Skip to main content Accessibility help
×
Home

Bisimulation proof methods in a path-based specification language for polynomial coalgebras

  • XIAO-CONG ZHOU (a1), YONG-JI LI (a1), WEN-JUN LI (a2), HAI-YAN QIAO (a1) and ZHONG-MEI SHU (a1)...

Abstract

What reasoning rules can be used for the deduction of bisimulation formulas in coalgebraic specifications is problematic because those rules used in algebraic specifications possibly cannot be applied to bisimulation formulas. Although some categorical bisimulation proof methods for coalgebras have been proposed, they are not based on specification languages of coalgebras so that they cannot be used as reasoning rules. In this paper, a specification language based on paths of polynomial functors is proposed to specify polynomial coalgebras. Paths of polynomial functors give detailed observations and transitions on the state space of coalgebras so that the techniques used in transition system specifications can be applied to such a path-based language. In particular, because bisimulations can be characterized by paths, the notions of progressions, respectful functions and faithful contexts can be defined based on paths, and then bisimulation up-to proof techniques, including bisimulation up-to bisimilarities and up-to contexts for transition systems can be transformed into reasoning rules in the language. Several examples illustrate how to reason syntactically about bisimulations in the language by using the rules induced by the bisimulation proof techniques.

Copyright

Footnotes

Hide All

Supported by the National Natural Science Foundation of China under Grant No. 60673050 and the Fundamental Research Funds for the Central Universities of China under Grant No. 11LGPY39.

Footnotes

References

Hide All
Bartels, F. (2003) Generalised coinduction. Mathematical Structures in Computer Science 13 321348.
Bonsangue, M., Rutten, J. and Silva, A. (2007) Regular expressions for polynomial coalgebras. Technical Report SEN-E0703, Centrum voor Wiskunde en Informatica (CWI).
Bonsangue, M., Rutten, J. and Silva, A. (2009) An algebra for Kripke polynomial coalgebras. In: Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society, Los Angeles, CA, USA4958.
Capretta, V. (2010) Bisimulations generated from corecursive equations. In: Mislove, M. and Selinger, P. (eds.) Proceedings of the 26th Conference on the Mathematical Foundations of Programming Semantics, Elsevier Electronic Notes in Theoretical Computer Science 265 245258.
Capretta, V. (2011) Coalgebras in functional programming and type theory. Theoretical Computer Science 412 50065024.
Cîrstea, C. (2000) Integrating Observations and Computations in the Specification of State-based, Dynamical Systems, Ph.D. thesis, University of Oxford.
Goldblatt, R. (2001) A calculus of terms for coalgebras of polynomial functors. In: Corradini, A., Lenisa, M. and Montanari, U. (eds.) Proceedings of Coalgebraic Methods in Computer Science. Elsevier Electronic Notes in Theoretical Computer Science 44 (1)161184.
Goldblatt, R. (2003a) Equational logic of polynomial coalgebras. In: Balbiani, P., Suzuki, N.-Y., Wolter, F. and Zakharyaschev, M. (eds.) Advances in Modal Logic, volume 4, King's College Publications 149184.
Goldblatt, R. (2003b) Observational ultraproducts of polynomial coalgebras. Annals of Pure and Applied Logic 123 235290.
Goldblatt, R. (2006) A modal proof theory for final polynomial coalgebras. Theoretical Computer Science 360 122.
Hughes, J. and Jacobs, B. (2004) Simulations in coalgebra. Theoretical Computer Science 327 (1–2)71108.
Jacobs, B. (1998) Coalgebraic reasoning about classes in object-oriented languages. In: Proceedings of Coalgebraic Methods in Computer Science. Elsevier Electronic Notes in Theoretical Computer Science 11 231242.
Jacobs, B. (1999) The temporal logic of coalgebras via Galois algebras, Technical Report CSI-R9906, Computer Science Institution, University of Nijmegen.
Jacobs, B. (2000) Towards a duality result in coalgebraic modal logic. In: Reichel, H. (ed.) Proceedings of Coalgebraic Methods in Computer Science. Elsevier Electronic Notes in Theoretical Computer Science 33 160195.
Jacobs, B. (2001) Many-sorted coalgebraic modal logic: a model-theoretical study. Theoretical Informatics and Applications 35 (1)3159.
Jacobs, B. (2002) Exercises in coalgebraic specification. In: Backhouse, R., Crole, R. and Gibbons, J. (eds.) Algebraic and Coalgebraic Methods in the Mathematics of Program Construction. Springer Lecture Notes in Computer Science 2297 237280.
Jacobs, B. (2005) Introduction to Coalgebra: Towards Mathematics of States and Observations. Book Draft, Available at http://www.cs.ru.nl/~bart.
Kupke, C. and Pattinson, D. (2011) Coalgebraic semantics of modal logics: An overview. Theoretical Computer Science 412 50705094.
Kupke, C. and Venema, Y. (2008). Coalgebraic automata theory: basic results. Logical Methods in Computer Science 4 143.
Kurz, A. (2001) Specifying coalgebras with modal logic. Theoretical Computer Science 260 (1–2)119138.
Lenisa, M. (1999) From set-theoretic coinduction to coalgebraic coinduction: Some results, some problems. In: Jacobs, B. and Rutten, J. (eds.) Coalgebraic Methods in Computer Science. Electronic Notes in Theoretical Computer Science 19 121.
Luo, L. (2006) An effective coalgebraic bisimulation proof method. In: Proceedings of the 8th Workshop on Coalgebraic Methods in Computer Science. Elsevier Electronic Notes in Theoretical Computer Science 164 105119.
Moss, L. (1999) Coalgebraic logic. Annals of Pure and Applied Logic 96 (1–3)277317.
Pattinson, D. (2003) Coalgebraic modal logic: Soundness, completeness and decidability of local consequence. Theoretical Computer Science 309 (1–3)177193.
Rößiger, M. (1999) Languages for coalgebras on datafunctors. In: Jacobs, B. and Rutten, J. (eds.) Proceedings of Coalgebraic Methods in Computer Science. Elsevier Electronic Notes in Theoretical Computer Science 19 3960.
Rößiger, M. (2001) From modal logic to terminal coalgebras. Theoretical Computer Science 260 209228.
Rothe, J., Tews, H. and Jacobs, B. (2001) The coalgebraic class specification language CCSL. Journal of Universal Computer Science 7 (2)175193.
Rutten, J. (2000) Universal coalgebra: A theory of systems. Theoretical Computer Science 249 380.
Rutten, J. (2001) Elements of stream calculus (an extensive exercise in coinduction). In: Brooks, S. and Mislove, M. (eds.) Proceedings of Mathematical Foundations of Programming Semantics. Elsevier Electronic Notes in Theoretical Computer Science 45 166.
Sangiorgi, D. (1998) On the bisimulation proof method. Mathematical Structures in Computer Science 8 447479.
Sangiorgi, D. (2009) On the origins of bisimulation and coinduction. ACM Transactions on Programming Languages and Systems 31 (4) Article 15.
Sangiorgi, D. and Walker, D. (2001) The pi-Calculus: A Theory of Mobile Processes. Cambridge University Press.
Zhou, X.-C., Li, Y.-J., Li, W.-J., Qiao, H.-Y. and Shu, Z.-M. (2010) Bisimulation proof methods in a path-based specification language for polynomial coalgebras. In: Ueda, K. (eds.) Proceedings of ASIAN Symposium on Program Languages and Systems, APLAS'10. Springer Lecture Notes in Computer Science 6461 239254.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed