  The notion of weak truth-table reducibility plays an important role in
recursion theory. In this paper, we introduce an elaboration of this notion,
where a computable bound on the use function is explicitly specified. This
elaboration enables us to deal with the notion of asymptotic behavior in a
manner like in computational complexity theory, while staying in computability
theory. We apply the elaboration to sets which appear in the statistical
mechanical interpretation of algorithmic information theory. We demonstrate the
power of the elaboration by revealing a critical phenomenon, i.e., a phase
transition, in the statistical mechanical interpretation, which cannot be
captured by the original notion of weak truth-table reducibility.
