Lluís Godo

Monoidal t-norm based logic: towards a logic for left-continuous t-norms

Hajeks BL logic is the fuzzy logic capturing the tautologies of continuous t-norms and their residua. In this paper we investigate a weaker logic, MTL, which is intended to cope with the tautologies of left-continuous t-norms and their residua. The corresponding algebraic structures, MTL-algebras, are defined and completeness of MTL with respect to linearly ordered MTL-algebras is proved. Besides, several schematic extensions of MTL are also considered as well as their corresponding predicate calculi.

A complete many-valued logic with product-conjunction

A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguens implication) and the corresponding negation (Gödels negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.

Devising A Trust Model For Multi-Agent Interactions Using Confidence And Reputation

In open environments in which autonomous agents can break contracts, computational models of trust have an important role to play in determining who to interact with and how interactions unfold. To this end, we develop such a trust model, based on confidence and reputation, and show how it can be concretely applied, using fuzzy sets, to guide agents in evaluating past interactions and in establishing new contracts with one another.

A logical approach to interpolation based on similarity relations

Abstract One of the possible semantics of fuzzy sets is in terms of similarity; namely, a grade of membership of an item in a fuzzy set can be viewed as the degree of resemblance between this item and prototypes of the fuzzy set. In such a framework, an interesting question is how to devise a logic of similarity, where inference rules can account for the proximity between interpretations. The aim is to capture the notion of interpolation inside a logical setting. In this paper, we investigate how a logic of similarity dedicated to interpolation can be defined, by considering different natural consequence relations induced by the presence of a similarity relation on the set of interpretations. These consequence relations are axiomatically characterized in a way that parallels the characterization of nonmonotonic consequence relationships. It is shown how to reconstruct the similarity relation underlying a given family of consequence relations that obey the axioms. Our approach strikingly differs from the logics of indiscernibility, such as the rough-set logics, because emphasis is put on interpolation capabilities. Potential applications are fuzzy rule-based systems and fuzzy case-based reasoning, where notions of similarity play a crucial role.

On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic

The monoidal t-norm based logic MTL is obtained from Hájeks Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics.

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones only evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

On aggregation operators for ordinal qualitative information

In many fuzzy systems applications, values to be aggregated are of a qualitative nature. In that case, if one wants to compute some type of average, the most common procedure is to perform a numerical interpretation of the values, and then apply one of the well-known (the most suitable) numerical aggregation operators. However, if one wants to stick to a purely qualitative setting, choices are reduced to either weighted versions of max-min combinations or to a few existing proposals of qualitative versions of ordered weighted average (OWA) operators. In this paper, we explore the feasibility of defining a qualitative counterpart of the weighted mean operator without having to use necessarily any numerical interpretation of the values. We propose a method to average qualitative values, belonging to a (finite) ordinal scale, weighted with natural numbers, and based on the use of finite t-norms and t-conorms defined on the scale of values. Extensions of the method for other OWA-like and Choquet integral-type aggregations are also considered.

Negotiating using rewards

In situations where self-interested agents interact repeatedly, it is important that they are endowed with negotiation techniques that enable them to reach agreements that are profitable in the long run. To this end, we devise a novel negotiation algorithm that generates promises of rewards in future interactions, as a means of permitting agents to reach better agreements, in a shorter time, in the present encounter. Moreover, we thus develop a specific negotiation tactic based on this reward generation algorithm and show that it can achieve significantly bettter outcomes than existing benchmark tactics that do not use such inducements. Specifically, we show, via empirical evaluation, that our tactic can lead to a 26% improvement in the utility of deals that are made and that 21 times fewer messages need to be exchanged in order to achieve this under concrete settings.

A modal account of similarity-based reasoning

Abstract One of the goals of a variety of approximate reasoning models is to cope with inference patterns more flexible than those of classical reasoning. Among them, similarity-based reasoning aims at modeling notions of resemblance or proximity among propositions and consequence relations which make sense in such a setting. One way of proceeding is to equip the set of interpretations or possible worlds with a similarity relation S, that is, a reflexive, symmetric, and t-norm-transitive fuzzy relation. We explore a modal approach to similarity-based reasoning, and we define three multi-modal systems with similarity-based Kripke model semantics. A similarity-based Kripke model is a structure 〈W, S, ‖ ‖〉, in which W is the set of possible worlds, ‖ ‖ represents an assignment of possible worlds to atomic formulas, and S is a similarity function S: W × W → G, where G is a subset of the unit interval [0,1] such that 0,1 ϵ G. We provide soundness and completeness results for these systems with respect to some classes of the above structures.

Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results

In this paper we study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for Lukasiewicz logic and more recent results for Godel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants {r@?|r@?C}, for a suitable countable C@?[0,1], and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Lukasiewicz, Godel and Product components, (ii) the set of truth-constants covers all the unit interval in the sense that each component of the t-norm contains at least one value of C different from the bounds of the component, and (iii) the truth-constants in Lukasiewicz components behave as rational numbers.