Development of languages for specifying or modelling problems is an important direction in constraint modelling. To provide greater abstraction and modelling convenience, these languages are becoming more syntactically rich, leading to a variety of questions about their expressive power. In this paper, we consider the expressiveness of Essence, a specification language with a rich variety of syntactic features. We identify natural fragments of Essence that capture the complexity classes P, NP, all levels \(\Sigma_i^p\) of the polynomial-time hierarchy, and all levels k-NEXP of the nondeterministic exponential-time hierarchy. The union of these classes is the very large complexity class ELEMENTARY. One goal is to begin to understand which features play a role in the high expressive power of the language and which are purely features of convenience. We also discuss the formalization of arithmetic in Essence and related languages, a notion of capturing NP-search which is slightly different than that of capturing NP, and a conjectured limit to the expressive power of Essence. Our study is an application of descriptive complexity theory, and illustrates the value of taking a logic-based view of modelling and specification languages.
KeywordsExpressive power Abstraction Essence Constraint modelling languages Descriptive complexity Model expansion
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- 3.Denecker, M., & Ternovska, E. (2008). A logic of non-monotone inductive definitions. ACM Transactions on Computational Logic, 9(2).Google Scholar
- 7.Fagin, R. (1974). Generalized first-order spectra and polynomial-time recognizable sets. In R. Karp (Ed.), Complexity and computation, SIAM-AMS Proc. (Vol. 7, pp. 43–73). Providence: AMS.Google Scholar
- 8.Flener, P., Pearson, J., & Agren, M. (2004). Introducing ESRA, a relational language for modelling combinatorial problems. In M. Bruynooghe (Ed.), Logic based program synthesis and transformation: 13th international symposium (LOPSTR ’03), revised selected papers, lecture notes in computer science (Vol. 3018, pp. 214–232). Heidelberg: Springer.Google Scholar
- 9.Friedman, H. M. (1999). Some decision problems of enormous complexity. In Proceedings, 14th symposium on logic in computer science (LICS ’99) (pp. 2–12).Google Scholar
- 10.Frisch, A. M., Grum, M., Jefferson, C., Martinez Hernandez, B., & Miguel, I. (2005). The essence of Essence: A constraint language for specifying combinatorial problems. In Proc., fourth international workshop on modelling and reformulating constraint satisfaction problems (pp. 73–88) (October).Google Scholar
- 11.Frisch, A. M., Grum, M., Jefferson, C., Martinez Hernandez, B., & Miguel, I. (2007). The design of Essence: A constraint language for specifying combinatorial problems. In Proc., twentieth international joint conference on artificial intelligence (IJCAI ’07) (pp. 80–87).Google Scholar
- 12.Frisch, A. M., Harvey, W., Jefferson, C., Martínez Hernández, B., & Miguel, I. (2008). Essence: A constraint language for specifying combinatorial problems. Constraints, 13(3).Google Scholar
- 13.Frisch, A. M., Jefferson, C., Martínez Hernández, B., & Miguel, I. (2005). The rules of constraint modelling. In Proc., 19th international joint conference on artificial intelligence (pp. 109–116).Google Scholar
- 14.Frisch, A. M., Jefferson, C., Martínez Hernández, B., & Miguel, I. (2007). Symmetry in the generation of constraint models. In Proceedings of the international symmetry conference.Google Scholar
- 16.Gebser, M., Schaub, T., & Thiele, S. (2007). Gringo: A new grounder for answer set programming. In C. Baral, G. Brewka, & J. S. Schlipf (Eds.), Proc., 9th international conference on logic programming and nonmonotonic reasoning (LPNMR ’07), lecture notes in computer science (Vol. 4483, pp. 266–271). Heidelberg: Springer.CrossRefGoogle Scholar
- 18.Grädel, E., (2007). Finite model theory and descriptive complexity, chapter 3. E. Grädel, P. G. Kolaitis, L. Libkin, M. Marx, J. Spencer, M. Y. Vardi, et al. (Eds.), Finite model theory and its applications (pp. 125–230). Heidelberg: Springer.Google Scholar
- 22.Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S., et al. (2006). The DLV system for knowledge representation and reasoning. ACM Transactions on Computational Logic, 7(3).Google Scholar
- 24.Mancini, T., & Cadoli, M. (2005). Detecting and breaking symmetries by reasoning on problem specifications. In Proc., 6th international symposium on abstraction, reformulation and approximation (SARA 2005), lecture notes in computer science (Vol. 3607, pp. 165–181). Heidelberg: Springer.Google Scholar
- 25.Mariën, M., Wittocx, J., & Denecker, M. (2006). The IDP framework for declarative problem solving. In E. Giunchiglia, V. Marek, D. Mitchell, & E. Ternovska (Eds.), Proc., search and logic: Answer set programming and SAT (LaSh ’06) (pp. 19–34).Google Scholar
- 26.Marriott, K., Nethercote, N., Rafeh, R., Stuckey, P. J., de la Banda, M. G., & Wallace, M. (2008). The design of the zinc modelling language. Constraints, 13(3).Google Scholar
- 27.Mills, P., Tsang, E. P. K., Williams, R., Ford, J., & Borrett, J. (1998). EaCL 1.0: An easy abstract constraint programming language. Technical report CSM-321, University of Essex (December).Google Scholar
- 28.Mitchell, D. G., & Ternovska, E. (2005). A framework for representing and solving NP search problems. In Proc. of the 20th national conf. on artificial intelligence (AAAI ’05) (pp. 430–435).Google Scholar
- 29.Mitchell, D., Ternovska, E., Hach, F., & Mohebali, R. (2006). Model expansion as a framework for modelling and solving search problems. Technical report TR 2006-24, School of Computing Science, Simon Fraser University.Google Scholar
- 31.Syrjänen, T. (2000). Lparse 1.0 user’s manual. Available from: http://www.tcs.hut.fi/Software/smodels/.
- 32.Van Hentenryck, P. (1999). The OPL optimization programming language. Cambridge: MIT.Google Scholar
- 33.Vardi, M. Y. (1982). The complexity of relational query languages. In 14th ACM symp. on theory of computing. Heidelberg: Springer.Google Scholar