Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-22T15:30:35.460Z Has data issue: false hasContentIssue false
  • English
  • Français

On relation between constraint answer set programming and satisfiability modulo theories*

Published online by Cambridge University Press:  28 June 2017

YULIYA LIERLER  and
BENJAMIN SUSMAN
YULIYA LIERLER
Affiliation:
Department of Computer Science, University of Nebraska at Omaha, Omaha, NE 68182, USA (e-mails: ylierler@unomaha.edu, bsusman@unomaha.edu)
BENJAMIN SUSMAN
Affiliation:
Department of Computer Science, University of Nebraska at Omaha, Omaha, NE 68182, USA (e-mails: ylierler@unomaha.edu, bsusman@unomaha.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

Constraint answer set programming is a promising research direction that integrates answer set programming with constraint processing. It is often informally related to the field of satisfiability modulo theories. Yet, the exact formal link is obscured as the terminology and concepts used in these two research areas differ. In this paper, we connect these two research areas by uncovering the precise formal relation between them. We believe that this work will boost the cross-fertilization of the theoretical foundations and the existing solving methods in both areas. As a step in this direction, we provide a translation from constraint answer set programs with integer linear constraints to satisfiability modulo linear integer arithmetic that paves the way to utilizing modern satisfiability modulo theories solvers for computing answer sets of constraint answer set programs.

Keywords

constraint answer set programmingconstraint satisfaction processingsatisfiability modulo theories
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 17 , Issue 4 , July 2017 , pp. 559 - 590
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

This is an extended version of the paper that appeared at IJCAI-2016 (Lierler and Susman 2016)

References

Balduccini, M. 2009. Representing constraint satisfaction problems in answer set programming. In Proc. of ICLP Workshop on Answer Set Programming and Other Computing Paradigms (ASPOCP). URL: https://www.mat.unical.it/ASPOCP09/ Google Scholar
Balduccini, M. 2013. ASP with non-Herbrand partial functions: A language and system for practical use. Theory and Practice of Logic Programming 13, 547–561.Google Scholar
Balduccini, M. and Lierler, Y. 2017. Constraint answer set solver EZCSP and why integration schemas matter. Theory and Practice of Logic Programming, in this issue.Google Scholar
Barrett, C., Conway, C. L., Deters, M., Hadarean, L., Jovanović, D., King, T., Reynolds, A. and Tinelli, C. 2011. CVC4. In Proc. of 23rd International Conference on Computer Aided Verification (CAV'11), Lecture Notes in Computer Science, vol. 6806. Springer International Publishing.CrossRefGoogle Scholar
Barrett, C., Fontaine, P. and Tinelli, C. 2015. The SMT-LIB standard: Version 2.5. Technical Report, Department of Computer Science, The University of Iowa. URL: www.SMT-LIB.org Google Scholar
Barrett, C., Sebastiani, R., Seshia, S. and Tinelli, C. 2008. Satisfiability modulo theories. In Handbook of Satisfiability, Biere, A., Heule, M., van Maaren, H. and Walsch, T., Eds. IOS Press, 737–797.Google Scholar
Barrett, C. and Tinelli, C. 2014. Satisfiability modulo theories. In Handbook of Model Checking, Clarke, E., Henzinger, T. and Veith, H., Eds. Springer International Publishing.Google Scholar
Bartholomew, M. and Lee, J. 2012. Stable models of formulas with intensional functions. In Proc. of International Conference on Principles of Knowledge Representation and Reasoning (KR).Google Scholar
Bomanson, J., Gebser, M., Janhunen, T., Kaufmann, B. and Schaub, T. 2015. Answer set programming modulo acyclicity. In Proc. of 13th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR), Lexington, KY, USA, Calimeri, F., Ianni, G. and Truszczynski, M., Eds. Springer International Publishing, Cham, 143–150.Google Scholar
Calimeri, F., Cozza, S., Ianni, G. and Leone, N. 2008. Computable functions in ASP: Theory and implementation. In Proc. of International Conference on Logic Programming (ICLP), 407–424.Google Scholar
Carlsson, M. and Fruehwirth, T. 2014. SICStus PROLOG User's Manual 4.3. Books On Demand - Proquest.Google Scholar
Clark, K. 1978. Negation as failure. In Logic and Data Bases, Gallaire, H. and Minker, J., Eds. Plenum Press, New York, 293–322.Google Scholar
De Moura, L. and Bjørner, N. 2008. Z3: An efficient SMT solver. In Proc. of the Theory and Practice of Software, 14th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, 337–340.Google Scholar
Denecker, M. and Vennekens, J. 2007. Well-founded semantics and the algebraic theory of non-monotone inductive definitions. In Proc. of 9th International Conference Logic Programming and Nonmonotonic Reasoning (LPNMR), Tempe, AZ, USA, Baral, C., Brewka, G. and Schlipf, J. S., Eds. Lecture Notes in Computer Science, vol. 4483. Springer International Publishing, 84–96.Google Scholar
Eiter, T., Fink, M., Krennwallner, T. and Redl, C. 2012. Conflict-driven ASP solving with external sources. Theory and Practice of Logic Programming 12, 4–5, 659–679.Google Scholar
Elkabani, I., Pontelli, E. and Son, T. C. 2004. Smodels with CLP and its applications: A simple and effective approach to aggregates in ASP. In Proc. of International Conference on Logic Programming (ICLP), 73–89.Google Scholar
Erdem, E. and Lifschitz, V. 2001. Fages' theorem for programs with nested expressions. In Proc. of International Conference on Logic Programming (ICLP), 242–254.Google Scholar
Fages, F. 1994. Consistency of Clark's completion and existence of stable models. Journal of Methods of Logic in Computer Science 1, 51–60.Google Scholar
Ferraris, P. and Lifschitz, V. 2005. Weight constraints as nested expressions. Theory and Practice of Logic Programming 5, 45–74.Google Scholar
Gebser, M., Janhunen, T. and Rintanen, J. 2014. SAT modulo graphs: Acyclicity. In Proc. of 14th European Conference Logics in Artificial Intelligence (JELIA), Funchal, Madeira, Portugal, Fermé, E. and Leite, J., Eds. Springer International Publishing, Cham, 137–151.Google Scholar
Gebser, M., Ostrowski, M. and Schaub, T. 2009. Constraint answer set solving. In Proc. of 25th International Conference on Logic Programming (ICLP). Springer International Publishing, 235–249.Google Scholar
Gebser, M., Schaub, T. and Thiele, S. 2007. Gringo: A new grounder for answer set programming. In Proc. of 9th International Conference on Logic Programming and Nonmonotonic Reasoning, 266–271.Google Scholar
Gelfond, M. and Przymusinska, H. 1996. Towards a theory of elaboration tolerance: Logic programming approach. International Journal of Software Engineering and Knowledge Engineering 6, 1, 89–112.Google Scholar
IBM 2009. IBM ILOG AMPL Version 12.1 User's Guide. IBM. URL: http://www.ibm.com/software/commerce/optimization/cplex-optimizer/ Google Scholar
Janhunen, T., Liu, G. and Niemela, I. 2011. Tight integration of non-ground answer set programming and satisfiability modulo theories. In Proc. of 1st Workshop on Grounding and Transformations for Theories with Variables.Google Scholar
King, T., Barrett, C. and Tinelli, C. 2014. Leveraging linear and mixed integer programming for SMT. In Proc. of 14th Conference on Formal Methods in Computer-Aided Design (FMCAD'14). FMCAD, Austin, TX, 24:139–24:146.Google Scholar
Lee, J. and Meng, Y. 2013. Answer set programming modulo theories and reasoning about continuous changes. In Proc. of 23rd International Joint Conference on Artificial Intelligence (IJCAI'13), Beijing, China, August 3–9, 2013.Google Scholar
Lierler, Y. 2014. Relating constraint answer set programming languages and algorithms. Artificial Intelligence 207C, 1–22.Google Scholar
Lierler, Y. and Susman, B. 2016. Constraint answer set programming versus satisfiability modulo theories. In Proc. of 25th International Joint Conference on Artificial Intelligence (IJCAI), 1181–1187.Google Scholar
Lierler, Y. and Truszczynski, M. 2011. Transition systems for model generators – a unifying approach. In Proc. Theory and Practice of Logic Programming, 27th International Conference on Logic Programming (ICLP'11), Special Issue 11, Issue 4–5.Google Scholar
Lifschitz, V. 2012. Logic programs with intensional functions. In Proc. of International Conference on Principles of Knowledge Representation and Reasoning (KR).Google Scholar
Lifschitz, V., Tang, L. R. and Turner, H. 1999. Nested expressions in logic programs. Annals of Mathematics and Artificial Intelligence 25, 369–389.Google Scholar
Liu, G., Janhunen, T. and Niemela, I. 2012. Answer set programming via mixed integer programming. In Proc. of 13th International Conference on Principles of Knowledge Representation and Reasoning (KR).Google Scholar
Marriott, K. and Stuckey, P. J. 1998. Programming with Constraints: An Introduction. MIT Press.Google Scholar
Mellarkod, V. S., Gelfond, M. and Zhang, Y. 2008. Integrating answer set programming and constraint logic programming. Annals of Mathematics and Artificial Intelligence 53, 1, 251–287.Google Scholar
Niemelä, I. 2008. Stable models and difference logic. Annals of Mathematics and Artificial Intelligence 53, 313–329.Google Scholar
Niemelä, I. and Simons, P. 2000. Extending the Smodels system with cardinality and weight constraints. In Logic-Based Artificial Intelligence, Minker, J., Ed. Kluwer, 491–521.Google Scholar
Nieuwenhuis, R. and Oliveras, A. 2005. DPLL(T) with exhaustive theory propagation and its application to difference logic. In Proc. of 17th International Conference on Computer Aided Verification (CAV'05), Lecture Notes in Computer Science, vol. 3576. Springer International Publishing.Google Scholar
Oikarinen, E. and Janhunen, T. 2006. Modular equivalence for normal logic programs. In Proc. of 17th European Conference on Artificial Intelligence (ECAI'06), Brewka, G., Coradeschi, S., Perini, A. and Traverso, P., Eds. IOS Press, Amsterdam, The Netherlands, 412–416.Google Scholar
Susman, B. and Lierler, Y. 2016. SMT-based constraint answer set solver EZSMT (system description). In Proc. of 32th International Conference on Logic Programming (ICLP). Dagstuhl Publishing, OpenAccess Series in Informatics (OASIcs).Google Scholar
Wielemaker, J., Schrijvers, T., Triska, M. and Lager, T. 2012. SWI-Prolog. Theory and Practice of Logic Programming 12, 1–2, 67–96.Google Scholar
Zhou, N. 2012. The language features and architecture of B-Prolog. Theory and Practice of Logic Programming 12, 1–2, 189–218.Google Scholar