__
TR21-134 | 19th August 2021 11:21
__

#### A refinement of the Meyer-McCreight Union Theorem

**Abstract:**
For a function $t : 2^\star \to 1^\star$, let $C_t$ be the set of problems decidable on input $x$ in time at most $t(x)$ almost everywhere. The Union Theorem of Meyer and McCreight asserts that any union $\bigcup_{i < \omega} C_{t_i}$ for a uniformly recursive sequence of bounds $t_i$ is equal to $C_L$ for some single recursive function $L$. In particular the class PTIME of polynomial-time relations can be expressed as $C_L$ for some total recursive function $L : 2^\star \to 1^\star$. By controlling the complexity of the construction, we show that in fact PTIME equals $C_L$ for some $L$ computable in quasi-polynomial time.