257 research outputs found

    Toward a probability theory for product logic: states, integral representation and reasoning

    Full text link
    The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense.Comment: 27 pages, 1 figur

    Aggregated fuzzy answer set programming

    Get PDF
    Fuzzy Answer Set programming (FASP) is an extension of answer set programming (ASP), based on fuzzy logic. It allows to encode continuous optimization problems in the same concise manner as ASP allows to model combinatorial problems. As a result of its inherent continuity, rules in FASP may be satisfied or violated to certain degrees. Rather than insisting that all rules are fully satisfied, we may only require that they are satisfied partially, to the best extent possible. However, most approaches that feature partial rule satisfaction limit themselves to attaching predefined weights to rules, which is not sufficiently flexible for most real-life applications. In this paper, we develop an alternative, based on aggregator functions that specify which (combination of) rules are most important to satisfy. We extend upon previous work by allowing aggregator expressions to define partially ordered preferences, and by the use of a fixpoint semantics

    Logics for Non-Cooperative Games with Expectations

    Get PDF
    International audienceWe introduce the logics E(G) for reasoning about probabilistic expectation over classes G of games with discrete polynomial payoff functions represented by finite-valued Lukasiewicz formulas and provide completeness and complexity results. In addition, we introduce a new class of games where players’ expected payoff functions are encoded by E(G)-formulas. In these games each player’s aim is to randomise her strategic choices in order to affect the other players’ expectations over an outcome as well as their own. We offer a logical and computational characterisation of this new class of games

    Logics for Non-Cooperative Games with Expectations

    Get PDF
    We introduce the logics E(G) for reasoning about probabilistic expectation over classes G of games with discrete polynomial payoff functions represented by finite-valued Lukasiewicz formulas and provide completeness and complexity results. In addition, we introduce a new class of games where players' expected payoff functions are encoded by E(G)-formulas. In these games each player's aim is to randomise her strategic choices in order to affect the other players' expectations over an outcome as well as their own. We offer a logical and computational characterisation of this new class of games.Godo acknowledges support from the Spanish projects EdeTRI (TIN2012-39348-C02-01) and AT (CONSOLIDER CSD 2007-0022). Marchioni acknowledges support from the Marie Curie Project NAAMSI (FP7-PEOPLE-2011-IEF).Peer Reviewe

    Lukasiewicz logic and Riesz spaces

    Full text link
    We initiate a deep study of {\em Riesz MV-algebras} which are MV-algebras endowed with a scalar multiplication with scalars from [0,1][0,1]. Extending Mundici's equivalence between MV-algebras and ℓ\ell-groups, we prove that Riesz MV-algebras are categorically equivalent with unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent with the class of commutative unital C∗^*-algebras. The propositional calculus RL{\mathbb R}{\cal L} that has Riesz MV-algebras as models is a conservative extension of \L ukasiewicz ∞\infty-valued propositional calculus and it is complete with respect to evaluations in the standard model [0,1][0,1]. We prove a normal form theorem for this logic, extending McNaughton theorem for \L ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in RL{\mathbb R}{\cal L} and we relate them with the analogue of de Finetti's coherence criterion for RL{\mathbb R}{\cal L}.Comment: To appear in Soft Computin

    Measures induced by units

    Full text link
    The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop -equivalently, in the enveloping lattice-ordered abelian group- amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoop H induces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional on H. Since H is representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals -in this context usually called states- amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group of H), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.Comment: 24 pages, 1 figure. Revised version according to the referee's suggestions. Examples added, proof of Lemma 2.6 simplified, Section 7 expanded. To appear in the Journal of Symbolic Logi
    • …
    corecore