Abstract
The class of regular propositional schemata, discovered by Aravantinos et al. [4], is a major advancement towards more expressive classes of inductive theorems with a decidable satisfiability problem. Though more expressive than previously known decidable classes outlined by Kapur & Giesl[17], it still requires the burdensome restriction of induction with only one free parameter. In general, unrestricted usage of multiple free parameters in schematic formulae is undecidable for satisfiability [2]. In later work, Aravantinos et al. [6] introduced normalized clause sets which have a decision procedure for satisfiability and allow for restricted usage of multiple parameters. In our work, we investigate classes of propositional schemata which allow for multiple free parameters and are more expressive than regular schemata. Specifically, the classes we investigate have a decision procedure for satisfiability testing without requiring the additional theoretical machinery of normalized clause sets. Thus, allowing one to avoid conversion to CNF formulae. Both of the classes we introduce, linked schemata and pure overlap schemata use the machinery introduced in the earlier works of Aravantinos et al.[4] with only a slight change to the decision procedure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aczel, P.: An introduction to inductive definitions. In: Barwise, J. (ed.) Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics, vol. 90, pp. 739–782. Elsevier (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.: A decidable class of nested iterated schemata. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 293–308. Springer, Heidelberg (2010)
Aravantinos, V., Caferra, R., Peltier, N.: Decidability and undecidability results for propositional schemata. J. Artif. Int. Res. 40(1), 599–656 (2011)
Aravantinos, V., Caferra, R., Peltier, N.: Linear temporal logic and propositional schemata, back and forth. In: Proceedings of the 2011 Eighteenth International Symposium on Temporal Representation and Reasoning, TIME 2011, pp. 80–87. IEEE Computer Society, Washington, DC (2011)
Aravantinos, V., Echenim, M., Peltier, N.: A resolution calculus for first-order schemata. Fundamenta Informaticae (2013)
Baaz, M.: Note on the generalization of calculations. Theoretical Computer Science 224(1-2), 3–11 (1999)
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)
Baaz, M., Zach, R.: Short proofs of tautologies using the schema of equivalence. In: Börger, E., Gurevich, Y., Meinke, K. (eds.) CSL 1993. LNCS, vol. 832, pp. 33–35. Springer, Heidelberg (1994)
Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Springer (1997)
Bradfield, J.C.: The modal mu-calculus alternation hierarchy is strict. In: Montanari, U., Sassone, V. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 233–246. Springer, Heidelberg (1996)
Comon, H.: Inductionless induction. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning (in 2 volumes), pp. 913–962. Elsevier and MIT Press (2001)
Cooper, D.: Theorem proving in arithmetic without multiplication. Machine Intelligence (1972)
Corcoran, J.: Schemata: The concept of schema in the history of logic. Bulletin of Symbolic Logic (2), 219–240
Dunchev, C., Leitsch, A., Rukhaia, M., Weller, D.: Ceres for first-order schemata. CoRR, abs/1303.4257 (2013)
Gentzen, G.: Fusion of several complete inductions. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen. Studies in Logic and the Foundations of Mathematics, vol. 55, pp. 309–311. Elsevier (1969)
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)
Kapur, D., Subramaniam, M.: Extending decision procedures with induction schemes. In: McAllester, D. (ed.) CADE-17. LNCS, vol. 1831, pp. 324–345. Springer, Heidelberg (2000)
Krajíček, J., Pudlák, P.: The number of proof lines and the size of proofs in first order logic. Archive for Mathematical Logic 27(1), 69–84 (1988)
Orevkov, V.P.: Proof schemata in Hilbert-type axiomatic theories. Journal of Soviet Mathematics 55(2), 1610–1620 (1991)
Parikh, R.J.: Some results on the length of proofs. Transactions of the American Mathematical Society 177, 29–36 (1973)
Takeuti, G.: Proof Theory. Studies in logic and the foundations of mathematics, vol. 81. American Elsevier Pub. (1975)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Cerna, D. (2014). A Tableaux-Based Decision Procedure for Multi-parameter Propositional Schemata. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds) Intelligent Computer Mathematics. CICM 2014. Lecture Notes in Computer Science(), vol 8543. Springer, Cham. https://doi.org/10.1007/978-3-319-08434-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-08434-3_6
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08433-6
Online ISBN: 978-3-319-08434-3
eBook Packages: Computer ScienceComputer Science (R0)