Degrees of Undecidability in Term Rewriting

  • Jörg Endrullis
  • Herman Geuvers
  • Hans Zantema
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5771)


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 \(\mathrm{\Pi}^{0}_{2}\)-complete, that is, of the same level as uniform halting of Turing machines. In this paper we show two exceptions. Weak confluence is \({\mathrm{\Sigma}}^{0}_{1}\)-complete, and therefore essentially easier than ground weak confluence which is \(\mathrm{\Pi}^{0}_{2}\)-complete.

The most surprising result is on dependency pair problems: we prove this to be \(\mathrm{\Pi}^{1}_{1}\)-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 \(\mathrm{\Pi}^{0}_{2}\)-complete again, just like the original termination problem for which dependency pair analysis was developed.


Turing Machine Dependency Pair Weak Ground Term Rewrite System Arithmetical Hierarchy 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jörg Endrullis
    • 1
  • Herman Geuvers
    • 2
    • 3
  • Hans Zantema
    • 2
    • 3
  1. 1.Vrije Universiteit AmsterdamThe Netherlands
  2. 2.Radboud Universiteit NijmegenThe Netherlands
  3. 3.Technische Universiteit EindhovenThe Netherlands

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