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Parametric Heap Usage Analysis for Functional Programs ∗
Leena Unnikrishnan Scott D. Stoller
Computer Science Department
Stony Brook University
{leena,stoller}@cs.stonybrook.edu
Abstract
This paper presents an analysis that derives a formula describing
the worst-case live heap space usage of programs in a functional
language with automated memory management (garbage collec-
tion). First, the given program is automatically transformed into
bound functions that describe upper bounds on the live heap space
usage and other related space metrics in terms of the sizes of func-
tion arguments. The bound functions are simplified and rewritten
to obtain recurrences, which are then solved to obtain the desired
formulas characterizing the worst-case space usage. These recur-
rences may be difficult to solve due to uses of the maximum opera-
tor. We give methods to automatically solve categories of such re-
currences. Our analysis determines and exploits monotonicity and
monotonicity-like properties of bound functions to derive upper
bounds on heap usage, without considering behaviors of the pro-
gram that cannot lead to maximal space usage.
Categories and Subject Descriptors F.3.2 [Logics and Meanings
of Programs]: Semantics of Programming Languages—Program
analysis
General Terms Languages, Performance, Verification
Keywords Live Heap Space Analysis, Functional Languages,
Garbage Collection, Recurrence Relations
1. Introduction
Analysis of the time and space requirements of computer programs
is important for virtually all computer applications, especially in
embedded systems, real-time systems, and interactive systems. The
importance of time analysis for real-time and embedded systems is
reflected in the long tradition of research on worst-case execution
time (WCET) analysis. Space usage is also critical in many real-
time and embedded systems, due to the limited amount of memory
and the potential for severe consequences if the system fails due
to insufficient memory. Due in part to the difficulty of predicting
the space usage of programs that use dynamic memory allocation,
real-time and embedded software typically use only statically al-
located data structures. However, this approach has disadvantages.
∗ This work was supported in part by ONR under Grant N00014-07-1-0928
and NSF under Grants CCF-0613913, CNS-0627447, CNS-0831298, and
CNS-0509230.
Permission to make digital or hard copies of all or part of this work for personal or
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ISMM’09, June 19–20, 2009, Dublin, Ireland.
Copyright c 2009 ACM 978-1-60558-347-1/09/06...$5.00
First, it can lead to poor utilization of memory, when one data struc-
ture is full and another has space to spare. Second, it requires ex-
plicit memory management: the programmer must keep track of
which array entries are in use and when it is safe to mark an occu-
pied entry as available; this can be difficult for entries that may be
used by multiple parts of the program and referenced from multi-
ple data structures. Of course, these are some of the reasons for the
growing popularity of languages with automatic memory manage-
ment (Java, C#, Python, Ruby, etc.) in non-embedded systems. Ver-
sions of these languages suitable for embedded systems exist and
are being promoted, e.g., Java Platform, Micro Edition (JavaME)
and Microsoft .NET Micro Framework. While advances in real-
time garbage collection (e.g., [5]) play an essential role in making
such languages practical, analyses that can accurately predict the
worst-case space usage of programs written in garbage-collected
languages are also important for their adoption in embedded sys-
tems [13].
Space analysis is important for determining space requirements
and for accurate execution-time analysis. For example, analysis
of worst-case execution time in real-time systems often uses loop
bounds or recursion depths, both of which are commonly deter-
mined by the size of the data being processed. Space analysis can
also help determine the timing effects of memory-related events
such as memory allocation, garbage collection, cache misses, and
page faults.
This paper describes a general approach for automatic accurate
analysis of heap space usage, specifically, the maximum size of live
data on the heap during execution. In other words, the analysis de-
termines heap usage of programs in the presence of perfect garbage
collection where garbage is collected as soon as it is created. This
result is the minimum amount of heap space needed to run a pro-
gram, no matter which garbage collection scheme is used. Limiting
a program to this “minimum amount of heap” in the presence of
imperfect garbage collection schemes, simply means that garbage
collection needs to be performed intermittently to free up space for
allocation. The analysis can easily be modified to determine related
metrics, such as space usage when garbage collection is performed
only at fixed points in the program. It can also be adapted to ana-
lyze the space usage of continuously running processes with cyclic
behavior. Our analysis is designed for a functional language. This
is a simplification that allows us to focus on the fundamental is-
sues first, deferring the complications needed to handle imperative
updates.
Our approach starts with a program P written in a functional
language with garbage collection. We construct, for every function
f in P, bound functions that describe worst-case live heap space
usage of f and other measures—such as the size of the result of
f—necessary to determine the space usage of the program. The
inputs to bound functions for f are the sizes of inputs to f. Bound
functions are simplified and rewritten to recurrence relations, which
are solved into closed form expressions that describe live heap

space usage and other measures in terms of sizes of inputs to the
corresponding original functions. The basic version of our analysis
is limited to programs that manipulate non-nested lists, i.e., lists in
which every element is a value of a primitive data type. Section 12
describes how to extend the analysis to accommodate other data
types.
Solving recurrence relations can be difficult. The availability
of increasingly sophisticated recurrence solvers [21, 6], is an as-
set to our analysis. We use methods such as elimination of redun-
dant arguments from recursive bound functions and recurrences to
improve solvability. This may even eliminate dependence on, and
hence the need to solve, certain recurrences. Since we are interested
in worst-case heap space usage, bound functions and their recur-
rences may contain the max operator, which is typically not han-
dled (or not handled exactly) by existing recurrence solvers such as
[21, 6]. We use templates based on pattern matching and inductive
reasoning to solve such recurrences. Results of some bound func-
tions are used as arguments to others. Maximizing the results of the
former bound functions leads to maximal live heap space usage for
the program only if the latter bound functions are monotonically in-
creasing with respect to their arguments. Monotonicity properties
of bound functions are ascertained from their closed form solutions,
if known, and using templates for bound functions still expressed
as recurrences.
2. Language
We formalize the analysis for a first-order, call-by-value functional
language that has literal values of primitive types (e.g., Boolean
and integer constants), operations on primitive types (e.g., addition
and subtraction), data constructors (e.g., null for the empty list, and
cons to construct lists), testers (which test whether a value is built
using a particular constructor), selectors (which extract parts of data
structures), conditionals, bindings (i.e., local variable declarations),
and function calls. A program is a set of function definitions of
the form f(v1, ..., vn) = e, where an expression e is given by the
grammar
e ::= v variable
| l literal
| cons(e1, e2) constructor application
| prim(e1, . . . , en) primitive operation
| null?(e) tester
| car(e) first element of list
| cdr(e) tail of list
| if p then e1 else e2 conditional expression
| let v1 = e1, . . . , vn = en in e binding expression
| f(e1, . . . , en) function application
Predicates in conditional expressions are given by the grammar
p ::= v | l | prim(p1, . . . , pn) | null?(p) | car(p) | cdr(p)
The basic version of our analysis supports recursive functions
but not mutual recursion. Section 12 describes how to handle mu-
tual recursion.
Our analysis makes use of type information, which may be
obtained from type declarations or from automated type inference.
type(v) denotes the type of variable v. rtype(f) denotes the return
type of function f. The allowed types are Bool, Int and List.
3. Definitions
Data Sizes. The size of an empty list is 0. The sizes of all other
atomic objects (booleans and numbers) are their values. The size of
a list is its length.
Argument-expression (arg-exp). An arg-exp is an expression
whose result is an argument to a function call. In a function ap-
plication f(e1, . . . , en) of the function f(v1, . . . , vn), expressions
e1, . . . , en are arg-exps of the call to f. For i ∈ [1, n], ei corre-
sponds to parameter vi of f.
Argument-uses (arg-uses) of functions. An arg-use of a func-
tion f is an occurrence of an application of f in an arg-exp. For
example, g(1 + f(x), 2) contains an arg-use of f which corre-
sponds to the first parameter of g. Arg-uses of expressions, e.g.,
g(if p then e1 else e2), are defined similarly.
Argument-users (arg-users) of functions. In a function applica-
tion f(e1, . . . , en), if ei contains an arg-use of a function g, then f
is called an arg-user of g. For example, in f(1 + g(x), 2), f is an
arg-user of g. Arg-users of expressions are defined similarly.
Dependence. A function f is dependent on a function g if there
is a path from f to g in the call graph of f.
Monotonicity. Consider function f : X1 × X2 . . . × Xm → Y ,
where X1, ..., Xm, and Y are sets of whole numbers, integers or
real numbers. f is monotonically increasing with respect to its ith
argument, if
∀ a, b ∈ Xi, c1 ∈ X1, . . . , cm ∈ Xm. a ≤ b ⇒
f(c1, c2, . . . , a, . . . , cm) ≤ f(c1, c2, . . . , b, . . . , cm)
where a and b are the i’th argument of f. We use the terms mono-
tonic and monotonically increasing interchangeably.
4. Bound Functions
Consider function f(v1, ..., vn) = ef in input program P. We de-
fine three bound functions for f. Bound functions of f take as ar-
guments, the sizes of arguments v1, ..., vn of f. We call these size
arguments and denote them by v1s , ..., vns . The live heap space
bound function of f, Sf , called simply the S function of f, de-
scribes the worst-case heap usage of f over all possible combina-
tions and values of arguments v1, ..., vn having sizes v1s , ..., vns ,
respectively.
Sf (v1s , . . . , vns )
= upper bound (min heap to evaluate f(a1, ..., an))
all args a1,...,anto f
s.t. ∀i∈[1,n]. |ai|=vis
The worst-case heap usage corresponds to the maximum size of
the live heap seen during the evaluation of f. This is also the
minimum heap space required to evaluate f on arguments of sizes
v1s , ..., vns , no matter which garbage collection scheme is used. A
heap object is live as long as it can be reached from the program
through function arguments or the reference to the result of the
most recently evaluated expression. The live heap space usage of
program P is described by Sg, where g is the entry function of P.
The new result-space bound function of f, also called the N
function of f, denoted Nf , describes the newly-allocated space
in the result of f in terms of the sizes of f’s arguments, and the
amounts of newly-allocated space in these arguments. For function
call f(e1, ..., en), Nf returns the number of new heap cells in the
result r of the call. A heap cell in r is new if it is created after
the start of evaluation of f(e1, ..., en), i.e., created in the body of
f or in one of the arg-exps e1, ..., en. So, in addition to its size
arguments, Nf also has newsize arguments, denoted v1w , ..., vnw ,
representing the number of new heap cells in each argument. For
f(e1, ..., en), viw is the number of heap cells in vi created during
evaluation of expression ei. viw is meaningful only if vi is a list;
if not, then there are no heap cells in vi and viw is not required or

used.
Nf (v1s , . . . , vns , v1w , . . . , vnw )
= upper bound

amount of new heap in the result of
f(a1, ..., an)

s.t. ∀i∈[1,n]. |ai|=vis
|new heap in ai|≤viw
The size bound function of f, also called the R function of f,
denoted Rf , describes the size of the result of f in terms of the
sizes of f’s arguments.
Rf (v1s , ..., vns ) = upper bound |f(a1, . . . , an)|
s.t. ∀i∈[1,n]. |ai|=vis
The bodies of Sf , Nf , and Rf are obtained by applying transfor-
mations S, N, and R, respectively, to the body ef of f. The defi-
nitions of transformations S, Nand R. are provided in Sections 5,
6, and 7.
S [
[e]
]: Computes an upper bound on the live heap required for
evaluation of e.
N [
[e]
]: Computes an upper bound on the number of new heap cells
in result r of e. If e occurs in the body of function f, then a heap
cell in r is new if it is created in e, or if it is new in v1, ..., vn,
which can easily be determined using v1w , ..., vnw . Since there is
no imperative update in our target language, any node that points to
another node must be newer than the latter. So, for a list argument
vi, only the first viw nodes are new.
R [
[e]
]: Computes an upper bound on the size of the result of e.
R [
[e]
] is boolean if e returns a boolean and is numeric if e returns a
number or a list.
5. Live Heap Space Bound Functions
Transformation S, defined in Figure 1, is used to derive live heap
space bound functions. In the S transformation of conditionals, if
the predicate contains car then its truth value is unknown, because
the analysis does not track the values of list elements. The heap
space required to evaluate the conditional is the maximum of that
required for the branches (the grammar in Section 2 implies that
predicates in conditional expressions do not perform heap alloca-
tion). If the predicate does not contain car, then we maintain the
conditional structure, and transform the predicate and the branches.
Such predicates are often tests of recursive functions that control re-
cursion, e.g., base case tests or tests that determine the next recur-
sion step. The transformed conditionals in bound functions even-
tually (after simplification) become multiple cases in the definition
of a recurrence. The original predicate p in the conditional expres-
sion is transformed into a predicate over size arguments. N and R
transformations of conditionals are similar.
In function application f(e1, ..., en), arg-exps e1, ..., en are
evaluated in order, followed by the call to f. The heap usage of
f(e1, ..., en) is the maximum of the heap usages of the arg-exps
e1, ..., en and the heap usage of f when called with arguments
whose sizes are those of the results of e1, ..., en. When ei, i ∈
[2, n], is evaluated, the results of the previous arg-exps are live. So,
the maximum size of the live heap during evaluation of ei is the sum
of the newly-allocated space in the results of previous arg-exps and
S [
[ei]
]. N [
[ei]
] [0/vw] is the amount of new heap in vi that is allo-
cated in ei. Newsize arguments are neither required nor applicable
and they are substituted with 0s. The function call is evaluated after
all arg-exps, and this takes space Sf (R [
[e1]
] , ..., R [
[en]
]).
See Figure 4 for examples of functions in an input program and
corresponding S functions.
For f(v1, ..., vn) = e, Sf (v1s , ..., vns ) = S [
[e]
].
S [
[l]
] = S [
[nil]
] = S [
[v]
] = 0
S [
[cons(e1, e2)]
] = 1 + max(S [
[e1]
] , S [
[e2]
])
S [
[prim(e1, ..., en)]
] = max(S [
[e1]
] , ..., S [
[en]
])
S [
[null?(e)]
] = S [
[car(e)]
] = S [
[cdr(e)]
] = S [
[e]
]
S [
[if p then e1 else e2 ]
]
=

max(S [
[e1]
] , S [
[e2]
]), if p contains car
if R [
[p]
] then S [
[e1]
] else S [
[e2]
] , otherwise
S [
[let v1 = e1, . . . , vn = en in e]
]
= max(S [
[e1]
] ,
N [
[e1]
] [0/vw] + S [
[e2]
] ,
.
.
.
N [
[e1]
] [0/vw] + . . . + N [
[en−1]
] [0/vw] + S [
[en]
] ,
N [
[e1]
] [0/vw] + . . . + N [
[en]
] [0/vw]+
S [
[e]
] [R [
[e1]
] /v1s , ..., R [
[en]
] /vns ])
S [
[f(e1, ..., en)]
]
= max(S [
[e1]
] ,
N [
[e1]
] [0/vw] + S [
[e2]
] ,
.
.
.
N [
[e1]
] [0/vw]... + N

e(n−1)

[0/vw] + S [
[en]
] ,
N [
[e1]
] [0/vw]... + N [
[en]
] [0/vw]+
Sf (R [
[e1]
] , ..., R [
[en]
]))
Figure 1. Transformation S. e[0/vw] substitutes every newsize
argument vw in e with 0.
6. New Result-Space Bound Functions
Transformation N, defined in Figure 2, produces new result space-
bound functions. If function f in the input program returns prim-
itive data, then its result uses no heap space and Nf returns 0. N
functions exploit the fact that, in the absence of imperative update,
newer heap cells reference older heap cells. So, if a list v con-
tains vw new heap cells, it is the first vw cells of v that are new.
Newsize arguments are decremented in recursive calls to N func-
tions if corresponding size arguments are, e.g., N [
[f(cdr(ls))]
] is
Nf (lss − 1, lsw − 1). Recursion often proceeds until lss = 0,
which implies decrementing lss from lsw. Only the first lsw of lss
cells are new. So, all decrements to newsize arguments are enclosed
by applications of max(0, ...). See Figure 4 for examples of func-
tions in an input program and corresponding N functions.
N functions may sometimes yield recurrences that are diffi-
cult to solve, because of the large number of arguments (size and
newsize arguments) and the presence of max(0, ...) expressions.
If Nf cannot be solved, we redefine it as Nf (v1s , ..., vns ) =
Rf (v1s , ..., vns ). In other words, instead of determining how much
of the list returned by f is new, we over-approximate, saying that
the whole list is new. In all our examples, these simpler N func-
tions, wherever used, do not introduce overestimations in the anal-
ysis result. The redefinition does not lose accuracy in practice, be-
cause our analysis determines worst-case space usage, and in the
worst case, most functions either build completely new lists or the
arguments given to functions are completely new; in both cases, all
cells in the result lists are new. This is in keeping with the following
observation:
Nf (v1s , ..., vns , v1s , ..., vns ) = Rf (v1s , ..., vns )
where the values of the newsize arguments of Nf equal those of the
corresponding size arguments.

For f(v1, ..., vn) = e,
Nf (v1s , ..., vns , v1w , ..., vnw ) =

0, if rtype(f) 6= List
N [
[e]
] , otherwise
N [
[l]
] = N [
[nil]
] = 0
N [
[v]
] =

0, if type(v) 6= List
vw, otherwise
N [
[cons(e1, e2)]
] = 1 + N [
[e2]
]
N [
[prim(e1, ..., en)]
] = 0
N [
[null?(e)]
] = N [
[car(e)]
] = 0
N [
[cdr(e)]
] = max(0, N [
[e]
] − 1)
N [
[if p then e1 else e2]
]
=

max(N [
[e1]
] , N [
[e2]
if R [
[p]
] then N [
[e1]
] else N [
[e2]
] , otherwise
N [
[let v1 = e1, . . . , vn = en in e]
]
= (N [
[e]
]) [R [
[e1]
] /v1s , ..., R [
[en]
] /vns ,
N [
[e1]
] /v1w , ..., N [
[en]
] /vnw ]
N [
[f(e1, . . . , en)]
]
=

0, if rtype(f) 6= List
Nf (R [
[e1]
] , ..., R [
[en]
] , N [
[e1]
] , ..., N [
[en]
]), otherwise
Figure 2. Transformation N.
7. Size Bound Functions
Transformation R in Figure 3 produces size bound functions; recall
that Rf bounds the size of the result (return value) of f. See figure
4 for examples of functions in an input program and corresponding
R functions. From the definitions of transformations S, N, and R,
it can be seen that results of R functions are used by S and N
functions only as the values of size arguments. Also, recall that we
are interested only in the maximal value of the input program’s S
function. Consider the R transformation of conditionals. Predicates
containing car are concluded as being undeterminable and in such
cases, the size of the result of the conditional is the maximum of
the sizes of e1 and e2, max(R [
[e1]
] , R [
[e2]
]). Ideally, we should
return that size which helps S functions determine the worst-case
(maximal) space usage of the input program. But it is impossible
to know which size to return, in the general case. Further, it is not
tractable to return all sizes. Thus, the use of max in the definition
of bound functions is a necessary approximation, and based on the
somewhat reasonable expectation that the maximum size leads to
maximum space-usage results of dependent S and N functions.
We perform checks, namely MC1, MC2, and MC3 described later
in this section, to verify if this is indeed the case, i.e., if the use
of max in R functions results in dependent S and N functions
determining true upper bounds on space measures. Transformation
R, along with checks MC1, MC2, and MC3, ensure that all R
functions, including ones whose definitions do not contain max
but are dependent on R functions that do, uniformly return upper
bounds (maximums) on size measures.
Composite recurrences. A composite recurrence is a recurrence
(or a definition of a bound function) which contains an application
of max in which at least one arg-exp contains a recursive call to
the function defined by the recurrence. A regular recurrence is one
that does not contain such occurrences of max. The R function
Rappend of append, which appends one list to another, is regu-
lar, while Rlls, the R function of lesserorequal-list is composite.
lesserorequal-list, an auxiliary function of quick-sort, takes a list ls
and a cutoff x as arguments and returns a new list containing the
For f(v1, ..., vn) = e, Rf (v1s , ..., vns ) = R [
[e]
]
R [
[l]
] = l, R [
[nil]
] = 0, R [
[v]
] = vs
R [
[cons(e1, e2)]
] = 1 + R [
[e2]
]
R [
[prim(e1, ..., en)]
] = prim(R [
[e1]
] , ..., R [
[en]
])
R [
[null?(e)]
] = eq?(R [
[e]
] , 0)
R [
[car(e)]
] = uk, R [
[cdr(e)]
] = R [
[e]
] − 1
R [
[if p then e1 else e2]
]
=

max(R [
[e1]
] , R [
[e2]
if R [
[p]
] then R [
[e1]
] else R [
[e2]
] , otherwise
R [
[let v1 = e1, . . . , vn = en in e]
]
= (R [
[e]
]) [R [
[e1]
] /v1s , . . . , R [
[en]
] /vns ]
R [
[f(e1, . . . , en)]
] = Rf (R [
[e1]
] , . . . , R [
[en]
])
Figure 3. Transformation R.
elements of ls that are less than or equal to x.
Rappend(n, m) =

m if n = 0
1 + Rappend(n − 1, m) otherwise
Rlls(n) =

0 if n = 0
max(1 + Rlls(n − 1), Rlls(n − 1)) otherwise
Given recurrence T, we define recurrence Tall
that has the same
arguments as T and returns the set of results obtained by using the
value of each argument of max in place of the max expression in
T. Note that this is done recursively, not only at the top-level call.
T, on the other hand, uses only the largest of the arguments to max
at every recursion level.
Intuitively, T captures only the maximum value of the size or
space measure for a function in the input program, while Tall
cap-
tures all possible values of the size or space measure for that func-
tion. We can generate the definition of Tall
from the definition of
T, by replacing each occurrence of max with union and replac-
ing every function call F(. . .) with Fall
(. . .); note that upper case
letters (F, G, H) are used to denote bound functions. These re-
placements in the definition of T need to be accompanied by mod-
ifications that ensure type-safety, for instance, the replacement of
primitive operators and functions with ones that are overloaded to
handle regular argument values as well as sets of argument values.
For example, for sets X and Y of argument values to f, f(X, Y ) =
{f(x, y) | x ∈ X, y ∈ Y }, and f(x, Y ) = {f(x, y) | y ∈ Y }.
Recurrences Rlls and Rall
lls and their solutions are shown below.
Rlls(n) =

0 if n = 0
max(1 + Rlls(n − 1), Rlls(n − 1)) otherwise
= n
Rall
lls (n) =

{0} if n = 0
∪(1 + Rall
lls (n − 1), Rall
lls (n − 1)) otherwise
= {n, n − 1, . . . , 0}
A composite recurrence T is simple if the solution of Tall
is
a singleton set for all evaluations of its arguments; otherwise, we
say that T is complex. If Rf is a simple composite recurrence, then
the size of the result of f is uniquely determined by the sizes of its
arguments. If Rf is a complex composite recurrence, then the size
of the result of f depends on the actual elements in list arguments
of f. The solution of Rall
lls is not a singleton set and therefore, Rlls
is complex. The R function Rins of insert which inserts a number

insertion-sort insert
insertion-sort(ls) = if null?(ls) then nil
else insert(car(ls),
insertion-sort(cdr(ls)))
insert(x, ls) = if null?(ls) then cons(x, nil)
else if lessoreq?(x, car(ls))
then cons(x, ls)
else cons(car(ls), insert(x, cdr(ls)))
Sis(lss) = if lss = 0 then 0
else max(Sis(lss − 1),
Nis(lss − 1) + Sins(Ris(lss − 1)))
=

0 if lss = 0
max(Sis(lss − 1), 2lss − 1) otherwise
=

0 if lss = 0
2lss − 1 otherwise
Sins(lss) = if lss = 0 then 1
else max(1, 1 + Sins(lss − 1))
=

1 if lss = 0
1 + Sins(lss − 1) otherwise
= lss + 1
Nis(lss) = if lss = 0 then 0 else Nins(Ris(lss − 1))
= lss
Nins(lss) = Rins(lss) = 1 + lss
Ris(lss) = if lss = 0 then 0 else Rins(Ris(lss − 1))
=

0 if lss = 0
1 + Ris(lss − 1) otherwise
= lss
Rins(lss) = if lss = 0 then 1 + 0
else max(1 + lss, 1 + Rins(lss − 1))
=

1 if lss = 0
max(1 + lss, 1 + Rins(lss − 1)) otherwise
= 1 + lss
Figure 4. Entry function insertion-sort and auxiliary function insert of program insertion-sort, followed by their bound functions, recur-
rences (if any) obtained from bound functions, and closed form solutions of recurrences.
into a sorted list is a simple composite recurrence, as shown below.
Rins(n) =

1 if n = 0
max(1 + n, 1 + Rins(n − 1)) otherwise
Rall
ins(n) =

{1}, if n = 0
∪({1 + n}, 1 + Rall
ins(n − 1)) otherwise
= {1 + n}
We determine if composite recurrences are simple or complex using
templates. This is described in Section 9.
Complex R functions. An R function is complex if it is defined
by a complex composite recurrence or it depends on another com-
plex R function or its solution contains an application of max over
closed forms; otherwise, we say that the R function is simple. R
functions Rappend and Rins are simple, while Rlls is complex.
Rqs, shown below, is a complex R function even though the recur-
rence defining it is regular because of the dependence on complex
R function Rlls.
Rqs(lss)
=

lss if lss = 0 or lss = 1
1 + Rqs(Rlls(lss − 1)) + Rqs(Rgls(lss − 1)) otherwise
8. Conditions on Uses of Complex R Functions
We observe that for an R function F that does not depend on any
complex R functions (in other words, does not call any R function
or depends only on simple R functions), F = max o Fall
. Con-
sider the following definition of a bound function G that depends
on a complex R function F, for which F = max o Fall
.
G(. . .) = . . . F(e1, . . . , en) . . .
From the discussion in Section 7, the desired result of G is that
of max o Gall
, whether G is an S or N or R function. For
this to be true, in place of F(e1, . . . , en) in the body of G, we
should use the specific size s ∈ Fall
(e1, . . . , en) that maximizes
the result of evaluation of the body of G. This desired size s is
not necessarily the maximum element in Fall
(e1, . . . , en), i.e.,
it is not always the same as the result of F(e1, . . . , en) (since
F = max o Fall
). Consider function G1(v) = v − F1(v) as an
example. The expression ‘v−F1(v)’ is the result of multiple rounds
of simplification, and this explains the occurrence of the result of
an R function in a non-recursive expression that defines G1 which
could be an S or N function. The result of max o Gall
1 , which
is the desired result of G1, is obtained by replacing ‘F1(v)’ in the
body of G with the minimum element of Fall
1 (v). Thus, the derived
definitions of bound functions such as G1 do not necessarily yield
the desired upper bounds because of the use of max in called R
functions. So, for every complex R function F for which F =
max o Fall
, we perform the following checks to ensure that the
definitions of bound functions dependent on F yield upper bounds
on relevant space and size measures.
MC1 All expressions containing calls to F are monotonically in-
creasing with respect to the calls to F (e.g., 1+F(. . .) is mono-
tonic with respect to the call to F, but n − (1 + F(. . .))) is
not. This check ensures that values of size arguments of S and
N functions, determined using expressions containing applica-
tions of complex R functions, are maximized.
MC2 For every arg-use of F by a function H, occurring in the body of
G, the solution of H should be monotonically increasing with
respect to the argument corresponding to the arg-use of F. An
example from quick-sort:
Nqs(lss, lsw)
= . . . Napp(. . . Rlls(lss − 1), . . . Rgls(lss − 1))
qs, app, lls, and gls are abbreviations for quick-sort, append,
lesserorequal-list, and greater-list, respectively. Rlls and Rgls
are complex R functions. So, the above definition of Nqs is
correct only if Napp is monotonically increasing with respect
to both its arguments (which it is, because Napp(ls1s , ls2w ) =
ls1s + ls2w ).
MC3 For every arg-use of F by function G, occurring in the body of
G, we need to ascertain from the definition of G, that the use of
the result of F (in other words, max o Fall
) yields the desired

maximum value, namely the result of max o Gall
. An example
from quick-sort:
Sqs(lss)
=



lss if lss = 0 or lss = 1
max(2lss−1
+ lss − 1 + Sqs(Rgls(lss − 1)),
3.2lss−1
− 2), otherwise
The definition of Sqs contains the recursive call Sqs(Rgls(lss−
1)). Rgls is a complex R function. The result of Rgls (equal to
that of max o Rall
gls) can be used only if we can ascertain that
this will result in the body of Sqs evaluating to max o Sall
qs .
Templates are used to determine such properties (see . Tem-
plates described in Section 10) are used to determine if MC3
holds for a given recurrence G. They do so by checking if
certain requirements, such as the following, are met: G con-
tains only safe patterns of recursion, base cases of G are in-
creasing, and closed form expressions added to results of re-
cursive calls to G are monotonically increasing (this pertains
to “2lss−1
+ lss − 1” in the definition of Sqs). These require-
ments ensure that recurring on a large argument value (in place
of the result of the complex R function) leads to a greater re-
sult of evaluation of the body of G, than recurring on a small
argument value.
Monotonicity properties in MC1 and MC2 follow directly from
monotonicity properties of primitive operations, e.g., +, × are
monotonic with respect to their arguments, −, ÷ are monotonic
with respect to their first arguments but not with respect to their
second arguments, and modulo is not monotonic with respect to
either of its two arguments.
Consider a bound function G such that for every R function F
that G depends on, F = max o Fall
. If checks MC1, MC2, and
MC3 are satisfied by all complex R functions that G is dependent
on, then G = max o Gall
. These checks are performed in the
analysis for all bound functions. This includes the case when G
is an R function. So, by induction, the previous argument holds
for all bound functions because for any called R function F, F =
max o Fall
.
An example of a recurrence T whose body contains an arg-use
of a complex R function Rf by T, and which does not satisfy MC3
is:
Rall
f (m) = {m − 1, . . . , 0}, Rf (m) = m − 1
T(n, m) =

n, if m = 0
T(n − 1, Rf (m)), otherwise
= n − m
diff(ls1, ls2) is an example of an input function from which a
bound function similar to T might be derived. diff(ls1, ls2) returns
the tail list of ls1 of size |ls1| − |ls2|. A variation of diff, with
similar bound functions, returns a copy of the tail list.
diff(ls1, ls2) = if null?(ls2) then ls1
else diff(cdr(ls1), cdr(ls2))
The need for checks MC2 and MC3 stems from the use of
R function results as arguments to other bound functions. Check
MC1 simply ensures that size arguments to bound functions are,
uniformly, maximums of sizes seen during program execution,
rather than an arbitrary size from amongst the possible sizes. These
checks are not relevant to applications of S and N functions. Re-
sults of S functions are never used as arguments to bound func-
tions. Results of N functions may sometimes be values of newsize
arguments (of N functions). We argue that all N functions are, by
construction, monotonically increasing with respect to newsize ar-
guments. These arguments are used only to determine the number
of new heap cells in an argument that, for example, may be part
of the result list. For example, for f(ls) = cons(1, cdr(ls)), the
N function is Nf (lss, lsw) = 1 + max(0, (lsw − 1)). In particu-
lar, newsize arguments to N functions only get added to the result;
non-monotonic operators are never applied to them. Further, new-
size arguments have no counterparts in the original program, unlike
size arguments of numeric variables where operators applied to the
latter may translate into the same operators applied to the former
in the definitions of bound functions. Predicates of conditionals
could result in recurrences that are non-monotonic with respect to
the arguments in the predicates; however, newsize arguments do
not occur in the predicates of conditional expressions after trans-
formation by N (note from the definition of N that predicates are
transformed by R). Arg-exps of functions are transformed by R to
obtain size arguments to corresponding bound functions. So, new-
size arguments are never used as size arguments with respect to
which bound functions can be non-monotonic.
9. Solving Composite Recurrences
We developed a set of templates to solve composite recurrences.
The templates are designed to match composite recurrences of the
forms that commonly arise in bound functions, such as from the S
transformation of function applications (see Figure 1). The applica-
bility of a template is defined by a pattern that the recurrence must
match, together with some requirements that must be satisfied by
the constants and expressions in the recurrence. Several templates
solve composite recurrences by reducing the given composite re-
currence into a regular recurrence containing a single argument e
of the max application, such that the evaluation of e is the maxi-
mum at the relevant base cases (different from the base cases of the
given recurrence), as well as the inductive case. Other templates
work by reducing composite recurrences with multiple max ap-
plications into component composite recurrences which are recur-
sively solved. A complete list of templates is available in [24].
We now describe a template for solving composite recurrences
of the form
T(n) =

c1, . . . , cj+1, if n = i, . . . , i + j, respectively
max(e1(n), e2(n) + aT(n − b)), otherwise
where i, j, c1, ..., cj+1, a, b are constants, and e1(n) and e2(n) are
closed forms in n. T has j + 1 base cases and one recursive
case. The requirements for applicability of the template include
a few conditions that ensure the recurrence is well-defined, e.g.,
0 b ≤ (j +1). The remaining requirements define three separate
cases in which a solution can be ascertained. In case (a), we also
conclude that the recurrence is simple, and therefore this case is
checked first. Let E1 be e1(n), and let E2 be e2(n) + aT(n − b).
Case (a): If the following conditions hold,
A1: e1(n) = e2(n) + ac(n−b−i+1), ∀n ∈ [i + j + 1, i + j + b]
A2: e1(n) = e2(n) + ae1(n − b), ∀n (i + j + b)
then T is simple, and the solution of T is:
T(n) =

e1(n), otherwise
Informally, this case applies when all unfoldings of T, using all
combinations of arg-exps E1 and E2 of max, yield the same
value. A1 ensures this for the smallest non-base cases of n (in
the range [i + j + 1, i + j + b]) for which T(n) uses a base
value of T; this corresponds to the base case of the proof of case
(a). A2 performs the necessary check for the inductive case: for
n (i + j + b), the value of E1 equals that of E2, unfolded once
using E1. For example, consider the R function Rre of remove-
element in selection-sort.
Rre(lss) =

0, if lss = 1
max(lss − 1, 1 + Rre(lss − 1)), otherwise

Case (a) applies and gives the solution Rre(lss) = lss − 1.
Case (b): If the following conditions hold,
B1: e1(n) ≥ e2(n) + ac(n−b−i+1), ∀n ∈ [i + j + 1, i + j + b]
B2: e1(n) ≥ e2(n) + ae1(n − b), ∀n (i + j + b)
then T is complex, and the solution of T is:
T(n) =

c1, . . . , cj+1, n = i, . . . , i + j, respectively
e1(n), otherwise
The solution is based on the observation that, under these condi-
tions, E1 always yields a larger value than E2.
Case (c): Define T0
as
T0
(n) =

e2(n) + aT0
(n − b), otherwise
Suppose the solution of T0
is
T0
(n) =

c1, . . . , cj+1, n = i, . . . , i + j, respectively
e3(n), otherwise
If e1(n) ≤ e3(n) for n (i + j), then T is complex and has
the same solution as T0
. The solution is based on the observation
that, under these conditions, arg-exp E2 of max always yields the
maximal value.
10. Performing Test MC3
Test MC3 is performed on bound function recurrences that contain
arg-uses of complex R functions, where the arg-users are the defin-
ing bound functions. Consider such a bound function G:
G(v) = . . . G(F(v)) . . .
where F is a complex R function. We consider a single-argument
function for simplicity. The discussion is easily extrapolated for
functions with multiple arguments. Test MC3 checks if using the
result of F(v) in the definition of G yields max(Gall
(v)), which
is the desired result of G.
An example of such a bound function is Rqs in Section 7. If Rqs
does not satisfy MC3, then the analysis cannot proceed without
an alternate strategy, e.g., if G is an N function, then it can be
redefined without newsize arguments, as discussed in Section 6.
This simplification may allow test MC3 to succeed or even make it
unnecessary.
Test MC3 is difficult to check in the general case. We developed
templates that perform the test by matching recurrences against
safe patterns and ensuring certain properties such as the following:
base values of the recurrence are monotonically increasing, base
values are less than values of the recurrence at the smallest non-
base cases, and closed form expressions in the recurrence definition
contain only monotonically increasing operators. Templates are
constructed such that, when checking MC3 for aforementioned G,
it is sufficient to provide its definition using just the solution of F,
instead of the set of elements of Fall
. A complete list of templates
is available in [24].
One of the templates that test MC3 matches recurrences of the
form
T(n) =

c1, ..., cj+1, n = i, ..., i + j, respectively
e(n) + aT(f(n)), otherwise
where i, j, c1, ..., cj+1, a are constants, and e(n) is a closed form
in n. The given recurrence is required to meet the following condi-
tions (we elide general well-formedness requirements that are or-
thogonal to MC3).
R1: c1 ≤ ... ≤ cj+1, i.e., base cases are monotonically increas-
ing.
R2: cj+1 ≤ e(i + j + 1) + aT(f(i + j + 1)), i.e., the largest
base value is not greater than the value of T at the smallest
non-base case.
R3: ∀n (i + j). f(n) n. For example, this holds for
f(n) = n − b and f(n) = n/2 (integer division).
R4: f is monotonically increasing w.r.t. n. For example, this
holds for f(n) = n − b and f(n) = n/2.
R5: ∀n (i + j), e(n) 0 and e(n) is monotonically
increasing w.r.t. n.
It is easy to see that if R1-R5 are satisfied by a recurrence G,
then MC3 is satisfied for G. The above template matches Rqs
which after evaluation of the calls to Rlls and Rgls is
Rqs(lss) =



0, if lss = 0
1, if lss = 1
1 + 2Rqs(lss − 1), otherwise
11. Examples
We applied the analysis to several list-processing programs, namely
list reversal, insertion sort, selection sort, merge sort, quicksort, and
programs that compute longest common subsequence, string edit
distance, and binomial coefficients efficiently using dynamic pro-
gramming. In the latter set of programs, lists (instead of arrays)
are used to store results of subproblems. Figure 5 shows the re-
sults of our analysis. The closed forms derived by the analysis in-
clude linear and quadratic polynomials, and exponential formulae.
A complete listing of example programs, their bound functions, re-
currence relations, resulting closed forms, and how these were de-
rived, appears in [24].
The accuracy (tightness) of the analysis results was confirmed
by comparing with results from the analysis in [25] for the same
programs. The closed form solutions derived by our analysis are
exact maximums of live heap space usage of all example programs
except quicksort. Our analysis gives a loose bound for quicksort
because it does not recognize that the sizes of the results of func-
tions lesserorequal-list (this is the function abbreviated as lls in
Section 7) and greater-list (which is similar but returns the sublist
containing elements greater than the cutoff) are correlated, specifi-
cally, that when both functions are applied to the same list of length
n with the same cutoff, the sizes of their results sums to n. We
verified that this is the only source of inaccuracy in the analysis
of quicksort by manually modifying the relevant bound functions
of quicksort to take this invariant into account; the analysis then
produces a quadratic polynomial that accurately describes the heap
usage of quicksort.
Figure 4 shows how live heap space analysis works for an
example program, namely insertion-sort. It lists the functions of
insertion-sort, corresponding S, N, and R functions, recurrences
obtained by simplifying the bound functions, and closed form so-
lutions of the recurrences.
12. Extensions to the Analysis
Our analysis can be extended to data types other than lists by defin-
ing an appropriate R function for each data type and introducing
corresponding size arguments to bound functions. Some data types,
such as arrays, are similar to lists and can be handled easily. Binary
trees are more interesting. Their size can be measured by height and
number of nodes, so we define two R functions for binary trees,
Rheight and Rnodes , and for each binary-tree argument v of each
function f in the original program, we introduce arguments vh and

Examples Entry Function S Function of Entry Function
Derived Heap
Usage Complexity
Tight?
reverse rev(ls) Srev(lss) =

0 if lss = 0
2lss − 1, otherwise
Linear Yes
insertion sort is(ls) Sis(lss) =

0 if lss = 0
Linear Yes
selection sort ss(ls) Sss(lss) =
lss(lss+1)
2
Quadratic Yes
merge sort ms(ls) Sms(lss) =

0 if lss = 0
Linear Yes
quick sort qs(ls) Sqs(lss) =

0 if lss = 0, 1
3.2lss−1 − 2, otherwise
Exponential No
longest common subsequence lcs(ls1, ls2) Slcs(ls1s , ls2s ) =



1 if ls1s = 0 or ls2s = 0
ls2s + 2 if ls1s = 1
2ls2s + 2, otherwise
Linear Yes
string edit distance se(str1, str2) Sse(str1s , str2s ) =



str1s + str2s + 1,
if str1s = 0 or str2s = 0
2str1s + 2str2s + 1, otherwise
Linear Yes
binomial coefficient bc(n, m) Sbc(ns, ms) =



1 if ms = ns
ms + 2 if ms = ns − 1
2ms + 2, otherwise
Linear Yes
Figure 5. Parametric heap space usage bounds derived for some example programs.
vn to the bound functions of f, representing the height and number
of nodes, respectively, of the tree passed to v.
In the presence of these new data types, bound functions may
need to make approximations, e.g., that each branch of a binary
tree v has at most min(2vh −1, vn −1) nodes. We can mitigate, if
not eliminate, the effects of such approximations by using type-
specific invariants in the analysis, e.g., vn ≤ 2vh+1
− 1 and
Rnodes [
[left(e)]
] + Rnodes [
[right(e)]
] = Rnodes [
[e]
], where left(e)
and right(e) return the subtrees of e, respectively. Note that the
metric Rnodes is useful for all data types, including nested lists.
The current R functions are, in fact, Rlength functions; for a nested
list ls, Rlength describes the length of the top-level list of ls.
S and N functions, and the S and N transformations, also
need to be specialized to new data types. For some data types,
S and N functions can take any one of several size measures
as arguments, depending on which size measure yields the most
compact and accurate formulas representing live heap usage and
new result-space usage, respectively. The analysis may try different
size measures and then determine which yields the best results.
The current version of our analysis does not allow mutual recur-
sion in the input programs. Such mutual recursion typically leads
to mutual recursion in the bound functions. Mutual recursion can
sometimes be handled by converting it into self-recursion (i.e., or-
dinary recursion): if f and g are mutually recursive, and unfolding
f a few times yields a definition of f that recursively calls only
f and not g, then f is in fact self-recursive. Our analysis can be
extended to first use such unfolding to try to convert mutual re-
cursion into self-recursion. If this does not succeed, then standard
mathematical techniques for solving mutual recurrences are used,
if applicable.
13. Related Work
There is a large amount of work on analyzing program cost or re-
source complexities, but the majority of it is on time analysis, e.g.,
[26, 19, 22, 23, 20, 15]. Analysis of stack space and heap alloca-
tion are similar to time analysis [20]. Analysis of live heap space
is different because live heap space increases and decreases dur-
ing program execution, while time always increases. Further, live
heap space analysis needs to determine when allocations become
garbage.
Wegbreit [26] proposed deriving closed form expressions de-
scribing running times of Lisp programs in terms of sizes of inputs,
much like we do for live heap space, by deriving recursive functions
describing running times, converting them into difference equa-
tions and solving the same. Analyses that produce such formulas
are sometimes called parametric analyses. Probabilities attached to
ambiguous predicates (that require information other than sizes of
inputs) enable the derivation of mean running times, in addition to
best-case and worst-case running times. The prototype implemen-
tation is limited to simple Lisp programs. [19] is similar to [26]
but derives asymptotic, rather than exact, upper bounds on running
times; also, the user is required to aid the analysis in converting the
recursive function into a closed form expression. These two papers
describe only limited methods to handle their respective equiva-
lents of composite recurrences, and neither of them use the notion
of monotonicity requirements.
Most of the work related to analysis of space is on analysis of
cache behavior, e.g., [27, 14], much of which is at a lower language
level, for compiler generated code, while our analyses are at source
level and can serve many purposes. Live heap analysis is a first
step towards analyzing cache behavior in the presence of garbage
collection.
Several type systems [18, 17, 12, 11] have been proposed for
reasoning about space and time bounds, and some of them include
implementations of type checkers [18, 12]. They require program-
mers to annotate their programs with cost functions as types. Fur-
thermore, some programs must be rewritten to have feasible types
[18, 17]. [9] proposes a loop-detecting algorithm to identify meth-
ods and instructions that execute an unbounded number of times,
thereby detecting possibly unbounded memory usage. [4, 7] ensure
acceptable memory use in bytecode, by verifying specified resource
annotations or memory consumption policies.
[16] proposes a method to obtain linear bounds on heap space
consumption of functional programs using type derivations and
linear programming. While programs are not required to be linearly

typed, restrictions are made on sharing of heap cells. Our analysis
is not restricted to linear bounds and does not restrict sharing of
heap cells.
Several methods have been developed to automatically infer
heap space bounds for Java-like imperative languages and byte-
code. Imperative update poses special challenges for determining
liveness. Chin et al. [10] propose a method to infer stack and heap
usage bounds for an assembly-like language. “dispose” commands
that reclaim unused heap space, are explicitly inserted into input
programs [11] at points where objects are no longer live. Recur-
sive constraint abstractions derived from these modified input pro-
grams describe stack and heap usages of the programs. Fixpoint
analysis of constraint abstractions using a Presburger solver yields
closed form expressions for stack and heap usages. Our analy-
sis applies to a high-level language and is not limited to Pres-
burger formulae, so it can derive both linear and non-linear bounds.
The alias/uniqueness analysis used to determine liveness and insert
“dispose” commands appropriately, is an over-approximation of the
behaviour of an ideal garbage collector. In the absence of impera-
tive update, live heap analysis is able to determine liveness exactly.
As a result, our bounds on heap usage are tighter than those of [10].
While their analysis has been applied to several benchmarks, all ex-
cept one of the benchmarks use few heap objects.
The heap analysis in [8] is also targeted to an imperative lan-
guage but is able to derive polynomial, not just linear, heap bounds.
Our analysis is not limited to a complexity category, although it is
limited by the solvability of the encountered recurrences. Region-
based memory management is used to model garbage collection;
objects are allocated in regions associated with methods and re-
gions are garbage collected at the end of methods. As in the above
discussion, this is an over-approximation of heap usage compared
to our analysis, which is, in the absence of imperative update, able
to model “ideal” garbage collection (objects are garbage collected
as soon as they become dead). The maximum memory occupied by
a region configuration is modelled as a parametric polynomial op-
timization problem. This is then solved using Bernstein basis. The
analysis does not handle recursions.
Albert et al. propose a two-part analysis of heap space usage for
Java bytecode [2]. The first part constructs cost equations character-
izing the amount of heap space allocated by each method and then
uses the technique in [1] to obtain closed-form upper bounds on the
total amount of allocation. The second part takes garbage collec-
tion into account in a simple way by using escape analysis to iden-
tify allocations that do not escape from the static scope containing
the allocation statement, and then ignoring these allocations when
computing the total amount of heap allocation. Solving the result-
ing cost equations provides an upper bound on what they call the
active heap space for each method. On the one hand, their analysis
handles various language features that ours does not, most notably
imperative update. On the other hand, their analysis cannot easily
be extended to compute what our analysis computes—the maxi-
mum live heap space, which they call the memory high-watermark.
The “active heap space” computed by their analysis only increases
(like running time); live heap space is different, because it increases
and decreases. They write: “Analysis for finding upper bounds on
the memory high-watermark cannot be directly done using cost re-
lations as introducing decrements in the equations requires com-
puting lower bounds” [2, Section 6]. To illustrate the difference
between live space and “active heap space”, consider the program
f(n) = len(g(n)) + len(g(n))
g(n) = if n = 0 then nil else cons(n, g(n − 1))
len(l) = if null?(l) then 0 else 1 + len(cdr(l))
Our analysis yields Sf (ns) = ns, i.e., the maximum live heap
space used by f is n cons cells, reflecting that the cells allocated by
the first call to g(n) become garbage before the second call to g(n).
The method of [1, 2] does not recognize this (since cells allocated
in g escape it) and yields a bound of 2n on the space usage of f.
Albert et al.’s paper in the same proceedings analyzes live heap
space (as opposed to total heap allocation and active heap space)
in the presence of garbage collection [3]. The methods used in our
analysis to solve composite recurrences can be enhanced by the
techniques in [1] to solve non-deterministic cost relations and those
in [15] to handle disjunctive invariants.
14. Conclusions and Future Work
In summary, this paper describes a method that aims to determine
worst-case live heap space usage of functional programs automati-
cally and accurately using source-level program analysis and trans-
formations. Specific technical contributions include a framework
for deriving bound (S, N, and R) functions that describe the live
heap space usage of functional programs, the identification and for-
mulation of conditions (tests MC1, MC2, and MC3) that are nec-
essary to ensure soundness of the analysis, and the templates used
to solve recurrences and check part of the monotonicity condition
(namely, MC3). The results reported in Section 11 suggest that the
analysis usually produces tight bounds in practice. Although these
example programs are small, they have non-trivial patterns of heap
allocation and garbage collection. In part, this is because they are
functional programs and must allocate new data structures instead
of updating existing data structures.
Future work on live heap analysis for functional programs in-
cludes modifying the analysis to reflect the effect of optimization of
tail recursion, incorporating techniques for analysis of higher-order
functions [23, 20], and evaluating the accuracy and scalability of
the analysis more extensively. Another important direction for fu-
ture work is to modify the analysis to handle imperative update.
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