Verifying Time Complexity of Deterministic Turing Machines

TL;DR

This paper characterizes the boundary of algorithmically verifying the time complexity of deterministic Turing machines, establishing decidability for certain bounds and undecidability for others, including polynomial and exponential times.

Contribution

It provides a precise characterization of when the time complexity verification problem is decidable or undecidable for Turing machines.

Findings

Verification is decidable for functions T(n)=o(n log n).

Verification is undecidable for functions T(n)≥(n+1) and T(n)=Ω(n log n).

Verification reduces to checking a specific condition for multi-tape machines.

Abstract

We show that, for all reasonable functions $T(n)=o(n\log n)$, we can algorithmically verify whether a given one-tape Turing machine runs in time at most $T(n)$. This is a tight bound on the order of growth for the function $T$ because we prove that, for $T(n)\geq(n+1)$ and $T(n)=\Omega(n\log n)$, there exists no algorithm that would verify whether a given one-tape Turing machine runs in time at most $T(n)$. We give results also for the case of multi-tape Turing machines. We show that we can verify whether a given multi-tape Turing machine runs in time at most $T(n)$ iff $T(n_0)< (n_0+1)$ for some $n_0\in\mathbb{N}$. We prove a very general undecidability result stating that, for any class of functions $\mathcal{F}$ that contains arbitrary large constants, we cannot verify whether a given Turing machine runs in time $T(n)$ for some $T\in\mathcal{F}$. In particular, we cannot verify whether a Turing machine runs in constant, polynomial or exponential time.