Degrees of undecidability in term rewriting

J. Endrullis, J.H. Geuvers, H. Zantema

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

11 Citations (Scopus)
2 Downloads (Pure)

Abstract

Undecidability of various properties of first order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This 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 also quantification over function variables is allowed. In this paper we consider properties of first order term rewriting systems and classify them in this hierarchy. Most of the standard properties are ¿0/2-complete, that is, of the same level as uniform halting of Turing machines. In this paper we show two exceptions. Weak confluence is S0/1 -complete, and therefore essentially easier than ground weak confluence which is ¿0/2-complete. The most surprising result is on dependency pair problems: we prove this to be ¿1/1-complete, which means that this property exceeds the arithmetical hierarchy and is essentially analytic. A minor variant, dependency pair problems with minimality flag, turns out be ¿0/2-complete again, just like the original termination problem for which dependency pair analysis was developed.
Original languageEnglish
Title of host publicationComputer Science Logic (23rd international workshop, CSL 2009, 18th annual conference of the EACSL, Coimbra, Portugal, September 7-11, 2009. Proceedings)
EditorsE. Grädel, R. Kahle
Place of PublicationBerlin
PublisherSpringer
Pages255-270
ISBN (Print)978-3-642-04026-9
DOIs
Publication statusPublished - 2009

Publication series

NameLecture Notes in Computer Science
Volume5771
ISSN (Print)0302-9743

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