Abstract In this work, we present polyhedral semantics as a means to tractably approximate Łukasiewicz infinitely-valued logic (Ł∞). As Ł∞ is an expressive multivalued propositional logic whose decision problem is NP-complete, we show how to to obtain an approximation for this problem providing a family of multivalued logics over the same language as Ł∞. Each element of the family is associated to a polynomial-time linear program, thus providing a tractable way of deciding each intermediate step. We also investigate properties of the logic system derived from polyhedral semantics and the details of an algorithm for the approximation process.

Abstract A Nash equilibrium is a strategy profile of a game in which none of the players involved has any gain by changing alone his/her own strategy. Given a two-player game, we show a codification of all of its Nash equilibria into Łukasiewicz infinitely-valued logic, that is, we derive a propositional theory in this logic whose models codify all the Nash equilibria. Based on such propositional theory, we derive a polynomial reduction from the problem of computing a Nash equilibrium to the problem of satisfiability of (sets of) formulas of Łukasiewicz infinitely-valued logic. These applications of logic to game theory lead to new methods for computing Nash equilibria.

For functional representation in an algebraizable logic we mean a representation of the algebras of formulas of the logic by means of (possibly real-valued) functions. Functional representations have shown to be a key tool for the study of non-classical logics, since they allow to regard formulas as functions and, by means of them, to approach the study of typical proof theoretical properties of the logics by means of their functional semantics. In the realm of (algebraizable) fuzzy logics, possibly the most well-known result in this respect is McNaughton theorem that shows formulas of the infinite-valued Lukasiewicz calculus to correspond, up to logical equivalence, to real valued continuous and piecewise linear functions. The functional representation for many-valued logics has been very recently shown in a paper by the second author to have an impact outside the purely logical realm and, in particular, they can be applied to study properties of artificial neural networks. In this contribution, we will provide a functional representation for the probability modal logic FP(L) that builds on Lukasiewicz calculus by adding to it a unary operator P that reads “it is probable that”. While the logic FP(L) is not algebraizable, at least not in the usual sense due to Blok and Pigozzi, we still can provide a functional representation result for its modal formulas. In order to do so, we adapt the usual universal algebraic methods to this peculiar setting, and moreover we make use of some techniques developed in a recent paper by two of the authors where a class of purely algebraic models for FP(L) based on de Finetti's coherence criterion have been introduced and studied. Our contribution will present two ways of providing a functional representation of the algebras of formulas of the modal logic FP(L): a local one, that relies on de Finetti's coherence argument; and a global one that, instead, relies on probability distributions on a finite domain.

Abstract This work concerns the representation of a class of continuous functions into Logic, so that one may automatically reason about properties of these functions using logical tools. Rational McNaughton functions may be implicitly represented by logical formulas in Łukasiewicz Infinitely-valued Logic by constraining the set of allowed valuations; such a restriction contemplates only those valuations that satisfy specific formulas. This work investigates two approaches to such depiction, called representation modulo satisfiability. Furthermore, a polynomial-time algorithm that builds this representation is presented, producing a pair of formulas consisting of the representative formula and the constraining one, given as input a rational McNaughton function in a suitable encoding. An implementation of the algorithm is discussed.