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Reasoning on Schemata of Formulæ

  • Conference paper
Intelligent Computer Mathematics (CICM 2012)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7362))

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Abstract

A logic is presented for reasoning on iterated sequences of formulæ over some given base language. The considered sequences, or schemata, are defined inductively, on some algebraic structure (for instance the natural numbers, the lists, the trees etc.). A proof procedure is proposed to relate the satisfiability problem for schemata to that of finite disjunctions of base formulæ. It is shown that this procedure is sound, complete and terminating, hence the basic computational properties of the base language can be carried over to schemata.

This work has been partly funded by the project ASAP of the French Agence Nationale de la Recherche (ANR-09-BLAN-0407-01).

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References

  1. Aczel, P.: An Introduction to Inductive Definitions. In: Barwise, K.J. (ed.) Handbook of Mathematical Logic, pp. 739–782. North-Holland, Amsterdam (1977)

    Chapter  Google Scholar 

  2. Aravantinos, V., Caferra, R., Peltier, N.: A Schemata Calculus for Propositional Logic. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS, vol. 5607, pp. 32–46. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  3. Aravantinos, V., Caferra, R., Peltier, N.: Decidability and undecidability results for propositional schemata. Journal of Artificial Intelligence Research 40, 599–656 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Aravantinos, V., Peltier, N.: Schemata of SMT-Problems. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS, vol. 6793, pp. 27–42. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  5. Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: CERES: An analysis of Fürstenberg’s proof of the infinity of primes. Theor. Comput. Sci. 403(2-3), 160–175 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baelde, D., Miller, D., Snow, Z.: Focused Inductive Theorem Proving. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 278–292. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  7. Benzmüller, C., Paulson, L.C., Theiss, F., Fietzke, A.: LEO-II - A Cooperative Automatic Theorem Prover for Classical Higher-Order Logic (System Description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 162–170. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  8. Bouhoula, A., Kounalis, E., Rusinowitch, M.: SPIKE, an Automatic Theorem Prover. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 460–462. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  9. Bouhoula, A., Rusinowitch, M.: Implicit induction in conditional theories. Journal of Automated Reasoning 14, 14–189 (1995)

    Article  MathSciNet  Google Scholar 

  10. Boyer, R.S., Moore, J.S.: A Theorem Prover for a Computational Logic. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 1–15. Springer, Heidelberg (1990)

    Chapter  Google Scholar 

  11. Bundy, A.: The automation of proof by mathematical induction. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 845–911. Elsevier and MIT Press (2001)

    Google Scholar 

  12. Comon, H.: Inductionless induction. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch. 14, pp. 913–962. North-Holland (2001)

    Google Scholar 

  13. Echenim, M., Peltier, N.: Reasoning on Schemata of Formulae. Technical report, CoRR, abs/1204.2990 (2012)

    Google Scholar 

  14. Giesl, J., Kapur, D.: Decidable Classes of Inductive Theorems. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 469–484. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  15. Gupta, A., Fisher, A.L.: Parametric Circuit Representation Using Inductive Boolean Functions. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 15–28. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  16. Kapur, D., Musser, D.: Proof by consistency. Artificial Intelligence 31 (1987)

    Google Scholar 

  17. Lenzi, G.: A New Logical Characterization of Büchi Automata. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 467–477. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  18. Park, D.M.: Finiteness is Mu-ineffable. Theoretical Computer Science 3, 173–181 (1976)

    Article  MathSciNet  Google Scholar 

  19. Paulin-Mohring, C.: Inductive Definitions in the system Coq - Rules and Properties. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 328–345. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

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Echenim, M., Peltier, N. (2012). Reasoning on Schemata of Formulæ. In: Jeuring, J., et al. Intelligent Computer Mathematics. CICM 2012. Lecture Notes in Computer Science(), vol 7362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31374-5_21

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  • DOI: https://doi.org/10.1007/978-3-642-31374-5_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31373-8

  • Online ISBN: 978-3-642-31374-5

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