Abstract
Answer Set Programming (ASP) has become an increasingly popular formalism for declarative problem solving. Among the huge body of theoretical results, investigations of different equivalence notions between logic programs play a fundamental role for understanding modularity and optimization. While strong equivalence between two programs holds if they can be faithfully replaced by each other in any context (facts and rules), uniform equivalence amounts to equivalent behavior of programs under any set of facts. Both notions (as well as several variants thereof) have been extensively studied. However, the somewhat reverse notion of equivalence which holds if two programs are equivalent under the addition of any set of proper rules (i.e., all rules except facts) has not been investigated yet. In this paper, we close this gap and give a thorough study of this notion, which we call rule equivalence (RE), and its parameterized version where we allow facts over a given restricted alphabet to appear in the context. RE is thus a relationship between two programs whose input is (partially) fixed but where additional proper rules might still be added. Such a notion might be helpful in debugging of programs. We give full characterization results and a complexity analysis for the propositional case of RE. Moreover, we show that RE is decidable in the non-ground case.
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- 1.
Within SE-models, we denote interpretations \(\{ a_1,\ldots , a_n \}\) by juxtaposition \(a_1\cdots a_n\) of their elements.
- 2.
In fact, the programs we use here form a proper subclass of programs compared to [11], where, e.g., also double negation is allowed. However, for our purpose it is sufficient to consider this weaker class.
- 3.
The concept of saturation refers to a programming technique, where reasons for a candidate answer set I to be ruled out are not explicitly stated via constraints, but in terms of rules which ensure that a certain model \(J\subset I\) of the program’s reduct with respect to I exists, see, e.g., [10].
References
Brewka, G., Eiter, T., Truszczyński, M.: Answer set programming at a glance. Communications of the ACM 54(12), 92–103 (2011)
Eiter, T., Fink, M., Tompits, H., Woltran, S.: Simplifying Logic Programs Under Uniform and Strong Equivalence. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 87–99. Springer, Heidelberg (2003)
Eiter, T., Fink, M.: Uniform Equivalence of Logic Programs under the Stable Model Semantics. In: Palamidessi, C. (ed.) ICLP 2003. LNCS, vol. 2916, pp. 224–238. Springer, Heidelberg (2003)
Eiter, T., Fink, M., Pührer, J., Tompits, H., Woltran, S.: Model-based recasting in answer-set programming. J. Appl. Non-Classical Logics 23(1–2), 75–104 (2013). http://dx.org/10.1080/11663081.2013.799318
Eiter, T., Fink, M., Tompits, H., Woltran, S.: Strong and uniform equivalence in answer-set programming: characterizations and complexity results for the non-ground case. In: Proceedings of the 20th National Conference on Artificial Intelligence (AAAI 2005), pp. 695–700. AAAI Press (2005)
Eiter, T., Fink, M., Woltran, S.: Semantical characterizations and complexity of equivalences in answer set programming. ACM Trans. Comput. Log. 8(3), 1–53 (2007). http://doi.acm.org/10.1145/1243996.1244000
Fink, M.: A general framework for equivalences in answer-set programming by countermodels in the logic of here-and-there. Theory Pract. Logic Programm. 11(2–3), 171–202 (2011)
Inoue, K., Sakama, C.: Equivalence of logic programs under updates. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 174–186. Springer, Heidelberg (2004)
Janhunen, T., Oikarinen, E., Tompits, H., Woltran, S.: Modularity aspects of disjunctive stable models. J. Artif. Intell. Res. (JAIR) 35, 813–857 (2009). http://dx.org/10.1613/jair.2810
Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S., Scarcello, F.: The DLV system for knowledge representation and reasoning. ACM Trans. Comput. Log. 7(3), 499–562 (2006)
Lifschitz, V., Tang, L., Turner, H.: Nested expressions in logic programs. Ann. Math. Artif. Intell. 25(3–4), 369–389 (1999)
Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Trans. Comput. Logic 2(4), 526–541 (2001)
Oikarinen, E., Janhunen, T.: Modular equivalence for normal logic programs. In: Proceedings of the 17th European Conference on Artificial Intelligence (ECAI 2006), pp. 412–416. IOS Press (2006)
Pearce, D.J., Valverde, A.: Uniform equivalence for equilibrium logic and logic programs. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 194–206. Springer, Heidelberg (2003)
Sagiv, Y.: Optimizing datalog programs. In: Minker, J. (ed.) Foundations of Deductive Databases and Logic Programming, pp. 659–698. Morgan Kaufmann, USA (1988)
Truszczynski, M., Woltran, S.: Relativized hyperequivalence of logic programs for modular programming. TPLP 9(6), 781–819 (2009). http://dx.org/10.1017/S1471068409990159
Turner, H.: Strong equivalence made easy: nested expressions and weight constraints. Theor. Pract. Logic Program. 3(4–5), 602–622 (2003)
Woltran, S.: Characterizations for relativized notions of equivalence in answer set programming. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 161–173. Springer, Heidelberg (2004)
Woltran, S.: A common view on strong, uniform, and other notions of equivalence in answer-set programming. TPLP 8(2), 217–234 (2008). http://dx.org/10.1017/S1471068407003250
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This work was supported by the Austrian Science Fund (FWF) projects P25607 and Y698.
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Bliem, B., Woltran, S. (2016). Equivalence Between Answer-Set Programs Under (Partially) Fixed Input. In: Gyssens, M., Simari, G. (eds) Foundations of Information and Knowledge Systems. FoIKS 2016. Lecture Notes in Computer Science(), vol 9616. Springer, Cham. https://doi.org/10.1007/978-3-319-30024-5_6
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