Defeasibility in Answer Set Programs via Argumentation Theories

Abstract

Abstract. Defeasible reasoning has been studied extensively in the last two decades and many different and dissimilar approaches are currently on the table. This multitude of ideas has made the field hard to navigate and the different techniques hard to compare. Our earlier work on Logic Programming with Defaults and Argumentation Theories (LPDA) introduced a degree of unification into the approaches that rely on the well-founded semantics. The present work takes this idea further and introduces ASPDA-a unifying framework for defeasibility of disjunctive logic programs under the Answer Set Programming (ASP). Since the well-founded and the answer set semantics underlie almost all existing approaches to defeasible reasoning in Logic Programming, LPDA and ASPDA together capture most of those approaches. In addition to AS-PDA, we obtained a number of interesting and non-trivial results. First, we show that ASPDA is reducible to ordinary ASP programs, albeit at the cost of exponential blowup in the number of rules. Second, we study reducibility of ASPDA to the non-disjunctive case and show that head-cycle-free ASPDA programs reduce to the non-disjunctive casesimilarly to head-cycle-free ASP programs, but through a more complex transformation. The blowup in the program size is linear in this case.

A Polynomial Reduction from ASPDA to ASP

“…We briefly review syntax and semantics of ASPDA, for details we refer to [2]. The language assumes a set of atoms; in [2] this set is not fixed, here we assume it to consist of first-order or propositional atomic formulas.…”

“…ASPDA is a framework for expressing defeasibility in Answer Set Programs via socalled argumentation theories, proposed by Wan, Kifer, and Grosof in [2]. ASPDA programs provide a very general means for defeating literals and rules and capture several earlier proposals for defeasibility in logic programming.…”

“…ASPDA programs provide a very general means for defeating literals and rules and capture several earlier proposals for defeasibility in logic programming. In [2], the authors also provide a reduction from ASPDA to ASP (in the sense of [1]). However, this reduction can easily lead to ASP programs that are exponentially larger than the original ASPDA programs.…”

“…In this paper, we show that this exponential behavior is not necessary, by providing an alternative reduction. Different to the reduction in [2], ours introduces new symbols and also needs a concept of rule identifier, which make proving correctness of the reduction slightly more cumbersome. However, this reduction immediately provides complexity results for ASPDA, in particular showing that (virtually all) computational tasks over ASPDA programs have the same complexity as those over ASP programs.…”

“…ASPDA is a framework for expressing defeasibility in Answer Set Programs via so-called argumentation theories, proposed by Wan, Kifer, and Grosof in [2]. The authors describe a reduction from ASPDA to plain Answer Set Programming, which however exponentially inflate programs.…”