Paraconsistent Semantics for Extended Fuzzy Logic Programs via Approximation Fixpoint Theory [Extended Version]

2026-05-06

AI Summary

This work addresses fuzzy logic programs incorporating both “failure-as-negation” and strong negation by introducing approximation fixed-point theory into this domain for the first time, thereby establishing a unified formal semantic framework. The proposed approach not only subsumes and generalizes several existing semantics but also systematically yields a family of novel semantics endowed with desirable theoretical properties. By providing a rigorous yet flexible foundation for reasoning in the presence of dual negation mechanisms, the framework significantly enhances the expressive power and non-monotonic reasoning capabilities of fuzzy logic programs.

Abstract

In logic programming, negation can be interpreted in various ways. Probably best known is the concept of "negation as failure", where "$\mathit{not}\, p$" is true if we have no evidence for $p$. On the other hand, strong negation requires that we have evidence for $p$ being false. Defining semantics for logic programs containing both kinds of negation is a challenging task, and this becomes even more challenging when combining this with other extensions of logic programming, e.g. fuzziness. In this work, we use the framework of approximating fixpoint theory to formulate well-behaved semantics for fuzzy logic programs containing both "by-failure" and strong negation. We show that this framework generalizes several existing semantics as well as giving rise to a host of new semantics.

Problem

• paraconsistent semantics
• fuzzy logic programs
• negation as failure
• strong negation
• approximation fixpoint theory

Innovation

• paraconsistent semantics
• fuzzy logic programs
• approximation fixpoint theory
• negation as failure
• strong negation