A Free Logic for Stable Models with Partial Intensional Functions (original) (raw)

Abstract

In this paper we provide a new logical characterisation of stable models with partial functions that consists in a free-logic extension of Quantified Equilibrium Logic (QEL). In so-called "free" logics, terms may denote objects that are outside the domain of quantification, something that can be immediately used to capture partial functions. We show that this feature can be naturally accommodated in the monotonic basis of QEL (the logic of Quantified Here-and-There, QHT) by allowing variable quantification domains that depend on the world where the formula is being interpreted. The paper provides two main contributions: (i) a correspondence with Cabalar's semantics for stable models with partial functions; and (ii) a Gentzen system for free QHT, the monotonic basis of free QEL.

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References (27)

  1. for quantified formulas: The Gentzen system works over the domain V , a denumerable set of variables (or parameters); that is, the introduction of quan- tifiers is always made from variables, not from terms of the original language. In the following rules, y is a fresh variable, i.e. a variable which does not occur free in Γ ∪ ∆ and τ ∈ T erms(C ∪ V, F ):
  2. Mario Alviano, Francesco Calimeri, Günther Charwat, Minh Dao-Tran, Carmine Dodaro, Giovambattista Ianni, Thomas Krennwallner, Martin Kronegger, Jo- hannes Oetsch, Andreas Pfandler, Jörg Pührer, Christoph Redl, Francesco Ricca, Patrik Schneider, Martin Schwengerer, Lara Katharina Spendier, Johannes Peter Wallner, and Guohui Xiao. The fourth answer set programming competition: Pre- liminary report. In Pedro Cabalar and Tran Cao Son, editors, Proc. of the 12th Intl. Conf. on Logic Programming and Nonmonotonic Reasoning (LPNMR 2013), Corunna, Spain, September 15-19, volume 8148 of Lecture Notes in Computer Sci- ence, pages 42-53. Springer, 2013.
  3. Marcello Balduccini. A "conservative" approach to extending answer set program- ming with nonherbrand functions. In Esra Erdem, Joohyung Lee, Yuliya Lierler, and David Pearce, editors, Correct Reasoning, pages 24-39. Springer-Verlag, 2012.
  4. Michael Bartholomew and Joohyung Lee. Stable models of formulas with in- tensional functions. In Proceedings of International Conference on Principles of Knowledge Representation and Reasoning (KR'12), pages 2-12, 2012.
  5. Michael Bartholomew and Joohyung Lee. On the stable model semantics for inten- sional functions. In Proceedings of International Conference on Logic Programming (ICLP'13), 2013.
  6. Gerhard Brewka, Thomas Eiter, and Miroslaw Truszczynski. Answer set program- ming at a glance. Commun. ACM, 54(12):92-103, 2011.
  7. Tyler Burge. Truth and singular terms. Nous, 8(4):309-325, 1974.
  8. Pedro Cabalar. Functional answer set programming. Theory and Practice of Logic Programming, 10(2-3):203-233, 2011.
  9. Pedro Cabalar and David Lorenzo. Logic programs with functions and default values. In Proc. of the 9th European Conf. on Logics in AI (JELIA'04) (LNCS 3229), pages 294-306, 2004.
  10. Pedro Cabalar, David Pearce, and Agustín Valverde. A revised concept of safety for general answer set programs. In Esra Erdem, Fangzhen Lin, and Torsten Schaub, editors, Proc. of the 10th Intl. Conf. on Logic Programming and Nonmonotonic Reasoning (LPNMR 2009), Potsdam, Germany, September 14-18, volume 5753 of Lecture Notes in Computer Science, pages 58-70. Springer, 2009.
  11. Francesco Calimeri, Susanna Cozza, Giovambattista Ianni, and Nicola Leone. Com- putable functions in ASP: Theory and implementation. In 24th Intl. Conf. on Logic Programming, volume 5366 of Lecture Notes in Computer Science, pages 407-424. Springer-Verlag, 2008.
  12. Francesco Calimeri, Susanna Cozza, Giovambattista Ianni, and Nicola Leone. An ASP system with functions, lists, and sets. In 10th Intl. Conf. on Logic Program- ming and Nonmonotonic Reasoning, volume 5753 of Lecture Notes in Computer Science, pages 483-489. Springer-Verlag, 2009.
  13. Luis Fariñas del Cerro, David Pearce, and Agustín Valverde. FQHT: The logic of stable models for logic programs with intensional functions. In Proceedings of International Joint Conference on Artificial Intelligence (IJCAI'13), 2013.
  14. P. Ferraris, J. Lee, and V. Lifschitz. A new perspective on stable models. In Proc. of the International Joint Conference on Artificial Intelligence (IJCAI'07), pages 372-379, 2007.
  15. Dov M. Gabbay, David Pearce, and Agustín Valverde. Interpolable formulas in equilibrium logic and answer set programming. Journal of Artificial Intelligence Research (JAIR), 42:917-943, 2011.
  16. Michael Gelfond and Vladimir Lifschitz. The stable model semantics for logic programming. In Proc. of the 5th Intl. Conf. on Logic Programming, pages 1070- 1080, 1988.
  17. Michael Hanus. The integration of functions into logic programming: from theory to practice. Journal of Logic Programming, 19,20:583-628, 1994.
  18. Arend Heyting. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pages 42-56, 1930.
  19. Vladimir Lifschitz. Logic programs with intensional functions. In Proceedings of In- ternational Conference on Principles of Knowledge Representation and Reasoning (KR'12), 2012.
  20. Vladimir Lifschitz, David Pearce, and Agustín Valverde. A characterization of strong equivalence for logic programs with variables. In Proc. of the 9th Intl. Conf. on Logic Programming and Nonmonotonic Reasoning (LPNMR'07), pages 188-200, 2007.
  21. Fangzhen Lin and Yisong Wang. Answer set programming with functions. In Proc. of the 11th Intl. Conf. on Principles of Knowledge Representation and Reasoning (KR'08), 2008.
  22. V. Marek and M. Truszczyński. Stable models and an alternative logic program- ming paradigm. In The Logic Programming Paradigm: a 25-Year Perspective, pages 169-181. Springer-Verlag, 1999.
  23. I. Niemelä. Logic programs with stable model semantics as a constraint program- ming paradigm. Annals of Mathematics and Artificial Intelligence, 25:241-273, 1999.
  24. David Pearce. A new logical characterisation of stable models and answer sets. In Non monotonic extensions of logic programming. Proc. NMELP'96. (LNAI 1216). Springer-Verlag, 1996.
  25. David Pearce and Agustín Valverde. Towards a first order equilibrium logic for nonmonotonic reasoning. In Proc. of the 9th European Conf. on Logics in AI (JELIA'04), pages 147-160, 2004.
  26. David Pearce and Agustín Valverde. Synonymous theories and knowledge repre- sentations in answer set programming. Journal of Computer and System Sciences, 78(1):86-104, 2012.
  27. Carl J. Posy. A free IPC is a natural logic: Strong completeness for some intuition- istic free logics. Topoi, 1(1-2):30-43, 1982.