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Noetherianity and Combination Problems

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Frontiers of Combining Systems (FroCoS 2007)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4720))

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Abstract

In abstract algebra, a structure is said to be Noetherian if it does not admit infinite strictly ascending chains of congruences. In this paper, we adapt this notion to first-order logic by defining the class of Noetherian theories. Examples of theories in this class are Linear Arithmetics without ordering and the empty theory containing only a unary function symbol. Interestingly, it is possible to design a non-disjoint combination method for extensions of Noetherian theories. We investigate sufficient conditions for adding a temporal dimension to such theories in such a way that the decidability of the satisfiability problem for the quantifier-free fragment of the resulting temporal logic is guaranteed. This problem is firstly investigated for the case of Linear time Temporal Logic and then generalized to arbitrary modal/temporal logics whose propositional relativized satisfiability problem is decidable.

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References

  1. Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: On a rewriting approach to satisfiability procedures: extension, combination of theories and an experimental appraisal. In: Gramlich, B. (ed.) FroCoS 2005. LNCS (LNAI), vol. 3717, pp. 65–80. Springer, Heidelberg (2005)

    Google Scholar 

  2. Baader, F., Lutz, C., Sturm, H., Wolter, F.: Fusions of description logics and abstract description systems. Journal of A.I. Research 16, 1–58 (2002)

    MATH  MathSciNet  Google Scholar 

  3. Baumgartner, P., Furbach, U., Petermann, U.: A unified approach to theory reasoning. Research Report 15–92, Universität Koblenz-Landau (1992)

    Google Scholar 

  4. Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  5. Bonacina, M.P., Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Decidability and undecidability results for Nelson-Oppen and rewrite-based decision procedures. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 513–527. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  6. Bryant, R.E., Lahiri, S.K., Seshia, S.A.: Modeling and verifying systems using a logic of counter arithmetic with lambda expressions and uninterpreted functions. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 78–92. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. Chang, C.-C., Keisler, J.H.: Model Theory, 3rd edn. North-Holland, Amsterdam-London (1990)

    Book  MATH  Google Scholar 

  8. Le Chenadec, P.: Canonical Forms in Finitely Presented Algebras. Research Notes in Theoretical Computer Science. Pitman-Wiley (1986)

    Google Scholar 

  9. Degtyarev, A., Fisher, M., Konev, B.: Monodic temporal resolution. ACM Transaction on Computational Logic 7(1), 108–150 (2006)

    Article  MathSciNet  Google Scholar 

  10. Ebbinghaus, H.-D., Flum, J., Thomas, W.: Mathematical logic. In: Undergraduate Texts in Mathematics, 2nd edn., Springer, Heidelberg (1994)

    Google Scholar 

  11. Finger, M., Gabbay, D.M.: Adding a temporal dimension to a logic system. Journal of Logic, Language, and Information 1(3), 203–233 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gabbay, D.M., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-Dimensional Modal Logics: Theory and Applications. Studies in Logic and the Foundations of Mathematics, vol. 148. North-Holland, Amsterdam (2003)

    MATH  Google Scholar 

  13. Ghilardi, S.: Model theoretic methods in combined constraint satisfiability. Journal of Automated Reasoning 33(3-4), 221–249 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Combination methods for satisfiability and model-checking of infinite-state systems. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603, pp. 362–378. Springer, Heidelberg (2007)

    Google Scholar 

  15. Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Combination methods for satisfiability and model-checking of infinite-state systems. Technical Report RI313-07, Università degli Studi di Milano (2007) Available at http://homes.dsi.unimi.it/~zucchell/publications/techreport/GhiNiRaZu-RI313-07.pdf

  16. Ghilardi, S., Nicolini, E., Zucchelli, D.: A comprehensive framework for combined decision procedures. ACM Transactions on Computational Logic (to appear)

    Google Scholar 

  17. Hodkinson, I.M., Wolter, F., Zakharyaschev, M.: Decidable fragment of first-order temporal logics. Annals of Pure and Applied Logic 106(1–3), 85–134 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kirchner, H., Ranise, S., Ringeissen, C., Tran, D.-K.: On superposition-based satisfiability procedures and their combination. In: Van Hung, D., Wirsing, M. (eds.) ICTAC 2005. LNCS, vol. 3722, pp. 594–608. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  19. Lamport, L.: The temporal logic of actions. ACM Transactions on Programming Languages and Systems 16(3), 872–923 (1994)

    Article  Google Scholar 

  20. Lassez, J.-L., Maher, M.J.: On Fourier’s algorithm for linear arithmetic constraints. Journal of Automated Reasoning 9(3), 373–379 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lassez, J.-L., McAloon, K.: A canonical form for generalized linear constraints. Journal of Symbolic Computation 13(1), 1–24 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  22. MacLane, S., Birkhoff, G.: Algebra, 3rd edn. Chelsea Publishing, New York (USA) (1988)

    MATH  Google Scholar 

  23. Manna, Z., Pnueli, A.: Temporal Verification of Reactive Systems: Safety. Springer, Heidelberg (1995)

    Google Scholar 

  24. Merz, S.: Isabelle/TLA (1999), Available at http://isabelle.in.tum.de/library/HOL/TLA

  25. Merz, S.: On the logic of TLA. Computing and Informatics 22, 351–379 (2003)

    MATH  MathSciNet  Google Scholar 

  26. Minsky, M.L.: Recursive unsolvability of Post’s problem of “tag” and other topics in the theory of Turing machines. Annals of Mathematics 74(3), 437–455 (1961)

    Article  MathSciNet  Google Scholar 

  27. Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Transaction on Programming Languages and Systems 1(2), 245–257 (1979)

    Article  MATH  Google Scholar 

  28. Nicolini, E.: Combined decision procedures for constraint satisfiability. PhD thesis, Dipartimento di Matematica, Università degli Studi di Milano (2007)

    Google Scholar 

  29. Plaisted, D.A.: A decision procedure for combination of propositional temporal logic and other specialized theories. Journal of Automated Reasoning 2(2), 171–190 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ranise, S., Tinelli, C.: Satisfiability modulo theories. IEEE Magazine on Intelligent Systems 21(6), 71–81 (2006)

    Google Scholar 

  31. Tinelli, C., Harandi, M.T.: A new correctness proof of the Nelson-Oppen combination procedure. In: Proc. of FroCoS 1996, Applied Logic, pp. 103–120. Kluwer Academic Publishers, Dordrecht (1996)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Milano, Italia

    Silvio Ghilardi & Daniele Zucchelli

  2. LORIA & INRIA-Lorraine, Nancy, France

    Enrica Nicolini, Silvio Ranise & Daniele Zucchelli

Authors
  1. Silvio Ghilardi
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  2. Enrica Nicolini
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  3. Silvio Ranise
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  4. Daniele Zucchelli
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Boris Konev Frank Wolter

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Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D. (2007). Noetherianity and Combination Problems. In: Konev, B., Wolter, F. (eds) Frontiers of Combining Systems. FroCoS 2007. Lecture Notes in Computer Science(), vol 4720. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74621-8_14

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  • DOI: https://doi.org/10.1007/978-3-540-74621-8_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74620-1

  • Online ISBN: 978-3-540-74621-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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