License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSCD.2016.23
URN: urn:nbn:de:0030-drops-59972
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Kop, Cynthia ; Grue Simonsen, Jakob

Complexity Hierarchies and Higher-Order Cons-Free Rewriting

LIPIcs-FSCD-2016-23.pdf (0.6 MB)


Constructor rewriting systems are said to be cons-free if, roughly,
constructor terms in the right-hand sides of rules are subterms of
constructor terms in the left-hand side; the computational intuition
is that rules cannot build new data structures. It is well-known that
cons-free programming languages can be used to characterize
computational complexity classes, and that cons-free first-order term
rewriting can be used to characterize the set of polynomial-time
decidable sets.

We investigate cons-free higher-order term rewriting systems, the
complexity classes they characterize, and how these depend on the
order of the types used in the systems. We prove that, for every k >=
1, left-linear cons-free systems with type order k characterize
E^kTIME if arbitrary 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
possible rule overlaps with no assumptions about reduction strategy,
(ii) results for such term rewriting systems have previously only been
obtained for k = 1, and with additional syntactic restrictions on top
of cons-freeness and left-linearity.

Our results are apparently among the first implicit characterizations
of the hierarchy E^1TIME != E^2TIME != .... Our work confirms prior
results that having full non-determinism (via overlaps of 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
such as admitting product types or non-left-linear rules.

BibTeX - Entry

  author =	{Cynthia Kop and Jakob Grue Simonsen},
  title =	{{Complexity Hierarchies and Higher-Order Cons-Free Rewriting}},
  booktitle =	{1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-010-1},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{52},
  editor =	{Delia Kesner and Brigitte Pientka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-59972},
  doi =		{10.4230/LIPIcs.FSCD.2016.23},
  annote =	{Keywords: higher-order term rewriting, implicit complexity, cons-freeness, ETIME hierarchy}

Keywords: higher-order term rewriting, implicit complexity, cons-freeness, ETIME hierarchy
Collection: 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)
Issue Date: 2016
Date of publication: 17.06.2016

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