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

TL;DR

This paper develops a paraconsistent semantics for extended fuzzy logic programs using approximation fixpoint theory, accommodating both negation as failure and strong negation.

Contribution

It introduces a unified framework that generalizes existing semantics and supports new semantics for fuzzy logic programs with multiple negation types.

Findings

Framework generalizes several existing semantics.

Supports well-behaved semantics for fuzzy logic with negation.

Enables combining negation as failure and strong negation.

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.