The possibilistic horn non-clausal knowledge bases

Abstract

Non-clausal deduction in classical logic is one of the oldest areas in artificial intelligence. It first appeared in the sixties and consequently a large body of research has been devoted to it. Within the last decades, computing with non-clausal formulas has been considered in several fields, and in particular, in answer set programming, wherein non-clausal or nested logic programs were conceived in 1999.Possibilistic logic is the most extended approach to handle uncertain and partially inconsistent information. Here, we generalize some well-known clausal outcomes in possibilistic reasoning to the non-clausal setting, concretely the objective of our proposal is: (i) to extend available insights from clausal to non-clausal form; (ii) to show that possibilistic reasoning admits feasible classes also at the non-clausal level; (iii) to combine the high expressiveness of non-clausal possibilistic logic with the highest efficient (polynomial) reasoning mechanisms; and (iii) to suggest that some meaningful subclasses of possibilistic nested programs can be efficiently processed.Firstly, we define the class of Possibilistic Horn Non-Clausal formulas, or H‾Σ, which covers the classes: possibilistic Horn and propositional Horn-NC. H‾Σ is shown to be non-clausal, analogous to the standard Horn class.Secondly, we define Possibilistic Non-Clausal Unit-Resolution, or URΣ, and prove that URΣ correctly computes the inconsistency degree of Horn-NC bases. URΣ is formulated in a clausal-like manner, which eases its understanding, formal proofs and future extension towards full non-clausal resolution.Thirdly, we prove that computing the inconsistency degree of Horn-NC bases takes polynomial time. Although there already exist tractable classes in possibilistic logic, all of them are clausal, and thus, H‾Σ turns out to be the first characterized polynomial non-clausal class within possibilistic reasoning.We discuss that our approach serves as a starting point to developing uncertain non-clausal reasoning on the basis of both methodologies: DPLL and resolution.