
4.2.6. Quasiquotation
(quasiquote hqq templatei)
`hqq templatei

syntax
syntax

“Backquote” or “quasiquote” expressions are useful for
constructing a list or vector structure when most but not
all of the desired structure is known in advance. If no
commas appear within the hqq templatei, the result of
evaluating `hqq templatei is equivalent to the result of
evaluating ’hqq templatei. If a comma appears within
the hqq templatei, however, the expression following the
comma is evaluated (“unquoted”) and its result is inserted
into the structure instead of the comma and the expression. If a comma appears followed immediately by an atsign (@), then the following expression must evaluate to
a list; the opening and closing parentheses of the list are
then “stripped away” and the elements of the list are inserted in place of the comma at-sign expression sequence.
A comma at-sign should only appear within a list or vector
hqq templatei.
`(list ,(+ 1 2) 4)
=⇒ (list 3 4)
(let ((name ’a)) `(list ,name ’,name))
=⇒ (list a (quote a))
`(a ,(+ 1 2) ,@(map abs ’(4 -5 6)) b)
=⇒ (a 3 4 5 6 b)
`(( foo ,(- 10 3)) ,@(cdr ’(c)) . ,(car ’(cons)))
=⇒ ((foo 7) . cons)
`#(10 5 ,(sqrt 4) ,@(map sqrt ’(16 9)) 8)
=⇒ #(10 5 2 4 3 8)

Quasiquote forms may be nested. Substitutions are made
only for unquoted components appearing at the same nesting level as the outermost backquote. The nesting level increases by one inside each successive quasiquotation, and
decreases by one inside each unquotation.

The two notations `hqq templatei and (quasiquote
hqq templatei) are identical in all respects. ,hexpressioni
is identical to (unquote hexpressioni), and ,@hexpressioni
is identical to (unquote-splicing hexpressioni). The external syntax generated by write for two-element lists
whose car is one of these symbols may vary between implementations.
(quasiquote (list (unquote (+ 1 2)) 4))
=⇒ (list 3 4)
’(quasiquote (list (unquote (+ 1 2)) 4))
=⇒ `(list ,(+ 1 2) 4)
i.e., (quasiquote (list (unquote (+ 1 2)) 4))

Unpredictable behavior can result if any of the symbols
quasiquote, unquote, or unquote-splicing appear in positions within a hqq templatei otherwise than as described
above.

4.3. Macros
Scheme programs can define and use new derived expression types, called macros. Program-defined expression
types have the syntax
(hkeywordi hdatumi ...)

where hkeywordi is an identifier that uniquely determines
the expression type. This identifier is called the syntactic
keyword, or simply keyword, of the macro. The number of
the hdatumis, and their syntax, depends on the expression
type.
Each instance of a macro is called a use of the macro. The
set of rules that specifies how a use of a macro is transcribed
into a more primitive expression is called the transformer
of the macro.
The macro definition facility consists of two parts:
• A set of expressions used to establish that certain identifiers are macro keywords, associate them with macro
transformers, and control the scope within which a
macro is defined, and
• a pattern language for specifying macro transformers.
The syntactic keyword of a macro may shadow variable
bindings, and local variable bindings may shadow keyword
bindings. All macros defined using the pattern language
are “hygienic” and “referentially transparent” and thus
preserve Scheme’s lexical scoping [14, 15, 2, 7, 9]:

Revised5 Scheme

14

• If a macro transformer inserts a binding for an identifier (variable or keyword), the identifier will in effect be
renamed throughout its scope to avoid conflicts with
other identifiers. Note that a define at top level may
or may not introduce a binding; see section 5.2.
• If a macro transformer inserts a free reference to an
identifier, the reference refers to the binding that was
visible where the transformer was specified, regardless
of any local bindings that may surround the use of the
macro.
4.3.1. Binding constructs for syntactic keywords
Let-syntax and letrec-syntax are analogous to let and
letrec, but they bind syntactic keywords to macro transformers instead of binding variables to locations that contain values. Syntactic keywords may also be bound at top
level; see section 5.3.
(let-syntax hbindingsi hbodyi)

syntax

Syntax: hBindingsi should have the form

of the letrec-syntax expression with macros whose keywords are the hkeywordis, bound to the specified transformers. Each binding of a hkeywordi has the hbindingsi
as well as the hbodyi within its region, so the transformers
can transcribe expressions into uses of the macros introduced by the letrec-syntax expression.
(letrec-syntax
((my-or (syntax-rules ()
((my-or) #f)
((my-or e) e)
((my-or e1 e2 ...)
(let ((temp e1))
(if temp
temp
(my-or e2 ...)))))))
(let ((x #f)
(y 7)
(temp 8)
(let odd?)
(if even?))
(my-or x
(let temp)
(if y)
y)))
=⇒ 7

((hkeywordi htransformer speci) . . . )

Each hkeywordi is an identifier, each htransformer speci
is an instance of syntax-rules, and hbodyi should be a
sequence of one or more expressions. It is an error for a
hkeywordi to appear more than once in the list of keywords
being bound.
Semantics: The hbodyi is expanded in the syntactic environment obtained by extending the syntactic environment
of the let-syntax expression with macros whose keywords
are the hkeywordis, bound to the specified transformers.
Each binding of a hkeywordi has hbodyi as its region.
(let-syntax ((when (syntax-rules ()
((when test stmt1 stmt2 ...)
(if test
(begin stmt1
stmt2 ...))))))
(let ((if #t))
(when if (set! if ’now))
if))
=⇒ now
(let ((x ’outer))
(let-syntax ((m (syntax-rules () ((m) x))))
(let ((x ’inner))
(m))))
=⇒ outer

4.3.2. Pattern language
A htransformer speci has the following form:
(syntax-rules hliteralsi hsyntax rulei . . . )
Syntax: hLiteralsi is a list of identifiers and each
hsyntax rulei should be of the form
(hpatterni htemplatei)

The hpatterni in a hsyntax rulei is a list hpatterni that
begins with the keyword for the macro.
A hpatterni is either an identifier, a constant, or one of the
following
(hpatterni ...)
(hpatterni hpatterni ... . hpatterni)
(hpatterni ... hpatterni hellipsisi)
#(hpatterni ...)
#(hpatterni ... hpatterni hellipsisi)

and a template is either an identifier, a constant, or one of
the following
(helementi ...)
(helementi helementi ... . htemplatei)
#(helementi ...)

syntax

Syntax: Same as for let-syntax.

where an helementi is a htemplatei optionally followed by
an hellipsisi and an hellipsisi is the identifier “...” (which
cannot be used as an identifier in either a template or a
pattern).

Semantics: The hbodyi is expanded in the syntactic environment obtained by extending the syntactic environment

Semantics: An instance of syntax-rules produces a new
macro transformer by specifying a sequence of hygienic

(letrec-syntax hbindingsi hbodyi)

4. Expressions 15
rewrite rules. A use of a macro whose keyword is associated
with a transformer specified by syntax-rules is matched
against the patterns contained in the hsyntax ruleis, beginning with the leftmost hsyntax rulei. When a match is
found, the macro use is transcribed hygienically according
to the template.
An identifier that appears in the pattern of a hsyntax rulei
is a pattern variable, unless it is the keyword that begins
the pattern, is listed in hliteralsi, or is the identifier “...”.
Pattern variables match arbitrary input elements and are
used to refer to elements of the input in the template. It
is an error for the same pattern variable to appear more
than once in a hpatterni.
The keyword at the beginning of the pattern in a
hsyntax rulei is not involved in the matching and is not
considered a pattern variable or literal identifier.
Rationale: The scope of the keyword is determined by the
expression or syntax definition that binds it to the associated
macro transformer. If the keyword were a pattern variable or
literal identifier, then the template that follows the pattern
would be within its scope regardless of whether the keyword
were bound by let-syntax or by letrec-syntax.

Identifiers that appear in hliteralsi are interpreted as literal
identifiers to be matched against corresponding subforms
of the input. A subform in the input matches a literal
identifier if and only if it is an identifier and either both its
occurrence in the macro expression and its occurrence in
the macro definition have the same lexical binding, or the
two identifiers are equal and both have no lexical binding.
A subpattern followed by ... can match zero or more elements of the input. It is an error for ... to appear in
hliteralsi. Within a pattern the identifier ... must follow
the last element of a nonempty sequence of subpatterns.
More formally, an input form F matches a pattern P if and
only if:
• P is a non-literal identifier; or
• P is a literal identifier and F is an identifier with the
same binding; or
• P is a list (P1 . . . Pn ) and F is a list of n forms that
match P1 through Pn , respectively; or
• P is an improper list (P1 P2 . . . Pn . Pn+1 ) and
F is a list or improper list of n or more forms that
match P1 through Pn , respectively, and whose nth
“cdr” matches Pn+1 ; or
• P is of the form (P1 . . . Pn Pn+1 hellipsisi) where
hellipsisi is the identifier ... and F is a proper list
of at least n forms, the first n of which match P1
through Pn , respectively, and each remaining element
of F matches Pn+1 ; or

• P is a vector of the form #(P1 . . . Pn ) and F is a
vector of n forms that match P1 through Pn ; or
• P is of the form #(P1 . . . Pn Pn+1 hellipsisi) where
hellipsisi is the identifier ... and F is a vector of n or
more forms the first n of which match P1 through Pn ,
respectively, and each remaining element of F matches
Pn+1 ; or
• P is a datum and F is equal to P in the sense of the
equal? procedure.
It is an error to use a macro keyword, within the scope of
its binding, in an expression that does not match any of
the patterns.
When a macro use is transcribed according to the template
of the matching hsyntax rulei, pattern variables that occur
in the template are replaced by the subforms they match
in the input. Pattern variables that occur in subpatterns
followed by one or more instances of the identifier ... are
allowed only in subtemplates that are followed by as many
instances of .... They are replaced in the output by all
of the subforms they match in the input, distributed as
indicated. It is an error if the output cannot be built up
as specified.
Identifiers that appear in the template but are not pattern
variables or the identifier ... are inserted into the output
as literal identifiers. If a literal identifier is inserted as a
free identifier then it refers to the binding of that identifier
within whose scope the instance of syntax-rules appears.
If a literal identifier is inserted as a bound identifier then
it is in effect renamed to prevent inadvertent captures of
free identifiers.
As an example, if let and cond are defined as in section 7.3
then they are hygienic (as required) and the following is not
an error.
(let ((=> #f))
(cond (#t => ’ok)))

=⇒ ok

The macro transformer for cond recognizes => as a local
variable, and hence an expression, and not as the top-level
identifier =>, which the macro transformer treats as a syntactic keyword. Thus the example expands into
(let ((=> #f))
(if #t (begin => ’ok)))

instead of
(let ((=> #f))
(let ((temp #t))
(if temp (’ok temp))))

which would result in an invalid procedure call.

Revised5 Scheme

16

5.

Program structure

5.1. Programs

5.2.1. Top level definitions
At the top level of a program, a definition
(define hvariablei hexpressioni)

A Scheme program consists of a sequence of expressions,
definitions, and syntax definitions. Expressions are described in chapter 4; definitions and syntax definitions are
the subject of the rest of the present chapter.
Programs are typically stored in files or entered interactively to a running Scheme system, although other
paradigms are possible; questions of user interface lie outside the scope of this report. (Indeed, Scheme would still be
useful as a notation for expressing computational methods
even in the absence of a mechanical implementation.)
Definitions and syntax definitions occurring at the top level
of a program can be interpreted declaratively. They cause
bindings to be created in the top level environment or modify the value of existing top-level bindings. Expressions
occurring at the top level of a program are interpreted imperatively; they are executed in order when the program
is invoked or loaded, and typically perform some kind of
initialization.
At the top level of a program (begin hform1 i . . . ) is
equivalent to the sequence of expressions, definitions, and
syntax definitions that form the body of the begin.

5.2. Definitions
Definitions are valid in some, but not all, contexts where
expressions are allowed. They are valid only at the top
level of a hprogrami and at the beginning of a hbodyi.
A definition should have one of the following forms:
• (define hvariablei hexpressioni)
• (define (hvariablei hformalsi) hbodyi)
hFormalsi should be either a sequence of zero or more
variables, or a sequence of one or more variables followed by a space-delimited period and another variable (as in a lambda expression). This form is equivalent to
(define hvariablei
(lambda (hformalsi) hbodyi)).

• (define (hvariablei . hformali) hbodyi)
hFormali should be a single variable. This form is
equivalent to
(define hvariablei
(lambda hformali hbodyi)).

has essentially the same effect as the assignment expression
(set! hvariablei hexpressioni)

if hvariablei is bound. If hvariablei is not bound, however,
then the definition will bind hvariablei to a new location
before performing the assignment, whereas it would be an
error to perform a set! on an unbound variable.
(define add3
(lambda (x) (+ x 3)))
(add3 3)
(define first car)
(first ’(1 2))

=⇒

6

=⇒

1

Some implementations of Scheme use an initial environment in which all possible variables are bound to locations,
most of which contain undefined values. Top level definitions in such an implementation are truly equivalent to
assignments.
5.2.2. Internal definitions
Definitions may occur at the beginning of a hbodyi (that
is, the body of a lambda, let, let*, letrec, let-syntax,
or letrec-syntax expression or that of a definition of an
appropriate form). Such definitions are known as internal
definitions as opposed to the top level definitions described
above. The variable defined by an internal definition is
local to the hbodyi. That is, hvariablei is bound rather
than assigned, and the region of the binding is the entire
hbodyi. For example,
(let ((x 5))
(define foo (lambda (y) (bar x y)))
(define bar (lambda (a b) (+ (* a b) a)))
(foo (+ x 3)))
=⇒ 45

A hbodyi containing internal definitions can always be converted into a completely equivalent letrec expression. For
example, the let expression in the above example is equivalent to
(let ((x 5))
(letrec ((foo (lambda (y) (bar x y)))
(bar (lambda (a b) (+ (* a b) a))))
(foo (+ x 3))))

Just as for the equivalent letrec expression, it must be
possible to evaluate each hexpressioni of every internal definition in a hbodyi without assigning or referring to the
value of any hvariablei being defined.
Wherever an internal definition may occur (begin
hdefinition1 i . . . ) is equivalent to the sequence of definitions that form the body of the begin.

6. Standard procedures 17

5.3. Syntax definitions
Syntax definitions are valid only at the top level of a
hprogrami. They have the following form:
(define-syntax hkeywordi htransformer speci)
hKeywordi is an identifier, and the htransformer speci
should be an instance of syntax-rules. The top-level syntactic environment is extended by binding the hkeywordi
to the specified transformer.
There is no define-syntax analogue of internal definitions.
Although macros may expand into definitions and syntax
definitions in any context that permits them, it is an error
for a definition or syntax definition to shadow a syntactic
keyword whose meaning is needed to determine whether
some form in the group of forms that contains the shadowing definition is in fact a definition, or, for internal definitions, is needed to determine the boundary between the
group and the expressions that follow the group. For example, the following are errors:
(define define 3)

top-level binding that has not been introduced by a definition has an unspecified effect on the behavior of the built-in
procedures.

6.1. Equivalence predicates
A predicate is a procedure that always returns a boolean
value (#t or #f). An equivalence predicate is the computational analogue of a mathematical equivalence relation
(it is symmetric, reflexive, and transitive). Of the equivalence predicates described in this section, eq? is the finest
or most discriminating, and equal? is the coarsest. Eqv?
is slightly less discriminating than eq?.
(eqv? obj1 obj2 )

procedure

The eqv? procedure defines a useful equivalence relation
on objects. Briefly, it returns #t if obj1 and obj2 should
normally be regarded as the same object. This relation is
left slightly open to interpretation, but the following partial specification of eqv? holds for all implementations of
Scheme.
The eqv? procedure returns #t if:

(begin (define begin list))

• obj1 and obj2 are both #t or both #f.
(let-syntax
((foo (syntax-rules ()
((foo (proc args ...) body ...)
(define proc
(lambda (args ...)
body ...))))))
(let ((x 3))
(foo (plus x y) (+ x y))
(define foo x)
(plus foo x)))

6.

Standard procedures

This chapter describes Scheme’s built-in procedures. The
initial (or “top level”) Scheme environment starts out with
a number of variables bound to locations containing useful
values, most of which are primitive procedures that manipulate data. For example, the variable abs is bound to
(a location initially containing) a procedure of one argument that computes the absolute value of a number, and
the variable + is bound to a procedure that computes sums.
Built-in procedures that can easily be written in terms of
other built-in procedures are identified as “library procedures”.
A program may use a top-level definition to bind any variable. It may subsequently alter any such binding by an
assignment (see 4.1.6). These operations do not modify
the behavior of Scheme’s built-in procedures. Altering any

• obj1 and obj2 are both symbols and
(string=? (symbol->string obj1)
(symbol->string obj2))
=⇒ #t
Note: This assumes that neither obj1 nor obj2 is an “uninterned symbol” as alluded to in section 6.3.3. This report does not presume to specify the behavior of eqv? on
implementation-dependent extensions.

• obj1 and obj2 are both numbers, are numerically equal
(see =, section 6.2), and are either both exact or both
inexact.
• obj1 and obj2 are both characters and are the same
character according to the char=? procedure (section 6.3.4).
• both obj1 and obj2 are the empty list.
• obj1 and obj2 are pairs, vectors, or strings that denote
the same locations in the store (section 3.4).
• obj1 and obj2 are procedures whose location tags are
equal (section 4.1.4).
The eqv? procedure returns #f if:
• obj1 and obj2 are of different types (section 3.2).

Revised5 Scheme

18

• one of obj1 and obj2 is #t but the other is #f.
• obj1 and obj2 are symbols but
(string=? (symbol->string obj1 )
(symbol->string obj2 ))
=⇒ #f

• one of obj1 and obj2 is an exact number but the other
is an inexact number.
• obj1 and obj2 are numbers for which the = procedure
returns #f.
• obj1 and obj2 are characters for which the char=? procedure returns #f.
• one of obj1 and obj2 is the empty list but the other is
not.
• obj1 and obj2 are pairs, vectors, or strings that denote
distinct locations.
• obj1 and obj2 are procedures that would behave differently (return different value(s) or have different side
effects) for some arguments.
’a ’a)
=⇒
’a ’b)
=⇒
2 2)
=⇒
’() ’())
=⇒
100000000 100000000) =⇒
(cons 1 2) (cons 1 2))=⇒
(lambda () 1)
(lambda () 2))
=⇒
(eqv? #f ’nil)
=⇒
(let ((p (lambda (x) x)))
(eqv? p p))
=⇒
(eqv?
(eqv?
(eqv?
(eqv?
(eqv?
(eqv?
(eqv?

#t
#f
#t
#t
#t
#f
#f
#f
#t

The following examples illustrate cases in which the above
rules do not fully specify the behavior of eqv?. All that
can be said about such cases is that the value returned by
eqv? must be a boolean.
(eqv? "" "")
(eqv? ’#() ’#())
(eqv? (lambda (x)
(lambda (x)
(eqv? (lambda (x)
(lambda (y)

x)
x))
x)
y))

(define gen-counter
(lambda ()
(let ((n 0))
(lambda () (set! n (+ n 1)) n))))
(let ((g (gen-counter)))
(eqv? g g))
=⇒ #t
(eqv? (gen-counter) (gen-counter))
=⇒ #f
(define gen-loser
(lambda ()
(let ((n 0))
(lambda () (set! n (+ n 1)) 27))))
(let ((g (gen-loser)))
(eqv? g g))
=⇒ #t
(eqv? (gen-loser) (gen-loser))
=⇒ unspecified
(letrec ((f (lambda () (if (eqv? f g) ’both ’f)))
(g (lambda () (if (eqv? f g) ’both ’g))))
(eqv? f g))
=⇒ unspecified
(letrec ((f (lambda () (if (eqv? f g) ’f ’both)))
(g (lambda () (if (eqv? f g) ’g ’both))))
(eqv? f g))
=⇒ #f

Since it is an error to modify constant objects (those returned by literal expressions), implementations are permitted, though not required, to share structure between
constants where appropriate. Thus the value of eqv? on
constants is sometimes implementation-dependent.
(eqv? ’(a) ’(a))
(eqv? "a" "a")
(eqv? ’(b) (cdr ’(a b)))
(let ((x ’(a)))
(eqv? x x))

=⇒
=⇒
=⇒

unspecified
unspecified
unspecified

=⇒

#t

Rationale: The above definition of eqv? allows implementations latitude in their treatment of procedures and literals: implementations are free either to detect or to fail to detect that
two procedures or two literals are equivalent to each other, and
can decide whether or not to merge representations of equivalent
objects by using the same pointer or bit pattern to represent
both.

=⇒
=⇒

unspecified
unspecified

(eq? obj1 obj2 )

=⇒

unspecified

=⇒

unspecified

Eq? is similar to eqv? except that in some cases it is capable
of discerning distinctions finer than those detectable by
eqv?.

The next set of examples shows the use of eqv? with procedures that have local state. Gen-counter must return a
distinct procedure every time, since each procedure has its
own internal counter. Gen-loser, however, returns equivalent procedures each time, since the local state does not
affect the value or side effects of the procedures.

procedure

Eq? and eqv? are guaranteed to have the same behavior on
symbols, booleans, the empty list, pairs, procedures, and
non-empty strings and vectors. Eq?’s behavior on numbers
and characters is implementation-dependent, but it will always return either true or false, and will return true only
when eqv? would also return true. Eq? may also behave
differently from eqv? on empty vectors and empty strings.

6. Standard procedures 19
(eq? ’a ’a)
(eq? ’(a) ’(a))
(eq? (list ’a) (list ’a))
(eq? "a" "a")
(eq? "" "")
(eq? ’() ’())
(eq? 2 2)
(eq? #\A #\A)
(eq? car car)
(let ((n (+ 2 3)))
(eq? n n))
(let ((x ’(a)))
(eq? x x))
(let ((x ’#()))
(eq? x x))
(let ((p (lambda (x) x)))
(eq? p p))

=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒

#t
unspecified
#f
unspecified
unspecified
#t
unspecified
unspecified
#t

=⇒

unspecified

=⇒

#t

=⇒

#t

=⇒

#t

Rationale: It will usually be possible to implement eq? much
more efficiently than eqv?, for example, as a simple pointer comparison instead of as some more complicated operation. One
reason is that it may not be possible to compute eqv? of two
numbers in constant time, whereas eq? implemented as pointer
comparison will always finish in constant time. Eq? may be used
like eqv? in applications using procedures to implement objects
with state since it obeys the same constraints as eqv?.

(equal? obj1 obj2 )

library procedure

Equal? recursively compares the contents of pairs, vectors,
and strings, applying eqv? on other objects such as numbers and symbols. A rule of thumb is that objects are
generally equal? if they print the same. Equal? may fail
to terminate if its arguments are circular data structures.
(equal? ’a ’a)
(equal? ’(a) ’(a))
(equal? ’(a (b) c)
’(a (b) c))
(equal? "abc" "abc")
(equal? 2 2)
(equal? (make-vector 5 ’a)
(make-vector 5 ’a))
(equal? (lambda (x) x)
(lambda (y) y))

=⇒
=⇒

#t
#t

=⇒
=⇒
=⇒

#t
#t
#t

=⇒

#t

=⇒

unspecified

6.2. Numbers
Numerical computation has traditionally been neglected
by the Lisp community. Until Common Lisp there was
no carefully thought out strategy for organizing numerical
computation, and with the exception of the MacLisp system [20] little effort was made to execute numerical code
efficiently. This report recognizes the excellent work of the
Common Lisp committee and accepts many of their recommendations. In some ways this report simplifies and
generalizes their proposals in a manner consistent with the
purposes of Scheme.

It is important to distinguish between the mathematical numbers, the Scheme numbers that attempt to model
them, the machine representations used to implement the
Scheme numbers, and notations used to write numbers.
This report uses the types number, complex, real, rational,
and integer to refer to both mathematical numbers and
Scheme numbers. Machine representations such as fixed
point and floating point are referred to by names such as
fixnum and flonum.
6.2.1. Numerical types
Mathematically, numbers may be arranged into a tower of
subtypes in which each level is a subset of the level above
it:
number
complex
real
rational
integer
For example, 3 is an integer. Therefore 3 is also a rational,
a real, and a complex. The same is true of the Scheme
numbers that model 3. For Scheme numbers, these types
are defined by the predicates number?, complex?, real?,
rational?, and integer?.
There is no simple relationship between a number’s type
and its representation inside a computer. Although most
implementations of Scheme will offer at least two different
representations of 3, these different representations denote
the same integer.
Scheme’s numerical operations treat numbers as abstract
data, as independent of their representation as possible.
Although an implementation of Scheme may use fixnum,
flonum, and perhaps other representations for numbers,
this should not be apparent to a casual programmer writing
simple programs.
It is necessary, however, to distinguish between numbers
that are represented exactly and those that may not be.
For example, indexes into data structures must be known
exactly, as must some polynomial coefficients in a symbolic
algebra system. On the other hand, the results of measurements are inherently inexact, and irrational numbers may
be approximated by rational and therefore inexact approximations. In order to catch uses of inexact numbers where
exact numbers are required, Scheme explicitly distinguishes
exact from inexact numbers. This distinction is orthogonal
to the dimension of type.
6.2.2. Exactness
Scheme numbers are either exact or inexact. A number is
exact if it was written as an exact constant or was derived
from exact numbers using only exact operations. A number

20

Revised5 Scheme

is inexact if it was written as an inexact constant, if it
was derived using inexact ingredients, or if it was derived
using inexact operations. Thus inexactness is a contagious
property of a number.
If two implementations produce exact results for a computation that did not involve inexact intermediate results,
the two ultimate results will be mathematically equivalent.
This is generally not true of computations involving inexact numbers since approximate methods such as floating
point arithmetic may be used, but it is the duty of each
implementation to make the result as close as practical to
the mathematically ideal result.
Rational operations such as + should always produce exact results when given exact arguments. If the operation
is unable to produce an exact result, then it may either
report the violation of an implementation restriction or it
may silently coerce its result to an inexact value. See section 6.2.3.
With the exception of inexact->exact, the operations described in this section must generally return inexact results
when given any inexact arguments. An operation may,
however, return an exact result if it can prove that the
value of the result is unaffected by the inexactness of its
arguments. For example, multiplication of any number by
an exact zero may produce an exact zero result, even if the
other argument is inexact.
6.2.3. Implementation restrictions
Implementations of Scheme are not required to implement
the whole tower of subtypes given in section 6.2.1, but
they must implement a coherent subset consistent with
both the purposes of the implementation and the spirit
of the Scheme language. For example, an implementation
in which all numbers are real may still be quite useful.
Implementations may also support only a limited range of
numbers of any type, subject to the requirements of this
section. The supported range for exact numbers of any
type may be different from the supported range for inexact numbers of that type. For example, an implementation
that uses flonums to represent all its inexact real numbers
may support a practically unbounded range of exact integers and rationals while limiting the range of inexact reals
(and therefore the range of inexact integers and rationals)
to the dynamic range of the flonum format. Furthermore
the gaps between the representable inexact integers and rationals are likely to be very large in such an implementation
as the limits of this range are approached.
An implementation of Scheme must support exact integers
throughout the range of numbers that may be used for
indexes of lists, vectors, and strings or that may result
from computing the length of a list, vector, or string. The
length, vector-length, and string-length procedures

must return an exact integer, and it is an error to use
anything but an exact integer as an index. Furthermore
any integer constant within the index range, if expressed
by an exact integer syntax, will indeed be read as an exact
integer, regardless of any implementation restrictions that
may apply outside this range. Finally, the procedures listed
below will always return an exact integer result provided all
their arguments are exact integers and the mathematically
expected result is representable as an exact integer within
the implementation:
+
quotient
max
numerator
lcm
truncate
expt

remainder
min
denominator
floor
round

*
modulo
abs
gcd
ceiling
rationalize

Implementations are encouraged, but not required, to support exact integers and exact rationals of practically unlimited size and precision, and to implement the above procedures and the / procedure in such a way that they always
return exact results when given exact arguments. If one of
these procedures is unable to deliver an exact result when
given exact arguments, then it may either report a violation of an implementation restriction or it may silently
coerce its result to an inexact number. Such a coercion
may cause an error later.
An implementation may use floating point and other approximate representation strategies for inexact numbers.
This report recommends, but does not require, that the
IEEE 32-bit and 64-bit floating point standards be followed
by implementations that use flonum representations, and
that implementations using other representations should
match or exceed the precision achievable using these floating point standards [12].
In particular, implementations that use flonum representations must follow these rules: A flonum result must be
represented with at least as much precision as is used to
express any of the inexact arguments to that operation. It
is desirable (but not required) for potentially inexact operations such as sqrt, when applied to exact arguments, to
produce exact answers whenever possible (for example the
square root of an exact 4 ought to be an exact 2). If, however, an exact number is operated upon so as to produce an
inexact result (as by sqrt), and if the result is represented
as a flonum, then the most precise flonum format available
must be used; but if the result is represented in some other
way then the representation must have at least as much
precision as the most precise flonum format available.
Although Scheme allows a variety of written notations for
numbers, any particular implementation may support only
some of them. For example, an implementation in which
all numbers are real need not support the rectangular and

6. Standard procedures 21
polar notations for complex numbers. If an implementation encounters an exact numerical constant that it cannot
represent as an exact number, then it may either report a
violation of an implementation restriction or it may silently
represent the constant by an inexact number.
6.2.4. Syntax of numerical constants
The syntax of the written representations for numbers is
described formally in section 7.1.1. Note that case is not
significant in numerical constants.
A number may be written in binary, octal, decimal, or hexadecimal by the use of a radix prefix. The radix prefixes
are #b (binary), #o (octal), #d (decimal), and #x (hexadecimal). With no radix prefix, a number is assumed to be
expressed in decimal.
A numerical constant may be specified to be either exact or
inexact by a prefix. The prefixes are #e for exact, and #i
for inexact. An exactness prefix may appear before or after
any radix prefix that is used. If the written representation
of a number has no exactness prefix, the constant may be
either inexact or exact. It is inexact if it contains a decimal
point, an exponent, or a “#” character in the place of a
digit, otherwise it is exact.
In systems with inexact numbers of varying precisions it
may be useful to specify the precision of a constant. For
this purpose, numerical constants may be written with an
exponent marker that indicates the desired precision of the
inexact representation. The letters s, f, d, and l specify
the use of short, single, double, and long precision, respectively. (When fewer than four internal inexact representations exist, the four size specifications are mapped onto
those available. For example, an implementation with two
internal representations may map short and single together
and long and double together.) In addition, the exponent
marker e specifies the default precision for the implementation. The default precision has at least as much precision
as double, but implementations may wish to allow this default to be set by the user.
3.14159265358979F0
Round to single — 3.141593
0.6L0
Extend to long — .600000000000000

6.2.5. Numerical operations
The reader is referred to section 1.3.3 for a summary of
the naming conventions used to specify restrictions on the
types of arguments to numerical routines. The examples
used in this section assume that any numerical constant
written using an exact notation is indeed represented as
an exact number. Some examples also assume that certain
numerical constants written using an inexact notation can

be represented without loss of accuracy; the inexact constants were chosen so that this is likely to be true in implementations that use flonums to represent inexact numbers.
(number? obj )
(complex? obj )
(real? obj )
(rational? obj )
(integer? obj )

procedure
procedure
procedure
procedure
procedure

These numerical type predicates can be applied to any kind
of argument, including non-numbers. They return #t if the
object is of the named type, and otherwise they return #f.
In general, if a type predicate is true of a number then
all higher type predicates are also true of that number.
Consequently, if a type predicate is false of a number, then
all lower type predicates are also false of that number.
If z is an inexact complex number, then (real? z) is true
if and only if (zero? (imag-part z)) is true. If x is an
inexact real number, then (integer? x) is true if and only
if (= x (round x)).
(complex? 3+4i)
(complex? 3)
(real? 3)
(real? -2.5+0.0i)
(real? #e1e10)
(rational? 6/10)
(rational? 6/3)
(integer? 3+0i)
(integer? 3.0)
(integer? 8/4)

=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒

#t
#t
#t
#t
#t
#t
#t
#t
#t
#t

Note: The behavior of these type predicates on inexact numbers is unreliable, since any inaccuracy may affect the result.
Note: In many implementations the rational? procedure will
be the same as real?, and the complex? procedure will be the
same as number?, but unusual implementations may be able
to represent some irrational numbers exactly or may extend the
number system to support some kind of non-complex numbers.

(exact? z)
(inexact? z)

procedure
procedure

These numerical predicates provide tests for the exactness
of a quantity. For any Scheme number, precisely one of
these predicates is true.
(= z1 z2 z3 . . . )
(< x1 x2 x3 . . . )
(> x1 x2 x3 . . . )
(<= x1 x2 x3 . . . )
(>= x1 x2 x3 . . . )

procedure
procedure
procedure
procedure
procedure

These procedures return #t if their arguments are (respectively): equal, monotonically increasing, monotonically decreasing, monotonically nondecreasing, or monotonically
nonincreasing.

22

Revised5 Scheme

These predicates are required to be transitive.
Note: The traditional implementations of these predicates in
Lisp-like languages are not transitive.
Note: While it is not an error to compare inexact numbers
using these predicates, the results may be unreliable because a
small inaccuracy may affect the result; this is especially true of
= and zero?. When in doubt, consult a numerical analyst.

((((/
(/

3 4)
3 4 5)
3)
3 4 5)
3)

=⇒
=⇒
=⇒
=⇒
=⇒

(abs x )

-1
-6
-3
3/20
1/3

library procedure

Abs returns the absolute value of its argument.
(zero? z)
(positive? x)
(negative? x)
(odd? n)
(even? n)

library
library
library
library
library

procedure
procedure
procedure
procedure
procedure

(abs -7)

=⇒

7

These numerical predicates test a number for a particular
property, returning #t or #f. See note above.

(quotient n1 n2 )
(remainder n1 n2 )
(modulo n1 n2 )

(max x1 x2 . . . )
(min x1 x2 . . . )

These procedures implement number-theoretic (integer) division. n2 should be non-zero. All three procedures return
integers. If n1 /n2 is an integer:

library procedure
library procedure

These procedures return the maximum or minimum of their
arguments.
(max 3 4)
(max 3.9 4)

=⇒
=⇒

4
4.0

; exact
; inexact

Note: If any argument is inexact, then the result will also be
inexact (unless the procedure can prove that the inaccuracy is
not large enough to affect the result, which is possible only in
unusual implementations). If min or max is used to compare
numbers of mixed exactness, and the numerical value of the
result cannot be represented as an inexact number without loss
of accuracy, then the procedure may report a violation of an
implementation restriction.

(+ z1 . . . )
(* z1 . . . )

procedure
procedure

These procedures return the sum or product of their arguments.
(+ 3 4)
(+ 3)
(+)
(* 4)
(*)

((((/
(/
(/

z1
z)
z1
z1
z)
z1

z2 )
z2 . . . )
z2 )
z2 . . . )

=⇒
=⇒
=⇒
=⇒
=⇒

7
3
0
4
1

procedure
procedure
optional procedure
procedure
procedure
optional procedure

With two or more arguments, these procedures return the
difference or quotient of their arguments, associating to the
left. With one argument, however, they return the additive
or multiplicative inverse of their argument.

(quotient n1 n2 )
(remainder n1 n2 )
(modulo n1 n2 )

procedure
procedure
procedure

=⇒ n1 /n2
=⇒ 0
=⇒ 0

If n1 /n2 is not an integer:
(quotient n1 n2 )
(remainder n1 n2 )
(modulo n1 n2 )

=⇒ nq
=⇒ nr
=⇒ nm

where nq is n1 /n2 rounded towards zero, 0 < |nr | < |n2 |,
0 < |nm | < |n2 |, nr and nm differ from n1 by a multiple of
n2 , nr has the same sign as n1 , and nm has the same sign
as n2 .
From this we can conclude that for integers n1 and n2 with
n2 not equal to 0,
(= n1 (+ (* n2 (quotient n1 n2 ))
(remainder n1 n2 )))
=⇒ #t

provided all numbers involved in that computation are exact.
(modulo 13 4)
(remainder 13 4)

=⇒
=⇒

1
1

(modulo -13 4)
(remainder -13 4)

=⇒
=⇒

3
-1

(modulo 13 -4)
(remainder 13 -4)

=⇒
=⇒

-3
1

(modulo -13 -4)
(remainder -13 -4)

=⇒
=⇒

-1
-1

(remainder -13 -4.0)

=⇒

-1.0

; inexact

6. Standard procedures 23
(gcd n1 . . . )
(lcm n1 . . . )

library procedure
library procedure

These procedures return the greatest common divisor or
least common multiple of their arguments. The result is
always non-negative.
(gcd 32 -36)
(gcd)
(lcm 32 -36)
(lcm 32.0 -36)
(lcm)

=⇒
=⇒
=⇒
=⇒
=⇒

4
0
288
288.0
1

(numerator q)
(denominator q)

; inexact

procedure
procedure

These procedures return the numerator or denominator of
their argument; the result is computed as if the argument
was represented as a fraction in lowest terms. The denominator is always positive. The denominator of 0 is defined
to be 1.
(numerator (/ 6 4))
=⇒ 3
(denominator (/ 6 4))
=⇒ 2
(denominator
(exact->inexact (/ 6 4))) =⇒ 2.0

(floor x )
(ceiling x )
(truncate x )
(round x )

procedure
procedure
procedure
procedure

These procedures return integers. Floor returns the
largest integer not larger than x. Ceiling returns the
smallest integer not smaller than x. Truncate returns the
integer closest to x whose absolute value is not larger than
the absolute value of x. Round returns the closest integer to x, rounding to even when x is halfway between two
integers.
Rationale: Round rounds to even for consistency with the default rounding mode specified by the IEEE floating point standard.
Note: If the argument to one of these procedures is inexact,
then the result will also be inexact. If an exact value is needed,
the result should be passed to the inexact->exact procedure.
(floor -4.3)
(ceiling -4.3)
(truncate -4.3)
(round -4.3)

=⇒
=⇒
=⇒
=⇒

-5.0
-4.0
-4.0
-4.0

(floor 3.5)
(ceiling 3.5)
(truncate 3.5)
(round 3.5)

=⇒
=⇒
=⇒
=⇒

3.0
4.0
3.0
4.0

(round 7/2)
(round 7)

=⇒
=⇒

4
7

(rationalize x y)

library procedure

Rationalize returns the simplest rational number differing from x by no more than y. A rational number r1 is
simpler than another rational number r2 if r1 = p1 /q1 and
r2 = p2 /q2 (in lowest terms) and |p1 | ≤ |p2 | and |q1 | ≤ |q2 |.
Thus 3/5 is simpler than 4/7. Although not all rationals
are comparable in this ordering (consider 2/7 and 3/5) any
interval contains a rational number that is simpler than every other rational number in that interval (the simpler 2/5
lies between 2/7 and 3/5). Note that 0 = 0/1 is the simplest rational of all.
(rationalize
(inexact->exact .3) 1/10) =⇒ 1/3
(rationalize .3 1/10)
=⇒ #i1/3

(exp z)
(log z)
(sin z)
(cos z)
(tan z)
(asin z)
(acos z)
(atan z)
(atan y x)

; exact
; inexact

procedure
procedure
procedure
procedure
procedure
procedure
procedure
procedure
procedure

These procedures are part of every implementation that
supports general real numbers; they compute the usual
transcendental functions. Log computes the natural logarithm of z (not the base ten logarithm). Asin, acos,
and atan compute arcsine (sin−1 ), arccosine (cos−1 ), and
arctangent (tan−1 ), respectively. The two-argument variant of atan computes (angle (make-rectangular x y))
(see below), even in implementations that don’t support
general complex numbers.
In general, the mathematical functions log, arcsine, arccosine, and arctangent are multiply defined. The value of
log z is defined to be the one whose imaginary part lies in
the range from −π (exclusive) to π (inclusive). log 0 is undefined. With log defined this way, the values of sin−1 z,
cos−1 z, and tan−1 z are according to the following formulæ:
p
sin−1 z = −i log(iz + 1 − z 2 )
cos−1 z = π/2 − sin−1 z
tan−1 z = (log(1 + iz) − log(1 − iz))/(2i)

; inexact
; exact

The above specification follows [27], which in turn
cites [19]; refer to these sources for more detailed discussion
of branch cuts, boundary conditions, and implementation
of these functions. When it is possible these procedures
produce a real result from a real argument.

24

Revised5 Scheme

(sqrt z)

procedure

Returns the principal square root of z. The result will have
either positive real part, or zero real part and non-negative
imaginary part.
(expt z1 z2 )

procedure

Returns z1 raised to the power z2 . For z1 6= 0
z1 z2 = ez2 log z1
0z is 1 if z = 0 and 0 otherwise.
(make-rectangular x1 x2 )
(make-polar x3 x4 )
(real-part z)
(imag-part z)
(magnitude z)
(angle z)

procedure
procedure
procedure
procedure
procedure
procedure

These procedures are part of every implementation that
supports general complex numbers. Suppose x1 , x2 , x3 ,
and x4 are real numbers and z is a complex number such
that
z = x1 + x2 i = x3 · eix4
Then
(make-rectangular x1 x2 )
(make-polar x3 x4 )
(real-part z)
(imag-part z)
(magnitude z)
(angle z)

=⇒
=⇒
=⇒
=⇒
=⇒
=⇒

z
z
x1
x2
|x3 |
xangle

(number->string z )
(number->string z radix )

procedure
procedure

Radix must be an exact integer, either 2, 8, 10, or 16. If
omitted, radix defaults to 10. The procedure number->
string takes a number and a radix and returns as a string
an external representation of the given number in the given
radix such that
(let ((number number)
(radix radix))
(eqv? number
(string->number (number->string number
radix)
radix)))

is true. It is an error if no possible result makes this expression true.
If z is inexact, the radix is 10, and the above expression
can be satisfied by a result that contains a decimal point,
then the result contains a decimal point and is expressed
using the minimum number of digits (exclusive of exponent
and trailing zeroes) needed to make the above expression
true [3, 5]; otherwise the format of the result is unspecified.
The result returned by number->string never contains an
explicit radix prefix.
Note: The error case can occur only when z is not a complex
number or is a complex number with a non-rational real or
imaginary part.

where −π < xangle ≤ π with xangle = x4 + 2πn for some
integer n.
Rationale:
Magnitude is the same as abs for a real argument, but abs must be present in all implementations, whereas
magnitude need only be present in implementations that support general complex numbers.

(exact->inexact z)
(inexact->exact z)

6.2.6. Numerical input and output

procedure
procedure

Exact->inexact returns an inexact representation of z.
The value returned is the inexact number that is numerically closest to the argument. If an exact argument has no
reasonably close inexact equivalent, then a violation of an
implementation restriction may be reported.
Inexact->exact returns an exact representation of z. The
value returned is the exact number that is numerically closest to the argument. If an inexact argument has no reasonably close exact equivalent, then a violation of an implementation restriction may be reported.
These procedures implement the natural one-to-one correspondence between exact and inexact integers throughout
an implementation-dependent range. See section 6.2.3.

Rationale: If z is an inexact number represented using flonums,
and the radix is 10, then the above expression is normally satisfied by a result containing a decimal point. The unspecified case
allows for infinities, NaNs, and non-flonum representations.

(string->number string)
(string->number string radix )

procedure
procedure

Returns a number of the maximally precise representation
expressed by the given string. Radix must be an exact
integer, either 2, 8, 10, or 16. If supplied, radix is a default
radix that may be overridden by an explicit radix prefix in
string (e.g. "#o177"). If radix is not supplied, then the
default radix is 10. If string is not a syntactically valid
notation for a number, then string->number returns #f.
(string->number
(string->number
(string->number
(string->number

"100")
"100" 16)
"1e2")
"15##")

=⇒
=⇒
=⇒
=⇒

100
256
100.0
1500.0

Note: The domain of string->number may be restricted by
implementations in the following ways. String->number is permitted to return #f whenever string contains an explicit radix
prefix. If all numbers supported by an implementation are real,

6. Standard procedures 25
then string->number is permitted to return #f whenever string
uses the polar or rectangular notations for complex numbers. If
all numbers are integers, then string->number may return #f
whenever the fractional notation is used. If all numbers are
exact, then string->number may return #f whenever an exponent marker or explicit exactness prefix is used, or if a #
appears in place of a digit. If all inexact numbers are integers,
then string->number may return #f whenever a decimal point
is used.

(boolean? obj )

library procedure

Boolean? returns #t if obj is either #t or #f and returns
#f otherwise.
(boolean? #f)
(boolean? 0)
(boolean? ’())

=⇒
=⇒
=⇒

#t
#f
#f

6.3.2. Pairs and lists

6.3. Other data types
This section describes operations on some of Scheme’s nonnumeric data types: booleans, pairs, lists, symbols, characters, strings and vectors.

6.3.1. Booleans
The standard boolean objects for true and false are written
as #t and #f. What really matters, though, are the objects
that the Scheme conditional expressions (if, cond, and,
or, do) treat as true or false. The phrase “a true value”
(or sometimes just “true”) means any object treated as
true by the conditional expressions, and the phrase “a false
value” (or “false”) means any object treated as false by the
conditional expressions.
Of all the standard Scheme values, only #f counts as false
in conditional expressions. Except for #f, all standard
Scheme values, including #t, pairs, the empty list, symbols, numbers, strings, vectors, and procedures, count as
true.
Note:
Programmers accustomed to other dialects of Lisp
should be aware that Scheme distinguishes both #f and the
empty list from the symbol nil.

Boolean constants evaluate to themselves, so they do not
need to be quoted in programs.
=⇒
=⇒
=⇒

#t
#f
’#f

(not obj )

#t
#f
#f

library procedure

Not returns #t if obj is false, and returns #f otherwise.
(not
(not
(not
(not
(not
(not
(not

#t)
3)
(list 3))
#f)
’())
(list))
’nil)

=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒

#f
#f
#f
#t
#f
#f
#f

A pair (sometimes called a dotted pair) is a record structure
with two fields called the car and cdr fields (for historical
reasons). Pairs are created by the procedure cons. The
car and cdr fields are accessed by the procedures car and
cdr. The car and cdr fields are assigned by the procedures
set-car! and set-cdr!.
Pairs are used primarily to represent lists. A list can be
defined recursively as either the empty list or a pair whose
cdr is a list. More precisely, the set of lists is defined as
the smallest set X such that
• The empty list is in X .
• If list is in X , then any pair whose cdr field contains
list is also in X .
The objects in the car fields of successive pairs of a list are
the elements of the list. For example, a two-element list
is a pair whose car is the first element and whose cdr is a
pair whose car is the second element and whose cdr is the
empty list. The length of a list is the number of elements,
which is the same as the number of pairs.
The empty list is a special object of its own type (it is not
a pair); it has no elements and its length is zero.
Note: The above definitions imply that all lists have finite
length and are terminated by the empty list.

The most general notation (external representation) for
Scheme pairs is the “dotted” notation (c1 . c2 ) where c1
is the value of the car field and c2 is the value of the cdr
field. For example (4 . 5) is a pair whose car is 4 and
whose cdr is 5. Note that (4 . 5) is the external representation of a pair, not an expression that evaluates to a
pair.
A more streamlined notation can be used for lists: the
elements of the list are simply enclosed in parentheses and
separated by spaces. The empty list is written () . For
example,
(a b c d e)

and
(a . (b . (c . (d . (e . ())))))

26

Revised5 Scheme

are equivalent notations for a list of symbols.

(car pair )

A chain of pairs not ending in the empty list is called an
improper list. Note that an improper list is not a list.
The list and dotted notations can be combined to represent
improper lists:

Returns the contents of the car field of pair . Note that it
is an error to take the car of the empty list.

(a b c . d)

is equivalent to

(car
(car
(car
(car

’(a b c))
’((a) b c d))
’(1 . 2))
’())

procedure

=⇒
=⇒
=⇒
=⇒

a
(a)
1
error

(a . (b . (c . d)))

Whether a given pair is a list depends upon what is stored
in the cdr field. When the set-cdr! procedure is used, an
object can be a list one moment and not the next:
(define x (list ’a ’b ’c))
(define y x)
y
(list? y)
(set-cdr! x 4)
x
(eqv? x y)
y
(list? y)
(set-cdr! x x)
(list? x)

=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒

(a b c)
#t
unspecified
(a . 4)
#t
(a . 4)
#f
unspecified
#f

Within literal expressions and representations of objects read by the read procedure, the forms ’hdatumi,
`hdatumi, ,hdatumi, and ,@hdatumi denote two-element lists whose first elements are the symbols quote,
quasiquote, unquote, and unquote-splicing, respectively. The second element in each case is hdatumi. This
convention is supported so that arbitrary Scheme programs may be represented as lists.
That is, according
to Scheme’s grammar, every hexpressioni is also a hdatumi
(see section 7.1.2). Among other things, this permits the
use of the read procedure to parse Scheme programs. See
section 3.3.
(pair? obj )

procedure

Pair? returns #t if obj is a pair, and otherwise returns #f.
(pair?
(pair?
(pair?
(pair?

’(a . b))
’(a b c))
’())
’#(a b))

=⇒
=⇒
=⇒
=⇒

(cons obj1 obj2 )

#t
#t
#f
#f

’a ’())
’(a) ’(b c d))
"a" ’(b c))
’a 3)
’(a b) ’c)

procedure

Returns the contents of the cdr field of pair . Note that it
is an error to take the cdr of the empty list.
(cdr ’((a) b c d))
(cdr ’(1 . 2))
(cdr ’())

(set-car! pair obj )

=⇒
=⇒
=⇒

(b c d)
2
error

procedure

Stores obj in the car field of pair . The value returned by
set-car! is unspecified.
(define (f) (list ’not-a-constant-list))
(define (g) ’(constant-list))
(set-car! (f) 3)
=⇒ unspecified
(set-car! (g) 3)
=⇒ error

(set-cdr! pair obj )

procedure

Stores obj in the cdr field of pair . The value returned by
set-cdr! is unspecified.
(caar pair )
(cadr pair )
.
.
.

library procedure
library procedure
..
.

(cdddar pair )
(cddddr pair )

library procedure
library procedure

These procedures are compositions of car and cdr, where
for example caddr could be defined by
(define caddr (lambda (x) (car (cdr (cdr x))))).

Arbitrary compositions, up to four deep, are provided.
There are twenty-eight of these procedures in all.
procedure

Returns a newly allocated pair whose car is obj1 and whose
cdr is obj2 . The pair is guaranteed to be different (in the
sense of eqv?) from every existing object.
(cons
(cons
(cons
(cons
(cons

(cdr pair )

=⇒
=⇒
=⇒
=⇒
=⇒

(a)
((a) b c d)
("a" b c)
(a . 3)
((a b) . c)

(null? obj )

library procedure

Returns #t if obj is the empty list, otherwise returns #f.
(list? obj )

library procedure

Returns #t if obj is a list, otherwise returns #f. By definition, all lists have finite length and are terminated by the
empty list.

6. Standard procedures 27
(list? ’(a b c))
=⇒
(list? ’())
=⇒
(list? ’(a . b))
=⇒
(let ((x (list ’a)))
(set-cdr! x x)
(list? x))
=⇒

(list obj . . . )

#t
#t
#f

#f

library procedure

(list-ref list k)

library procedure

Returns the kth element of list. (This is the same as the
car of (list-tail list k).) It is an error if list has fewer
than k elements.
(list-ref ’(a b c d) 2)
=⇒ c
(list-ref ’(a b c d)
(inexact->exact (round 1.8)))
=⇒ c

Returns a newly allocated list of its arguments.
(list ’a (+ 3 4) ’c)
(list)

=⇒
=⇒

(length list)

(a 7 c)
()

library procedure

Returns the length of list.
(length ’(a b c))
(length ’(a (b) (c d e)))
(length ’())

=⇒
=⇒
=⇒

(append list . . . )

3
3
0

library procedure

Returns a list consisting of the elements of the first list
followed by the elements of the other lists.
(append ’(x) ’(y))
(append ’(a) ’(b c d))
(append ’(a (b)) ’((c)))

=⇒
=⇒
=⇒

(x y)
(a b c d)
(a (b) (c))

The resulting list is always newly allocated, except that
it shares structure with the last list argument. The last
argument may actually be any object; an improper list
results if the last argument is not a proper list.
(append ’(a b) ’(c . d))
(append ’() ’a)

(reverse list)

=⇒
=⇒

(a b c . d)
a

library procedure

Returns a newly allocated list consisting of the elements of
list in reverse order.
(reverse ’(a b c))
=⇒ (c b a)
(reverse ’(a (b c) d (e (f))))
=⇒ ((e (f)) d (b c) a)

(list-tail list k)

library procedure

Returns the sublist of list obtained by omitting the first k
elements. It is an error if list has fewer than k elements.
List-tail could be defined by
(define list-tail
(lambda (x k)
(if (zero? k)
x
(list-tail (cdr x) (- k 1)))))

(memq obj list)
(memv obj list)
(member obj list)

library procedure
library procedure
library procedure

These procedures return the first sublist of list whose car
is obj , where the sublists of list are the non-empty lists
returned by (list-tail list k ) for k less than the length
of list. If obj does not occur in list, then #f (not the empty
list) is returned. Memq uses eq? to compare obj with the
elements of list, while memv uses eqv? and member uses
equal?.
(memq ’a ’(a b c))
(memq ’b ’(a b c))
(memq ’a ’(b c d))
(memq (list ’a) ’(b (a) c))
(member (list ’a)
’(b (a) c))
(memq 101 ’(100 101 102))
(memv 101 ’(100 101 102))

(assq obj alist)
(assv obj alist)
(assoc obj alist)

=⇒
=⇒
=⇒
=⇒

(a b c)
(b c)
#f
#f

=⇒
=⇒
=⇒

((a) c)
unspecified
(101 102)

library procedure
library procedure
library procedure

Alist (for “association list”) must be a list of pairs. These
procedures find the first pair in alist whose car field is obj ,
and returns that pair. If no pair in alist has obj as its car,
then #f (not the empty list) is returned. Assq uses eq? to
compare obj with the car fields of the pairs in alist, while
assv uses eqv? and assoc uses equal?.
(define e ’((a 1) (b 2) (c 3)))
(assq ’a e)
=⇒ (a 1)
(assq ’b e)
=⇒ (b 2)
(assq ’d e)
=⇒ #f
(assq (list ’a) ’(((a)) ((b)) ((c))))
=⇒ #f
(assoc (list ’a) ’(((a)) ((b)) ((c))))
=⇒ ((a))
(assq 5 ’((2 3) (5 7) (11 13)))
=⇒ unspecified
(assv 5 ’((2 3) (5 7) (11 13)))
=⇒ (5 7)
Rationale: Although they are ordinarily used as predicates,
memq, memv, member, assq, assv, and assoc do not have question
marks in their names because they return useful values rather
than just #t or #f.

28

Revised5 Scheme

6.3.3. Symbols
Symbols are objects whose usefulness rests on the fact that
two symbols are identical (in the sense of eqv?) if and only
if their names are spelled the same way. This is exactly the
property needed to represent identifiers in programs, and
so most implementations of Scheme use them internally for
that purpose. Symbols are useful for many other applications; for instance, they may be used the way enumerated
values are used in Pascal.
The rules for writing a symbol are exactly the same as the
rules for writing an identifier; see sections 2.1 and 7.1.1.
It is guaranteed that any symbol that has been returned as
part of a literal expression, or read using the read procedure, and subsequently written out using the write procedure, will read back in as the identical symbol (in the sense
of eqv?). The string->symbol procedure, however, can
create symbols for which this write/read invariance may
not hold because their names contain special characters or
letters in the non-standard case.
Note: Some implementations of Scheme have a feature known
as “slashification” in order to guarantee write/read invariance
for all symbols, but historically the most important use of this
feature has been to compensate for the lack of a string data
type.
Some implementations also have “uninterned symbols”, which
defeat write/read invariance even in implementations with
slashification, and also generate exceptions to the rule that two
symbols are the same if and only if their names are spelled the
same.

(symbol? obj )

procedure

The following examples assume that the implementation’s
standard case is lower case:
(symbol->string ’flying-fish)
=⇒
(symbol->string ’Martin)
=⇒
(symbol->string
(string->symbol "Malvina"))
=⇒

(string->symbol string)

"flying-fish"
"martin"

"Malvina"

procedure

Returns the symbol whose name is string. This procedure
can create symbols with names containing special characters or letters in the non-standard case, but it is usually
a bad idea to create such symbols because in some implementations of Scheme they cannot be read as themselves.
See symbol->string.
The following examples assume that the implementation’s
standard case is lower case:
(eq? ’mISSISSIppi ’mississippi)
=⇒ #t
(string->symbol "mISSISSIppi")
=⇒ the symbol with name "mISSISSIppi"
(eq? ’bitBlt (string->symbol "bitBlt"))
=⇒ #f
(eq? ’JollyWog
(string->symbol
(symbol->string ’JollyWog)))
=⇒ #t
(string=? "K. Harper, M.D."
(symbol->string
(string->symbol "K. Harper, M.D.")))
=⇒ #t

Returns #t if obj is a symbol, otherwise returns #f.
(symbol?
(symbol?
(symbol?
(symbol?
(symbol?
(symbol?

’foo)
(car ’(a b)))
"bar")
’nil)
’())
#f)

(symbol->string symbol )

=⇒
=⇒
=⇒
=⇒
=⇒
=⇒

#t
#t
#f
#t
#f
#f

6.3.4. Characters
Characters are objects that represent printed characters
such as letters and digits. Characters are written using the
notation #\hcharacteri or #\hcharacter namei. For example:
procedure

Returns the name of symbol as a string. If the symbol
was part of an object returned as the value of a literal
expression (section 4.1.2) or by a call to the read procedure, and its name contains alphabetic characters, then the
string returned will contain characters in the implementation’s preferred standard case—some implementations will
prefer upper case, others lower case. If the symbol was returned by string->symbol, the case of characters in the
string returned will be the same as the case in the string
that was passed to string->symbol. It is an error to apply
mutation procedures like string-set! to strings returned
by this procedure.

#\a
#\A
#\(
#\
#\space
#\newline

;
;
;
;
;
;

lower case letter
upper case letter
left parenthesis
the space character
the preferred way to write a space
the newline character

Case is significant in #\hcharacteri, but not in #\hcharacter
namei. If hcharacteri in #\hcharacteri is alphabetic, then
the character following hcharacteri must be a delimiter
character such as a space or parenthesis. This rule resolves
the ambiguous case where, for example, the sequence of

6. Standard procedures 29
characters “#\space” could be taken to be either a representation of the space character or a representation of the
character “#\s” followed by a representation of the symbol
“pace.”
Characters written in the #\ notation are self-evaluating.
That is, they do not have to be quoted in programs.
Some of the procedures that operate on characters ignore
the difference between upper case and lower case. The procedures that ignore case have “-ci” (for “case insensitive”)
embedded in their names.
(char? obj )

procedure

(char-alphabetic? char )
(char-numeric? char )
(char-whitespace? char )
(char-upper-case? letter )
(char-lower-case? letter )

library
library
library
library
library

procedure
procedure
procedure
procedure
procedure

These procedures return #t if their arguments are alphabetic, numeric, whitespace, upper case, or lower case characters, respectively, otherwise they return #f. The following remarks, which are specific to the ASCII character set,
are intended only as a guide: The alphabetic characters are
the 52 upper and lower case letters. The numeric characters are the ten decimal digits. The whitespace characters
are space, tab, line feed, form feed, and carriage return.

Returns #t if obj is a character, otherwise returns #f.
(char=? char1 char2 )
(char<? char1 char2 )
(char>? char1 char2 )
(char<=? char1 char2 )
(char>=? char1 char2 )

procedure
procedure
procedure
procedure
procedure

These procedures impose a total ordering on the set of
characters. It is guaranteed that under this ordering:
• The upper case characters are in order. For example,
(char<? #\A #\B) returns #t.
• The lower case characters are in order. For example,
(char<? #\a #\b) returns #t.
• The digits are in order. For example, (char<? #\0
#\9) returns #t.
• Either all the digits precede all the upper case letters,
or vice versa.
• Either all the digits precede all the lower case letters,
or vice versa.
Some implementations may generalize these procedures to
take more than two arguments, as with the corresponding
numerical predicates.
(char-ci=? char1 char2 )
(char-ci<? char1 char2 )
(char-ci>? char1 char2 )
(char-ci<=? char1 char2 )
(char-ci>=? char1 char2 )

library
library
library
library
library

procedure
procedure
procedure
procedure
procedure

These procedures are similar to char=? et cetera, but they
treat upper case and lower case letters as the same. For
example, (char-ci=? #\A #\a) returns #t. Some implementations may generalize these procedures to take more
than two arguments, as with the corresponding numerical
predicates.

(char->integer char )
(integer->char n)

procedure
procedure

Given a character, char->integer returns an exact integer representation of the character. Given an exact integer that is the image of a character under char->integer,
integer->char returns that character. These procedures
implement order-preserving isomorphisms between the set
of characters under the char<=? ordering and some subset
of the integers under the <= ordering. That is, if
(char<=? a b) =⇒ #t

and

(<= x y) =⇒ #t

and x and y are in the domain of integer->char, then
(<= (char->integer a)
(char->integer b))

=⇒

#t

(char<=? (integer->char x)
(integer->char y)) =⇒

#t

(char-upcase char )
(char-downcase char )

library procedure
library procedure

These procedures return a character char2 such that
(char-ci=? char char2 ). In addition, if char is alphabetic, then the result of char-upcase is upper case and
the result of char-downcase is lower case.
6.3.5. Strings
Strings are sequences of characters. Strings are written
as sequences of characters enclosed within doublequotes
("). A doublequote can be written inside a string only by
escaping it with a backslash (\), as in
"The word \"recursion\" has many meanings."

A backslash can be written inside a string only by escaping
it with another backslash. Scheme does not specify the
effect of a backslash within a string that is not followed by
a doublequote or backslash.

30

Revised5 Scheme

A string constant may continue from one line to the next,
but the exact contents of such a string are unspecified.

(string=? string1 string2 )
(string-ci=? string1 string2 )

The length of a string is the number of characters that it
contains. This number is an exact, non-negative integer
that is fixed when the string is created. The valid indexes
of a string are the exact non-negative integers less than
the length of the string. The first character of a string has
index 0, the second has index 1, and so on.

Returns #t if the two strings are the same length and contain the same characters in the same positions, otherwise
returns #f. String-ci=? treats upper and lower case letters as though they were the same character, but string=?
treats upper and lower case as distinct characters.

In phrases such as “the characters of string beginning with
index start and ending with index end ,” it is understood
that the index start is inclusive and the index end is exclusive. Thus if start and end are the same index, a null
substring is referred to, and if start is zero and end is the
length of string, then the entire string is referred to.

(string<? string1 string2 )
(string>? string1 string2 )
(string<=? string1 string2 )
(string>=? string1 string2 )
(string-ci<? string1 string2 )
(string-ci>? string1 string2 )
(string-ci<=? string1 string2 )
(string-ci>=? string1 string2 )

Some of the procedures that operate on strings ignore the
difference between upper and lower case. The versions that
ignore case have “-ci” (for “case insensitive”) embedded
in their names.
(string? obj )

procedure

Returns #t if obj is a string, otherwise returns #f.
(make-string k)
(make-string k char )

procedure
procedure

Make-string returns a newly allocated string of length k.
If char is given, then all elements of the string are initialized to char , otherwise the contents of the string are
unspecified.
(string char . . . )

library procedure

Returns a newly allocated string composed of the arguments.
(string-length string)

procedure

Returns the number of characters in the given string.
(string-ref string k)

procedure

k must be a valid index of string. String-ref returns
character k of string using zero-origin indexing.
(string-set! string k char )

procedure

k must be a valid index of string. String-set! stores char
in element k of string and returns an unspecified value.
(define (f) (make-string 3 #\*))
(define (g) "***")
(string-set! (f) 0 #\?)
=⇒ unspecified
(string-set! (g) 0 #\?)
=⇒ error
(string-set! (symbol->string ’immutable)
0
#\?)
=⇒ error

library procedure
library procedure

library
library
library
library
library
library
library
library

procedure
procedure
procedure
procedure
procedure
procedure
procedure
procedure

These procedures are the lexicographic extensions to
strings of the corresponding orderings on characters. For
example, string<? is the lexicographic ordering on strings
induced by the ordering char<? on characters. If two
strings differ in length but are the same up to the length
of the shorter string, the shorter string is considered to be
lexicographically less than the longer string.
Implementations may generalize these and the string=?
and string-ci=? procedures to take more than two arguments, as with the corresponding numerical predicates.
(substring string start end )

library procedure

String must be a string, and start and end must be exact
integers satisfying
0 ≤ start ≤ end ≤ (string-length string).
Substring returns a newly allocated string formed from
the characters of string beginning with index start (inclusive) and ending with index end (exclusive).
(string-append string . . . )

library procedure

Returns a newly allocated string whose characters form the
concatenation of the given strings.
(string->list string)
(list->string list)

library procedure
library procedure

String->list returns a newly allocated list of the characters that make up the given string. List->string returns
a newly allocated string formed from the characters in the
list list, which must be a list of characters. String->list
and list->string are inverses so far as equal? is concerned.
(string-copy string)

library procedure

Returns a newly allocated copy of the given string.

6. Standard procedures 31
(string-fill! string char )

library procedure

Stores char in every element of the given string and returns
an unspecified value.
6.3.6. Vectors
Vectors are heterogenous structures whose elements are indexed by integers. A vector typically occupies less space
than a list of the same length, and the average time required to access a randomly chosen element is typically
less for the vector than for the list.
The length of a vector is the number of elements that it
contains. This number is a non-negative integer that is
fixed when the vector is created. The valid indexes of a
vector are the exact non-negative integers less than the
length of the vector. The first element in a vector is indexed
by zero, and the last element is indexed by one less than
the length of the vector.
Vectors are written using the notation #(obj . . . ). For
example, a vector of length 3 containing the number zero
in element 0, the list (2 2 2 2) in element 1, and the
string "Anna" in element 2 can be written as following:
#(0 (2 2 2 2) "Anna")

Note that this is the external representation of a vector, not
an expression evaluating to a vector. Like list constants,
vector constants must be quoted:
’#(0 (2 2 2 2) "Anna")
=⇒ #(0 (2 2 2 2) "Anna")

(vector? obj )

procedure

(vector-ref ’#(1 1 2 3 5 8 13 21)
5)
=⇒ 8
(vector-ref ’#(1 1 2 3 5 8 13 21)
(let ((i (round (* 2 (acos -1)))))
(if (inexact? i)
(inexact->exact i)
i)))
=⇒ 13

(vector-set! vector k obj )

procedure

k must be a valid index of vector . Vector-set! stores obj
in element k of vector . The value returned by vector-set!
is unspecified.
(let ((vec (vector 0 ’(2 2 2 2) "Anna")))
(vector-set! vec 1 ’("Sue" "Sue"))
vec)
=⇒ #(0 ("Sue" "Sue") "Anna")
(vector-set! ’#(0 1 2) 1 "doe")
=⇒ error ; constant vector

(vector->list vector )
(list->vector list)

library procedure
library procedure

Vector->list returns a newly allocated list of the objects
contained in the elements of vector . List->vector returns
a newly created vector initialized to the elements of the list
list.
(vector->list ’#(dah dah didah))
=⇒ (dah dah didah)
(list->vector ’(dididit dah))
=⇒ #(dididit dah)

Returns #t if obj is a vector, otherwise returns #f.
(make-vector k )
(make-vector k fill )

procedure
procedure

Returns a newly allocated vector of k elements. If a second
argument is given, then each element is initialized to fill .
Otherwise the initial contents of each element is unspecified.
(vector obj . . . )

library procedure

Returns a newly allocated vector whose elements contain
the given arguments. Analogous to list.
(vector ’a ’b ’c)

=⇒

#(a b c)

(vector-fill! vector fill )

library procedure

Stores fill in every element of vector . The value returned
by vector-fill! is unspecified.

6.4. Control features
This chapter describes various primitive procedures which
control the flow of program execution in special ways. The
procedure? predicate is also described here.
(procedure? obj )

procedure

Returns #t if obj is a procedure, otherwise returns #f.
(vector-length vector )

procedure

Returns the number of elements in vector as an exact integer.
(vector-ref vector k )

procedure

k must be a valid index of vector . Vector-ref returns the
contents of element k of vector .

(procedure? car)
=⇒ #t
(procedure? ’car)
=⇒ #f
(procedure? (lambda (x) (* x x)))
=⇒ #t
(procedure? ’(lambda (x) (* x x)))
=⇒ #f
(call-with-current-continuation procedure?)
=⇒ #t

32

Revised5 Scheme

(apply proc arg1 . . . args)

procedure

Proc must be a procedure and args must be a list. Calls
proc with the elements of the list (append (list arg1
. . . ) args) as the actual arguments.
(apply + (list 3 4))

=⇒

7

(define compose
(lambda (f g)
(lambda args
(f (apply g args)))))
((compose sqrt *) 12 75)

=⇒

(map proc list1 list2 . . . )

30

library procedure

The lists must be lists, and proc must be a procedure taking
as many arguments as there are lists and returning a single
value. If more than one list is given, then they must all
be the same length. Map applies proc element-wise to the
elements of the lists and returns a list of the results, in
order. The dynamic order in which proc is applied to the
elements of the lists is unspecified.
(map cadr ’((a b) (d e) (g h)))
=⇒ (b e h)
(map (lambda (n) (expt n n))
’(1 2 3 4 5))
=⇒ (1 4 27 256 3125)
(map + ’(1 2 3) ’(4 5 6))

=⇒

(5 7 9)

(let ((count 0))
(map (lambda (ignored)
(set! count (+ count 1))
count)
’(a b)))
=⇒ (1 2) or (2 1)

(for-each proc list1 list2 . . . )

library procedure

The arguments to for-each are like the arguments to map,
but for-each calls proc for its side effects rather than for
its values. Unlike map, for-each is guaranteed to call proc
on the elements of the lists in order from the first element(s) to the last, and the value returned by for-each is
unspecified.
(let ((v (make-vector 5)))
(for-each (lambda (i)
(vector-set! v i (* i i)))
’(0 1 2 3 4))
v)
=⇒ #(0 1 4 9 16)

(force promise)

library procedure

Forces the value of promise (see delay, section 4.2.5). If
no value has been computed for the promise, then a value is

computed and returned. The value of the promise is cached
(or “memoized”) so that if it is forced a second time, the
previously computed value is returned.
(force (delay (+ 1 2)))
=⇒
(let ((p (delay (+ 1 2))))
(list (force p) (force p)))
=⇒

3

(3 3)

(define a-stream
(letrec ((next
(lambda (n)
(cons n (delay (next (+ n 1)))))))
(next 0)))
(define head car)
(define tail
(lambda (stream) (force (cdr stream))))
(head (tail (tail a-stream)))
=⇒

2

Force and delay are mainly intended for programs written
in functional style. The following examples should not be
considered to illustrate good programming style, but they
illustrate the property that only one value is computed for
a promise, no matter how many times it is forced.
(define count 0)
(define p
(delay (begin (set! count (+ count 1))
(if (> count x)
count
(force p)))))
(define x 5)
p
=⇒ a promise
(force p)
=⇒ 6
p
=⇒ a promise, still
(begin (set! x 10)
(force p))
=⇒ 6

Here is a possible implementation of delay and force.
Promises are implemented here as procedures of no arguments, and force simply calls its argument:
(define force
(lambda (object)
(object)))

We define the expression
(delay hexpressioni)

to have the same meaning as the procedure call
(make-promise (lambda () hexpressioni))

as follows
(define-syntax delay
(syntax-rules ()
((delay expression)
(make-promise (lambda () expression))))),

6. Standard procedures 33
where make-promise is defined as follows:
(define make-promise
(lambda (proc)
(let ((result-ready? #f)
(result #f))
(lambda ()
(if result-ready?
result
(let ((x (proc)))
(if result-ready?
result
(begin (set! result-ready? #t)
(set! result x)
result))))))))
Rationale: A promise may refer to its own value, as in the
last example above. Forcing such a promise may cause the
promise to be forced a second time before the value of the first
force has been computed. This complicates the definition of
make-promise.

Various extensions to this semantics of delay and force
are supported in some implementations:
• Calling force on an object that is not a promise may
simply return the object.
• It may be the case that there is no means by which
a promise can be operationally distinguished from its
forced value. That is, expressions like the following
may evaluate to either #t or to #f, depending on the
implementation:
(eqv? (delay 1) 1)
(pair? (delay (cons 1 2)))

=⇒
=⇒

unspecified
unspecified

• Some implementations may implement “implicit forcing,” where the value of a promise is forced by primitive procedures like cdr and +:

call-with-current-continuation. Except for continuations created by the call-with-values procedure, all continuations take exactly one value. The effect of passing no
value or more than one value to continuations that were
not created by call-with-values is unspecified.
The escape procedure that is passed to proc has unlimited
extent just like any other procedure in Scheme. It may be
stored in variables or data structures and may be called as
many times as desired.
The following examples show only the most common ways
in which call-with-current-continuation is used. If
all real uses were as simple as these examples, there
would be no need for a procedure with the power of
call-with-current-continuation.
(call-with-current-continuation
(lambda (exit)
(for-each (lambda (x)
(if (negative? x)
(exit x)))
’(54 0 37 -3 245 19))
#t))
=⇒ -3
(define list-length
(lambda (obj)
(call-with-current-continuation
(lambda (return)
(letrec ((r
(lambda (obj)
(cond ((null? obj) 0)
((pair? obj)
(+ (r (cdr obj)) 1))
(else (return #f))))))
(r obj))))))
(list-length ’(1 2 3 4))

=⇒

4

(list-length ’(a b . c))

=⇒

#f

Rationale:
(+ (delay (* 3 7)) 13)

=⇒

(call-with-current-continuation proc)

34

procedure

Proc must be a procedure of one argument. The procedure
call-with-current-continuation packages up the current continuation (see the rationale below) as an “escape
procedure” and passes it as an argument to proc. The escape procedure is a Scheme procedure that, if it is later
called, will abandon whatever continuation is in effect at
that later time and will instead use the continuation that
was in effect when the escape procedure was created. Calling the escape procedure may cause the invocation of before
and after thunks installed using dynamic-wind.
The escape procedure accepts the same number of arguments as the continuation to the original call to

A common use of call-with-current-continuation is for
structured, non-local exits from loops or procedure bodies, but
in fact call-with-current-continuation is extremely useful
for implementing a wide variety of advanced control structures.
Whenever a Scheme expression is evaluated there is a continuation wanting the result of the expression. The continuation
represents an entire (default) future for the computation. If the
expression is evaluated at top level, for example, then the continuation might take the result, print it on the screen, prompt
for the next input, evaluate it, and so on forever. Most of the
time the continuation includes actions specified by user code,
as in a continuation that will take the result, multiply it by the
value stored in a local variable, add seven, and give the answer
to the top level continuation to be printed. Normally these
ubiquitous continuations are hidden behind the scenes and programmers do not think much about them. On rare occasions,
however, a programmer may need to deal with continuations explicitly. Call-with-current-continuation allows Scheme pro-

34

Revised5 Scheme

grammers to do that by creating a procedure that acts just like
the current continuation.
Most programming languages incorporate one or more specialpurpose escape constructs with names like exit, return, or
even goto. In 1965, however, Peter Landin [16] invented a
general purpose escape operator called the J-operator. John
Reynolds [24] described a simpler but equally powerful construct in 1972. The catch special form described by Sussman
and Steele in the 1975 report on Scheme is exactly the same as
Reynolds’s construct, though its name came from a less general
construct in MacLisp. Several Scheme implementors noticed
that the full power of the catch construct could be provided by
a procedure instead of by a special syntactic construct, and the
name call-with-current-continuation was coined in 1982.
This name is descriptive, but opinions differ on the merits of
such a long name, and some people use the name call/cc instead.

(values obj . . .)

procedure

Delivers all of its arguments to its continuation. Except
for continuations created by the call-with-values procedure, all continuations take exactly one value. Values
might be defined as follows:
(define (values . things)
(call-with-current-continuation
(lambda (cont) (apply cont things))))

(call-with-values producer consumer )

procedure

Calls its producer argument with no values and a continuation that, when passed some values, calls the consumer
procedure with those values as arguments. The continuation for the call to consumer is the continuation of the call
to call-with-values.
(call-with-values (lambda () (values 4 5))
(lambda (a b) b))
=⇒ 5
(call-with-values * -)

=⇒

(dynamic-wind before thunk after )

-1

procedure

Calls thunk without arguments, returning the result(s) of
this call. Before and after are called, also without arguments, as required by the following rules (note that
in the absence of calls to continuations captured using
call-with-current-continuation the three arguments
are called once each, in order). Before is called whenever
execution enters the dynamic extent of the call to thunk
and after is called whenever it exits that dynamic extent.
The dynamic extent of a procedure call is the period between when the call is initiated and when it returns. In

Scheme, because of call-with-current-continuation,
the dynamic extent of a call may not be a single, connected
time period. It is defined as follows:
• The dynamic extent is entered when execution of the
body of the called procedure begins.
• The dynamic extent is also entered when execution is not within the dynamic extent and a
continuation is invoked that was captured (using
call-with-current-continuation) during the dynamic extent.
• It is exited when the called procedure returns.
• It is also exited when execution is within the dynamic
extent and a continuation is invoked that was captured
while not within the dynamic extent.
If a second call to dynamic-wind occurs within the dynamic
extent of the call to thunk and then a continuation is invoked in such a way that the after s from these two invocations of dynamic-wind are both to be called, then the after
associated with the second (inner) call to dynamic-wind is
called first.
If a second call to dynamic-wind occurs within the dynamic extent of the call to thunk and then a continuation is invoked in such a way that the befores from these
two invocations of dynamic-wind are both to be called,
then the before associated with the first (outer) call to
dynamic-wind is called first.
If invoking a continuation requires calling the before from
one call to dynamic-wind and the after from another, then
the after is called first.
The effect of using a captured continuation to enter or exit
the dynamic extent of a call to before or after is undefined.
(let ((path ’())
(c #f))
(let ((add (lambda (s)
(set! path (cons s path)))))
(dynamic-wind
(lambda () (add ’connect))
(lambda ()
(add (call-with-current-continuation
(lambda (c0)
(set! c c0)
’talk1))))
(lambda () (add ’disconnect)))
(if (< (length path) 4)
(c ’talk2)
(reverse path))))
=⇒ (connect talk1 disconnect
connect talk2 disconnect)

6. Standard procedures 35

6.5. Eval

6.6. Input and output
6.6.1. Ports

(eval expression environment-specifier )

procedure

Evaluates expression in the specified environment and returns its value. Expression must be a valid Scheme expression represented as data, and environment-specifier must
be a value returned by one of the three procedures described below. Implementations may extend eval to allow
non-expression programs (definitions) as the first argument
and to allow other values as environments, with the restriction that eval is not allowed to create new bindings
in the environments associated with null-environment or
scheme-report-environment.
(eval ’(* 7 3) (scheme-report-environment 5))
=⇒ 21
(let ((f (eval ’(lambda (f x) (f x x))
(null-environment 5))))
(f + 10))
=⇒ 20

(scheme-report-environment version)
(null-environment version)

procedure
procedure

Version must be the exact integer 5, corresponding to this
revision of the Scheme report (the Revised5 Report on
Scheme). Scheme-report-environment returns a specifier
for an environment that is empty except for all bindings defined in this report that are either required or both optional
and supported by the implementation. Null-environment
returns a specifier for an environment that is empty except
for the (syntactic) bindings for all syntactic keywords defined in this report that are either required or both optional
and supported by the implementation.
Other values of version can be used to specify environments
matching past revisions of this report, but their support is
not required. An implementation will signal an error if
version is neither 5 nor another value supported by the
implementation.
The effect of assigning (through the use of eval) a variable bound in a scheme-report-environment (for example car) is unspecified. Thus the environments specified
by scheme-report-environment may be immutable.
(interaction-environment)

optional procedure

This procedure returns a specifier for the environment that
contains implementation-defined bindings, typically a superset of those listed in the report. The intent is that this
procedure will return the environment in which the implementation would evaluate expressions dynamically typed
by the user.

Ports represent input and output devices. To Scheme, an
input port is a Scheme object that can deliver characters
upon command, while an output port is a Scheme object
that can accept characters.
(call-with-input-file string proc) library procedure
(call-with-output-file string proc) library procedure
String should be a string naming a file, and proc
should be a procedure that accepts one argument. For
call-with-input-file, the file should already exist; for
call-with-output-file, the effect is unspecified if the
file already exists. These procedures call proc with one argument: the port obtained by opening the named file for
input or output. If the file cannot be opened, an error is
signalled. If proc returns, then the port is closed automatically and the value(s) yielded by the proc is(are) returned.
If proc does not return, then the port will not be closed
automatically unless it is possible to prove that the port
will never again be used for a read or write operation.
Rationale:
Because Scheme’s escape procedures have unlimited extent, it is possible to escape from the current continuation but later to escape back in. If implementations
were permitted to close the port on any escape from the
current continuation, then it would be impossible to write
portable code using both call-with-current-continuation
and call-with-input-file or call-with-output-file.

(input-port? obj )
(output-port? obj )

procedure
procedure

Returns #t if obj is an input port or output port respectively, otherwise returns #f.
(current-input-port)
(current-output-port)

procedure
procedure

Returns the current default input or output port.
(with-input-from-file string thunk )
optional procedure
(with-output-to-file string thunk )
optional procedure
String should be a string naming a file, and proc should be
a procedure of no arguments. For with-input-from-file,
the file should already exist; for with-output-to-file,
the effect is unspecified if the file already exists. The
file is opened for input or output, an input or output
port connected to it is made the default value returned
by current-input-port or current-output-port (and is

36

Revised5 Scheme

used by (read), (write obj ), and so forth), and the thunk
is called with no arguments. When the thunk returns,
the port is closed and the previous default is restored.
With-input-from-file and with-output-to-file return(s) the value(s) yielded by thunk . If an escape procedure is used to escape from the continuation of these
procedures, their behavior is implementation dependent.

(open-input-file filename)

procedure

Takes a string naming an existing file and returns an input
port capable of delivering characters from the file. If the
file cannot be opened, an error is signalled.

(open-output-file filename)

procedure

Takes a string naming an output file to be created and
returns an output port capable of writing characters to a
new file by that name. If the file cannot be opened, an
error is signalled. If a file with the given name already
exists, the effect is unspecified.

(close-input-port port)
(close-output-port port)

procedure
procedure

Closes the file associated with port, rendering the port incapable of delivering or accepting characters. These routines have no effect if the file has already been closed. The
value returned is unspecified.

(read-char)
(read-char port)

procedure
procedure

Returns the next character available from the input port,
updating the port to point to the following character. If
no more characters are available, an end of file object is
returned. Port may be omitted, in which case it defaults
to the value returned by current-input-port.

(peek-char)
(peek-char port)

procedure
procedure

Returns the next character available from the input port,
without updating the port to point to the following character. If no more characters are available, an end of file
object is returned. Port may be omitted, in which case it
defaults to the value returned by current-input-port.
Note: The value returned by a call to peek-char is the same as
the value that would have been returned by a call to read-char
with the same port. The only difference is that the very next call
to read-char or peek-char on that port will return the value
returned by the preceding call to peek-char. In particular, a
call to peek-char on an interactive port will hang waiting for
input whenever a call to read-char would have hung.

(eof-object? obj )

procedure

Returns #t if obj is an end of file object, otherwise returns
#f. The precise set of end of file objects will vary among
implementations, but in any case no end of file object will
ever be an object that can be read in using read.

6.6.2. Input

(read)
(read port)

library procedure
library procedure

Read converts external representations of Scheme objects
into the objects themselves. That is, it is a parser for the
nonterminal hdatumi (see sections 7.1.2 and 6.3.2). Read
returns the next object parsable from the given input port,
updating port to point to the first character past the end
of the external representation of the object.
If an end of file is encountered in the input before any characters are found that can begin an object, then an end of
file object is returned. The port remains open, and further attempts to read will also return an end of file object.
If an end of file is encountered after the beginning of an
object’s external representation, but the external representation is incomplete and therefore not parsable, an error is
signalled.
The port argument may be omitted, in which case it defaults to the value returned by current-input-port. It is
an error to read from a closed port.

(char-ready?)
(char-ready? port)

procedure
procedure

Returns #t if a character is ready on the input port and
returns #f otherwise. If char-ready returns #t then the
next read-char operation on the given port is guaranteed
not to hang. If the port is at end of file then char-ready?
returns #t. Port may be omitted, in which case it defaults
to the value returned by current-input-port.
Rationale: Char-ready? exists to make it possible for a program to accept characters from interactive ports without getting
stuck waiting for input. Any input editors associated with such
ports must ensure that characters whose existence has been asserted by char-ready? cannot be rubbed out. If char-ready?
were to return #f at end of file, a port at end of file would
be indistinguishable from an interactive port that has no ready
characters.

6.6.3. Output

6. Standard procedures 37
(write obj )
(write obj port)

library procedure
library procedure

Writes a written representation of obj to the given port.
Strings that appear in the written representation are enclosed in doublequotes, and within those strings backslash
and doublequote characters are escaped by backslashes.
Character objects are written using the #\ notation. Write
returns an unspecified value. The port argument may be
omitted, in which case it defaults to the value returned by
current-output-port.
(display obj )
(display obj port)

library procedure
library procedure

Writes a representation of obj to the given port. Strings
that appear in the written representation are not enclosed
in doublequotes, and no characters are escaped within
those strings. Character objects appear in the representation as if written by write-char instead of by write.
Display returns an unspecified value. The port argument
may be omitted, in which case it defaults to the value returned by current-output-port.
Rationale: Write is intended for producing machine-readable
output and display is for producing human-readable output.
Implementations that allow “slashification” within symbols will
probably want write but not display to slashify funny characters in symbols.

(newline)
(newline port)

library procedure
library procedure

Writes an end of line to port. Exactly how this is done
differs from one operating system to another. Returns
an unspecified value. The port argument may be omitted, in which case it defaults to the value returned by
current-output-port.
(write-char char )
(write-char char port)

procedure
procedure

Writes the character char (not an external representation of the character) to the given port and returns an
unspecified value. The port argument may be omitted, in which case it defaults to the value returned by
current-output-port.
6.6.4. System interface
Questions of system interface generally fall outside of the
domain of this report. However, the following operations
are important enough to deserve description here.
(load filename)

optional procedure

Filename should be a string naming an existing file containing Scheme source code. The load procedure reads expressions and definitions from the file and evaluates them

sequentially. It is unspecified whether the results of the
expressions are printed. The load procedure does not
affect the values returned by current-input-port and
current-output-port. Load returns an unspecified value.
Rationale: For portability, load must operate on source files.
Its operation on other kinds of files necessarily varies among
implementations.

(transcript-on filename)
(transcript-off)

optional procedure
optional procedure

Filename must be a string naming an output file to be created. The effect of transcript-on is to open the named
file for output, and to cause a transcript of subsequent
interaction between the user and the Scheme system to
be written to the file. The transcript is ended by a call
to transcript-off, which closes the transcript file. Only
one transcript may be in progress at any time, though some
implementations may relax this restriction. The values returned by these procedures are unspecified.

38

7.

Revised5 Scheme

Formal syntax and semantics

7.1. Formal syntax

hsyntactic keywordi −→ hexpression keywordi
| else | => | define
| unquote | unquote-splicing
hexpression keywordi −→ quote | lambda | if
| set! | begin | cond | and | or | case
| let | let* | letrec | do | delay
| quasiquote

This section provides a formal syntax for Scheme written
in an extended BNF.

hvariablei −→ hany hidentifieri that isn’t
also a hsyntactic keywordii

This chapter provides formal descriptions of what has already been described informally in previous chapters of this
report.

All spaces in the grammar are for legibility. Case is insignificant; for example, #x1A and #X1a are equivalent. hemptyi
stands for the empty string.
The following extensions to BNF are used to make the description more concise: hthingi* means zero or more occurrences of hthingi; and hthingi+ means at least one hthingi.

hbooleani −→ #t | #f
hcharacteri −→ #\ hany characteri
| #\ hcharacter namei
hcharacter namei −→ space | newline
hstringi −→ " hstring elementi* "
hstring elementi −→ hany character other than " or \i
| \" | \\

7.1.1. Lexical structure
This section describes how individual tokens (identifiers,
numbers, etc.) are formed from sequences of characters.
The following sections describe how expressions and programs are formed from sequences of tokens.
hIntertoken spacei may occur on either side of any token,
but not within a token.
Tokens which require implicit termination (identifiers,
numbers, characters, and dot) may be terminated by any
hdelimiteri, but not necessarily by anything else.
The following five characters are reserved for future extensions to the language: [ ] { } |
htokeni −→ hidentifieri | hbooleani | hnumberi
| hcharacteri | hstringi
| ( | ) | #( | ’ | ` | , | ,@ | .
hdelimiteri −→ hwhitespacei | ( | ) | " | ;
hwhitespacei −→ hspace or newlinei
hcommenti −→ ; hall subsequent characters up to a
line breaki
hatmospherei −→ hwhitespacei | hcommenti
hintertoken spacei −→ hatmospherei*
hidentifieri −→ hinitiali hsubsequenti*
| hpeculiar identifieri
hinitiali −→ hletteri | hspecial initiali
hletteri −→ a | b | c | ... | z
hspecial initiali −→ ! | $ | % | & | * | / | : | < | =
| > | ? | ^ | _ | ~
hsubsequenti −→ hinitiali | hdigiti
| hspecial subsequenti
hdigiti −→ 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
hspecial subsequenti −→ + | - | . | @
hpeculiar identifieri −→ + | - | ...

hnumberi −→ hnum 2i| hnum 8i
| hnum 10i| hnum 16i
The following rules for hnum Ri, hcomplex Ri, hreal Ri,
hureal Ri, huinteger Ri, and hprefix Ri should be replicated for R = 2, 8, 10, and 16. There are no rules for
hdecimal 2i, hdecimal 8i, and hdecimal 16i, which means
that numbers containing decimal points or exponents must
be in decimal radix.
hnum Ri −→ hprefix Ri hcomplex Ri
hcomplex Ri −→ hreal Ri | hreal Ri @ hreal Ri
| hreal Ri + hureal Ri i | hreal Ri - hureal Ri i
| hreal Ri + i | hreal Ri - i
| + hureal Ri i | - hureal Ri i | + i | - i
hreal Ri −→ hsigni hureal Ri
hureal Ri −→ huinteger Ri
| huinteger Ri / huinteger Ri
| hdecimal Ri
hdecimal 10i −→ huinteger 10i hsuffixi
| . hdigit 10i+ #* hsuffixi
| hdigit 10i+ . hdigit 10i* #* hsuffixi
| hdigit 10i+ #+ . #* hsuffixi
huinteger Ri −→ hdigit Ri+ #*
hprefix Ri −→ hradix Ri hexactnessi
| hexactnessi hradix Ri
hsuffixi −→ hemptyi
| hexponent markeri hsigni hdigit 10i+
hexponent markeri −→ e | s | f | d | l
hsigni −→ hemptyi | + | hexactnessi −→ hemptyi | #i | #e
hradix 2i −→ #b
hradix 8i −→ #o
hradix 10i −→ hemptyi | #d

7. Formal syntax and semantics 39
hradix 16i −→ #x
hdigit 2i −→ 0 | 1
hdigit 8i −→ 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
hdigit 10i −→ hdigiti
hdigit 16i −→ hdigit 10i | a | b | c | d | e | f
7.1.2. External representations
hDatumi is what the read procedure (section 6.6.2) successfully parses. Note that any string that parses as an
hexpressioni will also parse as a hdatumi.
hdatumi −→ hsimple datumi | hcompound datumi
hsimple datumi −→ hbooleani | hnumberi
| hcharacteri | hstringi | hsymboli
hsymboli −→ hidentifieri
hcompound datumi −→ hlisti | hvectori
hlisti −→ (hdatumi*) | (hdatumi+ . hdatumi)
| habbreviationi
habbreviationi −→ habbrev prefixi hdatumi
habbrev prefixi −→ ’ | ` | , | ,@
hvectori −→ #(hdatumi*)
7.1.3. Expressions
hexpressioni −→ hvariablei
| hliterali
| hprocedure calli
| hlambda expressioni
| hconditionali
| hassignmenti
| hderived expressioni
| hmacro usei
| hmacro blocki
hliterali −→ hquotationi | hself-evaluatingi
hself-evaluatingi −→ hbooleani | hnumberi
| hcharacteri | hstringi
hquotationi −→ ’hdatumi | (quote hdatumi)
hprocedure calli −→ (hoperatori hoperandi*)
hoperatori −→ hexpressioni
hoperandi −→ hexpressioni
hlambda expressioni −→ (lambda hformalsi hbodyi)
hformalsi −→ (hvariablei*) | hvariablei
| (hvariablei+ . hvariablei)
hbodyi −→ hdefinitioni* hsequencei
hsequencei −→ hcommandi* hexpressioni
hcommandi −→ hexpressioni
hconditionali −→ (if htesti hconsequenti halternatei)
htesti −→ hexpressioni
hconsequenti −→ hexpressioni
halternatei −→ hexpressioni | hemptyi

hassignmenti −→ (set! hvariablei hexpressioni)
hderived expressioni −→
(cond hcond clausei+ )
| (cond hcond clausei* (else hsequencei))
| (case hexpressioni
hcase clausei+ )
| (case hexpressioni
hcase clausei*
(else hsequencei))
| (and htesti*)
| (or htesti*)
| (let (hbinding speci*) hbodyi)
| (let hvariablei (hbinding speci*) hbodyi)
| (let* (hbinding speci*) hbodyi)
| (letrec (hbinding speci*) hbodyi)
| (begin hsequencei)
| (do (hiteration speci*)
(htesti hdo resulti)
hcommandi*)
| (delay hexpressioni)
| hquasiquotationi
hcond clausei −→ (htesti hsequencei)
| (htesti)
| (htesti => hrecipienti)
hrecipienti −→ hexpressioni
hcase clausei −→ ((hdatumi*) hsequencei)
hbinding speci −→ (hvariablei hexpressioni)
hiteration speci −→ (hvariablei hiniti hstepi)
| (hvariablei hiniti)
hiniti −→ hexpressioni
hstepi −→ hexpressioni
hdo resulti −→ hsequencei | hemptyi
hmacro usei −→ (hkeywordi hdatumi*)
hkeywordi −→ hidentifieri
hmacro blocki −→
(let-syntax (hsyntax speci*) hbodyi)
| (letrec-syntax (hsyntax speci*) hbodyi)
hsyntax speci −→ (hkeywordi htransformer speci)

7.1.4. Quasiquotations
The following grammar for quasiquote expressions is not
context-free. It is presented as a recipe for generating an
infinite number of production rules. Imagine a copy of the
following rules for D = 1, 2, 3, . . .. D keeps track of the
nesting depth.
hquasiquotationi −→ hquasiquotation 1i
hqq template 0i −→ hexpressioni

40

Revised5 Scheme

hquasiquotation Di −→ `hqq template Di
| (quasiquote hqq template Di)
hqq template Di −→ hsimple datumi
| hlist qq template Di
| hvector qq template Di
| hunquotation Di
hlist qq template Di −→ (hqq template or splice Di*)
| (hqq template or splice Di+ . hqq template Di)
| ’hqq template Di
| hquasiquotation D + 1i
hvector qq template Di −→ #(hqq template or splice Di*)
hunquotation Di −→ ,hqq template D − 1i
| (unquote hqq template D − 1i)
hqq template or splice Di −→ hqq template Di
| hsplicing unquotation Di
hsplicing unquotation Di −→ ,@hqq template D − 1i
| (unquote-splicing hqq template D − 1i)
In hquasiquotationis, a hlist qq template Di can
times be confused with either an hunquotation
a hsplicing unquotation Di. The interpretation
hunquotationi or hsplicing unquotation Di takes
dence.

someDi or
as an
prece-

7.1.5. Transformers
htransformer speci −→
(syntax-rules (hidentifieri*) hsyntax rulei*)
hsyntax rulei −→ (hpatterni htemplatei)
hpatterni −→ hpattern identifieri
| (hpatterni*)
| (hpatterni+ . hpatterni)
| (hpatterni* hpatterni hellipsisi)
| #(hpatterni*)
| #(hpatterni* hpatterni hellipsisi)
| hpattern datumi
hpattern datumi −→ hstringi
| hcharacteri
| hbooleani
| hnumberi
htemplatei −→ hpattern identifieri
| (htemplate elementi*)
| (htemplate elementi+ . htemplatei)
| #(htemplate elementi*)
| htemplate datumi
htemplate elementi −→ htemplatei
| htemplatei hellipsisi
htemplate datumi −→ hpattern datumi
hpattern identifieri −→ hany identifier except ...i
hellipsisi −→ hthe identifier ...i

7.1.6. Programs and definitions
hprogrami −→ hcommand or definitioni*

hcommand or definitioni −→ hcommandi
| hdefinitioni
| hsyntax definitioni
| (begin hcommand or definitioni+ )
hdefinitioni −→ (define hvariablei hexpressioni)
| (define (hvariablei hdef formalsi) hbodyi)
| (begin hdefinitioni*)
hdef formalsi −→ hvariablei*
| hvariablei* . hvariablei
hsyntax definitioni −→
(define-syntax hkeywordi htransformer speci)

7.2. Formal semantics
This section provides a formal denotational semantics for
the primitive expressions of Scheme and selected built-in
procedures. The concepts and notation used here are described in [29]; the notation is summarized below:
h...i
s↓k
#s
s§t
s†k
t → a, b
ρ[x/i]
x in D
x|D

sequence formation
kth member of the sequence s (1-based)
length of sequence s
concatenation of sequences s and t
drop the first k members of sequence s
McCarthy conditional “if t then a else b”
substitution “ρ with x for i”
injection of x into domain D
projection of x to domain D

The reason that expression continuations take sequences
of values instead of single values is to simplify the formal
treatment of procedure calls and multiple return values.
The boolean flag associated with pairs, vectors, and strings
will be true for mutable objects and false for immutable
objects.
The order of evaluation within a call is unspecified. We
mimic that here by applying arbitrary permutations permute and unpermute, which must be inverses, to the arguments in a call before and after they are evaluated. This is
not quite right since it suggests, incorrectly, that the order
of evaluation is constant throughout a program (for any
given number of arguments), but it is a closer approximation to the intended semantics than a left-to-right evaluation would be.
The storage allocator new is implementation-dependent,
but it must obey the following axiom: if new σ ∈ L, then
σ (new σ | L) ↓ 2 = false.
The definition of K is omitted because an accurate definition of K would complicate the semantics without being
very interesting.
If P is a program in which all variables are defined before
being referenced or assigned, then the meaning of P is
E[[((lambda (I*) P’) hundefinedi . . . )]]

7. Formal syntax and semantics 41
where I* is the sequence of variables defined in P, P0 is the
sequence of expressions obtained by replacing every definition in P by an assignment, hundefinedi is an expression
that evaluates to undefined, and E is the semantic function
that assigns meaning to expressions.
7.2.1. Abstract syntax
K
I
E
Γ

∈
∈
∈
∈

Con
Ide
Exp
Com = Exp

Exp −→
|
|
|
|
|

constants, including quotations
identifiers (variables)
expressions
commands

K | I | (E0 E*)
(lambda (I*) Γ* E0 )
(lambda (I* . I) Γ* E0 )
(lambda I Γ* E0 )
(if E0 E1 E2 ) | (if E0 E1 )
(set! I E)

7.2.2. Domain equations
α
ν

∈
∈

φ


∈

σ
ρ
θ
κ

∈

∈

∈
∈
∈

L
N
T
Q
H
R
Ep
Ev
Es
M

=

=
=
=
=

F =
E =
S
U
C
K
A
X

=
=
=
=

locations
natural numbers
{false, true}
booleans
symbols
characters
numbers
L×L×T
pairs
L* × T
vectors
L* × T
strings
{false, true, null, undefined, unspecified}
miscellaneous
L × (E* → K → C) procedure values
Q + H + R + Ep + E v + Es + M + F
expressed values
L → (E × T)
stores
Ide → L
environments
S→A
command continuations
E* → C
expression continuations
answers
errors

7.2.3. Semantic functions
K:
E:
E* :
C:

Con → E
Exp → U → K → C
Exp* → U → K → C
Com* → U → C → C

E[[I]] = λρκ . hold (lookup ρ I)
(single(λ .  = undefined →
wrong “undefined variable”,
send  κ))
E[[(E0 E*)]] =
λρκ . E*(permute(hE0 i § E*))
ρ
(λ* . ((λ* . applicate (* ↓ 1) (* † 1) κ)
(unpermute *)))
E[[(lambda (I*) Γ* E0 )]] =
λρκ . λσ .
new σ ∈ L →
send (hnew σ | L,
λ*κ0 . #* = #I* →
tievals(λα* . (λρ0 . C[[Γ*]]ρ0 (E[[E0 ]]ρ0 κ0 ))
(extends ρ I* α*))
*,
wrong “wrong number of arguments”i
in E)
κ
(update (new σ | L) unspecified σ),
wrong “out of memory” σ
E[[(lambda (I* . I) Γ* E0 )]] =
λρκ . λσ .
new σ ∈ L →
send (hnew σ | L,
λ*κ0 . #* ≥ #I* →
tievalsrest
(λα* . (λρ0 . C[[Γ*]]ρ0 (E[[E0 ]]ρ0 κ0 ))
(extends ρ (I* § hIi) α*))
*
(#I*),
wrong “too few arguments”i in E)
κ
(update (new σ | L) unspecified σ),
wrong “out of memory” σ
E[[(lambda I Γ* E0 )]] = E[[(lambda (. I) Γ* E0 )]]
E[[(if E0 E1 E2 )]] =
λρκ . E[[E0 ]] ρ (single (λ . truish  → E[[E1 ]]ρκ,
E[[E2 ]]ρκ))
E[[(if E0 E1 )]] =
λρκ . E[[E0 ]] ρ (single (λ . truish  → E[[E1 ]]ρκ,
send unspecified κ))
Here and elsewhere, any expressed value other than undefined
may be used in place of unspecified.
E[[(set! I E)]] =
λρκ . E[[E]] ρ (single(λ . assign (lookup ρ I)

(send unspecified κ)))
E*[[ ]] = λρκ . κh i
E*[[E0 E*]] =
λρκ . E[[E0 ]] ρ (single(λ0 . E*[[E*]] ρ (λ* . κ (h0 i § *))))

Definition of K deliberately omitted.

C[[ ]] = λρθ . θ

E[[K]] = λρκ . send (K[[K]]) κ

C[[Γ0 Γ*]] = λρθ . E[[Γ0 ]] ρ (λ* . C[[Γ*]]ρθ)

42

Revised5 Scheme

7.2.4. Auxiliary functions
lookup : U → Ide → L
lookup = λρI . ρI
extends : U → Ide* → L* → U
extends =
λρI*α* . #I* = 0 → ρ,
extends (ρ[(α* ↓ 1)/(I* ↓ 1)]) (I* † 1) (α* † 1)
wrong : X → C

[implementation-dependent]

send : E → K → C
send = λκ . κhi
single : (E → C) → K
single =
λψ* . #* = 1 → ψ(* ↓ 1),
wrong “wrong number of return values”
new : S → (L + {error})

[implementation-dependent]

hold : L → K → C
hold = λακσ . send (σα ↓ 1)κσ
assign : L → E → C → C
assign = λαθσ . θ(update ασ)
update : L → E → S → S
update = λασ . σ[h, truei/α]
tievals : (L* → C) → E* → C
tievals =
λψ*σ . #* = 0 → ψh iσ,
new σ ∈ L → tievals (λα* . ψ(hnew σ | Li § α*))
(* † 1)
(update(new σ | L)(* ↓ 1)σ),
wrong “out of memory”σ
tievalsrest : (L* → C) → E* → N → C
tievalsrest =
λψ*ν . list (dropfirst *ν)
(single(λ . tievals ψ ((takefirst *ν) § hi)))
dropfirst = λln . n = 0 → l, dropfirst (l † 1)(n − 1)
takefirst = λln . n = 0 → h i, hl ↓ 1i § (takefirst (l † 1)(n − 1))
truish : E → T
truish = λ .  = false → false, true
permute : Exp* → Exp*
unpermute : E* → E*

[implementation-dependent]
[inverse of permute]

applicate : E → E* → K → C
applicate =
λ*κ .  ∈ F → ( | F ↓ 2)*κ, wrong “bad procedure”
onearg : (E → K → C) → (E* → K → C)
onearg =
λζ*κ . #* = 1 → ζ(* ↓ 1)κ,
wrong “wrong number of arguments”
twoarg : (E → E → K → C) → (E* → K → C)
twoarg =
λζ*κ . #* = 2 → ζ(* ↓ 1)(* ↓ 2)κ,
wrong “wrong number of arguments”

list : E* → K → C
list =
λ*κ . #* = 0 → send null κ,
list (* † 1)(single(λ . consh* ↓ 1, iκ))
cons : E* → K → C
cons =
twoarg (λ1 2 κσ . new σ ∈ L →
(λσ 0 . new σ 0 ∈ L →
send (hnew σ | L, new σ 0 | L, truei
in E)
κ
(update(new σ 0 | L)2 σ 0 ),
wrong “out of memory”σ 0 )
(update(new σ | L)1 σ),
wrong “out of memory”σ)
less : E* → K → C
less =
twoarg (λ1 2 κ . (1 ∈ R ∧ 2 ∈ R) →
send (1 | R < 2 | R → true, false)κ,
wrong “non-numeric argument to <”)
add : E* → K → C
add =
twoarg (λ1 2 κ . (1 ∈ R ∧ 2 ∈ R) →
send ((1 | R + 2 | R) in E)κ,
wrong “non-numeric argument to +”)
car : E* → K → C
car =
onearg (λκ .  ∈ Ep → hold ( | Ep ↓ 1)κ,
wrong “non-pair argument to car”)
cdr : E* → K → C

[similar to car]

setcar : E* → K → C
setcar =
twoarg (λ1 2 κ . 1 ∈ Ep →
(1 | Ep ↓ 3) → assign (1 | Ep ↓ 1)
2
(send unspecified κ),
wrong “immutable argument to set-car!”,
wrong “non-pair argument to set-car!”)
eqv : E* → K → C
eqv =
twoarg (λ1 2 κ . (1 ∈ M ∧ 2 ∈ M) →
send (1 | M = 2 | M → true, false)κ,
(1 ∈ Q ∧ 2 ∈ Q) →
send (1 | Q = 2 | Q → true, false)κ,
(1 ∈ H ∧ 2 ∈ H) →
send (1 | H = 2 | H → true, false)κ,
(1 ∈ R ∧ 2 ∈ R) →
send (1 | R = 2 | R → true, false)κ,
(1 ∈ Ep ∧ 2 ∈ Ep ) →
send ((λp1 p2 . ((p1 ↓ 1) = (p2 ↓ 1)∧
(p1 ↓ 2) = (p2 ↓ 2)) → true,
false)
(1 | Ep )
(2 | Ep ))
κ,

7. Formal syntax and semantics 43
(1 ∈ Ev ∧ 2 ∈ Ev ) → . . . ,
(1 ∈ Es ∧ 2 ∈ Es ) → . . . ,
(1 ∈ F ∧ 2 ∈ F) →
send ((1 | F ↓ 1) = (2 | F ↓ 1) → true, false)
κ,
send false κ)
apply : E* → K → C
apply =
twoarg (λ1 2 κ . 1 ∈ F → valueslist h2 i(λ* . applicate 1 *κ),
wrong “bad procedure argument to apply”)
valueslist : E* → K → C
valueslist =
onearg (λκ .  ∈ Ep →
cdrhi
(λ* . valueslist
*
(λ* . carhi(single(λ . κ(hi § *))))),
 = null → κh i,
wrong “non-list argument to values-list”)
cwcc : E* → K → C
[call-with-current-continuation]
cwcc =
onearg (λκ .  ∈ F →
(λσ . new σ ∈ L →
applicate 
hhnew σ | L, λ*κ0 . κ*i in Ei
κ
(update (new σ | L)
unspecified
σ),
wrong “out of memory” σ),
wrong “bad procedure argument”)
values : E* → K → C
values = λ*κ . κ*
cwv : E* → K → C
[call-with-values]
cwv =
twoarg (λ1 2 κ . applicate 1 h i(λ* . applicate 2 *))

((cond (test)) test)
((cond (test) clause1 clause2 ...)
(let ((temp test))
(if temp
temp
(cond clause1 clause2 ...))))
((cond (test result1 result2 ...))
(if test (begin result1 result2 ...)))
((cond (test result1 result2 ...)
clause1 clause2 ...)
(if test
(begin result1 result2 ...)
(cond clause1 clause2 ...)))))
(define-syntax case
(syntax-rules (else)
((case (key ...)
clauses ...)
(let ((atom-key (key ...)))
(case atom-key clauses ...)))
((case key
(else result1 result2 ...))
(begin result1 result2 ...))
((case key
((atoms ...) result1 result2 ...))
(if (memv key ’(atoms ...))
(begin result1 result2 ...)))
((case key
((atoms ...) result1 result2 ...)
clause clauses ...)
(if (memv key ’(atoms ...))
(begin result1 result2 ...)
(case key clause clauses ...)))))
(define-syntax and
(syntax-rules ()
((and) #t)
((and test) test)
((and test1 test2 ...)
(if test1 (and test2 ...) #f))))

7.3. Derived expression types
This section gives macro definitions for the derived expression types in terms of the primitive expression types (literal, variable, call, lambda, if, set!). See section 6.4 for
a possible definition of delay.
(define-syntax cond
(syntax-rules (else =>)
((cond (else result1 result2 ...))
(begin result1 result2 ...))
((cond (test => result))
(let ((temp test))
(if temp (result temp))))
((cond (test => result) clause1 clause2 ...)
(let ((temp test))
(if temp
(result temp)
(cond clause1 clause2 ...))))

(define-syntax or
(syntax-rules ()
((or) #f)
((or test) test)
((or test1 test2 ...)
(let ((x test1))
(if x x (or test2 ...))))))
(define-syntax let
(syntax-rules ()
((let ((name val) ...) body1 body2 ...)
((lambda (name ...) body1 body2 ...)
val ...))
((let tag ((name val) ...) body1 body2 ...)
((letrec ((tag (lambda (name ...)
body1 body2 ...)))
tag)

44

Revised5 Scheme
val ...))))
(define-syntax let*
(syntax-rules ()
((let* () body1 body2 ...)
(let () body1 body2 ...))
((let* ((name1 val1) (name2 val2) ...)
body1 body2 ...)
(let ((name1 val1))
(let* ((name2 val2) ...)
body1 body2 ...)))))

The following letrec macro uses the symbol <undefined>
in place of an expression which returns something that
when stored in a location makes it an error to try to obtain the value stored in the location (no such expression is
defined in Scheme). A trick is used to generate the temporary names needed to avoid specifying the order in which
the values are evaluated. This could also be accomplished
by using an auxiliary macro.
(define-syntax letrec
(syntax-rules ()
((letrec ((var1 init1) ...) body ...)
(letrec "generate temp names"
(var1 ...)
()
((var1 init1) ...)
body ...))
((letrec "generate temp names"
()
(temp1 ...)
((var1 init1) ...)
body ...)
(let ((var1 <undefined>) ...)
(let ((temp1 init1) ...)
(set! var1 temp1)
...
body ...)))
((letrec "generate temp names"
(x y ...)
(temp ...)
((var1 init1) ...)
body ...)
(letrec "generate temp names"
(y ...)
(newtemp temp ...)
((var1 init1) ...)
body ...))))
(define-syntax begin
(syntax-rules ()
((begin exp ...)
((lambda () exp ...)))))

The following alternative expansion for begin does not
make use of the ability to write more than one expression

in the body of a lambda expression. In any case, note that
these rules apply only if the body of the begin contains no
definitions.
(define-syntax begin
(syntax-rules ()
((begin exp)
exp)
((begin exp1 exp2 ...)
(let ((x exp1))
(begin exp2 ...)))))

The following definition of do uses a trick to expand the
variable clauses. As with letrec above, an auxiliary macro
would also work. The expression (if #f #f) is used to
obtain an unspecific value.
(define-syntax do
(syntax-rules ()
((do ((var init step ...) ...)
(test expr ...)
command ...)
(letrec
((loop
(lambda (var ...)
(if test
(begin
(if #f #f)
expr ...)
(begin
command
...
(loop (do "step" var step ...)
...))))))
(loop init ...)))
((do "step" x)
x)
((do "step" x y)
y)))

Example 45

NOTES

EXAMPLE
Integrate-system integrates the system

Language changes

yk0 = fk (y1 , y2 , . . . , yn ), k = 1, . . . , n

This section enumerates the changes that have been made
to Scheme since the “Revised4 report” [6] was published.

of differential equations with the method of Runge-Kutta.

• The report is now a superset of the IEEE standard
for Scheme [13]: implementations that conform to the
report will also conform to the standard. This required
the following changes:

The parameter system-derivative is a function that
takes a system state (a vector of values for the state variables y1 , . . . , yn ) and produces a system derivative (the values y10 , . . . , yn0 ). The parameter initial-state provides
an initial system state, and h is an initial guess for the
length of the integration step.

– The empty list is now required to count as true.
– The classification of features as essential or
inessential has been removed. There are now
three classes of built-in procedures: primitive, library, and optional. The optional procedures are
load, with-input-from-file, with-outputto-file, transcript-on, transcript-off, and
interaction-environment, and - and / with
more than two arguments. None of these are in
the IEEE standard.
– Programs are allowed to redefine built-in procedures. Doing so will not change the behavior of
other built-in procedures.
• Port has been added to the list of disjoint types.
• The macro appendix has been removed. High-level
macros are now part of the main body of the report.
The rewrite rules for derived expressions have been
replaced with macro definitions. There are no reserved
identifiers.
• Syntax-rules now allows vector patterns.
• Multiple-value returns, eval, and dynamic-wind have
been added.
• The calls that are required to be implemented in a
properly tail-recursive fashion are defined explicitly.
• ‘@’ can be used within identifiers. ‘|’ is reserved for
possible future extensions.

ADDITIONAL MATERIAL
The Internet Scheme Repository at
http://www.cs.indiana.edu/scheme-repository/
contains an extensive Scheme bibliography, as well as papers, programs, implementations, and other material related to Scheme.

The value returned by integrate-system is an infinite
stream of system states.
(define integrate-system
(lambda (system-derivative initial-state h)
(let ((next (runge-kutta-4 system-derivative h)))
(letrec ((states
(cons initial-state
(delay (map-streams next
states)))))
states))))

Runge-Kutta-4 takes a function, f, that produces a system
derivative from a system state. Runge-Kutta-4 produces
a function that takes a system state and produces a new
system state.
(define runge-kutta-4
(lambda (f h)
(let ((*h (scale-vector h))
(*2 (scale-vector 2))
(*1/2 (scale-vector (/ 1 2)))
(*1/6 (scale-vector (/ 1 6))))
(lambda (y)
;; y is a system state
(let* ((k0 (*h (f y)))
(k1 (*h (f (add-vectors y (*1/2 k0)))))
(k2 (*h (f (add-vectors y (*1/2 k1)))))
(k3 (*h (f (add-vectors y k2)))))
(add-vectors y
(*1/6 (add-vectors k0
(*2 k1)
(*2 k2)
k3))))))))
(define elementwise
(lambda (f)
(lambda vectors
(generate-vector
(vector-length (car vectors))
(lambda (i)
(apply f
(map (lambda (v) (vector-ref
vectors)))))))
(define generate-vector
(lambda (size proc)

v i))

46

Revised5 Scheme
(let ((ans (make-vector size)))
(letrec ((loop
(lambda (i)
(cond ((= i size) ans)
(else
(vector-set! ans i (proc i))
(loop (+ i 1)))))))
(loop 0)))))

(define add-vectors (elementwise +))
(define scale-vector
(lambda (s)
(elementwise (lambda (x) (* x s)))))

Map-streams is analogous to map: it applies its first argument (a procedure) to all the elements of its second argument (a stream).
(define map-streams
(lambda (f s)
(cons (f (head s))
(delay (map-streams f (tail s))))))

Infinite streams are implemented as pairs whose car holds
the first element of the stream and whose cdr holds a
promise to deliver the rest of the stream.

REFERENCES
[1] Harold Abelson and Gerald Jay Sussman with Julie
Sussman. Structure and Interpretation of Computer
Programs, second edition. MIT Press, Cambridge,
1996.
[2] Alan Bawden and Jonathan Rees. Syntactic closures.
In Proceedings of the 1988 ACM Symposium on Lisp
and Functional Programming, pages 86–95.
[3] Robert G. Burger and R. Kent Dybvig. Printing
floating-point numbers quickly and accurately. In
Proceedings of the ACM SIGPLAN ’96 Conference
on Programming Language Design and Implementation, pages 108–116.
[4] William Clinger, editor. The revised revised report
on Scheme, or an uncommon Lisp. MIT Artificial
Intelligence Memo 848, August 1985. Also published
as Computer Science Department Technical Report
174, Indiana University, June 1985.

(define head car)
(define tail
(lambda (stream) (force (cdr stream))))

[5] William Clinger. How to read floating point numbers
accurately. In Proceedings of the ACM SIGPLAN
’90 Conference on Programming Language Design
and Implementation, pages 92–101. Proceedings published as SIGPLAN Notices 25(6), June 1990.

The following illustrates the use of integrate-system in
integrating the system

[6] William Clinger and Jonathan Rees, editors. The
revised4 report on the algorithmic language Scheme.
In ACM Lisp Pointers 4(3), pages 1–55, 1991.

C

dvC
vC
= −iL −
dt
R

diL
= vC
dt
which models a damped oscillator.
L

(define damped-oscillator
(lambda (R L C)
(lambda (state)
(let ((Vc (vector-ref state 0))
(Il (vector-ref state 1)))
(vector (- 0 (+ (/ Vc (* R C)) (/ Il C)))
(/ Vc L))))))
(define the-states
(integrate-system
(damped-oscillator 10000 1000 .001)
’#(1 0)
.01))

[7] William Clinger and Jonathan Rees. Macros that
work. In Proceedings of the 1991 ACM Conference
on Principles of Programming Languages, pages 155–
162.
[8] William Clinger. Proper Tail Recursion and Space
Efficiency. To appear in Proceedings of the 1998 ACM
Conference on Programming Language Design and
Implementation, June 1998.
[9] R. Kent Dybvig, Robert Hieb, and Carl Bruggeman.
Syntactic abstraction in Scheme. Lisp and Symbolic
Computation 5(4):295–326, 1993.
[10] Carol Fessenden, William Clinger, Daniel P. Friedman, and Christopher Haynes. Scheme 311 version 4
reference manual. Indiana University Computer Science Technical Report 137, February 1983. Superseded by [11].
[11] D. Friedman, C. Haynes, E. Kohlbecker, and
M. Wand. Scheme 84 interim reference manual. Indiana University Computer Science Technical Report
153, January 1985.

References 47
[12] IEEE Standard 754-1985. IEEE Standard for Binary
Floating-Point Arithmetic. IEEE, New York, 1985.
[13] IEEE Standard 1178-1990. IEEE Standard for the
Scheme Programming Language. IEEE, New York,
1991.
[14] Eugene E. Kohlbecker Jr. Syntactic Extensions in
the Programming Language Lisp. PhD thesis, Indiana University, August 1986.
[15] Eugene E. Kohlbecker Jr., Daniel P. Friedman,
Matthias Felleisen, and Bruce Duba. Hygienic macro
expansion. In Proceedings of the 1986 ACM Conference on Lisp and Functional Programming, pages
151–161.
[16] Peter Landin. A correspondence between Algol 60
and Church’s lambda notation: Part I. Communications of the ACM 8(2):89–101, February 1965.
[17] MIT Department of Electrical Engineering and Computer Science. Scheme manual, seventh edition.
September 1984.
[18] Peter Naur et al. Revised report on the algorithmic language Algol 60. Communications of the ACM
6(1):1–17, January 1963.
[19] Paul Penfield, Jr. Principal values and branch cuts
in complex APL. In APL ’81 Conference Proceedings, pages 248–256. ACM SIGAPL, San Francisco, September 1981. Proceedings published as
APL Quote Quad 12(1), ACM, September 1981.
[20] Kent M. Pitman. The revised MacLisp manual (Saturday evening edition). MIT Laboratory for Computer Science Technical Report 295, May 1983.
[21] Jonathan A. Rees and Norman I. Adams IV. T: A
dialect of Lisp or, lambda: The ultimate software
tool. In Conference Record of the 1982 ACM Symposium on Lisp and Functional Programming, pages
114–122.
[22] Jonathan A. Rees, Norman I. Adams IV, and James
R. Meehan. The T manual, fourth edition. Yale
University Computer Science Department, January
1984.
[23] Jonathan Rees and William Clinger, editors. The
revised3 report on the algorithmic language Scheme.
In ACM SIGPLAN Notices 21(12), pages 37–79, December 1986.
[24] John Reynolds. Definitional interpreters for higher
order programming languages. In ACM Conference
Proceedings, pages 717–740. ACM, 1972.

[25] Guy Lewis Steele Jr. and Gerald Jay Sussman. The
revised report on Scheme, a dialect of Lisp. MIT Artificial Intelligence Memo 452, January 1978.
[26] Guy Lewis Steele Jr. Rabbit: a compiler for Scheme.
MIT Artificial Intelligence Laboratory Technical Report 474, May 1978.
[27] Guy Lewis Steele Jr. Common Lisp: The Language,
second edition. Digital Press, Burlington MA, 1990.
[28] Gerald Jay Sussman and Guy Lewis Steele Jr.
Scheme: an interpreter for extended lambda calculus. MIT Artificial Intelligence Memo 349, December
1975.
[29] Joseph E. Stoy. Denotational Semantics: The ScottStrachey Approach to Programming Language Theory. MIT Press, Cambridge, 1977.
[30] Texas Instruments, Inc. TI Scheme Language Reference Manual. Preliminary version 1.0, November
1985.

48

Revised5 Scheme

ALPHABETIC INDEX OF DEFINITIONS OF CONCEPTS,
KEYWORDS, AND PROCEDURES
The principal entry for each term, procedure, or keyword is
listed first, separated from the other entries by a semicolon.
! 5
’ 8; 26
* 22
+ 22; 5, 42
, 13; 26
,@ 13
- 22; 5
-> 5
... 5; 14
/ 22
; 5
< 21; 42
<= 21
= 21; 22
=> 10
> 21
>= 21
? 4
‘ 13
abs 22; 24
acos 23
and 11; 43
angle 24
append 27
apply 32; 8, 43
asin 23
assoc 27
assq 27
assv 27
atan 23
#b 21; 38
backquote 13
begin 12; 16, 44
binding 6
binding construct 6
boolean? 25; 6
bound 6
caar 26
cadr 26
call 9
call by need 13
call-with-current-continuation 33; 8, 34, 43
call-with-input-file 35
call-with-output-file 35
call-with-values 34; 8, 43
call/cc 34
car 26; 42

case 10; 43
catch 34
cdddar 26
cddddr 26
cdr 26
ceiling 23
char->integer 29
char-alphabetic? 29
char-ci<=? 29
char-ci<? 29
char-ci=? 29
char-ci>=? 29
char-ci>? 29
char-downcase 29
char-lower-case? 29
char-numeric? 29
char-ready? 36
char-upcase 29
char-upper-case? 29
char-whitespace? 29
char<=? 29
char<? 29
char=? 29
char>=? 29
char>? 29
char? 29; 6
close-input-port 36
close-output-port 36
combination 9
comma 13
comment 5; 38
complex? 21; 19
cond 10; 15, 43
cons 26
constant 7
continuation 33
cos 23
current-input-port 35
current-output-port 35
#d 21
define 16; 14
define-syntax 17
definition 16
delay 13; 32
denominator 23
display 37
do 12; 44
dotted pair 25
dynamic-wind 34; 33
#e 21; 38

Index 49
else 10
empty list 25; 6, 26
eof-object? 36
eq? 18; 10
equal? 19
equivalence predicate 17
eqv? 17; 7, 10, 42
error 4
escape procedure 33
eval 35; 8
even? 22
exact 17
exact->inexact 24
exact? 21
exactness 19
exp 23
expt 24
#f 25
false 6; 25
floor 23
for-each 32
force 32; 13
gcd 23
hygienic 13
#i 21; 38
identifier 5; 6, 28, 38
if 10; 41
imag-part 24
immutable 7
implementation restriction 4; 20
improper list 26
inexact 17
inexact->exact 24; 20
inexact? 21
initial environment 17
input-port? 35
integer->char 29
integer? 21; 19
interaction-environment 35
internal definition 16
keyword 13; 38
lambda 9; 16, 41
lazy evaluation 13
lcm 23
length 27; 20
let 11; 12, 15, 16, 43
let* 11; 16, 44
let-syntax 14; 16
letrec 11; 16, 44
letrec-syntax 14; 16

library 3
library procedure 17
list 27
list->string 30
list->vector 31
list-ref 27
list-tail 27
list? 26
load 37
location 7
log 23
macro 13
macro keyword 13
macro transformer 13
macro use 13
magnitude 24
make-polar 24
make-rectangular 24
make-string 30
make-vector 31
map 32
max 22
member 27
memq 27
memv 27
min 22
modulo 22
mutable 7
negative? 22
newline 37
nil 25
not 25
null-environment 35
null? 26
number 19
number->string 24
number? 21; 6, 19
numerator 23
numerical types 19
#o 21; 38
object 3
odd? 22
open-input-file 36
open-output-file 36
optional 3
or 11; 43
output-port? 35
pair 25
pair? 26; 6
peek-char 36
port 35
port? 6

50

Revised5 Scheme

positive? 22
predicate 17
procedure call 9
procedure? 31; 6
promise 13; 32
proper tail recursion 7
quasiquote 13; 26
quote 8; 26
quotient 22
rational? 21; 19
rationalize 23
read 36; 26, 39
read-char 36
real-part 24
real? 21; 19
referentially transparent 13
region 6; 10, 11, 12
remainder 22
reverse 27
round 23
scheme-report-environment 35
set! 10; 16, 41
set-car! 26
set-cdr! 26
setcar 42
simplest rational 23
sin 23
sqrt 24
string 30
string->list 30
string->number 24
string->symbol 28
string-append 30
string-ci<=? 30
string-ci<? 30
string-ci=? 30
string-ci>=? 30
string-ci>? 30
string-copy 30
string-fill! 31
string-length 30; 20
string-ref 30
string-set! 30; 28
string<=? 30
string<? 30
string=? 30
string>=? 30
string>? 30
string? 30; 6
substring 30
symbol->string 28; 7
symbol? 28; 6
syntactic keyword 6; 5, 13, 38

syntax definition 17
syntax-rules 14; 17
#t 25
tail call 7
tan 23
token 38
top level environment 17; 6
transcript-off 37
transcript-on 37
true 6; 10, 25
truncate 23
type 6
unbound 6; 8, 16
unquote 13; 26
unquote-splicing 13; 26
unspecified 4
valid indexes 30; 31
values 34; 9
variable 6; 5, 8, 38
vector 31
vector->list 31
vector-fill! 31
vector-length 31; 20
vector-ref 31
vector-set! 31
vector? 31; 6
whitespace 5
with-input-from-file 35
with-output-to-file 35
write 37; 13
write-char 37
#x 21; 39
zero? 22