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Expressive power and abstraction in Essence. (English) Zbl 1147.68428

Summary: 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.

MSC:

68N15 Theory of programming languages
68T20 Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.)
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[1] Cadoli, M., Ianni, G., Palopoli, L., Schaerf, A., & Vasile, D. (2000). NP-SPEC: An executable specification language for solving all problems in NP. Computer Languages, 26, 165–195. · Zbl 0995.68025 · doi:10.1016/S0096-0551(01)00010-8
[2] Cadoli, M., & Mancini, T. (2006). Automated reformulation of specifications by safe delay of constraints. Artificial Intelligence, 170(8–9), 779–801. · Zbl 1131.68096 · doi:10.1016/j.artint.2006.01.008
[3] Denecker, M., & Ternovska, E. (2008). A logic of non-monotone inductive definitions. ACM Transactions on Computational Logic, 9(2). · Zbl 1122.68623
[4] East, D., & Truszczynski, M. (2006). Predicate-calculus based logics for modeling and solving search problems. ACM Transactions on Computational Logic, 7(1), 38–83. · doi:10.1145/1119439.1119441
[5] Ebbinghaus, H.-D., & Flum, J. (1995). Finite model theory. Heidelberg: Springer. · Zbl 0841.03014
[6] Enderton, H. B. (2001). A mathematical introduction to logic. New York: Harcourt/Academic. · Zbl 0992.03001
[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. · Zbl 0303.68035
[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.
[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).
[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).
[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).
[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). · Zbl 1147.68424
[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).
[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.
[15] Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP -completeness. San Francisco: W. H. Freeman. · Zbl 0411.68039
[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.
[17] Grädel, E., (1992). Capturing complexity classes by fragments of second order logic. Theoretical Computer Science, 101, 35–57. · Zbl 0780.68040 · doi:10.1016/0304-3975(92)90149-A
[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.
[19] Grädel, E., & Gurevich, Y. (1998). Metafinite model theory. Information and Computation, 140(1), 26–81. · Zbl 0892.68033 · doi:10.1006/inco.1997.2675
[20] Immerman, N. (1986). Relational queries computable in polytime. Information and Control, 68, 86–104. · Zbl 0612.68086 · doi:10.1016/S0019-9958(86)80029-8
[21] Immerman, N. (1999). Descriptive complexity. New York: Springer. · Zbl 0918.68031
[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). · Zbl 1367.68308
[23] Libkin, L. (2004). Elements of finite model theory. Heidelberg: Springer. · Zbl 1060.03002
[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.
[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).
[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). · Zbl 1146.68352
[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).
[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).
[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.
[30] Papadimitriou, C. (1994). Computational complexity. Reading: Addison-Wesley. · Zbl 0833.68049
[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.
[33] Vardi, M. Y. (1982). The complexity of relational query languages. In 14th ACM symp. on theory of computing. Heidelberg: Springer.
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