Cynthia Kop ; Jakob Grue Simonsen - Complexity Hierarchies and Higher-order Cons-free Term Rewriting

lmcs:2573 - Logical Methods in Computer Science, August 7, 2017, Volume 13, Issue 3 -
Complexity Hierarchies and Higher-order Cons-free Term RewritingArticle

Authors: Cynthia Kop ; Jakob Grue Simonsen

    Constructor rewriting systems are said to be cons-free if, roughly, constructor terms in the right-hand sides of rules are subterms of the left-hand sides; the computational intuition is that rules cannot build new data structures. In programming language research, cons-free languages have been used to characterize hierarchies of computational complexity classes; in term rewriting, cons-free first-order TRSs have been used to characterize the class PTIME. We investigate cons-free higher-order term rewriting systems, the complexity classes they characterize, and how these depend on the type order of the systems. We prove that, for every K $\geq$ 1, left-linear cons-free systems with type order K characterize E$^K$TIME if unrestricted evaluation is used (i.e., the system does not have a fixed reduction strategy). The main difference with prior work in implicit complexity is that (i) our results hold for non-orthogonal term rewriting systems with no assumptions on reduction strategy, (ii) we consequently obtain much larger classes for each type order (E$^K$TIME versus EXP$^{K-1}$TIME), and (iii) results for cons-free term rewriting systems have previously only been obtained for K = 1, and with additional syntactic restrictions besides cons-freeness and left-linearity. Our results are among the first implicit characterizations of the hierarchy E = E$^1$TIME $\subsetneq$ E$^2$TIME $\subsetneq$ ... Our work confirms prior results that having full non-determinism (via overlapping rules) does not directly allow for characterization of non-deterministic complexity classes like NE. We also show that non-determinism makes the classes characterized highly sensitive to minor syntactic changes like admitting product types or non-left-linear rules.

    Volume: Volume 13, Issue 3
    Published on: August 7, 2017
    Accepted on: July 26, 2017
    Submitted on: August 7, 2017
    Keywords: Computer Science - Computational Complexity,Computer Science - Logic in Computer Science
      Source : OpenAIRE Graph
    • Higher-Order Rewriting for Intensional Properties of Programs and Circuits; Funder: European Commission; Code: 658162

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