Game-theoretic Model of Computation

The present paper introduces an intrinsic notion of “(effective) computability” in game semantics motivated by the fact that strategies in game semantics have been defined recursive if they are “computable in an extrinsic sense”, i.e., they are representable by partial recursive functions, and so it has been difficult to regard game semantics as an autonomous foundation of computation. As a consequence, we have formulated a general notion of “algorithms” under the name of effective strategies, giving rise to a mathematical model of computation in the same sense as Turing machines but beyond computation on natural numbers, e.g., higherorder one, solely in terms of games and strategies. It subsumes computation of the programming language PCF, and so it is in particular Turing complete. Notably, effective strategies have a natural notion of types (i.e., games) unlike Turing machines, while they are non-inductively defined as opposed to partial recursive functions as well as semantic in contrast with λ-calculi and combinatory logic. Thus, in a sense, we have captured a mathematical (or semantic) notion of computation (and computability) that is more general than the “classical ones” in a fundamental level. Exploiting the flexibility of game semantics, our game-theoretic model of computation is intended to give a mathematical foundation of various (constructive) logics and programming languages.