The Possibilistic Horn Non-Clausal Knowledge Bases

TL;DR

This paper introduces a new class of possibilistic non-clausal knowledge bases and a resolution method, enabling efficient reasoning directly on non-clausal formulas, which improves practical applicability in uncertain and inconsistent information scenarios.

Contribution

It defines the Possibilistic Horn Non-Clausal class and a novel non-clausal resolution method, demonstrating polynomial-time computation of inconsistency degrees.

Findings

Defines Possibilistic Horn Non-Clausal Knowledge Bases.

Introduces Possibilistic Non-Clausal Unit-Resolution method.

Proves polynomial-time complexity for inconsistency degree computation.

Abstract

Posibilistic logic is the most extended approach to handle uncertain and partially inconsistent information. Regarding normal forms, advances in possibilistic reasoning are mostly focused on clausal form. Yet, the encoding of real-world problems usually results in a non-clausal (NC) formula and NC-to-clausal translators produce severe drawbacks that heavily limit the practical performance of clausal reasoning. Thus, by computing formulas in its original NC form, we propose several contributions showing that notable advances are also possible in possibilistic non-clausal reasoning. {\em Firstly,} we define the class of {\em Possibilistic Horn Non-Clausal Knowledge Bases,} or $\mathcal{\overline{H}}_\Sigma$, which subsumes the classes: possibilistic Horn and propositional Horn-NC. $\mathcal{\overline{H}}_\Sigma $ is shown to be a kind of NC analogous of the standard Horn class. {\em Secondly}, we define {\em Possibilistic Non-Clausal Unit-Resolution,} or $ \mathcal{UR}_\Sigma $, and prove that $ \mathcal{UR}_\Sigma $ correctly computes the inconsistency degree of $\mathcal{\overline{H}}_\Sigma $members. $\mathcal{UR}_\Sigma $ had not been proposed before and is formulated in a clausal-like manner, which eases its understanding, formal proofs and future extension towards non-clausal resolution. {\em Thirdly}, we prove that computing the inconsistency degree of $\mathcal{\overline{H}}_\Sigma $ members takes polynomial time. Although there already exist tractable classes in possibilistic logic, all of them are clausal, and thus, $\mathcal{\overline{H}}_\Sigma $ turns out to be the first characterized polynomial non-clausal class within possibilistic reasoning.