# Extended Asp Tableaux And Rule Redundancy In Normal Logic Programs*

MATTI JÄRVISALO and EMILIA  OIKARINEN
Helsinki University of Technology (TKK)
Department of Information and Computer Science P.O. Boz 5400, FI-02015 TKK, Finland
(e-mail: matti.jarvisalo@tkk.fi, emilia.oikarinen@tkk.fi)
ssse

## Abstract

We introduce an extended tableau calculus for answer set programming (ASP). The proof system is based on the ASP tableaux defined in [Gebser&Schaub, ICLP 2006], with an added extension rule. We investigate the power of Extended ASP Tableaux both theoretically and empirically. We study the relationship of Extended ASP Tableaux with the Extended Resolution proof system defined by Tseitin for sets of clauses, and separate Extended ASP Tableaux from ASP Tableaux by giving a polynomial-length proof for a family of normal logic programs { Π n } for which ASP Tableaux has exponential-length minimal proofs with respect to n. Additionally, Extended ASP Tableaux imply interesting insight into the effect of program simplification on the lengths of proofs in ASP. Closely related to Extended ASP Tableaux, we empirically investigate the effect of redundant rules on the efficiency of ASP solving KEY WORDS : Answer set programming , tableau method, extension rule, proof complexity, problem structure

## 1 Introduction

Answer set programming (ASP) (Marek and Truszczy´nski 1999; Niemel`a 1999; Gelfond and Leone 2002; Lifschitz 2002; Baral 2003) is a declarative problem solving paradigm which has proven successful for a variety of knowledge representation and reasoning tasks
(see (Soininen et al. 2001; Nogueira et al. 2001; Erdem et al. 2006; Brooks et al. 2007)
for examples). The success has been brought forth by efficient solver implementations such as smodels (Simons et al. 2002), dlv (Leone et al. 2006), noMore++ (Anger et al. 2005), cmodels (Giunchiglia et al. 2006), assat (Lin and Zhao 2004), and clasp (Gebser et al. 2007).

However, there has been an evident lack of theoretical studies into the reasons for the efficiency of ASP solvers Solver implementations and their inference techniques can be seen as deterministic implementations of the underlying rule-based proof systems . A solver implements
:Xiv:0809.3204v1  [cs.AI] 18 Sep 200