@inbook {5108,
title = {On possibilistic modal logics defined over MTL-chains},
booktitle = {Petr H{\'a}jek on Mathematical Fuzzy Logic},
number = {6},
year = {2015},
pages = {225-244},
publisher = {Springer},
organization = {Springer},
edition = {Franco Montagna},
abstract = {In this paper we revisit a 1994 paper by H{\'a}jek et al. where a modal logic over a finitely-valued Lukasiewicz logic is defined to capture possibilistic reasoning. In this paper we go further in two aspects: first, we generalize the approach in the sense of considering modal logics over an arbitrary finite MTL-chain, and second, we consider a different possibilistic semantics for the necessity and possibility modal operators. The main result is a completeness proof that exploits similar techniques to the ones involved in H{\'a}jek et al.{\textquoteright}s previous work.},
keywords = {Fuzzy Logic, Many-valued Modal Logic, Possibitistic Logic},
url = {http://link.springer.com/chapter/10.1007/978-3-319-06233-4_11},
author = {F{\`e}lix Bou and Francesc Esteva and Llu{\'\i}s Godo},
editor = {Franco Montagna}
}
@article {5359,
title = {On the relationship between fuzzy autoepistemic logic and fuzzy modal logics of belief},
journal = {Fuzzy Sets and Systems},
volume = {276},
year = {2015},
month = {10/2015},
pages = {74-99},
abstract = {Autoepistemic logic is an important formalism for nonmonotonic reasoning originally intended to model an ideal rational agent reflecting upon his own beliefs. Fuzzy autoepistemic logic is a generalization of autoepistemic logic that allows to represent an agent{\textquoteright}s rational beliefs on gradable propositions. It has recently been shown that, in the same way as autoepistemic logic generalizes answer set programming, fuzzy autoepistemic logic generalizes fuzzy answer set programming as well. Besides being related to answer set programming, autoepistemic logic is also closely related to several modal logics. To investigate whether a similar relationship holds in a fuzzy logical setting, we firstly generalize the main modal logics for belief to the setting of finitely-valued Lukasiewicz logic with truth constants L^c_k, and secondly we relate them with fuzzy autoepistemic logics. Moreover, we show that the problem of satisfiability checking in these logics is NP-complete. Finally, we generalize Levesque{\textquoteright}s results on stable expansions, belief sets, and {\textquotedblleft}only knowing{\textquotedblright} operators to our setting, and provide a complete axiomatization for a logic of {\textquotedblleft}only knowing{\textquotedblright} in the L^c_k framework.},
keywords = {Autoepistemic logic, Fuzzy Logic, Logic of only knowing, Lukasiewicz logic, Nonmonotonic modal logic},
url = {http://www.sciencedirect.com/science/article/pii/S0165011415001116},
author = {Marjon Blondeel and Tommaso Flaminio and Steven Schockaert and Llu{\'\i}s Godo and Martine de Cock}
}
@conference {55803,
title = {Valuations in Nilpotent Minimum Logic},
booktitle = {2015 IEEE International Symposium on Multiple-Valued Logic},
year = {2015},
month = {May},
keywords = {Algebra, Boolean algebra, Euler characteristic, finite distributive lattices, formal logic, Fuzzy Logic, idempotent Euler characteristics, Lattices, Nilpotent Minimum logic, NM algebra, NM algebras, NM logic, NM propositional logic, NM- logic, valuation},
doi = {10.1109/ISMVL.2015.19},
author = {Pietro Codara and Diego Valota}
}
@article {4143,
title = {A graded BDI agent model to represent and reason about preferences},
journal = {Artificial Intelligence},
volume = {175},
year = {2011},
pages = {1468-1478},
publisher = {Elsevier},
abstract = {In this research note, we introduce a graded BDI agent development framework, g-BDI for short, that allows to build agents as multi-context systems that reason about three fundamental and \emph{graded} mental attitudes (i.e. beliefs, desires and intentions). We propose a sound and complete logical framework for them and some logical extensions to accommodate slightly different views on desires.},
keywords = {BDI agents, bipolar preferences, Fuzzy Logic, multi-context systems, uncertainty},
author = {Ana Casali and Llu{\'\i}s Godo and Carles Sierra}
}
@article {3854,
title = {Extending possibilistic logic over Godel logic},
journal = {International Journal of Approximate Reasoning},
volume = {52},
year = {2011},
pages = {63-75},
publisher = {Elsevier},
abstract = {In this paper we present several fuzzy logics trying to capture different notions of necessity (in the sense of Possibility theory) for Godel logic formulas. Based on different characterizations of necessity measures on fuzzy sets, a group of logics with Kripke style semantics are built over a restricted language, namely a two level language composed of non-modal and modal formulas, the latter moreover not allowing for nested applications of the modal operator N. Completeness and some computational complexity results are shown.},
keywords = {Fuzzy Logic, Godel Logic, Necessity Measures, Posibilistic Logic},
author = {Pilar Dellunde and Llu{\'\i}s Godo and Enrico Marchioni}
}
@conference {3835,
title = {Base Belief Change for finitary monotonic logics},
booktitle = {Current Topics in Artificial Intelligence},
number = {5988},
year = {2010},
pages = {82-92},
publisher = {Springer},
organization = {Springer},
abstract = {We slightly improve on characterization results already in the literature for base revision. We show that in order to axiomatically characterize revision operators in a logic the only conditions this logic is required to satisfy are: finitarity and monotonicity. A characterization of limiting cases of revision operators, full meet and maxichoice, is also offered. In the second part of the paper, as a particular case, we focus on the class of graded fuzzy logics and distinguish two types of bases, naturally arising in that context, exhibiting di?erent behavior.},
keywords = {base revision, finitary monotonic logic, Fuzzy Logic, partial meet revision operators},
author = {Pere Pardo and Pilar Dellunde and Llu{\'\i}s Godo},
editor = {P. Meseguer, L. Mandow, R.M. Gasca}
}
@conference {4094,
title = {Modalities and many-valued: modeling uncertainty measures and similarity-based reasoning and application to Fuzzy description Logics .},
booktitle = {LINZ seminar},
year = {2010},
month = {9/02/2010},
pages = {45-48},
address = {LINZ (Austria)},
abstract = {The talk will began remembering results on papers \cite{hge, geh, ghe} where the authors model uncertainty measures as fuzzy modalities on a many-valued residuated logics. the basic idea is that uncertainty measures is not a truth degree but could be interpreted as a truth degree of the fuzzy sentence "the probability of $\varphi$". The main properties being that the language does not allow nested modalities and their semantic is not defined as generalized Kripke models in strict sense. We will give the example of probabilities (possibilities, belief functions) over classical propositions and its generalizations to uncertainty measures on many-valued (fuzzy) propositions. But the main goal of the talk is modal many-valued fuzzy logics defined by Kripke models and some applications. Modal many-valued logics is a topic that has deserved few attention until the nineties. The first known papers are the papers \cite{Fi92a, Fi92b} of Fitting where the author defines a modal system over a logic of a finite Heyting algebra and give a complete axiomatization of them. In its late paper Fitting justify it in a very nice an elegant semantics based on a multi-agent system each one using classical logic and having a preference relation. He defined the modal operators based on semantics and his definitions is the ones used in later papers on the topic as generalizations modal operators on many-valued systems based on Kripke semantics (with many-valued worlds and many-valued accessibility relations). In these approaches Modal many-valued language is built taking the language of the many-valued logic $(\land, \lor, \&, \to_\&, \neg_\&, 0, 1)$ plus at least one the usual modal operators (necessity $\Box$ and possibility $\Diamond$) and its semantic is defined by generalized Kripke models taking many-valued evaluation in each world $w \in W$ and many-valued accessibility relations $S: W \times W \to [0,1]$. The evaluation of modal operators are given (following Fitting \cite {Fi92a, Fi92b}) by \begin{itemize} \item $V ( \Box \varphi, w ) = {\displaystyle \bigwedge} \{ R(w,w{\textquoteright}) \to V(\varphi,w{\textquoteright}): w{\textquoteright} \in W \}$. \item $V ( \Diamond \varphi, w ) = {\displaystyle \bigvee} \{ R(w,w{\textquoteright}) \& V(\varphi,w{\textquoteright}): w{\textquoteright} \in W \}$. \end{itemize} There are different attempts and approaches to motivate and study modal logic formalisms based on Kripke semantics to the many-valued setting. Roughly speaking we can classify the approaches in three groups depending how the corresponding Kripke frames look like, in the sense of how many-valuedness affects the worlds and the accesibility relations. Next we describe these three groups and comment about our work in each of them. A first group (see e.g. \cite{ChFr92, EsGaGoRo97,Su97}) is formed by those logical systems whose class of Kripke frames are such that their {\em worlds are classical} (i.e. they follow the rules of classical logic) but their {\em accessibility relations are many-valued}, with values in some suitable lattice $A$. In such a case, the usual approach to capture the many-valuedness of an accessibility relation $R: W \times W \to A$ is by considering the induced set of classical accessibility relations $\{ R_a \mid a \in A \}$ defined by the different level-cuts of $R$, i.e. $ \langle w, w{\textquoteright} {\r a}ngle \in R_a$ iff $R(w, w{\textquoteright}) \geq a$. At the syntactical level, one then introduces as many (classical) necessity operators $\Box_a$ (or possibility operators $\Diamond_a$) as elements $a$ of the lattice $A$, interpreted by (classical) relations $R_a$. Therefore, in this kind of approach, one is led to a multi-modal language but where (both modal and non modal) formulas are Boolean in each world. In this setting we will present as example the similarity-based reasoning developed in \cite{DuEsGaGo95, EsGaGo94, EsGaGoRo97, DuPrEsGa97, EsGoRoVe10}. The starting point is the paper by Ruspini \cite{Ru91} about a possible semantics for fuzzy set theory. He develops the idea that we could represent a fuzzy concept by its set of prototypical elements (which will have fully membership to the corresponding fuzzy set) together with a similarity relation giving the degree of similarity of each element of the universe to the closest prototype. This degree is taken then as the membership to the fuzzy set. From this basic idea, we will see three graded entailments (approximate, proximity and strong) that can be represented in a multi-modal systems with frames where the (graded) accessibility is given by a fuzzy similarity relation on pairs of worlds, and for which we have proved completeness in several cases \cite{EsGaGoRo98}. A second group of approaches are the ones whose corresponding Kripke frames have {\em many-valued worlds}, evaluating propositional variables in a suitable lattice of truth-values $A$, but with {\em classical accessibility relations} (see e.g. \cite{HaHa96,Haj98,HaTe06}). In this case, we have languages with only one necessity and/or possibility operator ($\Box, \Diamond$), but whose truth-evaluation rules in the worlds is many-valued, so modal (and non-modal formulas) are many-valued. Finally, a third group of approaches are {\em fully many-valued}, in the sense that in their Kripke frames, both worlds and accessibility relations are many-valued, again over a suitable lattice $A$. In that case, some approaches (like \cite{Fi92a,Fi92b,KouNoPe02,CaRo10}) have a language with a single necessity/possibility operator ($\Box, \Diamond$), and some (like \cite{Mi05, BoEsGoRo09}) consider a multi-modal language with a family of indexed operators $\Box_a$ and $\Diamond_a$ for each $a \in A$, interpreted in the Kripke models via the level-cuts $R_a$ of a many-valued accessibility relations $R$. Actually, these two kinds of approaches are not always equivalent, in the sense that the operator $\Box$ and the set of operators $\{ \Box_a \mid a \in A\}$ are not always interdefinable (or analogously with the possibility operators). In this setting we will present recent results a summary \cite{BoEsGoRo09,CaRo10} on minimum modal many-valued logic over the logic of a finite residuated lattice and modal G{\"o}del logic respectively. Next we will sketch what Fuzzy description logic could be following the proposal of H{\'a}jek in \cite{Ha05,Ha06} and developed in \cite{GaArEs10}. Finally we propose a research proposal which main goal to be the study of n-graded Description Logics (depending of the underlying logic and the expressiveness of the description language we want), a topic for which we have at hand many results: canonical completeness of first order finite-valued residuated logic, modal many-valued results, decidability results and many possible reasoning algorithms.},
keywords = {Fuzzy Logic, Modal many valued, Similarity reasoning},
author = {Francesc Esteva},
editor = {Petr Cintula, Erich P. Klement, Lawrence Stout}
}