Computation-by-Interaction with Effects

A successful approach in the semantics of programming languages is to model programs by interaction dialogues. While dialogues are most often considered abstract mathematical objects, it has also been argued that they are useful for actual computation. A manual implementation of interaction dialogues can be complicated, however. To address this issue, we consider a general method for extending a given language with a metalanguage that supports the implementation of dialogues. This method is based on the construction by Dal Lago and the author of the programming language INTML, which applies interaction dialogues to sublinear space computation. We show that only few assumptions on the programming languages are needed to implement a useful INTML-like metalanguage. We identify a weak variant of the Enriched Effect Calculus (EEC) of Egger, Møgelberg & Simpson as a convenient setting for capturing the structure needed for the construction of the metalanguage. In particular, function types are not needed for the construction and iteration by means of a Conway operator is sufficient. By using EEC we show how computational effects can be accounted for in the implementation of interaction dialogues. In game semantics and related areas of programming language semantics there is a long tradition of modelling programs by interaction dialogues. Programs are modelled as entities that may engage in a dialogue with their environment. The interpretation of a program explains what kinds of queries it can receive and how it may answer. Large programs are composed of smaller ones that interact with each other, so that the whole execution of a program may be considered an interaction process. The question/answer dialogues that make up such models tend to have very concrete nature, which has lead to interesting applications, for example in algorithmic game semantics. The premise of this paper is that interaction dialogues are useful not only for interpreting programming languages, but also as an actual implementation method. There are many examples where dialogues have been used for the implementation of programs, e.g. [17, 6, 13]. Two recent examples provide the main motivation for the work reported here. First, Ghica introduces the Geometry of Synthesis [6] as a method of hardware synthesis. His approach is to construct a game model by implementing interaction dialogues by digital circuits and then to interpret a variant of Idealized Algol in the thus constructed game model. With this approach one can write a program in an high-level language (Idealized Algol) and by interpretation in the game model have it translated to a low-level language for digital circuits (Verilog). In this way, the implementation of dialogues is used as a method for hardware synthesis. A similar example has been studied by Dal Lago and the author in the context of computation with sublinear space [13]. There the problem is how to write programs