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A denotational semantics for equilibrium logic*

Published online by Cambridge University Press:  03 September 2015

FELICIDAD AGUADO ,
PEDRO CABALAR ,
DAVID PEARCE ,
GILBERTO PÉREZ  and
CONCEPCIÓN VIDAL
FELICIDAD AGUADO
Affiliation:
Department of Computer Science, University of Corunna, SPAIN (e-mail: aguado@udc.es, cabalar@udc.es, gperez@udc.es, eicovima@udc.es)
PEDRO CABALAR
Affiliation:
Department of Computer Science, University of Corunna, SPAIN (e-mail: aguado@udc.es, cabalar@udc.es, gperez@udc.es, eicovima@udc.es)
DAVID PEARCE
Affiliation:
Universidad Politécnica de Madrid, SPAIN (e-mail: david.pearce@upm.es)
GILBERTO PÉREZ
Affiliation:
Department of Computer Science, University of Corunna, SPAIN (e-mail: aguado@udc.es, cabalar@udc.es, gperez@udc.es, eicovima@udc.es)
CONCEPCIÓN VIDAL
Affiliation:
Department of Computer Science, University of Corunna, SPAIN (e-mail: aguado@udc.es, cabalar@udc.es, gperez@udc.es, eicovima@udc.es)
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Abstract

In this paper we provide an alternative semantics for Equilibrium Logic and its monotonic basis, the logic of Here-and-There (also known as Gödel's G3 logic) that relies on the idea of denotation of a formula, that is, a function that collects the set of models of that formula. Using the three-valued logic G3 as a starting point and an ordering relation (for which equilibrium/stable models are minimal elements) we provide several elementary operations for sets of interpretations. By analysing structural properties of the denotation of formulas, we show some expressiveness results for G3 such as, for instance, that conjunction is not expressible in terms of the other connectives. Moreover, the denotational semantics allows us to capture the set of equilibrium models of a formula with a simple and compact set expression. We also use this semantics to provide several formal definitions for entailment relations that are usual in the literature, and further introduce a new one called strong entailment. We say that α strongly entails β when the equilibrium models of α ∧ γ are also equilibrium models of β ∧ γ for any context γ. We also provide a characterisation of strong entailment in terms of the denotational semantics, and give an example of a sufficient condition that can be applied in some cases.

Keywords

Answer Set ProgrammingEquilibrium Logic
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 15 , Special Issue 4-5: 31st International Conference on Logic Programming , July 2015 , pp. 620 - 634
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

*

This research was partially supported by Spanish MEC project TIN2013-42149-P.

References

Aguado, F., Ascariz, P., Cabalar, P., Pérez, G. and Vidal, C. 2015. Formal verification for ASP: a case study using the PVS theorem prover. In Proc. of the 15th Intl. Conf. on Computational and Mathematical Methods in Science and Engineering CMMSE 2015 (July 3-7). Rota, Spain.Google Scholar
Aguado, F., Cabalar, P., Diéguez, M., Pérez, G. and Vidal, C. 2013. Temporal equilibrium logic: a survey. Journal of Applied Non-Classical Logics 23, 1–2, 2–24.CrossRefGoogle Scholar
Brewka, G., Eiter, T. and Truszczynski, M. 2011. Answer set programming at a glance. Communications of the ACM 54, 12, 92–103.Google Scholar
Cabalar, P. and Ferraris, P. 2007. Propositional theories are strongly equivalent to logic programs. Theory and Practice of Logic Programming 7, 6, 745–759.Google Scholar
Cabalar, P., Odintsov, S., Pearce, D. and Valverde, A. 2007. Partial equilibrium logic. Annals of Mathematics and Artificial Intelligence 50, 3–4, 305–331.CrossRefGoogle Scholar
Cabalar, P., Pearce, D. and Valverde, A. 2005. Reducing propositional theories in equilibrium logic to logic programs. In Progress in Artificial Intelligence, Proc. of the 12th Portuguese Conf. on Artificial Intelligence, EPIA'05, Covilhã, Portugal, December 5-8, 2005, Bento, C., Cardoso, A., and Dias, G., Eds. Lecture Notes in Computer Science, vol. 3808. Springer, 4–17.Google Scholar
Cabalar, P., Pearce, D. and Valverde, A. 2007. Minimal logic programs. In Proc. of the 23rd Intl. Conf. on Logic Programming, ICLP 2007 (September 8-13). Porto, Portugal, 104–118.Google Scholar
Delgrande, J. P., Schaub, T., Tompits, H. and Woltran, S. 2008. Belief revision of logic programs under answer set semantics. In Proc. of the 11th Intl. Conf. on Principles of Knowledge Representation and Reasoning KR 2008 (September 16-19). Sydney, Australia, 411–421.Google Scholar
Eiter, T., Fink, M., Pührer, J., Tompits, H., and Woltran, S. 2013. Model-based recasting in answer-set programming. Journal of Applied Non-Classical Logics 23, 1–2, 75–104.Google Scholar
Ferraris, P. 2005. Answer sets for propositional theories. In Logic Programming and Nonmonotonic Reasoning, 8th International Conference, LPNMR 2005, Diamante, Italy, September 5-8, 2005, Proceedings. 119–131.Google Scholar
Ferraris, P., Lee, J. and Lifschitz, V. 2007. A new perspective on stable models. In Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI'07) (January 6-12, 2007), Veloso, M. M., Ed. Hyderabad, India, 372–379.Google Scholar
Fink, M. 2011. A general framework for equivalences in answer-set programming by countermodels in the logic of here-and-there. TPLP 11, 2–3, 171–202.Google Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Logic Programming: Proc. of the Fifth International Conference and Symposium (Volume 2), Kowalski, R. A. and Bowen, K. A., Eds. MIT Press, Cambridge, MA, 1070–1080.Google Scholar
Gödel, K. 1932. Zum intuitionistischen aussagenkalkül. Anzeiger der Akademie der Wissenschaften Wien, mathematisch, naturwissenschaftliche Klasse 69, 65–66.Google Scholar
Harrison, A., Lifschitz, V., Valverde, A. and Pearce, D. 2014. Infinitary equilibrium logic. In Working Notes of Workshop on Answer Set Programming and Other Computing Paradigms (ASPOCP).Google Scholar
Heyting, A. 1930. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 42–56.Google Scholar
Kozen, D. 1983. Results on the propositional μ-calculus. Theoretical Computer Science 27, 3, 333–354.Google Scholar
Lifschitz, V., Pearce, D. and Valverde, A. 2001. Strongly equivalent logic programs. ACM Trans. Comput. Log. 2, 4, 526–541.CrossRefGoogle Scholar
Marek, V. and Truszczyński, M. 1999. Stable models and an alternative logic programming paradigm. Springer-Verlag, 169–181.Google Scholar
Nicholson, T. and Foo, N. 1989. A denotational semantics for prolog. ACM Transactions on Programming Languages and Systems 11, 4 (Oct.), 650–665.Google Scholar
Niemelä, I. 1999. Logic programs with stable model semantics as a constraint programming paradigm. Annals of Mathematics and Artificial Intelligence 25, 241–273.Google Scholar
Odintsov, S. P. and Pearce, D. 2005. Routley semantics for answer sets. In 8th Intl. Cond. on Logic Programming and Nonmonotonic Reasoning, LPNMR 2005 (September 5-8). Diamante, Italy, 343–355.Google Scholar
Owre, S., Rushby, J. M. and Shankar, N. 1992. Pvs: A prototype verfication system. In 11th International Conference on Automted Deduction (CADE). Lecture Notes in Artificial Intelligence 607, 748–752.Google Scholar
Pearce, D. 1996. A new logical characterisation of stable models and answer sets. In Non monotonic extensions of logic programming. Proc. NMELP'96. (LNAI 1216). Springer-Verlag.Google Scholar
Pearce, D. 2006. Equilibrium logic. Annals of Mathematics and Artificial Intelligence 47, 1–2, 3–41.Google Scholar
Pearce, D. and Valverde, A. 2004a. Towards a first order equilibrium logic for nonmonotonic reasoning. In Proc. of the 9th European Conf. on Logics in AI (JELIA'04). 147–160.Google Scholar
Pearce, D. and Valverde, A. 2004b. Uniform equivalence for equilibrium logic and logic programs. In Proc. of the 7th Intl. Conf. on Logic Programming and Nonmonotonic Reasoning, LPNMR 2004 (January 6-8). Fort Lauderdale, FL, USA, 194–206.Google Scholar
Scott, D. and Strachey, C. 1971. Toward a mathematical semantics for computer languages. Tech. Rep. PRG-6, Oxford Programming Research Group Technical Monograph.Google Scholar
Slota, M. and Leite, J. 2014. The rise and fall of semantic rule updates based on se-models. Theory and Practice of Logic Programming 14, 6, 869–907.Google Scholar
Woltran, S. 2008. A common view on strong, uniform, and other notions of equivalence in answer-set programming. Theory and Practice of Logic Programming 8, 2, 217–234.Google Scholar
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