「京都・ファジイ論理と非古典論理の一日ワークショップ」のお知らせ

logic 研究集会

JSPS日本・チェコ二国間学術交流事業に基づき、以下の要領で、6月14日に、京都でファジイ論理と非古典論理についての一日ワークショップを行います。皆様のご参加をお待ちしております。

Kyoto one-day workshop on fuzzy logics and non-classical logics

  • 10:00-11:00 Marta Bilkova "Modal logics arising from logical connections: some examples"
  • 11:00-12:00 Carles Noguera "General results on arithmetical complexity of first-order fuzzy logics."
  • 13:30-14:30 Yoshihiro Maruyama "Isbell-type adjunctions and dualities"
  • 14:30-15:30 Felix Bou "On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice"
  • 15:45-17:00 Libor Behounek "Lattice completions in Fuzzy Class Theory"

正式なタイムテーブルは、こちらでご紹介します。以下、要旨を紹介いたします。

Marta Bilkova : Modal Logics arising from logical connections: some examples

There are in general two ways of definig a modal logics (languages) of coalgebras of type given by a set functor T and treating them in a generic way: the first approach originated with L. Moss and it is based on cover modalities, the other is based on modalities arising from a logical connection. In both of the frameworks the classical modal logic K can be reconstructed and different other modal languages given by defined varying the functor T (the type of coalgebras) and certain parameters of the logical connection (e.g. varying the propositional part of the logics). We shall give examples of such logics, including some classical and intuitionistic modal logics.

Carles Noguera: General results on arithmetical complexity of first-order fuzzy logics

(Joint research with Franco Montagna)
All promiment examples of first-order predicate fuzzy logics are undecidable (see [4, 13]). This leads to the problem of the arithmetical complexity of their sets of tautologies and satisfiable sentences. An early contribution to such topic was that of Ragaz in [14] where he showed that the tautology problem for the [0, 1]-valued standard semantics of Lukasiewicz logic was Π2 -complete. H´ajek started addressing the problem in a systematic way already in Chapter 6 of his book [4] (which subsumed a couple of previous papers by himself ) with several results concerning the position in the arithmetical hierarchy of the sets of tautologies, positive tautologies, satisfiable sentences, and positively satisfiable sentences w.r.t. the standard semantics of the three basic fuzzy logics. Those results were subsequently extended to a rather complete study of arithmetical complexity problems for the standard semantics of the logic BL and its continuous t-norm based axiomatic extensions in a series of papers by H´ajek and the first author of the present work [5,11, 12]. The survey paper [6], besides collecting the mentioned results, provides a new study where the standard semantics is replaced by the so-called general semantics, i.e. the one given by models over arbitrary linearly ordered BL-algebras. Finally, the recent works [7,10, 2, 8] add some new knowledge on the matter by considering respectively fragments of continuous t-norm based logics, semantics based on linearly ordered complete BL-algebras, extensions of Lukasiewicz logic and G¨odel logics. Besides the standard and the general semantics, other kinds of semantics for first-order fuzzy logics, such as the ones given by algebras defined over the rational unit interval or over finite chains, have recently started receiving attention in the literature (see [1, 3]). This points to some new problems that had been neglected so far: What are the arithmetical complexities of the sets of (positive) tautologies and satisfiable sentences w.r.t. the rational and the finite-chain semantics? What are the relations of these sets with those corresponding with the general and the standard semantics?
The present contribution intends to expand the landscape of the studies on arithmetical complexity issues for fuzzy logics in two directions: (1) by considering the aforementioned rational and finite-chain semantics and their associated problems, and (2) by widening the framework of first-order fuzzy logics under focus to the classes of core and ∆-core fuzzy logics.1 These two layers amount to a much more general study encompassing (almost all) Roughly speaking, core and ∆-core fuzzy logics are good expansions of, respectively, MTL and MTL∆ and include in particular their axiomatic extensions and logics in enriched languages. Those classes have been the previous ones.
References

  • [1] P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna and C. Noguera. Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Annals of Pure and Applied Logic 160 (2009) 53–81.
  • [2] P. Cintula, P. H´ajek. Complexity issues in axiomatic extensions of Lukasiewicz logic. To appear in Journal of Logic and Computation, 2009.
  • [3] F. Esteva, L. Godo and C. Noguera. First-order t-norm based fuzzy logics with truth-constants: distinguished semantics and completeness properties. Annals of Pure and Applied Logic, 2009.
  • [4] P. H´ajek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic-Studia Logica Library, Kluwer, Dordrecht/Boston/London, 1998.
  • [5] P. H´ajek. Fuzzy logic and arithmetical hierarchy III, Studia Logica 68 (2001) 129–142.
  • [6] P. H´ajek. Arithmetical complexity of fuzzy predicate logics – a survey, Soft Computing 9 (2005) 935–941.
  • [7] P. H´ajek. On arithmetical complexity of fragments of prominent fuzzy predicate logics, Soft Computing 12 (2008) 335–340.
  • [8] P. H´ajek. Arithmetical complexity of fuzzy predicate logics – a survey II. Annals of Pure and Applied Logic 161 (2009) 212–219
  • [9] P. H´ajek and P. Cintula. On theories and models in fuzzy predicate logics, The Journal of Symbolic Logic 71 (2006) 863–880.
  • [10] P. H´ajek, F. Montagna. A note on the first-order logic of complete BL-chains, Mathematical Logic Quarterly 54 (2008) 435–448.
  • [11] F. Montagna. Three complexity problems in quantified fuzzy logic, Studia Logica 68 (2001) 143–152.
  • [12] F. Montagna. On the predicate logics of continuous t-norm BL-algebras, Archive for Mathematical Logic 44 (2005) 97–114.
  • [13] F. Montagna and H. Ono. Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo’s Logic MTL∀, Studia Logica 71 (2002) 227–245.
  • [14] M. E. Ragaz. Arithmetische Klassifikation von Formelnmengen der unend lichwertigen Logik. ETH Z¨urick Thesis, 1981. proved to be provide a good level of generality in the works [9, 1].

Yoshihiro Maruyama: Isbell-type adjunctions and dualities

In this talk, we study Isbell-type adjunctions and dualities for dcpos, preframes and continuous lattices, using the corresponding set-theoretical structures as dual spaces. We remark that preframes and continuous lattices are used as algebraic abstractions of convex structures (for preframes, see [1]; for continuous lattices, see [2]). Some of the results can be obtained in a categorical framework, while the others seem difficult to obtain in such a way.
References:

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

Coauthors: F. Esteva, L. Godo and R. Rodríguez
Link: http://arxiv.org/abs/0811.2107
This talk will deal with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We will 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 only evaluated in the idempotent elements and the ones evaluated in 0 and 1. We will 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 will also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

Libor Behounek: Lattice completions in Fuzzy Class Theory

First I shall present the definitions and basic properties of fuzzy bounds, maxima and minima, suprema and infima, and fuzzy lattices in Fuzzy Class Theory (FCT). Then I shall introduce two methods of fuzzy lattice completion (Dedekind and MacNeille), which coincide over classical logic, but differ in FCT. If time allows, I shall present a construction of fuzzy real numbers by fuzzy Dedekind completion of crisp rational numbers.