A Paraconsistent ASP-like Language with Tractable Model Generation

Answer Set Programming (ASP) is nowadays a dominant rule-based knowledge representation tool. Though existing ASP variants enjoy efficient implementations, generating an answer set remains intractable. The goal of this research is to define a new \asp-like rule language, 4SP, with tractable model generation. The language combines ideas of ASP and a paraconsistent rule language 4QL. Though 4SP shares the syntax of \asp and for each program all its answer sets are among 4SP models, the new language differs from ASP in its logical foundations, the intended methodology of its use and complexity of computing models. As we show in the paper, 4QL can be seen as a paraconsistent counterpart of ASP programs stratified with respect to default negation. Although model generation of well-supported models for 4QL programs is tractable, dropping stratification makes both 4QL and ASP intractable. To retain tractability while allowing non-stratified programs, in 4SP we introduce trial expressions interlacing programs with hypotheses as to the truth values of default negations. This allows us to develop a~model generation algorithm with deterministic polynomial time complexity. We also show relationships among 4SP, ASP and 4QL

An incremental algorithm for generating all minimal models

AbstractThe task of generating minimal models of a knowledge base is at the computational heart of diagnosis systems like truth maintenance systems, and of nonmonotonic systems like autoepistemic logic, default logic, and disjunctive logic programs. Unfortunately, it is NP-hard. In this paper we present a hierarchy of classes of knowledge bases, Ψ1,Ψ2,… , with the following properties: first, Ψ1 is the class of all Horn knowledge bases; second, if a knowledge base T is in Ψk, then T has at most k minimal models, and all of them may be found in time O(lk2), where l is the length of the knowledge base; third, for an arbitrary knowledge base T, we can find the minimum k such that T belongs to Ψk in time polynomial in the size of T; and, last, where K is the class of all knowledge bases, it is the case that ⋃i=1∞Ψi=K, that is, every knowledge base belongs to some class in the hierarchy. The algorithm is incremental, that is, it is capable of generating one model at a time

Reasoning with minimal models: efficient algorithms and applications

AbstractReasoning with minimal models is at the heart of many knowledge-representation systems. Yet it turns out that this task is formidable, even when very simple theories are considered. In this paper, we introduce the elimination algorithm, which performs, in linear time, minimal model finding and minimal model checking for a significant subclass of positive CNF theories which we call positive head-cycle-free (HCF) theories. We also prove that the task of minimal entailment is easier for positive HCF theories than it is for the class of all positive CNF theories. Finally, we show how variations of the elimination algorithm can be applied to allow queries posed on disjunctive deductive databases and disjunctive default theories to be answered in an efficient way

A Hierarchy of Tractable Subsets for Computing Stable Models

Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic. Unfortunately, it is NP-hard. In this paper we present a hierarchy of classes of knowledge bases,\Omega 2 ; :::, with the following properties: first,\Omega 1 is the class of all stratified knowledge bases; second, if a knowledge base \Pi is k , then \Pi has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in \Pi; third, for an arbitrary knowledge base \Pi, we can find the minimum k such that \Pi belongs in time polynomial in the size of \Pi; and, last, where K is the class of all knowledge bases, it is the case that i=1\Omega i = K, that is, every knowledge base belongs to some class in the hierarchy