2 Search Results for "Hofbauer, Dieter"

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DOI: 10.4230/LIPIcs.FSCD.2024.16

On the Complexity of the Small Term Reachability Problem for Terminating Term Rewriting Systems

Authors: Franz Baader and Jürgen Giesl

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Motivated by an application where we try to make proofs for Description Logic inferences smaller by rewriting, we consider the following decision problem, which we call the small term reachability problem: given a term rewriting system R, a term s, and a natural number n, decide whether there is a term t of size ≤ n reachable from s using the rules of R. We investigate the complexity of this problem depending on how termination of R can be established. We show that the problem is NP-complete for length-reducing term rewriting systems. Its complexity increases to N2ExpTime-complete (NExpTime-complete) if termination is proved using a (linear) polynomial order and to PSpace-complete for systems whose termination can be shown using a restricted class of Knuth-Bendix orders. Confluence reduces the complexity to P for the length-reducing case, but has no effect on the worst-case complexity in the other two cases.

Cite as

Franz Baader and Jürgen Giesl. On the Complexity of the Small Term Reachability Problem for Terminating Term Rewriting Systems. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{baader_et_al:LIPIcs.FSCD.2024.16,
  author =	{Baader, Franz and Giesl, J\"{u}rgen},
  title =	{{On the Complexity of the Small Term Reachability Problem for Terminating Term Rewriting Systems}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.16},
  URN =		{urn:nbn:de:0030-drops-203454},
  doi =		{10.4230/LIPIcs.FSCD.2024.16},
  annote =	{Keywords: Rewriting, Termination, Confluence, Creating small terms, Derivational complexity, Description Logics, Proof rewriting}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2019.21

Sparse Tiling Through Overlap Closures for Termination of String Rewriting

Authors: Alfons Geser, Dieter Hofbauer, and Johannes Waldmann

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Abstract

A strictly locally testable language is characterized by its set of admissible factors, prefixes and suffixes, called tiles. We over-approximate reachability sets in string rewriting by languages defined by sparse sets of tiles, containing only those that are reachable in derivations. Using the partial algebra defined by a tiling for semantic labeling, we obtain a transformational method for proving local termination. These algebras can be represented efficiently as finite automata of a certain shape. Using a known result on forward closures, and a new characterisation of overlap closures, we can automatically prove termination and relative termination, respectively. We report on experiments showing the strength of the method.

Cite as

Alfons Geser, Dieter Hofbauer, and Johannes Waldmann. Sparse Tiling Through Overlap Closures for Termination of String Rewriting. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{geser_et_al:LIPIcs.FSCD.2019.21,
  author =	{Geser, Alfons and Hofbauer, Dieter and Waldmann, Johannes},
  title =	{{Sparse Tiling Through Overlap Closures for Termination of String Rewriting}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{21:1--21:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.21},
  URN =		{urn:nbn:de:0030-drops-105282},
  doi =		{10.4230/LIPIcs.FSCD.2019.21},
  annote =	{Keywords: relative termination, semantic labeling, locally testable language, overlap closure}
}

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1 Baader, Franz
1 Geser, Alfons
1 Giesl, Jürgen
1 Hofbauer, Dieter
1 Waldmann, Johannes

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1 Theory of computation → Automated reasoning
1 Theory of computation → Complexity theory and logic
1 Theory of computation → Equational logic and rewriting
1 Theory of computation → Rewrite systems

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1 Confluence
1 Creating small terms
1 Derivational complexity
1 Description Logics
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