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Fuzzy Algebraic Theories

Authors Davide Castelnovo, Marino Miculan

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Author Details

Davide Castelnovo
  • Department of Mathematics, Computer Science and Physics, University of Udine, Italy
Marino Miculan
  • Department of Mathematics, Computer Science and Physics, University of Udine, Italy

Funding

  • Miculan, Marino: Supported by the Italian MIUR PRIN 2017FTXR7S IT MATTERS.

Cite AsGet BibTex

Davide Castelnovo and Marino Miculan. Fuzzy Algebraic Theories. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CSL.2022.13

Abstract

In this work we propose a formal system for fuzzy algebraic reasoning. The sequent calculus we define is based on two kinds of propositions, capturing equality and existence of terms as members of a fuzzy set. We provide a sound semantics for this calculus and show that there is a notion of free model for any theory in this system, allowing us (with some restrictions) to recover models as Eilenberg-Moore algebras for some monad. We will also prove a completeness result: a formula is derivable from a given theory if and only if it is satisfied by all models of the theory. Finally, leveraging results by Milius and Urbat, we give HSP-like characterizations of subcategories of algebras which are categories of models of particular kinds of theories.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • categorical logic
  • fuzzy sets
  • algebraic reasoning
  • equational axiomatisations
  • monads
  • Eilenberg-Moore algebras

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