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International Conference on Automated Reasoning with Analytic Tableaux and Related Methods

TABLEAUX 2017: Automated Reasoning with Analytic Tableaux and Related Methods pp 278–294Cite as

Integrating a Global Induction Mechanism into a Sequent Calculus

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10501))

Abstract

Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi contain a separate rule or axiom which reduces the important proof theoretic properties of the calculus. In such cases cut-elimination does not result in analytic proofs, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a generalization of LK-proofs able to simulate induction by linking proofs, indexed by a natural number, together. Using a global cut-elimination method a normal form can be reached which allows a schema of Herbrand Sequents to be produced, an essential step for proof analysis in the presence of induction. However, proof schema have only been studied in a limited context and are currently defined for a very particular proof structure based on a slight extension of the LK-calculus. The result is an opaque and complex formalization. In this paper, we introduce a calculus integrating the proof schema formalization and in the process we elucidate properties of proof schemata which can be used to extend the formalism.

D.M. Cerna—Partially supported by FWF under the project P 28789-N32.

M. Lettmann—Funded by FWF project W1255-N23.

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Notes

  1. 1.

    In this fragment the use of schematic constructors is restricted to one free parameter per formula.

  2. 2.

    A formal analysis has not be performed but through conversations with the authors and their collaborators a polynomial bound on the size of the produced refutations is expected.

  3. 3.

    A proof fulfilling the subformula property can be referred to as analytic. By subformula-like, we mean that the proof is non-analytic, but still allows the extraction of objects important for proof analysis which rely on analyticity.

  4. 4.

    See Orevkov’s proof [23] or Boolos’ proof [10].

  5. 5.

    Inductive eigenvariables are eigenvariables occurring in the context of an induction inference rule.

  6. 6.

    In general, the context is not empty. Since the rules, exempting the \(\curvearrowright \), are independent from the context, we can always adjust the context.

References

  1. Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. Log. Comput. 2(3), 297–347 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aravantinos, V., Caferra, R., Peltier, N.: A schemata calculus for propositional logic. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS (LNAI), vol. 5607, pp. 32–46. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02716-1_4

    Chapter  Google Scholar 

  3. Aravantinos, V., Caferra, R., Peltier, N.: A decidable class of nested iterated schemata. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS (LNAI), vol. 6173, pp. 293–308. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14203-1_25

    Chapter  Google Scholar 

  4. Aravantinos, V., Caferra, R., Peltier, N.: Decidability and undecidability results for propositional schemata. J. Artif. Intell. Res. 40(1), 599–656 (2011)

    MATH  MathSciNet  Google Scholar 

  5. Aravantinos, V., Caferra, R., Peltier, N.: Linear temporal logic and propositional schemata, back and forth. In: Proceedings of TIME 2011, pp. 80–87. IEEE (2011)

    Google Scholar 

  6. Aravantinos, V., Echenim, M., Peltier, N.: A resolution calculus for first-order schemata. Fundamenta Informaticae 125, 101–133 (2013)

    MATH  MathSciNet  Google Scholar 

  7. Aravantinos, V., Peltier, N.: Schemata of SMT-problems. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS (LNAI), vol. 6793, pp. 27–42. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22119-4_5

    Chapter  Google Scholar 

  8. Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: Ceres: an analysis of Fürstenberg’s proof of the infinity of primes. Theoret. Comput. Sci. 403(2–3), 160–175 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Baaz, M., Leitsch, A.: Cut-elimination and redundancy-elimination by resolution. J. Symb. Comput. 29, 149–176 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Boolos, G.: Don’t eliminate cut. J. Philos. Log. 13(4), 373–378 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brotherston, J.: Cyclic proofs for first-order logic with inductive definitions. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 78–92. Springer, Heidelberg (2005). doi:10.1007/11554554_8

    Chapter  Google Scholar 

  12. Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. J. Log. Comput. 21, 1177–1216 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cerna, D.: 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.) CICM 2014. LNCS (LNAI), vol. 8543, pp. 61–75. Springer, Cham (2014). doi:10.1007/978-3-319-08434-3_6

    Chapter  Google Scholar 

  14. Cerna, D.M.: Advances in schematic cut elimination. Ph.D. thesis, Technical University of Vienna (2015). http://media.obvsg.at/p-AC12246421-2001

  15. Cerna, D.M., Leitsch, A.: Schematic cut elimination and the ordered pigeonhole principle. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS, vol. 9706, pp. 241–256. Springer, Cham (2016). doi:10.1007/978-3-319-40229-1_17

    Google Scholar 

  16. Dunchev, C.: Automation of cut-elimination in proof schemata. Ph.D. thesis, Technical University of Vienna (2012)

    Google Scholar 

  17. Dunchev, C., Leitsch, A., Rukhaia, M., Weller, D.: Cut-elimination and proof schemata. J. Log. Lang. Comput. 8984, 117–136 (2013)

    MATH  MathSciNet  Google Scholar 

  18. 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, Amsterdam (1969)

    Chapter  Google Scholar 

  19. Hetzl, S., Leitsch, A., Weller, D., Woltzenlogel Paleo, B.: Herbrand sequent extraction. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) CICM 2008. LNCS, vol. 5144, pp. 462–477. Springer, Heidelberg (2008). doi:10.1007/978-3-540-85110-3_38

    Chapter  Google Scholar 

  20. Leitsch, A., Peltier, N., Weller, D.: Ceres for first-order schemata. J. Log. Comput. (2017, to appear)

    Google Scholar 

  21. Mcdowell, R., Miller, D.: Cut-elimination for a logic with definitions and induction. Theoret. Comput. Sci. 232, 91–119 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Miller, D.A.: A compact representation of proofs. Stud. Log. 46(4), 347–370 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  23. Orevkov, V.P.: Proof schemata in Hilbert-type axiomatic theories. J. Sov. Math. 55(2), 1610–1620 (1991)

    Article  MATH  Google Scholar 

  24. Robinson, R.M.: An essentially undecidable axiom system. In: Proceedings of the International Congress of Mathematics, pp. 729–730 (1950)

    Google Scholar 

  25. Takeuti, G.: Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 81. American Elsevier Publisher, Amsterdam (1975)

    Google Scholar 

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Authors and Affiliations

  1. Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria

    David M. Cerna

  2. Institute of Information Systems, Technische Universität Wien, Vienna, Austria

    Michael Lettmann

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  1. David M. Cerna
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  2. Michael Lettmann
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Correspondence to Michael Lettmann .

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Editors and Affiliations

  1. University of Manchester, Manchester, United Kingdom

    Renate A. Schmidt

  2. University of Brasília, Brasília D.F., Brazil

    Cláudia Nalon

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Cerna, D.M., Lettmann, M. (2017). Integrating a Global Induction Mechanism into a Sequent Calculus. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_17

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  • DOI: https://doi.org/10.1007/978-3-319-66902-1_17

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