Abstract
In the proof assistant Coq, one can model certain classes of graphs by coinductive types. The coinductive aspects account for infinite navigability already in finite but cyclic graphs, as in rational trees. Coq’s static checks exclude simple-minded definitions with lists of successors of a node. In previous work, we have shown how to mimic lists by a type of functions and built a Coq theory for such graphs. Naturally, these coinductive structures have to be compared by a bisimulation relation, and we defined it in a generic way.
However, there are many cases in which we would not like to distinguish between graphs that are constructed differently and that are thus not bisimilar, in particular if only the order of elements in the lists of successors is not the same. We offer a wider bisimulation relation that allows permutations. Technical problems arise with their specification since (1) elements have to be compared by a not necessarily decidable relation and (2) coinductive types are mixed with inductive ones. Still, a formal development has been carried out in Coq, by using its built-in language for proof automation.
Another extension of the original bisimulation relation based on cycle analysis provides indifference concerning the root node of the term graphs.
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References
Altenkirch, T.: A Formalization of the Strong Normalization Proof for System F in LEGO. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 13–28. Springer, Heidelberg (1993)
Berardi, S., Damiani, F., de’Liguoro, U. (eds.): TYPES 2008. LNCS, vol. 5497. Springer, Heidelberg (2009)
Berger, U.: From coinductive proofs to exact real arithmetic: theory and applications. Logical Methods in Computer Science 7(1) (2011)
Bertot, Y., Komendantskaya, E.: Using structural recursion for corecursion. In: Berardi et al. [2], pp. 220–236
Chlipala, A.: In Coq club thread “is Coq being too conservative?” (January 2010), https://sympa-roc.inria.fr/wws/arc/coq-club/2010-01/msg00089.html
Contejean, E.: Modeling Permutations in Coq for Coccinelle. In: Comon-Lundh, H., Kirchner, C., Kirchner, H. (eds.) Jouannaud Festschrift. LNCS, vol. 4600, pp. 259–269. Springer, Heidelberg (2007)
Coquand, T.: Infinite Objects in Type Theory. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 62–78. Springer, Heidelberg (1994)
Dams, C.: In Coq club thread “is Coq being too conservative?” (January 2010), https://sympa-roc.inria.fr/wws/arc/coq-club/2010-01/msg00085.html
Danielsson, N.A.: Beating the productivity checker using embedded languages. In: Bove, A., Komendantskaya, E., Niqui, M. (eds.) Proceedings Workshop on Partiality and Recursion in Interactive Theorem Provers. EPTCS, vol. 43, pp. 29–48 (2010)
Danielsson, N.A., Altenkirch, T.: Subtyping, Declaratively. In: Bolduc, C., Desharnais, J., Ktari, B. (eds.) MPC 2010. LNCS, vol. 6120, pp. 100–118. Springer, Heidelberg (2010)
Giménez, E., Castéran, P.: A tutorial on [co-]inductive types in Coq (2007), http://www.labri.fr/perso/casteran/RecTutorial.pdf
McBride, C., McKinna, J.: The view from the left. Journal of Functional Programming 14(1), 69–111 (2004)
Nakata, K., Uustalu, T.: Resumptions, weak bisimilarity and big-step semantics for while with interactive I/O: An exercise in mixed induction-coinduction. In: Aceto, L., Sobocinski, P. (eds.) SOS. EPTCS, vol. 32, pp. 57–75 (2010)
Nakata, K., Uustalu, T., Bezem, M.: A Proof Pearl with the Fan Theorem and Bar Induction - Walking through Infinite Trees with Mixed Induction and Coinduction. In: Yang, H. (ed.) APLAS 2011. LNCS, vol. 7078, pp. 353–368. Springer, Heidelberg (2011)
Niqui, M.: Coalgebraic reasoning in Coq: Bisimulation and the lambda-coiteration scheme. In: Berardi et al. [2], pp. 272–288
Picard, C., Matthes, R.: Coinductive graph representation: the problem of embedded lists. Electronic Communications of the EASST 39, 24 pages (2011)
Picard, C., Matthes, R.: Formalization in Coq for this article (2012), http://www.irit.fr/~Celia.Picard/Coq/Permutations/
Prince, R., Ghani, N., McBride, C.: Proving Properties about Lists Using Containers. In: Garrigue, J., Hermenegildo, M.V. (eds.) FLOPS 2008. LNCS, vol. 4989, pp. 97–112. Springer, Heidelberg (2008)
The Coq Development Team: The Coq Proof Assistant Reference Manual, http://coq.inria.fr
The Coq Development Team: The Coq Proof Assistant Standard Library, http://coq.inria.fr/stdlib/Coq.Sorting.Permutation.html
The Coq Development Team: The Coq Proof Assistant Standard Library, http://coq.inria.fr/stdlib/Coq.Sorting.PermutSetoid.html#permutation
Uustalu, T., Vene, V.: Least and greatest fixed points in intuitionistic natural deduction. Theoretical Computer Science 272, 315–339 (2002)
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Picard, C., Matthes, R. (2012). Permutations in Coinductive Graph Representation. In: Pattinson, D., Schröder, L. (eds) Coalgebraic Methods in Computer Science. CMCS 2012. Lecture Notes in Computer Science, vol 7399. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32784-1_12
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