# A Game-Semantic Model of Computation

## TL;DR

This paper introduces a new game-semantic model of computation based on viability, establishing Turing completeness for strategies and generalizing key computability theorems, offering a foundational perspective beyond natural number computation.

## Contribution

It develops a novel, intrinsic notion of computability called viability within game semantics, demonstrating Turing completeness and extending classical computability results.

## Key findings

- Viability is a new intrinsic notion of computability in game semantics.
- Viable strategies are shown to be Turing complete.
- Generalization of the smn-theorem and recursion theorem.

## Abstract

The present paper introduces a novel notion of `(effective) computability', called viability, of strategies in game semantics in an intrinsic (i.e., without recourse to the standard Church-Turing computability), non-inductive and non-axiomatic manner, and shows, as a main technical achievement, that viable strategies are Turing complete. Consequently, we have given a mathematical foundation of computation in the same sense as Turing machines but beyond computation on natural numbers, e.g., higher-order computation, in a more abstract fashion. As immediate corollaries, some of the well-known theorems in computability theory such as the smn-theorem and the first recursion theorem are generalized. Notably, our game-semantic framework distinguishes `high-level' computational processes that operate directly on mathematical objects such as natural numbers (not on their symbolic representations) and their `symbolic implementations' that define their `computability', which sheds new light on the very concept of computation. This work is intended to be a stepping stone towards a new mathematical foundation of computation, intuitionistic logic and constructive mathematics.