  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.
