Daniele Zucchelli

Towards SMT Model Checking of Array-Based Systems

We introduce the notion of array-based system as a suitable abstraction of infinite state systems such as broadcast protocols or sorting programs. By using a class of quantified-first order formulae to symbolically represent array-based systems, we propose methods to check safety (invariance) and liveness (recurrence) properties on top of Satisfiability Modulo Theories solvers. We find hypotheses under which the verification procedures for such properties can be fully mechanized.

Decision procedures for extensions of the theory of arrays

The theory of arrays, introduced by McCarthy in his seminal paper “Towards a mathematical science of computation,” is central to Computer Science. Unfortunately, the theory alone is not sufficient for many important verification applications such as program analysis. Motivated by this observation, we study extensions of the theory of arrays whose satisfiability problem (i.e., checking the satisfiability of conjunctions of ground literals) is decidable. In particular, we consider extensions where the indexes of arrays have the algebraic structure of Presburger arithmetic and the theory of arrays is augmented with axioms characterizing additional symbols such as dimension, sortedness, or the domain of definition of arrays. We provide methods for integrating available decision procedures for the theory of arrays and Presburger arithmetic with automatic instantiation strategies which allow us to reduce the satisfiability problem for the extension of the theory of arrays to that of the theories decided by the available procedures. Our approach aims to re-use as much as possible existing techniques so as to ease the implementation of the proposed methods. To this end, we show how to use model-theoretic, rewriting-based theorem proving (i.e., superposition), and techniques developed in the Satisfiability Modulo Theories communities to implement the decision procedures for the various extensions.

Combination Methods for Satisfiability and Model-Checking of Infinite-State Systems

Manna and Pnueli have extensively shown how a mixture of first-order logic (FOL) and discrete Linear time Temporal Logic (LTL) is sufficient to precisely state verification problems for the class of reactive systems. Theories in FOL model the (possibly infinite) data structures used by a reactive system while LTL specifies its (dynamic) behavior. In this paper, we derive undecidability and decidability results for both the satisfiability of (quantifier-free) formulae and the model-checking of safety properties by lifting combination methods for (non-disjoint) theories in FOL. The proofs of our decidability results suggest how decision procedures for the constraint satisfiability problem of theories in FOL and algorithms for checking the satisfiability of propositional LTL formulae can be integrated. This paves the way to employ efficient Satisfiability Modulo Theories solvers in the model-checking of infinite state systems. We illustrate our techniques on two examples.

A Comprehensive Framework for Combined Decision Procedures

We define a general notion of a fragment within higher order type theory; a procedure for constraint satisfiability in combined fragments is outlined, following Nelson-Oppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitable noetherianity conditions and allows an abstract version of a ‘Keisler-Shelah like’ isomorphism theorem. We show that this general decidability transfer result covers as special cases, besides applications which seem to be new, the recent extension of Nelson-Oppen procedure to non-disjoint signatures [16] and the fusion transfer of decidability of consistency of A-Boxes with respect to T-Boxes axioms in local abstract description systems [9]; in addition, it reduces decidability of modal and temporal monodic fragments [32] to their extensional and one-variable components.

Decidability and undecidability results for nelson-oppen and rewrite-based decision procedures

In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T1∪T2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable. In the second part of the paper we strengthen the Nelson-Oppen decidability transfer result, by showing that it applies to theories over disjoint signatures, whose satisfiability problem, in either arbitrary or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, complementing recent work on combination of theories in the rewrite-based approach to satisfiability.

A comprehensive combination framework

We define a general notion of a fragment within higher-order type theory; a procedure for constraint satisfiability in combined fragments is outlined, following Nelson-Oppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitable noetherianity conditions and admits an abstract version of a “Keisler-Shelah-like” isomorphism theorem. We show that this general decidability transfer result covers recent work on combination in first-order theories as well as in various intensional logics such as description, modal, and temporal logics.

Deciding extensions of the theory of arrays by integrating decision procedures and instantiation strategies

The theory of arrays, introduced by McCarthy in his seminal paper “Toward a mathematical science of computation”, is central to Computer Science. Unfortunately, the theory alone is not sufficient for many important verification applications such as program analysis. Motivated by this observation, we study extensions of the theory of arrays whose satisfiability problem (i.e. checking the satisfiability of conjunctions of ground literals) is decidable. In particular, we consider extensions where the indexes of arrays has the algebraic structure of Presburger Arithmetic and the theory of arrays is augmented with axioms characterizing additional symbols such as dimension, sortedness, or the domain of definition of arrays. We provide methods for integrating available decision procedures for the theory of arrays and Presburger Arithmetic with automatic instantiation strategies which allow us to reduce the satisfiability problem for the extension of the theory of arrays to that of the theories decided by the available procedures. Our approach aims to reuse as much as possible existing techniques so to ease the implementation of the proposed methods. To this end, we show how to use both model-theoretic and rewriting-based theorem proving (i.e., superposition) techniques to implement the instantiation strategies of the various extensions.

Noetherianity and Combination Problems

In abstract algebra, a structure is said to be Noetherian if it does not admit infinite strictly ascending chains of congruences. In this paper, we adapt this notion to first-order logic by defining the class of Noetherian theories. Examples of theories in this class are Linear Arithmetics without ordering and the empty theory containing only a unary function symbol. Interestingly, it is possible to design a non-disjoint combination method for extensions of Noetherian theories. We investigate sufficient conditions for adding a temporal dimension to such theories in such a way that the decidability of the satisfiability problem for the quantifier-free fragment of the resulting temporal logic is guaranteed. This problem is firstly investigated for the case of Linear time Temporal Logic and then generalized to arbitrary modal/temporal logics whose propositional relativized satisfiability problem is decidable.

A Decidability Result for the Model Checking of Infinite-State Systems

We present a decidability result for the model checking of a certain class of properties that can be conveniently expressed as ground formulae of a first-order temporal fragment. The decidability result is obtained by importing into the context of model-checking problems some techniques developed for the combination of decision procedures for the satisfiability of constraints. The general decidability result is then specialized for checking properties of particular interest, such as liveness and safety, and, for the latter case, a more optimized algorithm has been proposed.

Invited contribution : a comprehensive framework for combined decision procedures

We define a general notion of a fragment within higher order type theory; a procedure for constraint satisfiability in combined fragments is outlined, following Nelson-Oppen schema. The procedure is in general only sound, but it becomes terminating and complete when the shared fragment enjoys suitable noetherianity conditions and allows an abstract version of a ‘Keisler-Shelah like’ isomorphism theorem. We show that this general decidability transfer result covers as special cases, besides applications which seem to be new, the recent extension of Nelson-Oppen procedure to non-disjoint signatures [16] and the fusion transfer of decidability of consistency of A-Boxes with respect to T-Boxes axioms in local abstract description systems [9]; in addition, it reduces decidability of modal and temporal monodic fragments [32] to their extensional and one-variable components.