Witold Charatonik

Set constraints with projections are in NEXPTIME

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 (1993) and R. Gilleron, S. Tison, and M. Tommasi (1990) on decidability of negated set constraints and solves a problem that was open for several years.<<ETX>>

Negative set constraints with equality

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 (1993) and A. Aiken, D. Kozen, and E.L. Wimmers (1993). 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 paper by L. Bachmair, H. Ganzinger, and U. Waldmann (1993). We first give a new proof of decidability of systems of mixed (positive and negative) set constraints. We explicitly 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 projections and unrestricted diagonalization is in NEXPTIME.<<ETX>>

Finite-Control Mobile Ambients

We define a finite-control fragment of the ambient calculus, a formalism for describing distributed and mobile computations. A series of examples demonstrates the expressiveness of our fragment. In particular, we encode the choice-free, finite-control, synchronous ?-calculus. We present an algorithm for model checking this fragment against the ambient logic (without composition adjunct). This is the first proposal of a model checking algorithm for ambients to deal with recursively-defined, possibly nonterminating, processes. Moreover, we show that the problem is PSPACE-complete, like other fragments considered in the literature. Finite-control versions of other process calculi are obtained via various syntactic restrictions. Instead, we rely on a novel type system that bounds the number of active ambients and outputs in a process; any typable process has only a finite number of derivatives.

Model checking mobile ambients

We settle the complexity bounds of the model checking problem for the ambient calculus with public names against the ambient logic. We show that if either the calculus contains replication or the logic contains the guarantee operator, the problem is undecidable. In the case of the replication-free calculus and guarantee-free logic we prove that the problem is PSPACE-complete. For the complexity upper bound, we devise a new representation of processes that remains of polynomial size during process execution; this allows us to keep the model checking procedure in polynomial space. Moreover, we prove PSPACE-hardness of the problem for several quite simple fragments of the calculus and the logic; this suggests that there are no interesting fragments with polynomial-time model checking algorithms.

Set constraints with intersection

Set constraints are inclusions between expressions denoting sets of trees. The efficiency of their satisfiability test is a central issue in set-based program analysis, their main application domain. We introduce the class of set constraints with intersection (the only operators forming the expressions are constructors and intersection) and show that its satisfiability problem is DEXPTIME-complete. The complexity characterization continues to hold for negative set constraints with intersection (which have positive and negated inclusions). We reduce the satisfiability problem for these constraints to one over the interpretation domain of nonempty sets of trees. Set constraints with intersection over the domain of nonempty sets of trees enjoy the fundamental property of independence of negated conjuncts. This allows us to handle each negated inclusion separately by the entailment algorithm that we devise. We furthermore prove that set constraints with intersection are equivalent to the class of definite set constraints and thereby settle the complexity question of the historically first class for which the decidability question was solved.

On Name Generation and Set-Based Analysis in the Dolev-Yao Model

We study the control reachability problem in the Dolev-Yao model of cryptographic protocols when principals are represented by tail recursive processes with generated names. We propose a conservative approximation of the problem by reduction to a non-standard collapsed operational semantics and we introduce checkable syntactic conditions entailing the equivalence of the standard and the collapsed semantics. Then we introduce a conservative and decidable set-based analysis of the collapsed operational semantics and we characterize a situation where the analysis is exact.

Set-Based Analysis of Reactive Infinite-State Systems

We present an automated abstract verification method for infinite-state systems specified by logic programs (which are a uniform and intermediate layer to which diverse formalisms such as transition systems, pushdown processes and while programs can be mapped). We establish connections between: logic program semantics and CTL properties, set-based program analysis and pushdown processes, and also between model checking and constraint solving, viz. theorem proving. We show that set-based analysis can be used to compute supersets of the values of program variables in the states that satisfy a given CTL property.

Directional Type Inference for Logic Programs

We follow the set-based approach to directional types proposed by Aiken and Lakshman [1]. Their type checking algorithm works via set constraint solving and is sound and complete for given discriminative types. We characterize directional types in model-theoretic terms. We present an algorithm for inferring directional types. The directional type that we derive from a logic program P is uniformly at least as precise as any discriminative directional type of P, i.e., any directional type out of the class for which the type checking algorithm of Aiken and Lakshman is sound and complete. We improve their algorithm as well as their lower bound and thereby settle the complexity (Dexptime-complete) of the corresponding problem.

Set Constraints in Some Equational Theories

Set constraints are relations between sets of ground terms over a given alphabet. They give a natural formalism for many problems in program analysis, type inference, order-sorted unification, and constraint logic programming. In this paper we start studies of set constraints in the environment given by equational specifications. We show that in the case of associativity (i.e., in free monoids) as well as in the case of associativity and commutativity (i.e., in commutative monoids) the problem of consistency of systems of set constraints is undecidable; in linear nonerasing shallow theories the consistency of systems of positive set constraints is NEXPTIME-complete and in linear shallow theories the problem for positive and negative set constraints is decidable.

The independence property of a class of set constraints

We investigate a class of set constraints that is used for the type analysis of concurrent constraint programs. Its constraints are inclusions between first-order terms (without set operators) interpreted over non-empty sets of finite trees. We show that this class has the independence property. We give a polynomial algorithm for entailment. The independence property is a fundamental property of constraint systems. It says that the constraints cannot express disjunctions, or, equivalently, that negated conjuncts are independent from each other. As a consequence, the satisfiability of constraints with negated conjuncts can be directly reduced to entailment. On leave from University of Wroclaw, Poland. Partially supported by KBN grant 8 S503 022 07 and by Foundation for Polish Science.