arXiv subjects

Silvio Ghilardi

Publications and source records attributed to Silvio Ghilardi.

33 records · Page 2Linked to original sources

arXiv2019

Verification of Data-Aware Processes via Array-Based Systems (Extended Version)

We study verification over a general model of artifact-centric systems, to assess (parameterized) safety properties irrespectively of the initial database instance. We view such artifact systems as array-based systems, which allows us to check safety by adapting backward reachability, establishing for the first time a correspondence with model checking based on Satisfiability-Modulo-Theories (SMT). To do so, we make use of the model-theoretic machinery of model completion, which surprisingly turns out to be an effective tool for verification of relational systems, and represents the main original contribution of this paper. In this way, we pursue a twofold purpose. On the one hand, we reconstruct (restricted to safety) the essence of some important decidability results obtained in the literature for artifact-centric systems, and we devise a genuinely novel class of decidable cases. On the other, we are able to exploit SMT technology in implementations, building on the well-known MCMT model checker for array-based systems, and extending it to make all our foundational results fully operational.

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arXiv2019

Free Heyting Algebra Endomorphisms: Ruitenburg's Theorem and Beyond

Ruitenburg's Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N $\ge$ 0 such that f N +2 = f N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms between free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

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arXiv2018

Ruitenburg's Theorem via Duality and Bounded Bisimulations

For a given intuitionistic propositional formula A and a propositional variable x occurring in it, define the infinite sequence of formulae { A \_i | i$\ge$1} by letting A\_1 be A and A\_{i+1} be A(A\_i/x). Ruitenburg's Theorem [8] says that the sequence { A \_i } (modulo logical equivalence) is ultimately periodic with period 2, i.e. there is N $\ge$ 0 such that A N+2 $\leftrightarrow$ A N is provable in intuitionistic propositional calculus. We give a semantic proof of this theorem, using duality techniques and bounded bisimulations ranks.

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arXiv2018

Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version)

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the alge- braic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $μ$-calculus based on intuitionistic logic is trivial, every $μ$-formula being equiv- alent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $μ$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed- point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal.

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arXiv2017

Counter Simulations via Higher Order Quantifier Elimination: a preliminary report

Quite often, verification tasks for distributed systems are accomplished via counter abstractions. Such abstractions can sometimes be justified via simulations and bisimulations. In this work, we supply logical foundations to this practice, by a specifically designed technique for second order quantifier elimination. Our method, once applied to specifications of verification problems for parameterized distributed systems, produces integer variables systems that are ready to be model-checked by current SMT-based tools. We demonstrate the feasibility of the approach with a prototype implementation and first experiments.

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arXiv2017

Existentially Closed Brouwerian Semilattices

The variety of Brouwerian semilattices is amalgamable and locally finite, hence by well-known results due to W. H. Wheeler, it has a model completion (whose models are the existentially closed structures). In this paper, we supply for such a model completion a finite and rather simple axiomatization.

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arXiv2016

Monadic second order logic as the model companion of temporal logic

The main focus of this paper is on bisimulation-invariant MSO, and more particularly on giving a novel model-theoretic approach to it. In model theory, a model companion of a theory is a first-order description of the class of models in which all potentially solvable systems of equations and non-equations have solutions. We show that bisimulation-invariant MSO on trees gives the model companion for a new temporal logic, "fair CTL", an enrichment of CTL with local fairness constraints. To achieve this, we give a completeness proof for the logic fair CTL which combines tableaux and Stone duality, and a fair CTL encoding of the automata for the modal μ-calculus. Moreover, we also show that MSO on binary trees is the model companion of binary deterministic fair CTL.

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arXiv2016

A model-theoretic characterization of monadic second order logic on infinite words

Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary predicate symbols. Monadic second order logic over infinite words (S1S) can alternatively be described as a first-order logic interpreted in $\mathcal{P}(ω)$, the power set Boolean algebra of the natural numbers, equipped with modal operators for 'initial', 'next' and 'future' states. We prove that the first-order theory of this structure is the model companion of a class of algebras corresponding to the appropriate version of linear temporal logic (LTL) without until. The proof makes crucial use of two classical, non-trivial results from the literature, namely the completeness of LTL with respect to the natural numbers, and the correspondence between S1S-formulas and Büchi automata.

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arXiv2016

Counting Constraints in Flat Array Fragments

We identify a fragment of Presburger arithmetic enriched with free function symbols and cardinality constraints for interpreted sets, which is amenable to automated analysis. We establish decidability and complexity results for such a fragment and we implement our algorithms. The experiments run in discharging proof obligations coming from invariant checking and bounded model-checking benchmarks show the practical feasibility of our decision procedure.

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arXiv2016

Fixed-point elimination in the intuitionistic propositional calculus

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $μ$-calculus based on intuitionistic logic is trivial, every $μ$-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $μ$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal.

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arXiv2014

Monotonic Abstraction Techniques: from Parametric to Software Model Checking

Monotonic abstraction is a technique introduced in model checking parameterized distributed systems in order to cope with transitions containing global conditions within guards. The technique has been re-interpreted in a declarative setting in previous papers of ours and applied to the verification of fault tolerant systems under the so-called "stopping failures" model. The declarative reinterpretation consists in logical techniques (quantifier relativizations and, especially, quantifier instantiations) making sense in a broader context. In fact, we recently showed that such techniques can over-approximate array accelerations, so that they can be employed as a meaningful (and practically effective) component of CEGAR loops in software model checking too.

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arXiv2013

Abstraction and Acceleration in SMT-based Model-Checking for Array Programs

Abstraction (in its various forms) is a powerful established technique in model-checking; still, when unbounded data-structures are concerned, it cannot always cope with divergence phenomena in a satisfactory way. Acceleration is an approach which is widely used to avoid divergence, but it has been applied mostly to integer programs. This paper addresses the problem of accelerating transition relations for unbounded arrays with the ultimate goal of avoiding divergence during reachability analysis of abstract programs. For this, we first design a format to compute accelerations in this domain; then we show how to adapt the so-called 'monotonic abstraction' technique to efficiently handle complex formulas with nested quantifiers generated by the acceleration preprocessing. Notably, our technique can be easily plugged-in into abstraction/refinement loops, and strongly contributes to avoid divergence: experiments conducted with the MCMT model checker attest the effectiveness of our approach on programs with unbounded arrays, where acceleration and abstraction/refinement technologies fail if applied alone.

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arXiv2012

Quantifier-Free Interpolation of a Theory of Arrays

The use of interpolants in model checking is becoming an enabling technology to allow fast and robust verification of hardware and software. The application of encodings based on the theory of arrays, however, is limited by the impossibility of deriving quantifier- free interpolants in general. In this paper, we show that it is possible to obtain quantifier-free interpolants for a Skolemized version of the extensional theory of arrays. We prove this in two ways: (1) non-constructively, by using the model theoretic notion of amalgamation, which is known to be equivalent to admit quantifier-free interpolation for universal theories; and (2) constructively, by designing an interpolating procedure, based on solving equations between array updates. (Interestingly, rewriting techniques are used in the key steps of the solver and its proof of correctness.) To the best of our knowledge, this is the first successful attempt of computing quantifier- free interpolants for a variant of the theory of arrays with extensionality.

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arXiv2012

From Strong Amalgamability to Modularity of Quantifier-Free Interpolation

The use of interpolants in verification is gaining more and more importance. Since theories used in applications are usually obtained as (disjoint) combinations of simpler theories, it is important to modularly re-use interpolation algorithms for the component theories. We show that a sufficient and necessary condition to do this for quantifier-free interpolation is that the component theories have the 'strong (sub-)amalgamation' property. Then, we provide an equivalent syntactic characterization, identify a sufficient condition, and design a combined quantifier-free interpolation algorithm capable of handling both convex and non-convex theories, that subsumes and extends most existing work on combined interpolation.

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arXiv2010

Backward Reachability of Array-based Systems by SMT solving: Termination and Invariant Synthesis

The safety of infinite state systems can be checked by a backward reachability procedure. For certain classes of systems, it is possible to prove the termination of the procedure and hence conclude the decidability of the safety problem. Although backward reachability is property-directed, it can unnecessarily explore (large) portions of the state space of a system which are not required to verify the safety property under consideration. To avoid this, invariants can be used to dramatically prune the search space. Indeed, the problem is to guess such appropriate invariants. In this paper, we present a fully declarative and symbolic approach to the mechanization of backward reachability of infinite state systems manipulating arrays by Satisfiability Modulo Theories solving. Theories are used to specify the topology and the data manipulated by the system. We identify sufficient conditions on the theories to ensure the termination of backward reachability and we show the completeness of a method for invariant synthesis (obtained as the dual of backward reachability), again, under suitable hypotheses on the theories. We also present a pragmatic approach to interleave invariant synthesis and backward reachability so that a fix-point for the set of backward reachable states is more easily obtained. Finally, we discuss heuristics that allow us to derive an implementation of the techniques in the model checker MCMT, showing remarkable speed-ups on a significant set of safety problems extracted from a variety of sources.

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