Hostname: page-component-84b7d79bbc-tsvsl Total loading time: 0 Render date: 2024-07-29T00:18:17.299Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs

Published online by Cambridge University Press:  10 August 2018

ANGELOS CHARALAMBIDIS ,
PANOS RONDOGIANNIS  and
IOANNA SYMEONIDOU
ANGELOS CHARALAMBIDIS
Affiliation:
Department of Informatics and Telecommunications, University of Athens, Greece (e-mail: acharala@di.uoa.gr, prondo@di.uoa.gr, sioanna@di.uoa.gr)
PANOS RONDOGIANNIS
Affiliation:
Department of Informatics and Telecommunications, University of Athens, Greece (e-mail: acharala@di.uoa.gr, prondo@di.uoa.gr, sioanna@di.uoa.gr)
IOANNA SYMEONIDOU
Affiliation:
Department of Informatics and Telecommunications, University of Athens, Greece (e-mail: acharala@di.uoa.gr, prondo@di.uoa.gr, sioanna@di.uoa.gr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We define a novel, extensional, three-valued semantics for higher-order logic programs with negation. The new semantics is based on interpreting the types of the source language as three-valued Fitting-monotonic functions at all levels of the type hierarchy. We prove that there exists a bijection between such Fitting-monotonic functions and pairs of two-valued-result functions where the first member of the pair is monotone-antimonotone and the second member is antimonotone-monotone. By deriving an extension of consistent approximation fixpoint theory (Denecker et al. 2004) and utilizing the above bijection, we define an iterative procedure that produces for any given higher-order logic program a distinguished extensional model. We demonstrate that this model is actually a minimal one. Moreover, we prove that our construction generalizes the familiar well-founded semantics for classical logic programs, making in this way our proposal an appealing formulation for capturing the well-founded semantics for higher-order logic programs.

Keywords

Higher-Order Logic ProgrammingNegation in Logic ProgrammingApproximation Fixpoint Theory
Type
Original Article
Information
Theory and Practice of Logic Programming , Volume 18 , Special Issue 3-4: 34th International Conference on Logic Programming , July 2018 , pp. 421 - 437
Copyright
Copyright © Cambridge University Press 2018 

References

Bezem, M. 1999. Extensionality of simply typed logic programs. In Logic Programming: The 1999 International Conference, Las Cruces, New Mexico, USA, November 29 - December 4, 1999, Schreye, D. D., Ed. MIT Press, 395–410.Google Scholar
Bloom, S. L. and Ésik, Z. 1993. Iteration Theories - The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer.Google Scholar
Carayol, A. and Ésik, Z. 2016. An analysis of the equational properties of the well-founded fixed point. In Principles of Knowledge Representation and Reasoning: Proceedings of the Fifteenth International Conference, KR 2016, Cape Town, South Africa, April 25-29, 2016., Baral, C., Delgrande, J. P., and Wolter, F., Eds. AAAI Press, 533–536.Google Scholar
Charalambidis, A., Ésik, Z., and Rondogiannis, P. 2014. Minimum model semantics for extensional higher-order logic programming with negation. TPLP 14, 4–5, 725–737.Google Scholar
Charalambidis, A., Handjopoulos, K., Rondogiannis, P., and Wadge, W. W. 2013. Extensional higher-order logic programming. ACM Trans. Comput. Log. 14, 3, 21.Google Scholar
Charalambidis, A., Rondogiannis, P., and Symeonidou, I. 2017. Equivalence of two fixed-point semantics for definitional higher-order logic programs. Theor. Comput. Sci. 668, 27–42.Google Scholar
Davey, B. A. and Priestley, H. A. 2002. Introduction to Lattices and Order. Cambridge University Press.Google Scholar
Denecker, M., Bruynooghe, M., and Vennekens, J. 2012. Approximation fixpoint theory and the semantics of logic and answers set programs. In Correct Reasoning - Essays on Logic-Based AI in Honour of Vladimir Lifschitz, Erdem, E., Lee, J., Lierler, Y., and Pearce, D., Eds. Lecture Notes in Computer Science, vol. 7265. Springer, 178–194.Google Scholar
Denecker, M., Marek, V. and Truszczyński, M. 2000. Approximations, Stable Operators, Well-Founded Fixpoints and Applications in Nonmonotonic Reasoning. In: Logic-Based Artificial Intelligence. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston, MA, 127–144.Google Scholar
Denecker, M., Marek, V. W., and Truszczynski, M. 2004. Ultimate approximation and its application in nonmonotonic knowledge representation systems. Inf. Comput. 192, 1, 84–121.Google Scholar
Ésik, Z. 2015. Equational properties of stratified least fixed points (extended abstract). In Logic, Language, Information, and Computation - 22nd International Workshop, WoLLIC 2015, Bloomington, IN, USA, July 20-23, 2015, Proceedings, de Paiva, V., de Queiroz, R. J. G. B., Moss, L. S., Leivant, D., and de Oliveira, A. G., Eds. Lecture Notes in Computer Science, vol. 9160. Springer, 174–188.Google Scholar
Ésik, Z. and Rondogiannis, P. 2015. A fixed point theorem for non-monotonic functions. Theoretical Computer Science 574, 18–38.Google Scholar
Fitting, M. 2002. Fixpoint semantics for logic programming: a survey. Theor. Comput. Sci. 278, 1–2, 25–51.Google Scholar
Gelder, A. V., Ross, K. A., and Schlipf, J. S. 1991. The well-founded semantics for general logic programs. J. ACM 38, 3, 620–650.Google Scholar
Przymusinski, T. C. 1989. Every logic program has a natural stratification and an iterated least fixed point model. In Proceedings of the Eighth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, March 29-31, 1989, Philadelphia, Pennsylvania, USA. 11–21.Google Scholar
Rondogiannis, P. and Symeonidou, I. 2016. Extensional semantics for higher-order logic programs with negation. In Logics in Artificial Intelligence - 15th European Conference, JELIA 2016, Larnaca, Cyprus, November 9-11, 2016, Proceedings, Michael, L. and Kakas, A. C., Eds. Lecture Notes in Computer Science, vol. 10021. 447–462.Google Scholar
Rondogiannis, P. and Symeonidou, I. 2017. The intricacies of three-valued extensional semantics for higher-order logic programs. TPLP 17, 5–6, 974–991.Google Scholar
Tennent, R. D. 1991. Semantics of programming languages. Prentice Hall International Series in Computer Science. Prentice Hall.Google Scholar
van Emden, M. H. and Kowalski, R. A. 1976. The semantics of predicate logic as a programming language. J. ACM 23, 4, 733–742.Google Scholar
Wadge, W. W. 1991. Higher-order horn logic programming. In Logic Programming, Proceedings of the 1991 International Symposium, San Diego, California, USA, Oct. 28 - Nov 1, 1991, Saraswat, V. A. and Ueda, K., Eds. MIT Press, 289–303.Google Scholar
Supplementary material: PDF

Charalambidis et al. supplementary material

Charalambidis et al. supplementary material 1

Download Charalambidis et al. supplementary material(PDF)
PDF 254.4 KB