XML specifications often consist of a type definition (typically, a DTD) and a set of integrity constraints. It has been shown previously that such specifications can be inconsistent, and thus it is often desirable to check consistency at compile-time. It is known that for general keys and foreign keys, and DTDs, the consistency problem is undecidable; however, it becomes NP-complete when all keys are one-attribute (unary), and tractable, if no foreign keys are used.

this paper.

In this paper, we introduce the class of co-definite set constraints. This is a natural subclass of set constraints which, when satisfiable, have a greatest solution. It is practically motivated by the set-based analysis of logic programs with the greatest-model semantics. We present an algorithm solving co-definite set constraints and show that their satisfiability problem is DEXPTIME-complete. 1 Introduction Set constraints and set-based analysis form an established research topic. It combines theoretical investigations ranging from expressiveness and decidability to program semantics and domain theory, with direct practical applications to type inference, optimization and verification of imperative, functional, logic and reactive programs (see [1, 14, 20] for overviews). In set-based analysis, the problem of reasoning about runtime properties of programs is transferred to the problem of solving set constraints. The design of a system for a particular program analysis problem (for a...

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.

Typestate systems ensure many desirable properties of imperative programs, including initialization of object fields and correct use of stateful library interfaces. Abstract sets with cardinality constraints naturally generalize typestate properties: relationships between the typestates of objects can be expressed as subset and disjointness relations on sets, and elements of sets can be represented as sets of cardinality one. In addition, sets with cardinality constraints provide a natural language for specifying operations and invariants of data structures. Motivated by these program analysis applications, this paper presents new algorithms and new complexity results for constraints on sets and their cardinalities. We study several classes of constraints and demonstrate a trade-off between their expressive power and their complexity. Our first result concerns a quantifier-free fragment of Boolean Algebra with Presburger Arithmetic. We give a nondeterministic polynomial-time algorithm for reducing the satisfiability of sets with symbolic cardinalities to constraints on constant cardinalities, and give a polynomial-space algorithm for the resulting problem. The best previously existing algorithm runs in exponential space and nondeterministic exponential time. In a quest for more efficient fragments, we identify several subclasses of sets with cardinality constraints whose satisfiability is NP-hard. Finally, we identify a class of constraints that has polynomial-time satisfiability and entailment problems and can serve as a foundation for efficient program analysis. We give a system of rewriting rules for enforcing certain consistency properties of these constraints and show how to extract complete information from constraints in normal form. This result implies the soundness and completeness of our algorithms. 1.

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Abstract. Logics that involve collections (sets, multisets), and cardinality constraints are useful for reasoning about unbounded data structures and concurrent processes. To make such logics more useful in verification this paper extends them with the ability to compute direct and inverse relation and function images. We establish decidability and complexity bounds for the extended logics. 1

Automated reasoning about sets has received a considerable amount of interest in the literature. Techniques for such reasoning have been used in, for instance, analyses of programming languages, terminological logics and spatial reasoning. In this paper, we identify a new class of set constraints where checking satisfiability is tractable (i.e. polynomial-time). We show how to use this tractability result for constructing a new tractable fragment of intuitionistic logic. Furthermore, we prove NP-completeness of several other cases of reasoning about sets. 1 Introduction There has been considerable interest in formalisms for describing and reasoning about sets. We begin by describing some of these. The most well-studied class of set constraints is, probably, Herbrand set constraints. Such have been suggested as a formalism for describing relationships between sets of terms of a free algebra. A positive set constraint has the form X ` Y , where X and Y are set expressions. Examples of ...

The problem of computing with temporal information was early recognised within the area of artificial intelligence, most notably the temporal interval algebra by Allen has become a widely used formalism for representing and computing with qualitative knowledge about relations between temporal intervals. However, the computational properties of the algebra and related formalisms are known to be bad: most problems (like satisfiability) are NP-hard. This thesis contributes to finding restrictions (as weak as possible) on Allen's algebra and related temporal formalisms (the point-interval algebra and extensions of Allen's algebra for metric time) for which the satisfiability problem can be computed in polynomial time. Another research area utilising temporal information is that of reasoning about action, which treats the problem of drawing conclusions based on the knowledge about actions having been performed at certain time points (this amounts to solving the infamous frame problem). One ...

. In this paper we show that satisfiability of Tarskian set constraints (without recursion) can be decided in exponential time. This closes the gap left open by D.A. McAllester, R. Givan, C. Witty and D. Kozen in [14]. 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 [3], [11], [12], [15], [16], [18]. The systems of set constraints used for program analysis were considered as inclusion constraints over the Herbrand universe i.e. a solution consisted of a collection of subsets of the Herbrand universe. To distinguish them from set constraints studied here we shall call them the Herbrand set constraints. The satisfiability problem for Herbrand set constraints have been studied by many authors including N. Heintze and J. Jaffar [10], A. Aiken and E.L. Wimmers...