Live Data Structures in Logic Programs: Derivation by Means of Abstract Interpretation

224 Pages·1993·9.83 MB·English

Lecture Notes in Computer Science 675
Edited by G. Goos and J. Hartmanis
Advisory Board: W. Brauer D. Gries J. Stoer
Anne Mulkers
Live Data Structures
in Logic Programs
Derivation by Means of Abstract Interpretation
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Gerhard Goos Juris Hartmanis
Universit~it Karlsruhe Cornell University
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Author
Anne Mulkers
Department of Computer Science, K.U. Leuven
Celestijnenlaan 200 A, B-3001 Heverlee, Belgium
CR Subject Classification (1991): F.3.1, D.3.4, 1.2.2-3
ISBN 3-540-56694-5 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-56694-5 Springer-Verlag New York Berlin Heidelberg
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P r e f a c e
Abstract interpretation is a general approach for program analysis to discover
at compile time properties of the run-time behavior of programs, as a basis to
perform sophisticated compiler optimizations. Several frameworks of abstract
interpretation for logic programs have been presented 11, 25, 27, 43, 48, 49, 51,
55, 57, 65, 81, 82. A framework is a parameterized construction for the static
analysis of programs, together with theorems that ensure the soundness and
termination of the analysis. To complete the construction, an application specific
domain and primitive operations satisfying certain safety conditions must be
provided.
This book elaborates on an application for such a generic framework. The
framework used 11 belongs to the class of top-down abstract interpretation
methods and collects the information derived in an abstract AND-OR-graph that
represents the set of concrete proof trees that can possibly occur when executing
the source program. The starting point of the present work is the previously
developed application of integrated type and mode analysis 38. The purpose
of that application was to guide the compiler, based on a characterization of the
entry uses of the program, to generate code that is more specific for the calls
that can occur at run time.
In an a t tempt to give further guidance to the compiler, we address the prob-
lem of compile-time garbage collection, the purpose of which is to (partially) shift
run-time storage reclamation overhead to compile time. In applicative program-
ming languages, the programmer has no direct control over storage utilization,
and run-time garbage collection is necessary. Garbage collection involves a pe-
riodic disruption of the program execution, during which usually a marking and
compaction algorithm is employed. Such schemes are expensive in time. Our
research shows that at compile time useful and detailed information about the
liveness of term substructures can be deduced which the compiler can use to
improve the allocation of run-time structures. In fact, it provides a technique
to automatically introduce destructive assignments into logic languages in a safe
and transparent way, thereby reducing the rate at which garbage cells are cre-
ated. The resulting system gets near to the methods of storage allocation used
in imperative programming languages.
The global flow analysis to be performed on Prolog source programs in order
to derive the liveness of data structures is constructed in three layers. The
vI
first layer, consisting of the type and mode analysis, basically supplies the logical
terms to which variables can be bound. The two subsequent layers of the analysis
heavily rely on these descriptions of term values. The sharing analysis derives
how the representation of logical terms as structures in memory can be shared,
and the liveness analysis uses the sharing information to determine when a term
structure in memory can be live.
Acknowledgments
This book is based on my Ph.D. dissertation 59 conducted at the Department
of Computer Science of the K.U.Leuven, Belgium. The research presented has
been carried out as part of the RFO/AI/02 project of the Diensten voor de
programmable van he~ wetenschapsbeleid, which started in November 1987 and
was aimed at the study of implementation aspects of logic programming: 'Logic
as a basis for artificial intelligence: control and efficiency of deductive inferencing
and parallelism'.
I am indebted to Professor Maurice Bruynooghe, my supervisor, for giving
me the opportunity to work on the project and introducing me to the domain
of abstract interpretation, for sharing his experience in logic programming~ his
invaluable insights and guidance. I wish to thank Will Winsborough for many
helpful discussions, for his advice on the design of the abstract domain and
safety proofs and his generous support; Gerda Janssens for her encouragement
and support, and for allowing the use of the prototype for type analysis as the
starting point for implementing the liveness analysis; Professors Yves Willems
and Bart Demoen, for managing the RFO/AI/02 project and providing me with
optimal working facilities; Professor Marc Gobin, my second supervisor, and
Professors Baudouin Le Charlier and Danny De Schreye, for their interest and
helpful comments, and for serving on my Ph.D. thesis committee. I also want to
thank my family, friends and colleagues for their support and companionship.
Leuven, March 1993 Anne Mulkers
C o n t e n t s
In t r o d u c t i o n 1
Abstract I n t e r p r e t a t i o n 5
2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Abst rac t In te rp re ta t ion Framework . . . . . . . . . . . . . . . 7
2.2.1 Overview of the Framework . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Concrete and Abs t rac t Domains of Subs t i tu t ions . . . . . . . . 10
2.2.3 Pr imi t ive Opera t ions . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.4 Abst rac t In te rp re ta t ion Procedure . . . . . . . . . . . . . . . . 14
2.3 Example: In tegra ted Type and Mode Inference . . . . . . . . . 16
2.3.1 Rigid and In tegra ted Type Graphs . . . . . . . . . . . . . . . . 16
2.3.2 Type-graph Env i ronmen t s . . . . . . . . . . . . . . . . . . . . . 23
2.3.3 Pr imi t ive Opera t ions for Type-graph Env i ronmen t s . . . . . . 25
3 R e l a t e d W o r k 31
3.1 Aliasing and Pointer Analys is . . . . . . . . . . . . . . . . . . . 31
3.2 Reference Coun t ing and Liveness Analysis . . . . . . . . . . . . 38
3.3 Code Op t imiza t ion . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Sharing A n a l y s i s 47
4.1 Sharing Env i ronmen t s . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.1 Concrete Representa t ion of Shared St ructure . . . . . . . . . . 48
4.1.2 Abs t rac t Representa t ion of Shared St ructure . . . . . . . . . . 55
4.1.3 The Concrete and Abs t rac t Domains . . . . . . . . . . . . . . . 62
4.1.4 Order Rela t ion and U p p e r b o u n d Opera t ion . . . . . . . . . . . 66
4.2 Pr imi t ive Opera t ions . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1 Unif icat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1.1 X i : X j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1.2 X i : f ( X i l , . . . , X i j ) . . . . . . . . . . . . . . . . . . . . . . . 85
4.2.2 Procedure En t ry . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.3 Procedure Exit . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Eva lua t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.3.1 Example: i n s e r t / 3 . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.2 Relevance of Sharing Edges . . . . . . . . . . . . . . . . . . . . 114
v lll C O N T E N T S
4.3.3 Imprecis ion in the Sharing Analysis . . . . . . . . . . . . . . . 117
4.3.4 Efficiency of the Shar ing Analysis . . . . . . . . . . . . . . . . 123
L i v e n e s s A n a l y s i s 127
5.1 Liveness Env i ronmen t s . . . . . . . . . . . . . . . . . . . . . . 127
5.1.1 Concrete Representa t ion of Liveness In fo rmat ion . . . . . . . . 128
5.1.2 Abst rac t Representa t ion of Liveness In fo rmat ion . . . . . . . . 133
5.1.3 The Concrete and Abs t rac t Domains . . . . . . . . . . . . . . . 141
5.1.4 Order Rela t ion and Uppe rbound Opera t ion . . . . . . . . . . . 145
5.2 Pr imi t ive Opera t ions . . . . . . . . . . . . . . . . . . . . . . . 147
5.2.1 Unificat ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2.1.1 Xi = Xj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.2.1.2 X i : f ( X i , , . . . , X i j ) . . . . . . . . . . . . . . . . . . . . . . . 153
5.2.2 Procedure Ent ry . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.2.3 Procedure Exit . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.3 Eva lua t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3.1 Example: q s o r t / 3 . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.3.2 Precision of the Liveness Analysis . . . . . . . . . . . . . . . . 168
5.3.3 The Pract ical Usefulness of Liveness In fo rmat ion . . . . . . . . 171
6 C o n c l u s i o n 179
A p p e n d i x ; D e t a i l e d E x a m p l e s 183
A.1 List of Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
A.2 a p p e n d / 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A.3 n rev /2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
A.4 bu i ld t ree /2 and inse r t /3 . . . . . . . . . . . . . . . . . . . . . . 193
A.5 p e r m u t a t i o n / 2 and select/3 . . . . . . . . . . . . . . . . . . . . 196
A.6 sp l i t /3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
A.7 qsor t /2 and p a r t i t i o n / 4 . . . . . . . . . . . . . . . . . . . . . . 202
A.8 sameleaves/2 and profile/2 . . . . . . . . . . . . . . . . . . . . 205
A.9 sif t /2 and remove/3 . . . . . . . . . . . . . . . . . . . . . . . . 209
B i b l i o g r a p h y 213
Chapter 1
I n t r o d u c t i o n
In conventional languages, such as C or Pascal, the programmer explicitly con-
trols the utilization of memory by means of declarations and destructive assign-
ments. For example, when reversing a linear list L, the list cells of the original
list can be reused to construct the reversed list in the case that the original list
is no longer needed for further computat ions. It is up to the programmer to
decide whether he needs to preserve the old list intact and construct a reversed
list which has only the list elernertLs in common with the list L (e.g. Rev_L1 in
Figure 1.1), rather than reuse the list-constructor cells of L as well (e.g. Rev_L2).
_1 I -J I _1
-, , I , - , , I , -~ , I i
Rev_L2
/ \ " / \ " -, RevL 1
eJ
/
Figure 1.1: Reversing a linear list.
Applicative languages, in their pure form, do not have destructive assign-
ments. Also type declarations are often absent. The declarative nature of these
languages is often cited as an impor tant advantage, which allows programmers
to focus on the logic of the problems they have to solve, rather than on more
technical aspects such as search control and efficient memory usage. Unfortu-
nately, the performance of current implementat ions of applicative languages does
not compare well with procedural languages yet. To achieve better utilization of
memory, global flow analysis techniques are being developed that are concerned
with determining the type and liveness of da ta structures that are dynamical ly
2 C H A P T E R 1. I N T R O D U C T I O N
append (nil, _Y, _Y).
append(_E I _UJ,_Y,_E I _W) :- append(_U,_Y,_W).
nrev(nil, nil).
nrev(_E I _U, _Y) :-nrev(_U, _KU), append(_KU, ~, ~).
Program 1.h n r e v / 2 (Naive reverse)
created during program execution. Knowledge about the lifetime of da ta struc-
tures guides the compiler in the generation of target code to reuse heap storage
tha t is no longer accessible from program variables, i.e. to introduce destructive
operations and avoid the copying of data structures that have no subsequent
references.
In this book, we address the problem of liveness analysis for the class of pure
Horn clause logic programs. The language considered has a countable set of
variables (Vats), and countable sets of function and predicate symbols. A term
is a variable, a constant, or a compound term f ( Q , . . . , t , ~ ) where f is a n-ary
function symbol and the t~ are terms. An atom has the form p ( Q , . . . , tin) where
p is a m-ary predicate symbol and the ti are terms. A body is a (possibly empty)
finite conjunction of atoms, written A1, . . . , A,~. A clause consists of an a tom (its
head) and a body and is written A : - B. A program consists of a finite number
of clauses. A query or goalconsists of a body only, written 7- B. We assume that
the reader is acquainted with the basic terminology of logic programming and the
execution mechanism of Prolog which is based on unification and backtracking.
Features such as assert and retract are not considered, i.e. we assume that any
source code for the predicates that can be executed at run t ime is available to
the compiler.
The handling of da ta structures is very flexible in Prolog. Data manipulat ion
(record allocation as well as record access and parameter passing) is achieved
entirely via unification. An optimizing compiler can translate general unification
to more conventional memory manipulat ion operations if information is available
about the mode of use of the predicates. When at run t ime a compound term
becomes accessible for the first time, we can say the term is being constructed.
When a pat tern is matched against a compound te rm that is already accessible,
we can say the components of the term are being selected. Integrated type and
mode analysis in many cases allows to predict at compile t ime whether a unifi-
cation is a selection rather than a construction operation. Selection statements
in particular are good candidates to check for the possible creation of garbage
cells, i.e. cells that have no further references.
Consider the Prolog Program 1.1 for naive list reversal. We use the conven-
tion that variable names start with an underscore. If we assume that queries to
n r e v / 2 are restricted to have as first argument a list that is no longer referenced
after the call, and as second argument a free variable to return the output, then
it is possible to generate target code for this program that allocates no new
list-constructor cells, but rather reuses the list cells of the first argument. In-
deed, under the assumption, the integrated type and mode analysis will infer
tha t each call to the recursive clause of n r e v / 2 has as its first a rgument a list,
and as second argument a free variable. The unification of the call with the
clause head selects the head and tail of the first argument list. The principal
list-constructor cell of this list on the contrary has no subsequent references in
the clause following the unification of the call with the clause head. This means
that the compiler can recognize the principal list cell as garbage and generate
target code tha t reuses it. For instance, consider the caI1 to append/3 made by
the same clause. A single element list _E needs to be constructed. Instead of
allocating a new cell, the compiler can reuse the garbage cell that was detected.
Note that the problem is more complex if there may be multiple references to
the cells of the input list. Most implementat ions of unification unify a variable
and a compound structure by making the variable a reference to the structure -
not a copy of the structure. The representations in memory of the logical terms
to which variables can be bound typically share some of their structure: while
the denoted terms make up a forest of trees, their representations form a more
general directed acyclic graph. This is why in general the sharing analysis plays
a crucial part in the liveness analysis.
In the above example, we can also infer that , the first two arguments in
each call to append/3 will be lists and that the third argument will be a free
variable. Again, it is possible to detect that, after invocation of the recursive
clause of append/3 , the principal cell of the first argument is garbage and can be
reused to construct the value of the third (output) argument. Thus, all list con-
structions in this example can reuse garbage list ceils, eliminating all allocation
operations. Since the reused cells would otherwise be garbage, we have elim-
inated the garbage-collection overhead associated with the n r e v / 2 procedure.
Moreover, a compiler can detect that the element field of each reused list cell
already contains the value desired in the cells new use. The operations filling in
these car fields can be eliminated from the generated target code. The resulting
code closely resembles how a programmer using an imperative language would
solve the problem of reversing a linear list of linked records.
In the present work, we propose an abstract domain and operations to ana-
lyze the liveness of da ta structures within a framework of abstract interpretation.
Chapter 2 presents the principles of abstract interpretation for logic programs,
and the application of type and mode analysis on which the domain for liveness
analysis is based. In Chapter 3, we discuss work related to the application of
compile-t ime garbage collection in the context of both logic and functional pro-
gramming languages. In Chapter 4, we formalize an abstract interpretat ion for
analyzing how the terms to which program variables are bound at run time, can
share substructure in storage. We also augment the usual concrete semantics
with information about sharing of te rm structures and discuss whether any im-
plementat ion commitments are implied. As argued above, the sharing analysis
constitutes a prerequisite for the liveness analysis. The lat ter is presented in
Chapter 5. In both Chapter 4 and 5, the emphasis is mainly on the precision