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.
@InProceedings{castelnovo_et_al:LIPIcs.CSL.2022.13, author = {Castelnovo, Davide and Miculan, Marino}, title = {{Fuzzy Algebraic Theories}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.13}, URN = {urn:nbn:de:0030-drops-157332}, doi = {10.4230/LIPIcs.CSL.2022.13}, annote = {Keywords: categorical logic, fuzzy sets, algebraic reasoning, equational axiomatisations, monads, Eilenberg-Moore algebras} }
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