Viorica Sofronie-Stokkermans

On local reasoning in verification

We present a general framework which allows to identify complex theories important in verification for which efficient reasoning methods exist. The framework we present is based on a general notion of locality. We show that locality considerations allow us to obtain parameterized decidability and complexity results for many (combinations of) theories important in verification in general and in the verification of parametric systems in particular. We give numerous examples; in particular we show that several theories of data structures studied in the verification literature are local extensions of a base theory. The general framework we use allows us to identify situations in which some of the syntactical restrictions imposed in previous papers can be relaxed.

Automated Reasoning in Some Local Extensions of Ordered Structures

We give a uniform method for automated reasoning in several types of extensions of ordered algebraic structures (definitional extensions, extensions with boundedness axioms or with monotonicity axioms). We show that such extensions are local and, hence, efficient methods for hierarchical reasoning exist in all these cases.

Modular proof systems for partial functions with Evans equality

The paper presents a modular superposition calculus for the combination of first-order theories involving both total and partial functions. The modularity of the calculus is a consequence of the fact that all the inferences are pure-only involving clauses over the alphabet of either one, but not both, of the theories-when refuting goals represented by sets of pure formulae. The calculus is shown to be complete provided that functions that are not in the intersection of the component signatures are declared as partial. This result also means that if the unsatisfiability of a goal modulo the combined theory does not depend on the totality of the functions in the extensions, the inconsistency will be effectively found. Moreover, we consider a constraint superposition calculus for the case of hierarchical theories and show that it has a related modularity property. Finally, we identify cases where the partial models can always be made total so that modular superposition is also complete with respect to the standard (total function) semantics of the theories.

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators and Applications to the Semantics of Non-Classical Logics. Part II

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes.

Applications of Hierarchical Reasoning in the Verification of Complex Systems

In this paper we show how hierarchical reasoning can be used to verify properties of complex systems. Chains of local theory extensions are used to model a case study taken from the European Train Control System (ETCS) standard, but considerably simplified. We show how testing invariants and bounded model checking (for safety properties expressed by universally quantified formulae, depending on certain parameters of the systems) can automatically be reduced to checking satisfiability of ground formulae over a base theory.

Locality Results for Certain Extensions of Theories with Bridging Functions

We study possibilities of reasoning about extensions of base theories with functions which satisfy certain recursion (or homomorphism) properties. Our focus is on emphasizing possibilities of hierarchical and modular reasoning in such extensions and combinations thereof. We present practical applications in verification and cryptography.

Hierarchical and Modular Reasoning in Complex Theories: The Case of Local Theory Extensions

We present an overview of results on hierarchical and modular reasoning in complex theories. We show that for a special type of extensions of a base theory, which we call local, hierarchic reasoning is possible (i.e. proof tasks in the extension can be hierarchically reduced to proof tasks w.r.t. the base theory). Many theories important for computer science or mathematics fall into this class (typical examples are theories of data structures, theories of free or monotone functions, but also functions occurring in mathematical analysis). In fact, it is often necessary to consider complex extensions, in which various types of functions or data structures need to be taken into account at the same time. We show how such local theory extensions can be identified and under which conditions locality is preserved when combining theories, and we investigate possibilities of efficient modular reasoning in such theory combinations. We present several examples of application domains where local theories and local theory extensions occur in a natural way. We show, in particular, that various phenomena analyzed in the verification literature can be explained in a unified way using the notion of locality.

Interpolation in local theory extensions

In this paper we study interpolation in local extensions of a base theory. We identify situations in which it is possible to obtain interpolants in a hierarchical manner, by using a prover and a procedure for generating interpolants in the base theory as black-boxes. We present several examples of theory extensions in which interpolants can be computed this way, and discuss applications in verification, knowledge representation, and modular reasoning in combinations of local theories.

Verifying CSP-OZ-DC specifications with complex data types and timing parameters

We extend existing verification methods for CSP-OZ-DC to reason about real-time systems with complex data types and timing parameters. We show that important properties of systems can be encoded in well-behaved logical theories in which hierarchic reasoning is possible. Thus, testing invariants and bounded model checking can be reduced to checking satisfiability of ground formulae over a simple base theory. We illustrate the ideas by means of a simplified version of a case study from the European Train Control System standard.

Resolution-based decision procedures for the universal theory of some classes of distributive lattices with operators

We establish a link between the satisfiability of universal sentences with respect to classes of distributive lattices with operators and their satisfiability with respect to certain classes of relational structures. This justifies a method for structure-preserving translation to clause form of universal sentences in such classes of algebras. We show that refinements of resolution yield decision procedures for the universal theory of some such classes. In particular, we obtain exponential space and time decision procedures for the universal clause theory of (i) the class of all bounded distributive lattices with operators satisfying a set of (generalized) residuation conditions, and (ii) the class of all bounded distributive lattices with operators, and a doubly-exponential time decision procedure for the universal clause theory of the class of all Heyting algebras.