A Decidable Class of Nested Iterated Schemata (extended version)

Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called "iterated schemata", allow to express such patterns. Schemata extend propositional logic with indexed propositions, e.g. P_i, P_i+1, P_1, and with generalized connectives, e.g. /\i=1..n or i=1..n (called "iterations") where n is an (unbound) integer variable called a "parameter". The expressive power of iterated schemata is strictly greater than propositional logic: it is even out of the scope of first-order logic. We define a proof procedure, called DPLL*, that can prove that a schema is satisfiable for at least one value of its parameter, in the spirit of the DPLL procedure. However the converse problem, i.e. proving that a schema is unsatisfiable for every value of the parameter, is undecidable so DPLL* does not terminate in general. Still, we prove that it terminates for schemata of a syntactic subclass called "regularly nested". This is the first non trivial class for which DPLL* is proved to terminate. Furthermore the class of regularly nested schemata is the first decidable class to allow nesting of iterations, i.e. to allow schemata of the form /\i=1..n (/\j=1..n ...).Comment: 43 pages, extended version of "A Decidable Class of Nested Iterated Schemata", submitted to IJCAR 200

Undecidability of first-order modal and intuitionistic logics with two variables and one monadic predicate letter

We prove that the positive fragment of first-order intuitionistic logic in the language with two variables and a single monadic predicate letter, without constants and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without constants and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among them, QK, QT, QKB, QD, QK4, and QS4.Comment: Pre-final version of the paper published in Studia Logica,doi:10.1007/s11225-018-9815-

Fusions of Modal Logics Revisited

The fusion Ll ? Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halld?encompleteness are preserved under forming fusions of normal polyadic polymodal logics. Those problems remained open in [Fine & Schurz [3]] and [Kracht & Wolter [10]]. The paper defines the fusion `l ? `r of two classical modal consequence relations and proves that decidability transfers also in this case. Finally, these results are used to prove a general decidability result for modal logics based on superintuitionistic logics

A Complete Cyclic Proof System for Inductive Entailments in First Order Logic

International audienceIn this paper we develop a cyclic proof system for the problem of inclusion between the least sets of models of mutually recursive predicates, when the ground constraints in the inductive definitions are quantifier-free formulae of first order logic. The proof system consists of a small set of inference rules, inspired by a top-down language inclusion algorithm for tree automata [9]. We show the proof system to be sound, in general, and complete, under certain semantic restrictions involving the set of constraints in the inductive system. Moreover, we investigate the computational complexity of checking these restrictions, when the function symbols in the logic are given the canonical Herbrand interpretation

An Effective Tableau System for the Linear Time µ-Calculus

We present a tableau system for the model checking problem of the linear time µ-calculus. It improves the system of Stirling and Walker by simplifying the success condition for a tableau. In our system success for a leaf is determined by the path leading to it, whereas Stirling and Walker's method requires the examination of a potentially infinite number of paths extending over the whole tableau

Modelling opacity using petri nets

We consider opacity as a property of the local states of the secure (or high-level) part of the system, based on the observation of the local states of a low-level part of the system as well as actions. We propose a Petri net modelling technique which allows one to specify different information flow properties, using suitably defined observations of system behaviour. We then discuss expressiveness of the resulting framework and the decidability of the associated verification problems