Games as mathematics of logic and computation
Keywords: Theory of Computation
Recursion Theory
Denotational Semantics
Game Semantics
Foundations of Mathematics

Mathematical logic and theoretical computer science are the mathematical studies of logic and computation, respectively, which largely correspond to each other notably by the Curry-Howard isomorphism. However, logic and computation have been captured mainly syntactically, which may be criticized as conceptually unclear and mathematically cumbersome.
For this point, the field of semantics has been developed, i.e., its main aim is to explain logic and computation by mathematical and in particular syntax-independent concepts. However, logic and computation have not yet been completely captured semantically, where a main obstacle lies in the point that proofs and programs are central in these fields, and they are dynamic, intensional concepts, but mainstream mathematics has been concerned mainly with static, extensional objects such as sets and functions, lacking in dynamic, intensional structures.
Motivated by this foundational problem, the present thesis is written for the aim of establishing mathematically rigorous, syntax-independent, conceptually natural semantics of some central aspects of logic and computation, particularly their dynamics and intensionality, that have not been captured very well from a conceptual or mathematical viewpoint, for which our primary goal is to obtain a deeper understanding of logic and computation. Specifically, the thesis is concerned with concepts such as proof-normalization in logic (or reduction in computation), higher-order computability, and predicates in logic (or dependent types in computation).
Our approach is based on mathematical structures developed in the field of game semantics for they are mathematically rigorous, syntax-independent, conceptually natural and exible enough to model various kinds of logic and computation in a systematic manner; also, they have a good potential to capture dynamics and intensionality of logic and computation. More concretely, we generalize, for each concept mentioned above, an existing variant of games to capture the concept, where each case is confirmed by a certain soundness or sometimes completeness theorem. Since each of the generalizations is orthogonal to one another, we may combine them into a single framework that provides a unified view on these developments.
As technical achievements of the thesis, many of the generalizations give the first game-semantic interpretations of the corresponding concepts, for which we have introduced a number of novel mathematical structures and proved their properties. Also, conceptually, our games give dynamic, intensional semantics that explains various concepts and phenomena in a natural, intuitive yet mathematically precise manner, e.g., normalization of reduction, Π-, Σ- and Id-types as well as universes in Martin-Lof type theory, incompatibility of Σ-types and classical reasoning, and nonconstructivity of Univalence Axiom in homotopy type theory.
Last but not least, since our mathematical structures capture logic and computation more completely, well beyond conventional game semantics, one of their implications would be the semantics-first-view: Logic and computation may be understood abstractly and syntax-independently as mathematics, viz., games, rather than as syntactic entities to be analyzed by various semantics, which is our intention behind the title of the thesis. In particular, our game-semantic approach enables an analytic study of logic and computation, as opposed to more standard synthetic ones such as categorical semantics, in the sense that it reduces the two concepts to the highly primitive notions of '(dialogical) arguments' between prover and refuter mathematicians, and '(computational) processes' between a computational agent and an environment, respectively, giving a deeper understanding of the very notion of logic and computation. The former can be seen particularly as a unity of proof theory and model theory: It gives a syntax-independent formulation of proofs that is also a computational description of models (by which we mean what formal languages refer to in a loose sense), namely strategies, where provability and validity of a formula simply coincide as the existence of a winning strategy on the corresponding game. By this unity, the central questions of soundness and completeness in logic just disappear. Furthermore, consistency of the logic which our games and strategies embody is immediate. These points demonstrate certain technical advantages (in addition to the conceptual naturality mentioned above) of our game-semantic approach to logic.