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Heuristics for Hard ASP Programs ∗ Wolfgang Faber† and Nicola Leone and Francesco Ricca Department of Mathematics, University of Calabria, I-87030 Rende (CS), Italy Email: {faber,leone,ricca}@mat.unical.it Abstract Most of the optimization work on ASP systems has fo- cused on the efﬁcient evaluation of non-disjunctive programs We deﬁne a new heuristic hDS for ASP, and imple- (whose power is limited to NP/co-NP), whereas the opti- ment it in the (disjunctive) ASP system DLV. The mization of full (disjunctive) ASP programs has been treated new heuristic improves the evaluation of ΣP /ΠP - 2 2 in fewer works (e.g., in [Janhunen et al., 2000; Koch et al., hard ASP programs while maintaining the benign 2003]). In particular, we are not aware of any work concern- behaviour of the well-assessed heuristic of DLV ing heuristics for ΣP /ΠP -hard ASP programs. 2 2 on NP problems. We experiment with the new In this paper, we address the following two questions: heuristic on QBFs. hDS signiﬁcantly outperforms ◮ Can the heuristics of ASP systems be reﬁned to deal more the heuristic of DLV on hard 2QBF problems. efﬁciently with ΣP /ΠP -hard ASP programs? 2 2 We compare also the DLV system (with the new ◮ On hard ΣP /ΠP problems, can ASP systems compete with 2 2 heuristic hDS ) to three prominent QBF solvers. other AI systems, like QBF solvers? The results of the comparison, performed on in- stances used in the last QBF competition, indicate We deﬁne a new heuristic hDS for the (disjunctive) ASP that ASP systems can be faster than QBF systems system DLV, aiming at improving the evaluation of ΣP /ΠP - 2 2 on ΣP /ΠP -hard problems. hard ASP programs, but maintaining the benign behaviour 2 2 of the heuristic of DLV on NP problems. We experimen- tally compare hDS against the DLV heuristic on hard 2QBF 1 Introduction instances, showing a clear beneﬁt. We also experiment the Answer set programming (ASP) is a novel programming competitiveness of ASP w.r.t. QBF solvers on hard problems, paradigm, which has been recently proposed in the area of indicating that ASP systems are very competitive with QBF nonmonotonic reasoning and logic programming. The idea systems on ΣP /ΠP -hard problems. 2 2 of answer set programming is to represent a given computa- tional problem by a logic program whose answer sets corre- 2 Answer Set Computation and Heuristics spond to solutions, and then use an answer set solver to ﬁnd such a solution [Lifschitz, 1999]. The knowledge represen- We ﬁrst recall the main steps of the computational process tation language of ASP is very expressive in a precise math- performed by ASP systems, in particular the DLV system, ematical sense; in its general form, allowing for disjunction which will be used for the experiments. in rule heads and nonmonotonic negation in rule bodies, ASP An answer set program P in general contains variables. can represent every problem in the complexity class ΣP and 2 The ﬁrst step of a computation of an ASP system eliminates ΠP (under brave and cautious reasoning, respectively) [Eiter 2 these variables, then the following algorithm is invoked: et al., 1997]. Thus, ASP is strictly more powerful than SAT- Function ModelGenerator(I: Interpretation): Boolean; based programming, as it allows us to solve even problems begin which cannot be translated to SAT in polynomial time. The I := DetCons(I); high expressive power of ASP can be proﬁtably exploited in if I = L then return False; (* inconsistency *) AI, which often has to deal with problems of high complex- if no atom is undeﬁned in I then return IsAnswerSet(I); ity. For instance, problems in diagnosis and planning under Select an undeﬁned ground atom A according to a heuristic; incomplete knowledge are complete for the the complexity if ModelGenerator(I ∪ {A}) then return True; class ΣP or ΠP , and can be naturally encoded in ASP [Baral, else return ModelGenerator(I ∪ {not A}); 2 2 2002; Leone et al., 2001]. end; ∗ This work was partially supported by the European Commission Roughly, the Model Generator produces some “candidate” under projects IST-2002-33570 (INFOMIX) and IST-2001-37004 answer sets. The stability of each of them is subsequently ver- (WASP), and by FWF (Austrian Science Funds) under projects iﬁed by the function IsAnswerSet(I), which veriﬁes whether P16536-N04 and P17212-N04. the given “candidate” I is a minimal model of the GL- † Funded by APART of the Austrian Academy of Sciences. transformed program and outputs the model, if so. IsAnswer- Set(I) returns True if the computation should be stopped and 4 ASP vs QBF Solvers False otherwise. The main goal of this paper is to improve the performance The function DetCons() computes an extension of I with of ASP systems for problems located at the second level of the literals that can be deterministically inferred (or the set the polynomial hierarchy. One may wonder whether, on such of all literals L upon inconsistency). If DetCons does not ΣP /ΠP -hard problems, ASP systems are competitive with 2 2 detect any inconsistency, an atom A is selected according to other AI systems, like the QBF solvers. In order to give a a heuristic criterion and ModelGenerator is called on I ∪ {A} ﬁrst answer to this question, we have also performed a com- and on I ∪{not A}. The atom A plays the role of a branching parison with QBF solvers Quantor [Biere, 2004], Semprop variable of a SAT solver. [Letz, 2002], and yQuafﬂe [Zhang and Malik, 2002] on the The heuristic hU T , proposed in [Faber et al., 2001] is cur- set of all ΣP - and ΠP -complete QBF formulas of the last 2 2 rently employed in DLV. It is mostly based on the number of QBF competition. The results below report the number of in- UnsupportedTrue (UT) atoms (called MBTs in [Faber et al., stances solved within 660s and show that DLV (with heuristic 2001]), i.e., atoms which are true in the current interpretation hDS ) generally outperformed the QBF solvers. but miss a supporting rule, trying to minimize UT atoms and DLV (hDS ) Quantor Semprop yQuaf f le hence more likely arrive at supported models. Robot 27 (84%) 10 (31%) 13 (41%) 17 (53%) For hard ASP programs (i.e., non-HCF programs [Ben- Random 108 (100%) 14 (12%) 90 (83%) 53 (49%) Eliyahu and Dechter, 1994] – they express ΣP -complete 2 T ree 2 (100%) 2 (100%) 2 (100%) 2 (100%) KP H 1 (100%) 1 (100%) 1 (100%) 1 (100%) problems under brave reasoning), supported models are often T otal 137 (96%) 26 (18%) 105 (73%) 72 (50%) not answer sets. Moreover, answer-set checking is computa- tionally expensive (co-NP), and may consume a large portion References of the resources needed for computing an answer set. We therefore propose the new heuristic hDS , which tries [Baral, 2002] C. Baral. Knowledge Representation, Reason- in addition to maximize the degree of supportedness, the av- ing and Declarative Problem Solving. CUP, 2002. erage number of supporting rules for non-HCF true atoms. [Ben-Eliyahu and Dechter, 1994] R. Ben-Eliyahu and Intuitively, if all true atoms have many supporting rules in a R. Dechter. Propositional Semantics for Disjunctive Logic model M , then the elimination of an atom from the model Programs. AMAI, 12:53–87, 1994. would violate many rules, and it becomes less likely to ﬁnd a [Biere, 2004] A. Biere. Resolve and Expand. 2004. SAT’04. subset of M which is a model of the reduct P M , disproving [Eiter et al., 1997] T. Eiter, G. Gottlob, and H. Mannila. Dis- that M is an answer set. We deﬁne hDS as a reﬁnement of junctive Datalog. ACM TODS, 22(3):364–418, 1997. the heuristic hU T (i.e., A <hU T B ⇒ A <hDS B). In this [Faber et al., 2001] W. Faber, N. Leone, and G. Pfeifer. Ex- way, hDS keeps the behaviour of the well-assessed hU T on perimenting with Heuristics for Answer Set Programming. NP problems while, as we will see in Section 3, it sensibly In IJCAI 2001, pp. 635–640. improves over hU T on hard 2QBF problems (ΣP -complete). 2 [Gent and Walsh, 1999] I. Gent and T. Walsh. The QSAT Phase Transition. In AAAI, 1999. 3 Comparing hU T vs hDS : Experiments [Janhunen et al., 2000] T. Janhunen, I. Niemel¨ , P. Simons, a We generated randomly a data set of 100 2QBF formulas, fol- and J.-H. You. Partiality and Disjunctions in Stable Model lowing [Gent and Walsh, 1999], and used the ASP encoding Semantics. In KR 2000, 12-15, pp. 411–419. described in [Leone et al., 2005]. [Koch et al., 2003] C. Koch, N. Leone, and G. Pfeifer. En- Experiments were performed on a PentiumIV 1500 MHz hancing Disjunctive Logic Programming Systems by SAT machine with 256MB RAM running SuSe Linux 9.0. For ev- Checkers. Artiﬁcial Intelligence, 15(1–2):177–212, 2003. ery instance, we allowed a maximum running time of 7200 [Leone et al., 2001] N. Leone, R. Rosati, and F. Scarcello. seconds (two hours). The results of our experiments are dis- Enhancing Answer Set Planning. In IJCAI-01 Workshop played in the following graphs, in which a line stops when- on Planning under Uncertainty and Incomplete Informa- ever some instance was not solved within the time limit. tion, pp. 33–42, 2001. hUT hUT [Leone et al., 2005] N. Leone, G. Pfeifer, W. Faber, T. Eiter, hDS 1000 1000 hDS G. Gottlob, S. Perri, and F. Scarcello. The DLV System for Maximum Execution Time [s] Average Execution Time [s] 100 Knowledge Representation and Reasoning. ACM TOCL, 100 2005. To appear. 10 10 [Letz, 2002] R. Letz. Lemma and Model Caching in De- 1 1 cision Procedures for Quantiﬁed Boolean Formulas. In TABLEAUX 2002, pp. 160–175, Denmark, 2002. 0.1 0.1 [Lifschitz, 1999] V. Lifschitz. Answer Set Planning. In 0.01 0 20 40 60 80 100 0.01 0 20 40 60 80 100 ICLP’99, pp. 23–37. Number of propositional variables Number of propositional variables [Zhang and Malik, 2002] L. Zhang and S. Malik. Towards a Symmetric Treatment of Satisfaction and Conﬂicts in It is clear that the new heuristic hDS outperforms the Quantiﬁed Boolean Formula Evaluation. In CP 2002, pp. heuristic hU T in these experiments, advancing the “maximum 200–215, NY, USA, 2002. solvable-size” from 56 up to size 92, and reducing the average execution times of the smaller instances.

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