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Using Proofs by Coinduction to Find “Traditional” Proofs

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Algebra and Coalgebra in Computer Science (CALCO 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3629))

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Abstract

In the specific situation of formal reasoning concerned with “regular expression equivalence” we address instances of more general questions such as: how can coinductive argumentation be formalised logically and be applied effectively, as well as how is it linked to traditional forms of proof. For statements expressing that two regular expressions are language equivalent, we demonstrate that proofs by coinduction can be formulated in a proof system based on equational logic, where effective proof-search is possible. And we describe a proof-theoretic method for translating derivations in this proof system into a “traditional” axiom system: namely, into a “reverse form” of the axiomatisation of “regular expression equivalence” due to Salomaa. Hereby we obtain a coinductive completeness proof for the traditional proof system.

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References

  1. Brandt, M., Henglein, F.: Coinductive axiomatization of recursive type equality and subtyping. Fundamenta Informaticae 33, 1–30 (1998)

    MathSciNet  Google Scholar 

  2. Brzozowski, J.A.: Derivatives of regular expressions. Journal of the ACM 11, 481–494 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  3. Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, Boca Raton (1971)

    MATH  Google Scholar 

  4. Hüttel, H., Stirling, C.: Actions Speak Louder Than Words: Proving Bisimilarity for Context-Free Processes. Journ. of Logic and Computation 8(4), 485–509 (1998)

    Article  MATH  Google Scholar 

  5. Grabmayer, C.: Relating Proof Systems for Recursive Types, PhD thesis, Vrije Universiteit Amsterdam (2005), http://www.cs.vu.nl/~clemens/proefschrift.pdf

  6. Rutten, J.J.M.M.: Automata and Coinduction (an Exercise in Coinduction). In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 194–218. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  7. Salomaa, A.: Two complete axiom systems for the algebra of regular events. Journal of the ACM 13(1), 158–169 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  8. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

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© 2005 Springer-Verlag Berlin Heidelberg

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Grabmayer, C. (2005). Using Proofs by Coinduction to Find “Traditional” Proofs. In: Fiadeiro, J.L., Harman, N., Roggenbach, M., Rutten, J. (eds) Algebra and Coalgebra in Computer Science. CALCO 2005. Lecture Notes in Computer Science, vol 3629. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11548133_12

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  • DOI: https://doi.org/10.1007/11548133_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28620-2

  • Online ISBN: 978-3-540-31876-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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