Abstract:
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.
