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Closing the gap between runtime complexity and polytime computability
 In Proceedings of RTA 2010, volume 6 of LIPIcs
, 2010
"... Abstract. In earlier work, we have shown that for confluent term rewrite systems, innermost polynomial runtime complexity induces polytime computability of the functions defined. In this paper, we generalise this result to full rewriting. For that, we again exploit graph rewriting. We give a new pro ..."
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Abstract. In earlier work, we have shown that for confluent term rewrite systems, innermost polynomial runtime complexity induces polytime computability of the functions defined. In this paper, we generalise this result to full rewriting. For that, we again exploit graph rewriting. We give a new proof of the adequacy of graph rewriting for full rewriting that allows for a precise control of the resources copied. In sum we completely describe an implementation of rewriting on a Turing machine. We show that the runtime complexity with respect to rewrite systems is polynomially related to the runtime complexity on a Turing machine. Our result strengthens the evidence that the complexity of a rewrite system is truthfully represented through the length of derivations. Moreover our result allows the classification of deterministic as well as nondeterministic polytimecomputation based on runtime complexity analysis of rewrite systems. 1.
Derivational Complexity is an Invariant Cost Model ⋆
"... Abstract. We show that in the context of orthogonal term rewriting systems, derivational complexity is an invariant cost model, both in innermost and in outermost reduction. This has some interesting consequences for (asymptotic) complexity analysis, since many existing methodologies only guarantee ..."
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Abstract. We show that in the context of orthogonal term rewriting systems, derivational complexity is an invariant cost model, both in innermost and in outermost reduction. This has some interesting consequences for (asymptotic) complexity analysis, since many existing methodologies only guarantee bounded derivational complexity. 1
A Path Order for Rewrite Systems that Compute Exponential Time Functions ∗
"... In this paper we present a new path order for rewrite systems, the exponential path order EPO ⋆. Suppose a term rewrite system is compatible with EPO ⋆ , then the runtime complexity of this rewrite system is bounded from above by an exponential function. Furthermore, the class of function computed b ..."
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Cited by 1 (0 self)
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In this paper we present a new path order for rewrite systems, the exponential path order EPO ⋆. Suppose a term rewrite system is compatible with EPO ⋆ , then the runtime complexity of this rewrite system is bounded from above by an exponential function. Furthermore, the class of function computed by a rewrite system compatible with EPO ⋆ equals the class of functions computable in exponential time on a Turing machine.
A Formalization of Polytime Functions
 ITP 2011
, 2011
"... We present a deep embedding of Bellantoni and Cook’s syntactic characterization of polytime functions. We prove formally that it is correct and complete with respect to the original characterization by Cobham that required a bound to be proved manually. Compared to the paper proof by Bellantoni and ..."
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We present a deep embedding of Bellantoni and Cook’s syntactic characterization of polytime functions. We prove formally that it is correct and complete with respect to the original characterization by Cobham that required a bound to be proved manually. Compared to the paper proof by Bellantoni and Cook, we have been careful in making our proof fully contructive so that we obtain more precise bounding polynomials and more efficient translations between the two characterizations. Another difference is that we consider functions on bitstrings instead of functions on positive integers. This latter change is motivated by the application of our formalization in the context of formal security proofs in cryptography. Based on our core formalization, we have started developing a library of polytime functions that can be reused to build more complex ones.
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.