The Exact Hardness of Deciding Derivational and Runtime Complexity

For any class C of computable total functions satisfying some mild conditions, we prove that the following decision problems are complete for level Σ2 of the arithmetical hierarchy: (A) Deciding whether a term rewriting system (TRS for short) has runtime complexity bounded by a function in C. (B) Deciding whether a TRS has derivational complexity bounded by a function in C. In particular, the problems of deciding whether a TRS has polynomially (exponentially) bounded runtime complexity (respectively derivational complexity) are Σ2-complete. This places deciding polynomial derivational or runtime complexity of TRSs at the same level in the arithmetical hierarchy as deciding nontermination or nonconfluence of TRSs. We proceed to show that the related problem of deciding for a single computable function f whether a TRS has runtime complexity bounded from above by f is Π1-complete. We further prove that analysing the implicit complexity of TRSs is even more difficult: The problem of deciding whether a TRS accepts a language of terms accepted by some TRS with runtime complexity bounded by a function in C is Σ3-complete. All of our results are easily extended to the notion of minimal complexity (where the length of shortest reductions to normal form is considered) and remain valid under any computable reduction strategy. Finally, all results hold both for unrestricted TRSs and for the class of orthogonal TRSs. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes; F.2.2; F.4.1; F.4.2