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Levels of Undecidability in Rewriting
, 2011
"... Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarc ..."
Abstract

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Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2complete. The particular problem of local confluence turns out to be Π 0 2complete for ground terms, but only Σ 0 1complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.
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 Σ0 2 of the arithmetical hierarchy: (A) Deciding whether a term rewriting system (TRS for short) has runtime complexity bounded by a function in C. (B) ..."
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For any class C of computable total functions satisfying some mild conditions, we prove that the following decision problems are complete for level Σ0 2 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 Σ0 2complete. 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 runtimecomplete. We further prove that analysing the implicomplexity bounded from above by f is Π0 1 cit 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 Σ0 3complete. 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.