Interfaces as Games, Programs as Strategies
Peter Hancock and Anton Setzer developed the notion of interface to introduce interactive programming into dependent type theory. We generalise their notion and get an even simpler definition for interfaces. With this definition the relationship of interfaces to games becomes apparent. In fact from a game theoretical point of view interfaces are nothing other than special games. Programs are strategies for these games. There is an obvious notion of refinement which coincides exactly with the intuition. Interfaces together with the re.nement relation build a complete lattice. We can define several operators on interfaces: tensor, par, choice, bang etc. Every notion has a dual notion by interchanging the viewpoint of player and opponent. Identifying strategies by some kind of behavioural equivalence we conjecture to receive a linear category. All results so far can be achieved in intensional Martin-Löf Type Theory and are verified in the theorem prover Agda (with the exception of associativity of composition which is only proved on paper until now).
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- 3.Abbott, M., Altenkirch, T., Ghani, N., McBride, C.: \(\partial\) for data. Fundamenta Informatica 65(1-2) (2005)Google Scholar
- 4.Abbott, M.G.: Categories of Containers. PhD thesis, University of Leicester (2003)Google Scholar
- 5.Abramsky, S.: Semantics of Interaction: an introduction to Game Semantics. In: Dybjer, P., Pitts, A. (eds.) Proceedings of the 1996 CLiCS Summer School, pp. 1–31. Cambridge University Press, Cambridge (1997)Google Scholar
- 7.Hancock, P., Hyvernat, P.: Programming interfaces and basic topology. In: Proceedings of the second workshop of Formal Topology. Annals of Pure and Applied Logic (2004) (to appear)Google Scholar
- 9.Hancock, P., Setzer, A.: Specifying interactions with dependent types. In: Workshop on subtyping and dependent types in programming, Portugal, July 7 (2000); Electronic proceedingsGoogle Scholar
- 10.Hancock, P., Setzer, A.: Guarded induction and weakly final coalgebras in dependent type theory (extended version). In: Altenkirch, T., Hofmann, M., Hughes, J. (eds.) Dependently typed programming. Dagstuhl Seminar Proceedings 04381, Dagstuhl, Germany (2005)Google Scholar
- 11.Hancock, P., Setzer, A.: Interactive programs and weakly final coalgebras in dependent type theory. In: Crosilla, L., Schuster, P. (eds.) From sets and types to topology and analysis: Towards practicable foundations for constructive mathematics. Oxford Logic Guides, pp. 115–136. Oxford University Press, Oxford (2005)Google Scholar
- 13.Hyland, M.: Game Semantics. In: Dybjer, P., Pitts, A. (eds.) Proceedings of the 1996 CLiCS Summer School, pp. 131–184. Cambridge University Press, Cambridge (1997)Google Scholar
- 15.Hyvernat, P.: Synchronous Games, Simulations and λ-calculus. In: Ghica, D.R., McCusker, G. (eds.) Games for Logic and Programming Languages, GaLoP (ETAPS 2005), April 2005, pp. 1–15 (2005)Google Scholar
- 17.Michelbrink, M.: A generalisation of Hancock-Setzer Interfaces. Draft (2005), Available via, http://www.cs.swan.ac.uk/~csmichel/
- 18.Michelbrink, M.: Interfaces as functors, programs as coalgebras - a final coalgebra theorem in intensional type theory (2005) (submitted), Available via, http://www.cs.swan.ac.uk/~csmichel/
- 19.Michelbrink, M., Setzer, A.: State-dependent IO-monads in type theory. In: Birkedal, L. (ed.) Proceedings of the 10th Conference on Category Theory in Computer Science (CTCS 2004). Electronic Notes in Theoretical Computer Science, vol. 122, pp. 127–146. Elsevier, Amsterdam (2005)Google Scholar
- 20.Sambin, G.: The Basic Picture, a structure for topology (the Basic Picture, I) (2001)Google Scholar