Computation-Theoretic Clarification of the Phase Transition at Temperature \(T=1\)

Abstract

The notion of weak truth-table reducibility plays an important role in recursion theory (see e.g. Nies [27] and Downey and Hirschfeldt [14]). For any sets \(A, B\subset \mathbb {N}\), we say that A is weak truth-table reducible to B, denoted \(A\le _{wtt} B\), if there exist an oracle Turing machine M and a total recursive function \(g:\mathbb {N}\rightarrow \mathbb {N}\) such that (i) A is Turing reducible to B via M and (ii) on every input \(n\in \mathbb {N}\), the machine M only queries natural numbers at most g(n).