Implementing Explicit and Finding Implicit Sharing in
Embedded DSLs
Oleg Kiselyov
oleg@okmij.org

Aliasing, or sharing, is prominent in many domains, denoting that two differently-named objects
are in fact identical: a change in one object (memory cell, circuit terminal, disk block) is instantly
reflected in the other. Languages for modelling such domains should let the programmer explicitly
define the sharing among objects or expressions. A DSL compiler may find other identical expressions and share them, implicitly. Such common subexpression elimination is crucial to the efficient
implementation of DSLs. Sharing is tricky in embedded DSL, since host aliasing may correspond to
copying of the underlying objects rather than their sharing.
This tutorial summarizes discussions of implementing sharing in Haskell DSLs for automotive
embedded systems and hardware description languages. The technique has since been used in a
Haskell SAT solver and the DSL for music synthesis. We demonstrate the embedding in pure Haskell
of a simple DSL with a language form for explicit sharing. The DSL also has implicit sharing,
implemented via hash-consing. Explicit sharing greatly speeds up hash-consing. The seemingly
imperative nature of hash-consing is hidden beneath a simple combinator language. The overall
implementation remains pure functional and easy to reason about.

I think all DSLs suffer from the same problems: sharing and recursion. I’ve used wrappers for CSound, SuperCollider, MetaPost, they all have these problems. Henning Thielemann [19]

1

Introduction

We present implicit and explicit sharing in the original context of embedded domains-specific language
(DSL) compilers. The sharing implementation techniques have since found other uses, e.g., writing SAT
solvers [3]. Embedded compilers – typical for circuit description, embedded control systems or GPU
programming DSLs – complement the familiar embeddings of a DSL in a host language as a library
or an interpreter. For example, we may build a circuit model in Haskell using gate descriptions and
combinators provided by the DSL library. We may test the circuit in Haskell by running the model on
sample inputs. Eventually we have to produce a Verilog code or a Netlist to manufacture the circuit.
Likewise, a control system DSL program should eventually be compiled into machine code and burned
as firmware. An embedded compiler thus is a host language program that turns a DSL program into a
(lower-level) code.
One of the important tasks of a compiler is the so-called “common subexpression elimination” –
detecting subexpressions denoting the same computation and arranging for that computation to be performed once and the results shared. This optimization (significantly) improves both the running time and
compactness of the code and is particularly important for hardware description and firmware compilers.
As DSL implementers, it becomes our duty to detect duplicate subexpressions and have them shared.
We call this detection implicit sharing, to contrast with the sharing explicitly declared by users. Imperative languages, where it matters whether an expression or its result are duplicated, have to provide a
Olivier Danvy, Chung-chieh Shan (Eds.): IFIP Working Conference
on Domain-Specific Languages 2011 (DSL 2011).
EPTCS 66, 2011, pp. 210–225, doi:10.4204/EPTCS.66.11

c Oleg Kiselyov
This work is licensed under the
Creative Commons Attribution License.

Oleg Kiselyov

211

form (some sort of a local variable introduction) for the programmer to declare the sharing of expression’s result. In the present paper we limit ourselves to side-effect–free expressions such as arithmetic
expressions or combinational circuits. Explicit sharing is still important: sharing declarations can significantly reduce the search space for common subexpressions and prevent exponential explosions. We
shall cite several examples later. Sharing declarations also help human readers of the program, drawing
their attention to the ‘common’ computations.
A common pitfall in implementing explicit sharing is attempting to use the let form of the host
language to express sharing in the DSL code. §5 will show that although the let -form may speed up
some DSL interpretations, it leads to code duplication rather than sharing in the compilation result.
Before we get to that discussion, we introduce our running example in §2, describing an unsuccessful
attempt to detect sharing in the course of implementing real DSLs. Show-stopping was the expression
comparison, which, in pure language is structural and requires the full traversal of expressions. In §3 we
review the main approaches to speed-up the comparison, all relying on some sort of ‘pointer’ equality.
Our method of expression comparison and sharing detection is presented in §4. The method is a pure
veneer over hash-consing, alleviating the cost of comparison. Its main benefit is the facilitation of explicit
sharing, described in §5.
The code accompanying the paper is available online at http://okmij.org/ftp/tagless-final/
sharing/.

2

Detecting sharing: necessity and difficulty

Our running example is a show-stopping problem encountered by the implementer of an embedded DSL
compiler for an embedded control system, posed in [12]. The example illustrates the need and the
difficulty of common subexpression elimination. The problem was posed for arithmetic expressions,
which are part of nearly every DSL. Typically arithmetic expressions are embedded in Haskell as the
values of the datatype1
data Exp
= Add Exp Exp
| Variable String
| Constant Int
deriving (Eq, Ord, Show)
Here are the sample expressions in our DSL:
exp a = Add (Constant 10) (Variable ”i1 ”)
exp b = Add exp a (Variable ”i2 ”)
The implementer wanted to compile these DSL expressions into C or the machine code, using the
standard approach of finding common subexpressions and sharing them. To make the sharing explicit,
the expression tree is converted into a directed acyclic graph (DAG). The graph is then topologically
sorted and each subexpression is assigned a C operation or a machine instruction.
The first step, detecting identical subexpressions, was the most troublesome. The step is crucial since
common subexpressions are abound, being easy to create. We show two real-life examples, to be used
throughout the paper. The first example is the multiplication by a known integer constant. Our DSL
does not have the multiplication operation (8-bit CPUs rarely have the needed instruction). Nevertheless,
1 See

the file ExpI.hs in the accompaniment for the complete code.

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we can multiply a DSL expression by the known constant using repeated addition. Here is the standard
efficient procedure based on the recursive subdivision:
mul :
mul 0
mul 1
mul n
mul n

Int → Exp → Exp
= Constant 0
x =x
x | n ‘ mod‘ 2 == 0 = mul (n ‘div‘ 2) (Add x x)
x = Add x (mul (n−1) x)

The result of the sample expression
exp mul4 = mul 4 (Variable ”i1 ”)
shows two identical subexpressions of adding the variable i1 to itself:
Add (Add (Variable ”i1 ”) (Variable ”i1 ”)) (Add (Variable ”i1 ”) (Variable ”i1 ”))
The result of mul 8 (Variable ”i1 ”) shows twice as much duplication.
The other running example, from the domain of hardware description, is sklansky by Naylor [15],
with further credit to Sheeran and Axelsson. The example computes the running sum of given expressions; like the previous multiplication example, sklansky uses recursive subdivision to expose more
parallelism and reduce latency:
sklansky :
sklansky f
sklansky f
sklansky f
where
(left ,
left ’
right ’

(a → a → a) → [ a] → [ a]
[] = []
[ x] = [x]
xs = left ’ ++ [f (last left ’) r | r ← right ’]
right ) = splitAt (length xs ‘ div‘ 2) xs
= sklansky f left
= sklansky f right

The pretty-printed result of sklansky Add (map (Variable ◦show) [1..4]
[ ”v1”,”(v1+v2)”,”((v1+v2)+v3)”,”((v1+v2)+(v3+v4))”]
demonstrates the triplication of the subexpression v1+v2. The duplication should be eliminated when
we build the circuit.
In the process of converting expressions like exp mul4 to a DAG, the implementer [12] had to compare subexpressions. It is there he encountered a problem: in a pure language, we may only compare
datatype values structurally. General pointer comparison destroys referential transparency and parametricity. To check that the summands of the top-level addition in exp mul4 are identical, we have to
therefore traverse them completely. Such comparisons of two expression trees take more and more time
with larger programs (say, as we multiply by bigger integers). “As these trees grow in size, the equality
comparison in graph construction quickly becomes the bottleneck for DSL compilation. What’s worse,
the phase transition from tractable to intractable is very sharp. In one of my DSL programs, I made a
seemingly small change, and compilation time went from milliseconds to not-in-a-million-years.”[12].
His message, entitled “I love purity, but it’s killing me” was a cry for help: he was about to give up on
Haskell. He wondered how to dramatically speed-up comparisons, or find a better method of detecting
common subexpressions and sharing them.

Oleg Kiselyov

3

213

Pointer comparison

Sharing detection, specifically, fast identity comparison of expressions, is a common metaprogramming
problem and has been investigated extensively. This section reviews the main approaches, which are
all based on some sort of ‘pointer’ equality. They associate with an expression a unique datum, e.g., an
integer, admitting efficient comparison. For instance, we may define the data type of labeled expressions,
where each variant carries a unique integer label2 :
type Lab = Int
data ExpL
= AddL Lab ExpL ExpL
| VariableL Lab String
| ConstantL Lab Int
deriving Show
ExpL expressions are fast to compare, by comparing their labels. The full traversal of expressions is no
longer needed:
instance Eq ExpL where
e1 == e2 = label e1 == label e2
where
label (AddL p )
=p
label (ConstantL p ) = p
label (VariableL p ) = p
The first approach to building labeled expressions is manual labeling. For example, we construct our
sample expressions as follows, taking great care to pick unique labels:
expL a = AddL 3 (ConstantL 1 10) (VariableL 2 ”i1 ”)
expL b = AddL 4 expL a (VariableL 5 ”i2”)
Needless to say this approach is greatly error-prone, let alone tedious. When implementing the mul
example we stumble on another complication: the manual threading of a counter to generate unique
labels. Some sort of automation is direly needed.
A promising and increasingly popular way to hide and automate label assignment is Template Haskell
(see the thesis [1] for the extensive discussion). A DSL implemented in Template Haskell quotations can
hardly be called ‘embedded’ however. Another approach is the State monad hiding the counter used for
the generation of unique labels:
type ExpM = State Lab ExpL
new labelM : State Lab Lab
new labelM = do
p ← get
put (p+1)
return p
run expM : ExpM →ExpL
run expM m = evalState m 0
2 The

complete code for this section is in the file Ptrs.hs.

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The computation new labelM, or ‘gensym’, yields a unique label; the function run expM runs the
monadic computation returning the produced labeled expression.
To hide the labeling of expression’s constructors, we build expressions with ‘constructor functions’,
often called ‘smart constructors’:
varM : String → ExpM
varM v = new labelM >>=\p →return $ VariableL p v
constM : Int → ExpM
constM x = new labelM >>=\p →return $ ConstantL p x
addM : ExpL → ExpL → ExpM
addM x y = new labelM >>=\p →return $ AddL p x y
The sample expression looks awful
expM a = do
xv ← constM 10
yv ← varM ”i1”
addM xv yv
although appropriate combinators may improve the appearance. In fact, the multiplication example below is quite perspicuous: the labeling is well-hidden (compare with mul in §2). The occasional return
and the ‘call-by-value application’ (=<<) betray the effectful computation:
mulM : Int → ExpL → ExpM
mulM 0 = constM 0
mulM 1 x = return x
mulM n x | n ‘ mod‘ 2 == 0 = mulM (n ‘div‘ 2) =<<addM x x
mulM n x = addM x =<<mulM (n−1) x
The subexpression addM x x on the last-but-one line builds the addition expression from two identically named summands. The two occurrences of the Haskell variable x denote the same ExpL expression, with the same label. These identical summands are thus easily identifiable as shared. Indeed, the
multiplication-by-4 computation
expM mul4 = mulM 4 =<<varM ”i1”
when run, yields the labeled expression
AddL 2
(AddL 1 (VariableL 0 ”i1 ”) (VariableL 0 ”i1 ”))
(AddL 1 (VariableL 0 ”i1 ”) (VariableL 0 ”i1 ”))
with the clearly seen sharing: just look at the labels.
Exercise 1 Why we did not define addM as follows?
addM : ExpM →ExpM →ExpM
addM x y = do
xv ← x
yv ← y
p ← new labelM
return $ AddL p xv yv

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215

After all, it does let us write the sample expression expM a concisely:
expM a = addM (constM 10) (varM ”i1”)
Hint: implement mulM and see the result of expM mul4.
The monadic approach may seem adequate – to us, DSL implementers. It does not however feel
right to our users, domain experts. They are accustomed to writing and manipulating arithmetic expressions as familiar mathematical objects. Now they have to deal with effectful computations. O’Donnell,
summarizing the dissatisfaction of hardware designers with such a monadic DSL, wrote: “A more severe
problem is that the circuit specification is no longer a system of simultaneous equations, which can be
manipulated formally just by ’substituting equals for equals’. Instead, the specification is now a sequence
of computations that – when executed – will yield the desired circuit. It feels like writing an imperative
program to draw a circuit, instead of defining the circuit directly.”[17]. The hardware description DSL
Lava has tried the monadic approach and abandoned it.
The most popular approach to detect sharing is a so-called ‘observable sharing’ [6], which hides the
labeling further, to the point of breaking Haskell and using internal, unsafe operations of GHC. Fresh
label generation is such a benign effect. The breaking of the referential transparency is hardly noticeable,
one may argue [6, 16]. There are many variations of observable sharing [1, 6, 16]: some rely on IORef
cells as labels (they can be compared efficiently as pointers), some use gensym. We demonstrate the
latter, and define gensym, or new label, as a supposedly pure, ordinary Haskell function:
{−# NOINLINE counter #−}
counter = unsafePerformIO (newIORef 0)
new label : () → Int
new label () = unsafePerformIO $ do
p ← readIORef counter
writeIORef counter (p+1)
return p
The function is certainly not pure since each evaluation of new label () (should) yield a new result.
Exercise 2 Why do we need ‘NOINLINE’?
Smart constructors again hide the labeling. Now they have pure types, yielding DSL expressions
ExpL themselves rather than expression computations ExpM:
varU : String → ExpL
varU v = VariableL (new label ()) v
constU : Int → ExpL
constU x = ConstantL (new label ()) x
addU : ExpL → ExpL → ExpL
addU x y = AddL (new label ()) x y
The sample expressions look quite like those in §2, with no traces of labeling, with no leaking of
implementation details into the domain-specific abstraction:
expU a = addU (constU 10) (varU ”i1”)
expU b = addU expU a (varU ”i2”)

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The multiplication example also looks just like the pure version in §2, defining multiplication with
familiar mathematical equations.
mulU : Int → ExpL → ExpL
mulU 0 = constU 0
mulU 1 x = x
mulU n x | n ‘ mod‘ 2 == 0 = mulU (n ‘div‘ 2) (addU x x)
mulU n x = addU x (mulU (n−1) x)
The result of the multiplication by four:
expU mul4 = mulU 4 (varU ”i1”)
shows that identically-named expressions have identical labels and are hence clearly shared
AddL 0 (AddL 1 (VariableL 2 ”i1”) (VariableL 2 ”i1 ”))
(AddL 1 (VariableL 2 ”i1 ”) (VariableL 2 ”i1 ”))
One look at the labels is enough to see the sharing.
Breaking the referential transparency and lying to the compiler about effects of our functions may
come to haunt us however:
Exercise 3 Why cannot we η-reduce the smart constructors as follows?
constU = ConstantL (new label ())
varU = VariableL (new label ())
addU = AddL (new label ())
Hint: try it.
Observable sharing will no longer be used in this paper.

4

Pure Hash-Consing

Hash-consing is a well-established technique to identify and share structurally equal data [10]. Although
the name comes from Lisp, the technique has been first described prior to Lisp, in 1957 [9]. The technique
relies on the global mutable hash table mapping structured values to integer hashes, which are quick to
compare. Value constructors check the table to see if the equal value has been constructed already,
returning the found hash. We describe hash-consing for a DSL embedded in pure, safe Haskell2010.
The imperative details of hash-consing are hidden better in a final-tagless style of the DSL embedding,
described next. The complete code for this section is in the file ExpF.hs in the accompanying code.

4.1

Tagless-final embedding

In the tagless-final approach [5], embedded DSL expressions are built with ‘constructor functions’ such
as constant, variable , add rather than the data constructors Constant, Variable , Add that we have
seen in §2. The constructor functions yield a representation for the DSL expression being built. The
representation could be a string (for pretty-printing), an integer (for evaluator), etc. Since the same DSL
expression may be concretely represented in several ways, the constructor functions are polymorphic,
parameterized by the representation repr . In other words, the constructor functions are the members of
the type class

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217

class Exp repr where
constant : Int → repr Int
variable : String → repr Int
add
: repr Int → repr Int → repr Int
The apparent difference from the datatype Exp of §2 is the lower case of the ‘constructors’. We have
parameterized the representation by expression’s type, as common [5]. We did not have to, since the type
so far has been the same, Int. The parameterization by the type will come handy once we add boolean
expressions.
The sample expressions from §2 look almost the same:
exp a = add (constant 10) (variable ”i1 ”)
exp b = add exp a (variable ”i2 ”)
differing only in the lower case of the ‘constructors’.
The datatype Exp from §2 is one concrete (so-called ‘initial’) representation of the DSL expressions –
one instantiation of repr :
newtype ExpI t = ExpI Exp
instance Exp ExpI where
constant = ExpI ◦Constant
variable = ExpI ◦Variable
add (ExpI x) (ExpI y) = ExpI (Add x y)
Exercise 4 Why do we need the wrapper ExpI?
Interpreting repr as ExpI lets us pretty-print final-tagless expressions, thanks to the derived Show
instance for the data type Exp:
test shb = case exp b of ExpI e → e
−− Add (Add (Constant 10) (Variable ”i1”)) (Variable ”i2 ”)
The multiplication example is largely unchanged, modulo the lower-case of the constructors and the
type signature:
mul : Exp repr ⇒ Int → repr Int → repr Int
mul 0 = constant 0
···
exp mul4 = mul 4 (variable ”i1 ”)
The conversion to ExpI took the form of an instance of the class Exp providing the interpretation
for the expression primitives, as the values of the domain ExpI. We may write other interpretations, for
example, the evaluator, interpreting an expression as an element of the domain R
type REnv = [(String,Int)]
newtype R t = R{unR :REnv →t} −− A reader Monad
that is, an integer in the environment giving the values for the free variables occurring in the expression.
instance Exp R
constant x =
variable x =
add e1 e2 =

where
R (\ → x)
R (\env → maybe (error $ ”no var: ” ++ x) id $ lookup x env)
R (\env → unR e1 env +unR e2 env)

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The evaluator lets us test the multiplication-by-4 example:
test val4 = unR exp mul4 [(”i1”,5)] −− 20
Exercise 5 Add subtraction and negation to the language. Can we get by without changing the type
class Exp, that is, without breaking the existing code?

4.2

Detecting implicit sharing

Recall that our goal is to detect structurally equal subexpressions and share them, converting an expression tree into a DAG. The goal is closer if we construct an expression as a DAG to start with. We
represent the DAG as a collection of Nodes identified by NodeIds, which link the nodes:
type NodeId = Int
data Node = NConst Int
| NVar String
| NAdd NodeId NodeId
deriving (Eq,Ord,Show)
The Node data type resembles Exp from §2; however, Node is not a recursive data type and can be
compared in constant time. The mapping between Nodes and NodeIds is realized through a BiMap
interface:
data BiMap a −− abstract
lookup key : Ord a ⇒ a → BiMap a →Maybe Int
lookup val : Int → BiMap a →a
insert
: Ord a ⇒ a → BiMap a →(Int, BiMap a)
empty
: BiMap a
BiMap a establishes a bijection between the values of the type a and integers, with the operations to
retrieve the value given its key, to find the key for the existing value, and to extend the bijection with a
new association. The type a should at least permit equality comparison; in the present implementation,
we require a to be a member of Ord. BiMaps can be pretty-printed. Our DAG thus is as follows:
newtype DAG = DAG (BiMap Node) deriving Show
Having settled on the DAG implementation we now describe its construction, which is easier bottomup. As we construct a node for a subexpression, we check if the DAG already has the equal node. If so, we
return its NodeId; otherwise, we add the node to the DAG. This procedure is nothing but hash-consing.
In fact, it is quite close to Ershov’s original description of hash-consing [9]; our DAG representation is
also similar to his. The BiMap interface has exactly the right operations, to check for the presence of a
node and to insert the node, allocating a new NodeId. Since the DAG is being modified as new nodes are
built, the construction procedure is the State monad computation with the DAG as the state.
The bottom-up DAG construction maps well to computing a representation for a tagless-final expression, which is also evaluated bottom-up. The DAG construction can therefore be written as a tagless-final
interpreter, an instance of the type class Exp. The interpreter maps a tagless-final expression to the concrete representation that is a NodeId in the current DAG:
newtype N t = N{unN :State DAG NodeId}
run expN : N t → (NodeId, DAG)
run expN (N m) = runState m (DAG empty)

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219

The function run expN runs the DAG-construction interpreter and returns the node, as a reference within
a DAG. The construction algorithm is codified as follows
instance Exp N where
constant x = N(hashcons $ NConst x)
variable x = N(hashcons $ NVar x)
add e1 e2 = N(do
h1 ← unN e1
h2 ← unN e2
hashcons $ NAdd h1 h2)
with the auxiliary hashcons doing the hash-consing, inserting the Node in the DAG if it has not been
there already.
hashcons : Node →State DAG NodeId
hashcons e = do
DAG m ←get
case lookup key e m of
Nothing →let (k, m’) = insert e m
in put (DAG m’)  return k
Just k → return k
In §4.1 we have defined sample tagless-final expressions exp mul4 and exp mul8 for the multiplication by 4 and 8 and interpreted them in several ways, as an integer value and an expression tree. We now
interpret the very same expressions as DAGs: run expN exp mul4 produces the result
(2, DAG BiMap[(0,NVar ”i1”),(1,NAdd 0 0),(2,NAdd 1 1)])
whereas run expN exp mul8 gives
(3, DAG BiMap[(0,NVar ”i1”),(1,NAdd 0 0),(2,NAdd 1 1),(3,NAdd 2 2)])
A DAG is printed as the list of (NodeId,Node) associations. The sharing of the left and right summands
is patent,
The shown results are in fact netlists: a low-level representation of a circuit listing the gates and
their connections, used in circuit manufacturing. Since our BiMap allocated monotonically increasing
NodeIds, the resulting netlist comes out topologically sorted. Therefore, we can straightforwardly generate machine code after the standard register allocation.
Exercise 6 The method add in the Exp N instance looks quite like the function addM in Exercise 1. The
addM function was flawed. Why does add work?
The imperative nature of hash-consing is well-hidden behind the pure final-tagless interface. We
stress this point with the sklansky example, of computing the running sum of several expressions. The
function sklansky was defined in §2 with the signature sklansky : (a → a → a) → [ a] → [ a] . In
particular, sklansky Add applied to the list of four variables produced the following list of (prettyprinted) expressions
[ ”v1”,”(v1+v2)”,”((v1+v2)+v3)”,”((v1+v2)+(v3+v4))”]
We will use the very same sklansky to build a ‘DAG forest’: the list of nodes sharing children
within the same DAG. The result is emphatically not the list of isolated DAGs. Rather, we return the

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Sharing in embedded DSLs

list of NodeIds all referring to the same DAG structure, sharing the components not only within the
same expression but also across independent expressions. The result might seem impossible yet is easily
achievable: we build the list of DAG-constructing computations and then run them in sequence, with
the same DAG state:
test sklansky n = runState sk (DAG empty)
where
sk = sequence (map unN (sklansky add xs))
xs = map (variable ◦show) [1.. n]
Until the very end, the monadic nature of the DAG construction was hidden. We manipulated taglessfinal expressions as pure, mapping and passing them around without any regard for their possible effects.
Sharing will be detected nevertheless. The running sum for the list of four variables, test sklansky 4,
now reads
([0,2,4,7],
DAG BiMap[
(0, NVar ”1”),(1,NVar ”2”),(2,NAdd 0 1),
(3, NVar ”3”),(4,NAdd 2 3),
(5, NVar ”4”),(6,NAdd 3 5),
(7, NAdd 2 6)])
The repeated expression v1+v2, represented by NodeId 2, is built only once and referenced at three
places.
Exercise 7 Our current implementation of BiMap relies on a pair of finite maps. We could have used
a highly optimized hash table from the Haskell standard library. However, hash table operations are
performed in the IO monad. Can we accommodate such mutable hash tables without leaking the IO
monad, maintaining the form of the Exp N interpreter and the purity of tagless-final expressions?
We have demonstrated the sharing detection technique that represents a DSL program as a DAG,
eliminating multiple occurrences of common subexpressions. Alas, to find all these common subexpressions we have to examine the entire expression tree, which may take long time for large programs (large
circuits). The next section describes this problem and its solution by explicit sharing.

5

Explicit sharing

This section motivates the extension of the DSL with a syntactic form to explicitly indicate expression
sharing, and describes its implementation. This form lets the programmer state their view of a computation as ‘common’, to be executed once and its result shared. The programmer thus helps the DSL
compiler as well as the human readers of the code.
The case for the sharing form as part of the pure, non-imperative DSL is compelling but subtle.
At first blush, the host language let form seems sufficient – and it is, for some eDSL interpreters.
The tagless-final DSL embedding helps clarify the subtlety. Our running example will be the familiar
multiplication-by-4, written explicitly below:3
exp mul4 =
let x = variable ”i1 ” in
3

The complete code for this section is in the file ExpLet.hs.

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221

let y = add x x in
add y y
The two occurrences of the variable y refer to the same tagless-final expression (namely, add x x). Such
a representation of a repeated expression by a variable makes the code compact. One may also expect
that in the internal representation of exp mul4, the two arguments of add refer to the same run-time
object.
Exercise 8 Is there any guarantee, in the Haskell Report or the GHC documentation that two occurrences of the same variable refer to the common shared object rather than duplicated objects?
Recall that add x x is a Haskell computation producing a particular representation of the DSL expression. What is shared in exp mul4 is the computation rather than the representation. This sharing of
computations, along with the memoization inherent in GHC, speeds up DSL interpretations. It appears
therefore that the Haskell let is sufficient to express explicit sharing: it makes sharing easy to see in the
code and it speeds up interpretations. The following two tagless-final interpreters show that the Haskell
let may indeed speed up some interpretations, but not the others.
The first interpreter computes the size of an expression, in the number of its constructors:
newtype Size t = Size Int
instance Exp Size where
constant = Size 1
variable
= Size 1
add (Size x) (Size y) = Size (x+y+1)
The computed size of exp mul4 is 7. When the computation add x x is first referred to from y, the
computation will be performed and its result memoized. The second reference to the same computation
via y will use the already determined result. The call-by-need evaluation strategy of GHC performs
shared computations only once. Therefore, computing the size of even mul (2ˆ30) (variable ”i1 ”) is
instantaneous. Although that Haskell expression denotes a large DSL expression tree, the computation
over the tree is compactly represented and is fast to perform.
The other interpreter prints a DSL expression
newtype Print t = Print (IO ())
instance Exp Print where
constant = Print ◦ putStr ◦show
variable = Print ◦ putStr
add (Print x) (Print y) = Print (x  putStr ” + ”  y)
Printing exp mul4 gives i1 + i1 + i1 + i1 , with the subexpression i1 +i1 duplicated. The duplication is
no surprise since our DSL has no sharing form and hence no way to indicate the sharing in the print-out.
We stress that the printing of i1 +i1 was done two times. As before, the computation to print add x x
was shared, and yet it has evidently been performed twice.
Exercise 9 Why the Size t computations were memoized but Print t computations apparently were
not? Is there is something special about IO? What about other monads, such as State, implemented as
pure functions?
The Print interpreter will therefore take a long time to print mul (2ˆ30) (variable ”i1 ”), even if we
redirect the output to /dev/null. Thus for some DSL interpreters including the DAG-constructing interpreter and almost any other DSL compiler, the running time will be at least proportional to the size of the

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DSL expression rather to the size of the Haskell code to construct the expression. The compactly written
Haskell code may represent exponentially large DSL expressions.
Compacting DSL expressions themselves requires a sharing form in the DSL itself. We call the
form let and add it to our DSL by defining the type class ExpLet. (Tagless-final embedded DSLs are
extended by introducing a new type class that describes the syntax of the new syntactic form. Such a
change does not break the existing expressions written in the non-extended DSL.)
class ExpLet repr where
let : repr a → (repr a → repr b) → repr b
The let of DSL, like the let of Haskell, expresses sharing through a local variable binding; multiple
occurrences of the let -bound variable within the let body all refer to the same expression, the one to
which the variable is bound. The DAG constructing interpretation of let , described below, will make
clear that a let -bound variable refers to the result of the shared expression. The form let is a binding
form, and is embedded in Haskell using the higher-order abstract syntax, with Haskell’s λ -bound variable
representing the DSL local variable. As an example of let , we re-write exp mul4 indicating the sharing
explicitly:
exp mul4’ =
let (variable ”i1 ”) (\x →
let (add x x)
(\y→
add y y))
We tell the existing tagless-final interpreters how to deal with let . For example, the R interpreter
treats let as the flipped application:
instance ExpLet R where
let x f = f x
The evaluation of the sample expressions
val mul4 = unR exp mul4 [( ”i1 ”,5)] −− 20
val mul4’ = unR exp mul4’ [(”i1”,5)] −− 20
shows that exp mul4 with and without explicit sharing evaluate to the same results. After all, sharing (of
pure expressions) is an optimization and should not affect the results of DSL programs.
Exercise 10 Extend the Size interpreter to account for explicit sharing (that is, write the instance
ExpLet Size). The size of shared expressions should be counted only once.
To ‘see’ the sharing, we need a show-like function, or an interpreter of tagless-final expressions as
strings. Just strings will not suffice: to show sharing as let-expressions we need to generate local variable
names. The domain of the show-interpretation is hence S t:
type LetVarCount = Int
newtype S t = S{unS :LetVarCount →String}
Exercise 11 Write the tagless-final interpreter for S t, that is, the instances Exp S and ExpLet S.
The S interpreter shows exp mul4 as
i1 + i1 + i1 + i1
and exp mul4’ as

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let v0 = i1 in let v1 = v0 +v0 in v1 + v1
We tell the DAG-constructing interpreter N how to handle explicit sharing. Recall that the expression
let e (\x → body) states that multiple occurrences of the variable x in the body should refer to the
same shared expression e. In the N interpreter, the meaning of a DSL expression is the DAG-constructing
computation producing a NodeId. Sharing a computation across several places means performing the
computation once and replicating its result. This principle is codified as follows
instance ExpLet N where
let e f = N(do
x ← unN e
unN $ f (N (return x)))
The result of interpreting exp mul4’ as a DAG is identical to that of exp mul4: the two expressions
are indeed identical after the common subexpression elimination. In exp mul4’, sharing was explicitly
declared; in exp mul4 is had to be determined. The explicit sharing declaration makes the difference in
the resources spent to get the results rather than in the results themselves.
Larger examples will show the difference in the resources. To obtain the examples, we re-write the
mul generator to use the explicit sharing. The difference from mul is on the last-but-one line.
mul’
mul’
mul’
mul’
mul’

:
0
1
n
n

(ExpLet repr, Exp repr) ⇒ Int → repr Int → repr Int
= constant 0
x =x
x | n ‘ mod‘ 2 == 0 = let x (\x’ → mul’ (n ‘ div‘ 2) (add x’ x’))
x = add x (mul’ (n−1) x)

Exercise 12 There is some sharing left to discover, isn’t there? Modify mul’ to explicitly declare all
sharing.
Without explicit sharing, running the DAG construction run expN (mul n (variable ”i ”)) in GHCi
takes 0.09 secs for n equal to 212 , and 0.20 secs for n equal to 213 . With explicit sharing, running of
the run expN (mul’ n (variable ”i ”)) takes the same 0.01 secs for n equal to 212 , or 220 or even 230 .
The construction is so fast that its timing is lost in noise, even for the expression with 231 − 1 constructors
(most of which are fortunately shared).
The programmer does not have to explicitly declare all sharing. Some amount of sharing could be
left implicit, for the DAG constructor to discover. The declared sharing may significantly speed up the
detection of the implicit sharing. For example, the mul’ code did not explicitly declared all sharing, as
seen from the printout of (mul’ 15 (variable ”i ”)):
i + let v0 = i in v0 + v0 +
let v1 = v0 +v0 in v1 + v1 + let v2 = v1 +v1 in v2 + v2
The DAG construction will find the undeclared sharing, producing the DAG
(6, DAG BiMap[
(0, NVar ”i”),
(1, NAdd 0 0), (2,NAdd 1 1), (3,NAdd 2 2),
(4, NAdd 2 3), (5,NAdd 1 4), (6,NAdd 0 5)])
For a large example, run expN (mul (2ˆ30−1) (variable ”i ”)) finishes within the same 0.01 secs (although producing the twice as large DAG). The explicit sharing helps find the remaining implicit sharing.
Exercise 13 How to re-write the sklansky example with the explicit sharing?

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Conclusions and further reading

We have demonstrated the sharing detection technique based on the tagless-final embedding that interprets a DSL program as a DAG, eliminating multiple occurrences of common subexpressions. We have
argued for the extension of the DSL with the syntactic form let to declare sharing explicitly. The sharing declarations not only help the human readers of the code but also reliably, sometimes exponentially,
speed up DSL interpreters. Implicit (not stated but detected) and explicit sharing play well together:
the programmer does not have to identify all expressions to share; the declared sharing helps, often
significantly, to detect the implicit one.
The technique, illustrated on arithmetic expressions, is immediately applicable to hardware description eDSLs. The technique has also been used in a SAT solver [3] and in the audio synthesizer mescaline [13].
The standard, thorough reference for compiling embedded DSLs is Elliott et al. [8]. Sections 4 and
8.1 of the paper discuss the detection and representation of sharing, with the particular attention to the
placement of the target-code let -expressions to state the sharing in the target code. In the presence of
loops and conditionals, the semantics-preserving let -insertion is quite non-trivial, as the paper discusses
in detail. To transform an expression tree to a DAG, the paper relied on “non-declarative pointer manipulation”, or so-called “observable sharing”, which we illustrated in §3. Broadly, observable sharing
denotes any use of unsafePerformIO for the detection of sharing [16]. Gill [11] has demonstrated that
sharing at certain types is observable, in the IO monad. However, we have to resort to GHC-specific
StableNames and accept their unreliability. §12 of [11] describes the advantages and many precautions
of observable sharing.
Detecting sharing is crucial in hardware description languages, since the modeled circuits are general
graphs rather than trees. The concise review of the long history of representing sharing in hardware
description embedded DSLs is given in §2.4.1 of the thesis [1]. The thesis describes in more detail the
approaches we have touched upon in §3.
Generating code with let -expressions to show sharing also has long history. The subtle aspects and
the need for writing the generator in the continuation-passing (or monadic) styles or using control effects
have been observed long time ago in partial evaluation community [2]. See [4, §3.1] for the detailed
explanation of the problem specifically in the context of code generation.
Exercise 14 Add recursion or iteration to our DSL. Do we need to extend the DSL with a new syntactic
form (e.g., loop), or recursive definitions of Haskell will suffice?
Exercise 15 Add boolean expressions: true and false literals, conjunctions and disjunctions, integer
comparison. Statically detect errors like taking the disjunction of integers.
Exercise 16 Add the conditional operator to the DSL. Should control-flow be taken into account when
searching for common subexpressions and sharing them?
Acknowledgement
I am indebted to Chung-chieh Shan for encouragement and many helpful discussions.

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