Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-19T20:29:09.569Z Has data issue: false hasContentIssue false
  • English
  • Français

Connecting many-sorted theories

Published online by Cambridge University Press:  12 March 2014

Franz Baader  and
Silvio Ghilardi
Franz Baader
Affiliation:
Institut für Theoretische Informatik, Tu Dresden, Germany, E-mail: baader@tcs.inf.tu-dresden.de
Silvio Ghilardi
Affiliation:
Dipartimento di Scienze Dell’Informazione, Università Degli Studi di Milano, Italy, E-mail: ghilardi@dsi.unimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the so-called -connection approach for combining modal logics to an algebraic setting.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 535 - 583
Copyright
Copyright © Association for Symbolic Logic 2007

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.)

References

REFERENCES

[1]Ajili, Farid and Kirchner, Claude, A modular framework for the combination of symbolic and built-in constraints, Proceedings of the Fourteenth International Conference on Logic Programming (Leuven, Belgium) (Naish, Lee, editor), The MIT press, 1997, pp. 331–345.Google Scholar
[2]Baader, Franz and Ghilardi, Silvio, Connecting many-sorted structures and theories through adjoint functions, Proceedings of the 5th International Workshop on Frontiers of Combining Systems (FroCoS 2005) (Vienna, Austria) (Gramlich, B., editor), Lecture Notes in Artificial Intelligence, vol. 3717, Springer-Verlag, 2005, pp. 31–47.Google Scholar
[3]Baader, Franz and Ghilardi, Silvio, Connecting many-sorted theories, Proceedings of the 20th International Conference on Automated Deduction (CADE-05) (Tallinn (Estonia)), Lecture Notes in Artificial Intelligence, vol. 3632, Springer-Verlag, 2005, pp. 278–294.Google Scholar
[4]Baader, Franz, Ghilardi, Silvio, and Tinelli, Cesare, A new combination procedure for the word problem that generalizes fusion decidability results in modal logics, Proceedings of the Second International Joint Conference on Automated Reasoning (IJCAR’04) (Cork, Ireland), Lecture Notes in Artificial Intelligence, vol. 3097, Springer-Verlag, 2004, pp. 183–197.Google Scholar
[5]Baader, Franz, Ghilardi, Silvio, and Tinelli, Cesare, A new combination procedure for the word problem that generalizes fusion decidability results in modal logics, Information and Computation, vol. 204 (2006), no. 10, pp. 1413–1452.CrossRefGoogle Scholar
[6]Baader, Franz, Lutz, Carsten, Sturm, Holger, and Wolter, Frank, Fusions of description logics and abstract description systems, Journal of Artificial Intelligence Research, vol. 16 (2002), pp. 1–58.CrossRefGoogle Scholar
[7]Baader, Franz and Tinelli, Cesare, A new approach for combining decision procedures for the word problem, and its connection to the Nelson-Oppen combination method, Proceedings of the 14th International Conference on Automated Deduction (Townsville, Australia) (McCune, William, editor), Lecture Notes in Artificial Intelligence, vol. 1249, Springer-Verlag, 1997, pp. 19–33.Google Scholar
[8]Baader, Franz and Tinelli, Cesare, Deciding the word problem in the union of equational theories, Information and Computation, vol. 178 (2002), no. 2, pp. 346–390.CrossRefGoogle Scholar
[9]Blackburn, Patrick, de Rijke, Maarten, and Venema, Yde, Modal logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, Cambridge University Press, 2001.CrossRefGoogle Scholar
[10]Bonacina, Maria Paola, Ghilardi, Silvio, Nicolini, Enrica, Ranise, Silvio, and Zucchelli, Daniele, Decidability and undecidability results for Nelson-Oppen and rewrite-based decision procedures, Proceedings of the Third International Joint Conference on Automated Reasoning (Furbach, U. and Shankar, N., editors), Lecture Notes in Artificial Intelligence, vol. 4130, Springer-Verlag, 2006, pp. 513–527.Google Scholar
[11]Chang, Chen-Chung and Keisler, H. Jerome, Model theory, III rd ed., North-Holland, 1990.Google Scholar
[12]Domenjoud, Eric, Klay, Francis, and Ringeissen, Christophe, Combination techniques for non-disjoint equational theories, Proceedings of the 12th International Conference on Automated Deduction (Nancy, France) (Bundy, Alan, editor), Lecture Notes in Artificial Intelligence, vol. 814, Springer-Verlag, 1994, pp. 267–281.Google Scholar
[13]Fiorentini, Camillo and Ghilardi, Silvio, Combining word problems through rewriting in categories with products, Theoretical Computer Science, vol. 294 (2003), pp. 103–149.CrossRefGoogle Scholar
[14]Gallier, Jean H., Logic for computer science: Foundations of automatic theorem proving, Harper & Row, 1986.Google Scholar
[15]Ghilardi, Silvio, Reasoners’ cooperation and quantifier elimination, Technical Report 288-03, Dipartimento di Scienze dell’Informazione, Università degli Studi di Milano, 2003, Available on-line at http://homes.dsi.unimi.it/∼ghilardi/allegati/eqfin.ps.Google Scholar
[16]Ghilardi, Silvio, Model-theoretic methods in combined constraint satisfiability, Journal of Automated Reasoning, vol. 33 (2005), no. 3–4, pp. 221–249.Google Scholar
[17]Ghilardi, Silvio and Zawadowski, Marek, Sheaves, games and model completions, Trends in Logic, vol. 14, Kluwer Academic Publishers, 2002.CrossRefGoogle Scholar
[18]Hodges, Wilfrid, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[19]Kracht, Marcus and Wolter, Frank, Properties of independently axiomatizable bimodal logics, this Journal, vol. 56 (1991), no. 4, pp. 1469–1485.Google Scholar
[20]Kutz, Oliver, Lutz, Carsten, Wolter, Frank, and Zakharyaschev, Michael, E-connections of abstract description systems, Artificial Intelligence, vol. 156 (2004), pp. 1–73.CrossRefGoogle Scholar
[21]Makkai, Michael and Reyes, Gonzalo E., First-order categorical logic, Lecture Notes in Mathematics, vol. 611, Springer-Verlag, 1977.CrossRefGoogle Scholar
[22]Nelson, Greg, Combining satisfiability procedures by equality-sharing, Automated theorem proving: After 25 years (Bledsoe, W. W. and Loveland, D. W., editors), Contemporary Mathematics, vol. 29, American Mathematical Society, Providence, RI, 1984, pp. 201–211.Google Scholar
[23]Nelson, Greg and Oppen, Derek C., Simplification by cooperating decision procedures, ACM Transactions on Programming Languages and Systems, vol. 1 (1979), no. 2, pp. 245–257.CrossRefGoogle Scholar
[24]Nipkow, Tobias, Combining matching algorithms: The regular case, Journal of Symbolic Computation, vol. 12 (1991), pp. 633–653.CrossRefGoogle Scholar
[25]Oppen, Derek C., Complexity, convexity and combinations of theories, Theoretical Computer Science, vol. 12 (1980), pp. 291–302.CrossRefGoogle Scholar
[26]Pigozzi, Don, The join of equational theories, Colloquium Mathematicum, vol. 30 (1974), no. 1, pp. 15–25.CrossRefGoogle Scholar
[27]Presburger, Mojzesz, Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Comptes rendus du congrès de mathématiciens des pays slaves (Warsaw, Poland), 1929.Google Scholar
[28]Schmidt-Schauß, Manfred, Unification in a combination of arbitrary disjoint equational theories, Journal of Symbolic Computation, vol. 8 (1989), no. 1–2, pp. 51–100, Special issue on unification. Part II.CrossRefGoogle Scholar
[29]Spaan, Edith, Complexity of modal logics, Ph.D. thesis, Department of Mathematics and Computer Science, University of Amsterdam, The Netherlands, 1993.Google Scholar
[30]Tinelli, Cesare, Cooperation of background reasoners in theory reasoning by residue sharing, Journal of Automated Reasoning, vol. 30 (2003), no. 1, pp. 1–31.CrossRefGoogle Scholar
[31]Tinelli, Cesare and Ringeissen, Christophe, Unions of non-disjoint theories and combinations of satisfiability procedures, Theoretical Computer Science, vol. 290 (2003), no. 1, pp. 291–353.CrossRefGoogle Scholar
[32]Wolter, Frank, Fusions of modal logics revisited, Advances in modal logic (Kracht, M., de Rijke, M., Wansing, H., and Zakharyaschev, M., editors), CSLI, Stanford, CA, 1998, pp. 361–379.Google Scholar
[33]Zarba, Calogero, Combining multisets with integers, Proceedings of the 18th International Conference on Automated Deduction (CADE’18) (Voronkov, Andrei, editor), Lecture Notes in Artificial Intelligence, vol. 2392, Springer-Verlag, 2002, pp. 363–376.Google Scholar
[34]Zawadowski, Marek W., Descent and duality, Annals of Pure and Applied Logic, vol. 71 (1995), no. 2, pp. 131–188.CrossRefGoogle Scholar