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

EXISTENTIALLY CLOSED BROUWERIAN SEMILATTICES

The variety of Brouwerian semilattices is amalgamable and locally finite, hence by well-known results, it has a model completion (whose models are the existen- tially closed structures). In this paper, we supply a finite and rather simple axiomatization of the model completio

From Non-Disjoint Combination to Satisfiability and Model-Checking of Infinite State Systems

In the first part of our contribution, we review recent results on combined constraint satisfiability for first order theories in the non-disjoint signatures case: this is done mainly in view of the applications to temporal satisfiability and model-checking covered by the second part of our talk, but we also illustrate in more detail some case-study where non-disjoint combination arises. The first case deals with extensions of the theory of arrays where indexes are endowed with a Presburger arithmetic structure and a length expressing `dimension\u27 is added; the second case deals with the algebraic counterparts of fusion in modal logics. We then recall the basic features of the Nelson-Oppen method and investigate sufficient conditions for it to be complete and terminating in the non-disjoint signatures case: for completeness we rely on a model-theoretic $T_0$-compatibility condition (generalizing stable infiniteness) and for termination we impose a noetherianity requirement on positive constraints chains. We finally supply examples of theories matching these combinability hypotheses. In the second part of our contribution, we develop a framework for integrating first-order logic (FOL) and discrete Linear time Temporal Logic (LTL). Manna and Pnueli have extensively shown how a mixture of FOL and 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. Our framework for the integration is the following: we fix a theory $T$ in a first-order signature $Sigma$ and consider as a temporal model a sequence $cM_1, cM_2, dots$ of standard (first-order) models of $T$ and assume such models to share the same carrier (or, equivalently, the domain of the temporal model to be `constant\u27). Following Plaisted, we consider symbols from a subsignature $Sigma_r$ of $Sigma$ to be emph{rigid}, i.e. in a temporal model $cM_1, cM_2, dots$, the $Sigma_r$-restrictions of the $cM_i$\u27s must coincide. The symbols in $Sigmasetminus Sigma_r$ are called `flexible\u27 and their interpretation is allowed to change over time (free variables are similarly divided into `rigid\u27 and `flexible\u27). For model-checking, the emph{initial states} and the emph{transition relation} are represented by first-order formulae, whose role is that of (non-deterministically) restricting the temporal evolution of the model. In the quantifier-free case, we obtain sufficient conditions for %undecidability and decidability for both satisfiability and model-checking of safety properties emph{by lifting combination methods} for emph{non-disjoint} theories in FOL: noetherianity and $T_0$-compatibility (where $T_0$ is the theory axiomatizing the rigid subtheory) gives decidability of satisfiability, whereas $T_0$-compatibility and local finiteness give safety model-checking decidability. The proofs of these 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 some examples and discuss further work in the area

Light-Weight SMT-based Model Checking

AbstractRecently, the notion of an array-based system has been introduced as an abstraction of infinite state systems (such as mutual exclusion protocols or sorting programs) which allows for model checking of invariant (safety) and recurrence (liveness) properties by Satisfiability Modulo Theories (SMT) techniques. Unfortunately, the use of quantified first-order formulae to describe sets of states makes fix-point checking extremely expensive. In this paper, we show how invariant properties for a sub-class of array-based systems can be model-checked by a backward reachability algorithm where the length of quantifier prefixes is efficiently controlled by suitable heuristics. We also present various refinements of the reachability algorithm that allows it to be easily implemented in a client-server architecture, where a “light-weight” algorithm is the client generating proof obligations for safety and fix-point checks and an SMT solver plays the role of the server discharging the proof obligations. We also report on some encouraging preliminary experiments with a prototype implementation of our approach