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.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Brandt, M., Henglein, F.: Coinductive axiomatization of recursive type equality and subtyping. Fundamenta Informaticae 33, 1–30 (1998)
Brzozowski, J.A.: Derivatives of regular expressions. Journal of the ACM 11, 481–494 (1964)
Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, Boca Raton (1971)
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)
Grabmayer, C.: Relating Proof Systems for Recursive Types, PhD thesis, Vrije Universiteit Amsterdam (2005), http://www.cs.vu.nl/~clemens/proefschrift.pdf
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)
Salomaa, A.: Two complete axiom systems for the algebra of regular events. Journal of the ACM 13(1), 158–169 (1966)
Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge University Press, Cambridge (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
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)