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
Aczel, P.: An Introduction to Inductive Definitions. In: Barwise, K.J. (ed.) Handbook of Mathematical Logic, pp. 739–782. North-Holland, Amsterdam (1977)
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)
Aravantinos, V., Caferra, R., Peltier, N.: Decidability and undecidability results for propositional schemata. Journal of Artificial Intelligence Research 40, 599–656 (2011)
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)
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)
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)
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)
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)
Bouhoula, A., Rusinowitch, M.: Implicit induction in conditional theories. Journal of Automated Reasoning 14, 14–189 (1995)
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)
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)
Comon, H.: Inductionless induction. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch. 14, pp. 913–962. North-Holland (2001)
Echenim, M., Peltier, N.: Reasoning on Schemata of Formulae. Technical report, CoRR, abs/1204.2990 (2012)
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)
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)
Kapur, D., Musser, D.: Proof by consistency. Artificial Intelligence 31 (1987)
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)
Park, D.M.: Finiteness is Mu-ineffable. Theoretical Computer Science 3, 173–181 (1976)
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)
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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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