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).