Abstract
When combining languages for symbolic constraints, one is typically faced with the problem of how to treat “mixed” constraints. The two main problems are (1) how to define a combined solution structure over which these constraints are to be solved, and (2) how to combine the constraint solving methods for pure constraints into one for mixed constraints. The paper introduces the notion of a “free amalgamated product” as a possible solution to the first problem. Subsequently, we define so-called simply-combinable structures (SC-structures). For SC-structures over disjoint signatures, a canonical amalgamation construction exists, which for the subclass of strong SC-structures yields the free amalgamated product. The combination technique of [BaS92, BaS94a] can be used to combine constraint solvers for (strong) SC-structures over disjoint signatures into a solver for their (free) amalgamated product. In addition to term algebras modulo equational theories, the class of SC-structures contains many solution structures that have been used in constraint logic programming, such as the algebra of rational trees, feature structures, and domains consisting of hereditarily finite (wellfounded or non-wellfounded) nested sets and lists.
This work was supported by a DFG grant (SSP Deduktion) and by the EC Working Group CCL, EP6028.
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References
P. Aczel, “Non-well-founded Sets,” CSLI Lecture Notes 14, Stanford University, 1988.
H. Ait-Kaci, A. Podelski, and G. Smolka, “A feature-based constraint system for logic programming with entailment,” Theoretical Comp. Science 122, 1994, pp. 263–283.
F. Baader and K.U. Schulz, “Unification in the union of disjoint equational theories: Combining decision procedures,” in: Proc. CADE-11, LNAI 607, 1992, pp. 50–65.
F. Baader and K.U. Schulz, “Combination techniques and decision problems for disunification,” in: Proc. RTA-9S, LNCS 690, 1993.
F. Baader and K.U. Schulz, “Combination of Constraint Solving Techniques: An Algebraic Point of View,” Research Report CIS-Rep-94-75, University Munich, 1994; short version in: Proc. RTA'95, Springer LNCS 914, 1995.
F. Baader and K.U. Schulz, “On the Combination of Symbolic Constraints, Solution Domains, and Constraint Solvers,” Research Report CIS-Rep-94-82, University Munich, 1994. Long version of this paper, available via anonymous ftp from ftp.cis.uni-muenchen.de, directory “schulz”, file name “SCstructures.ps.z”.
A. Boudet, “Unification in a combination of equational theories: An efficient algorithm,” in: Proc. CADE-10, LNCS 449, 1990, pp. 292–307.
G. Cherlin, “Model Theoretic Algebra: Selected Topics,” Springer Lecture Notes in Mathematics 521, 1976.
A. Colmerauer, “An introduction to PROLOG III,” C. ACM 33, 1990, pp. 69–90.
A. Dovier, E.G. Omodeo, E. Pontellio, G.F. Rossi, “g: A Logic Programming language with finite sets,” in: Logic Programming: Proc. 8th International Conf., The MIT Press, 1991.
A. Dovier, G. Rossi, “Embedding extensional finite sets in CLP,” in: Proc. International Logic Programming Symposium, 1993, pp. 540–556.
M. Droste, R. Göbel, “Universal domains and the amalgamation property,” Math. Struct. in Comp. Science 3, pp. 137–159, 1993.
H. Kirchner and Ch. Ringeissen, “Combining symbolic constraint solvers on algebraic domains,” J. Symbolic Computation, 18(2), 1994, pp. 113–155.
M.J. Maher, “Complete axiomatizations of the algebras of finite, rational and infinite trees,” in: Proceedings of Third Annual Symposium on Logic in Computer Science, LICS'88, pp. 348–357, Edinburgh, Scotland, 1988. IEEE Computer Society.
A.I. Mal'cev, “Algebraic Systems,” Volume 192 of Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer-Verlag, Berlin, 1973.
K. Mukai, “Constraint Logic Programming and the Unification of Information,” doctoral thesis, Dept. of Comp. Science, Faculty of Engineering, Tokyo Institute of Technology, 1991.
G. Nelson, D.C. Oppen, “Simplification by Cooperating Decision Procedures,” ACM TOPLAS, Vol. 1, No. 2, October 1979, 245–257.
Ch. Ringeissen, “Unification in a combination of equational theories with shared constants and its application to primal algebras,” in: Proc. LPAR'92, LNCS 624, 1992.
W.C. Rounds, “Set Values for Unification Based Grammar Formalisms and Logic Programming,” Research Report CSLI-88-129, Stanford, 1988.
M. Schmidt-Schauß, “Unification Algebras: An Axiomatic Approach to Unification, Equation Solving and Constraint Solving,” SEKI-Report, SR-88-23, University of Kaiserslautern, 1988.
M. Schmidt-Schauß, “Unification in a combination of arbitrary disjoint equational theories,” J. Symbolic Computation 8, 1989, pp. 51–99.
G. Smolka, R. Treinen, “Records for Logic Programming,” J. of Logic Programming 18(3) (1994), pp. 229–258 556.
Y.M. Vazhenin and B.V. Rozenblat, “Decidability of the positive theory of a free countably generated semigroup,” Math. USSR Sbornik 44 (1983), pp. 109–116.
J.G. Williams, “Instantiation Theory: On the Foundation of Automated Deduction,” Springer LNCS 518, 1991.
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Baader, F., Schulz, K.U. (1995). On the combination of symbolic constraints, solution domains, and constraint solvers. In: Montanari, U., Rossi, F. (eds) Principles and Practice of Constraint Programming — CP '95. CP 1995. Lecture Notes in Computer Science, vol 976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60299-2_23
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DOI: https://doi.org/10.1007/3-540-60299-2_23
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