Meeting on Fuzzy Reasoning

Department of Mathematics, University of Aveiro, 14th Jan, 2019


Room: 11.3.21


    -- A 3 hours mini-course --

    Regivan Santiago, UFRN, Natal, BRASIL
    Approximate Reasoning
    ABSTRACT: The human reasoning has, sometimes, an approximate aspect. For example, when you say: "If the night will be cold, then most of my guests will not come to my party". In this context, you consider imprecise aspects which are not captured by the classical formalization of reasoning, in which the thresholds must be established beforehand (for example: below 10C it must be assumed that the night is cold).
    Logic can be understood as a discipline which studies the principles of reasoning. In particular, Fuzzy Logic provides a precise way to deal with approximate reasoning. I will assume the context in which approximate reasoning means a process in which imprecise conclusions are obtained from imprecise premisses. This process will be based on what is known as Fuzzy Logic.

    -- Two talks --

    • Daniel Figueiredo, UA, Aveiro, PORTUGAL
      Introducing fuzziness on reactive models
      ABSTRACT: The term reactivity is used by Gabbay to refer state transition systems whose accessibility relation is not fixed but can vary according to the edges crossed. On this work, the possibility of assign weights and, in particular, fuzzy measures to edges is considered.

    • Leandro Gomes, UM, Braga, PORTUGAL
      Algebraic and logic semantics for fuzzy computations
      ABSTRACT: Kleene algebra with tests (KAT) [2] was introduced as an algebraic structure to model and reason about classic imperative programs, i.e. sequences of discrete actions guarded by Boolean tests. Particularly, it was shown that KAT subsumes propositional Hoare logic. Consequently, the specialized syntax and deductive apparatus of Hoare logic can be replaced by the simple equational reasoning of KAT. However, current complex dynamic systems require more generic computing domains, namely probabilistic [5], continuous. [3] or fuzzy [1]. Such settings entail the need for computing paradigms able to deal with some sort of weighted program ex- ecutions. Moreover, assertions about these programs often have a graded outcome.
      In this context, the development of algebraic structures and logics to model weighted computations becomes a must. This work builds on such motivations to propose a number of formalisms able to capture some types of weighted computational domains. We start by presenting two generalisations of KAT able to express programs as weighted transitions and tests with outcomes in non necessarily bivalent truth spaces. Moreover, in analogy to Kozen’s encoding of Propositional Hoare Logic (PHL) in KAT [2], we discuss the encoding of a graded PHL in the introduced structures. Secondly, with the objective of defining semantics for these structures, we define some new algebras, built over GKAT/HKAT. Still in this context, the concurrent computational paradigm is addressed. Based on the models presented in this work, namely those defined on top of fuzzy semantics, we extend Synchronous Kleene algebra (SKA) [4] to the fuzzy domain, instantiating it with a specific class of fuzzy automata, and presenting an analogous construction of [4], regarding the synchronous product of two automata.
      Finally, as a concrete application for this process, we provide the first steps to develop a semantics for fuzzy imperative programming languages based on Fuzzy control system. We take the algebraic formalisms and their semantic structures presented in this work as a leverage for formally describe the behaviour of if − then − else statements typically found on these languages: the many-valued nature of the assertions evaluated entail that conditional statements may split into two (or more) execution branches. For a concrete illustration, we show a program written in Fuzzy Arden Syntax [6], a fuzzy programming language used in medicine.

      [1] L. A. Zadeh. Fuzzy languages and their relation to human and machine intelligence. Proc. Int. Conf. on Man and Computer, pages 148–179, 05 1996.
      [2] D. Kozen. On Hoare logic and Kleene algebra with tests. ACM Transactions on Computational Logic (TOCL), 1(212):1–14, 2000.
      [3] A. Platzer. Logical Analysis of Hybrid Systems - Proving Theorems for Complex Dynamics. Springer, 2010.
      [4] C. Prisacariu. Synchronous kleene algebra. 79(7):608–635, 2010. J. Log. Algebr. Program.,
      [5] R. Qiao, J. Wu, Y. Wang, and X. Gao. Operational semantics of probabilistic Kleene algebra with tests. Proceedings - IEEE Symposium on Computers and Communications, pages 706–713, 2008.
      [6] T. Vetterlein, H. Mandl, and K.-P. Adlassnig. Fuzzy arden syntax: A fuzzy programming language for medicine. Artificial Intelligence in Medicine, 49(1):1 – 10, 2010.
      (A joint work with Alexandre Madeira and Luís Soares Barbosa)


11:30-12:00 - Leandro Gomes
14:00-15:30 - Regivan Santiago (Part I)
15:30-16:00 - Daniel Figueiredo
16:00-17:30 - Regivan Santiago (Part II)

This work is financed by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project (02/SAICT/2017) and project UID/MAT/04106/2019 (at CIDMA).