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Introduction to set constraintbased program analysis
 Science of Computer Programming
, 1999
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Set Constraints: Results, Applications and Future Directions
 In Second Workshop on the Principles and Practice of Constraint Programming
"... . Set constraints are a natural formalism for many problems that arise in program analysis. This paper provides a brief introduction to set constraints: what set constraints are, why they are interesting, the current state of the art, open problems, applications and implementations. 1 Introduction ..."
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Cited by 74 (4 self)
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. Set constraints are a natural formalism for many problems that arise in program analysis. This paper provides a brief introduction to set constraints: what set constraints are, why they are interesting, the current state of the art, open problems, applications and implementations. 1 Introduction Set constraints are a natural formalism for describing relationships between sets of terms of a free algebra. A set constraint has the form X ` Y , where X and Y are set expressions. Examples of set expressions are 0 (the empty set), ff (a setvalued variable), c(X; Y ) (a constructor application), and the union, intersection, or complement of set expressions. Recently, there has been a great deal of interest in program analysis algorithms based on solving systems of set constraints, including analyses for functional languages [AWL94, Hei94, AW93, AM91, JM79, MR85, Rey69], logic programming languages [AL94, HJ92, HJ90b, Mis84], and imperative languages [HJ91]. In these algorithms, sets of...
The complexity of set constraints
, 1993
"... Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. We present several results on the computational complexity of solving systems of set constraints. The systems we study form a natural complexity hierar ..."
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Cited by 74 (11 self)
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Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. We present several results on the computational complexity of solving systems of set constraints. The systems we study form a natural complexity hierarchy depending on the form of the constraint language.
Set Constraints are the Monadic Class
, 1992
"... We investigate the relationship between set constraints and the monadic class of firstorder formulas and show that set constraints are essentially equivalent to the monadic class. From this equivalence we can infer that the satisfiability problem for set constraints is complete for NEXPTIME. Mor ..."
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Cited by 72 (0 self)
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We investigate the relationship between set constraints and the monadic class of firstorder formulas and show that set constraints are essentially equivalent to the monadic class. From this equivalence we can infer that the satisfiability problem for set constraints is complete for NEXPTIME. More precisely, we prove that this problem has a lower bound of NTIME(c n= log n ). The relationship between set constraints and the monadic class also gives us decidability and complexity results for certain practically useful extensions of set constraints, in particular "negative projections" and subterm equality tests.
Decidability of Systems of Set Constraints with Negative Constraints
 INFORMATION AND COMPUTATION
, 1994
"... Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. Recently, several algorithms for solving general systems of positive set constraints have appeared. In this paper we consider systems of mixed posi ..."
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Cited by 56 (10 self)
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Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. Recently, several algorithms for solving general systems of positive set constraints have appeared. In this paper we consider systems of mixed positive and negative constraints, which are considerably more expressive than positive constraints alone. We show that it is decidable whether a given such system has a solution. The proof involves a reduction to a numbertheoretic decision problem that may be of independent interest.
Set Constraints and SetBased Analysis
 In Proceedings of the Workshop on Principles and Practice of Constraint Programming, LNCS 874
, 1994
"... This paper contains two main parts. The first examines the set constraint calculus, discusses its history, and overviews the current state of known algorithms and related issues. Here we will also survey the uses of set constraints, starting from early work in (imperative) program analysis, to more ..."
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Cited by 39 (0 self)
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This paper contains two main parts. The first examines the set constraint calculus, discusses its history, and overviews the current state of known algorithms and related issues. Here we will also survey the uses of set constraints, starting from early work in (imperative) program analysis, to more recent work in logic and functional programming systems. The second part describes setbased analysis. The aim here is a declarative interpretation of what it means to approximate the meaning of a program in just one way: ignore dependencies between variables, and instead, reason about each variable as the set of its possible runtime values. The basic approach starts with some description of the operational semantics, and then systematically replaces descriptions of environments (mappings from program variables to values) by set environments (mappings from program variables to sets
Set constraints with projections are in NEXPTIME
, 1994
"... Systems of set constraints describe relations between sets of ground terms. They have been successfully used in program analysis and type inference. In this paper we prove that the problem of existence of a solution of a system of set constraints with projections is in NEXPTIME, and thus that it is ..."
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Cited by 36 (10 self)
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Systems of set constraints describe relations between sets of ground terms. They have been successfully used in program analysis and type inference. In this paper we prove that the problem of existence of a solution of a system of set constraints with projections is in NEXPTIME, and thus that it is NEXPTIMEcomplete. This extends the result of A. Aiken, D. Kozen, and E.L. Wimmers [3] and R. Gilleron, S. Tison, and M. Tommasi [10] on decidability of negated set constraints and solves a problem that was open for several years. 1 Introduction Set constraints have a form of inclusions between set expressions built over a set of setvalued variables, constants and function symbols. They have been used in program analysis and type inference algorithms for functional, imperative and logic programming languages [4], [5], [12], [13], [15], [16], [18]. Solving a system of set constraints is the main part of these algorithms, however until now the satisfiability problem for such constraints was...
Negative Set Constraints With Equality
 In Ninth Annual IEEE Symposium on Logic in Computer Science
, 1994
"... Systems of set constraints describe relations between sets of ground terms. They have been successfully used in program analysis and type inference. So far two proofs of decidability of mixed set constraints have been given: by R. Gilleron, S. Tison and M. Tommasi [12] and A. Aiken, D. Kozen, and E. ..."
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Cited by 35 (10 self)
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Systems of set constraints describe relations between sets of ground terms. They have been successfully used in program analysis and type inference. So far two proofs of decidability of mixed set constraints have been given: by R. Gilleron, S. Tison and M. Tommasi [12] and A. Aiken, D. Kozen, and E.L. Wimmers [3]. However, both these proofs are long, involved and do not seem to extend to more general set constraints. Our approach is based on a reduction of set constraints to the monadic class given in a recent paper by L. Bachmair, H. Ganzinger, and U. Waldmann [7]. We first give a new proof of decidability of systems of mixed positive and negative set constraints. We explicitely describe a very simple algorithm working in NEXPTIME and we give in all detail a relatively easy proof of its correctness. Then, we sketch how our technique can be applied to get various extensions of this result. In particular we prove that the problem of consistency of mixed set constraints with restricted p...
Logical aspects of set constraints
 in Proc. 1993 Conf. Computer Science Logic
, 1993
"... Abstract. Set constraints are inclusion relations between sets of ground terms over a ranked alphabet. They have been used extensively in program analysis and type inference. Here we present an equational axiomatization of the algebra of set constraints. Models of these axioms are called termset alg ..."
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Abstract. Set constraints are inclusion relations between sets of ground terms over a ranked alphabet. They have been used extensively in program analysis and type inference. Here we present an equational axiomatization of the algebra of set constraints. Models of these axioms are called termset algebras. They are related to the Boolean algebras with operators of Jonsson and Tarski. We also de ne a family of combinatorial models called topological term automata, which are essentially the term automata studied by Kozen,Palsberg, and Schwartzbach endowed with a topology such that all relevant operations are continuous. These models are similar to Kripke frames for modal or dynamic logic. We establish a Stone duality between termset algebras and topological term automata, and use this to derive a completeness theorem for a related multidimensional modal logic. Finally, weproveasmall model property by ltration, and argue that this result contains the essence of several algorithms appearing in the literature on set constraints. 1