. 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 set-valued 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...

. 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. 1 Introduction Systems of set constraints have received considerable attention as a formalism for expressing algorithms in program analysis and type inference. Many algorithms based on set constraints have been proposed and implemented, but very little is known about the computational complexity of solving systems of set constraints. In this paper we present complexity results for a natural hierarchy of decision problems involving set constraints. Set constraints are formal inclusions between expressions representing subsets of T \Sigma , the set of ground terms over a finite ranked alphabet \Sigma . A positive set constraint is an in...

We present an algorithm for automatic type checking of logic programs with respect to directional types that describe both the structure of terms and the directionality of predicates. The type checking problem is reduced to a decidable problem on systems of inclusion constraints over set expressions. We discuss some properties of the reduction algorithm, complexity, and present a proof of correctness. 1 1 Introduction Most logic programming languages are untyped. In Prolog, for example, it is considered meaningful to apply any n-ary predicate to any n-tuple of terms. However, it is generally accepted that static type checking has great advantages in detecting programming errors early and for generating efficient executable code. Motivated at least in part by the success of type systems for procedural and functional languages, there is currently considerable interest in finding appropriate definitions of type and welltyping for logic languages. This paper explores the type checki...

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 set-based 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

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...

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 NEXPTIME-complete. 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 set-valued 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...

this paper.

: This paper is a follow-up on previous work by the author on subtyping with (set-theoretic) union, intersection and recursive types. Previously, it was shown how types may be encoded as regular tree expressions/set constraints. This gave rise to a sound and complete decision procedure for type inclusion. The result was, however, limited to a rather specific type language. In the work reported on here, we generalize the result and develop a general technique for deriving subtyping algorithms for type languages with union, intersection and recursive types. We present separate requirements for obtaining a subtyping algorithm which is respectively sound and complete. In this way we obtain a generic strategy for implementing the subtype relation for a broad class of very expressive type languages. Key-words: Type theory, Regular tree expressions, Set constraints, Algorithms, Semantics, Ideal model, Intersection types, Union types, Recursive types (R'esum'e : tsvp) The author is partially...

. Set constraints is a suitable formalism for static analysis of programs. However, it is known that the complexity of set constraint problems in the most general cases is very high (NEXPTIME-completeness of the satisfiability test). Lots of works are involved in finding more tractable subclasses. In this paper, we investigate two classes of set constraints shown to be useful for program analysis: the first one is an extension of definite set constraints including the main feature of quantified set expressions. We will show that the satisfiability problem for this class is EXPTIME- complete. The second one concerns constraints of the form X ` exp, where exp is built with function symbols, the intersection and union connectives and projection operators. The dual aspects of those two classes allows to find a common approach for solving both of them. This approach uses as basic tool tree automata, which are suitable both for computation and representing the solution of those solving prob...

We present a new constraint system called INES. Its constraints are conjunctions of inclusions t1 `t2 between first-order terms (without set operators) which are interpreted over non-empty sets of trees. The existing systems of set constraints can express INES constraints only if they include negation. Their satisfiability problem is NEXPTIME-complete. We present an incremental algorithm that solves the satisfiability problem of INES constraints in cubic time. We intend to apply INES constraints for type analysis for a concurrent constraint programming language.