A game-semantic model of computation

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 , non-axiomatic manner and shows, as a main technical achievement, tha...

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Bibliographic Details
Published in:Research in the mathematical sciences Vol. 6; no. 1; pp. 1 - 79
Main Author: Yamada, Norihiro
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.01.2019
Springer Nature B.V
Subjects:
Applications of Mathematics
Computation
Computational Mathematics and Numerical Analysis
Game theory
Mathematical analysis
Mathematics
Mathematics and Statistics
Number theory
Semantics
Theorems
Turing machines
Viability
Secondary 68Q05
Primary 03D10
ISSN:2522-0144, 2197-9847
Online Access:Get full text
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Summary: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 , 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 toward a new mathematical foundation of computation, intuitionistic logic and constructive mathematics.
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ISSN:2522-0144
2197-9847
DOI:10.1007/s40687-018-0163-z