## Degrees of undecidability in term rewriting (2009)

Venue: | Proceedings of Computer 30 Logic (CSL09), volume 5771 of Lecture Notes in Computer Science |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Endrullis09degreesof,

author = {Jörg Endrullis and Herman Geuvers and Hans Zantema},

title = {Degrees of undecidability in term rewriting},

booktitle = {Proceedings of Computer 30 Logic (CSL09), volume 5771 of Lecture Notes in Computer Science},

year = {2009},

pages = {255--270},

publisher = {Springer}

}

### OpenURL

### Abstract

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 Π 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 Σ 0 1-complete, and therefore essentially easier than ground weak confluence which is Π 0 2-complete. The most surprising result is on dependency pair problems: we prove this to be Π 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 Π 0 2-complete again, just like the original termination problem for which dependency pair analysis was developed. 1

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Citation Context ...etical relation. Our Contribution. We investigate the complexity, of the following properties of first order TRSs: – confluence (CR), – weak confluence (WCR), – finiteness of dependency pair problems =-=[2,5]-=- (DP), and – finiteness of dependency pair problems with minimality flag [5] (DP min ). In this paper we pinpoint the precise complexities of these properties in terms of the arithmetic and analytic h... |

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Citation Context ...0 2 Π0 2 Π 0 2 Σ 0 1 Π 0 2 Π 1 1 Π 0 2 single term Σ 0 1 Σ 0 1 Π 0 2 Π 0 2 Σ 0 1 Σ 0 1 Π 1 1 − The contributions of this paper are encircled. The non-encircled uniform properties have been studied in =-=[7]-=- and [12]. For the complexity of these properties for single terms we refer to [3], an extended version of the present paper. We deepen the study of [12] and find surprisingly that weak ground conflue... |

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Citation Context ...etical relation. Our Contribution. We investigate the complexity, of the following properties of first order TRSs: – confluence (CR), – weak confluence (WCR), – finiteness of dependency pair problems =-=[2,5]-=- (DP), and – finiteness of dependency pair problems with minimality flag [5] (DP min ). In this paper we pinpoint the precise complexities of these properties in terms of the arithmetic and analytic h... |

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Citation Context ...ation problems) we show it is of the same level as termination itself. We emphasize that the variant without minimality flag is commonly used: they arise for example from Haskell termination problems =-=[4]-=-, and transformations on dependency pair problems that do not preserve the minimality flag. For relations →1, →2 we write →1 / →2 for →∗ 2 ·→1. ForTRSsR, S instead of SN(→R,ɛ/→S) we shortly write SN(R... |

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Citation Context ...Π 0 2 -complete. Proof. For proving Π0 2-hardness we reduce the totality problem to confluence. Let M be an arbitrary Turing machine. We consider the TRS SM defined above. We employ type introduction =-=[1]-=-: we assign sort γ0 to Γ ∪{⊲} and sort γ1 to every symbol in {run, T}∪Q; the obtained many-sorted TRS is confluent if and only if SM is. Note that the terms of sort γ0 are normal forms and for terms o... |

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Citation Context ...operty SN(Stop/S) states that every infinite S-reduction contains only finitely many root steps. This is the same as the property SN ω when restricting to finite terms; for the definition of SN ω see =-=[14]-=- (basically, it states that in any infinite reduction the position of the contracted redex moves to infinity). However, when extending to infinite terms it still holds that for the TRS S in the proof ... |

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Citation Context ... Π 0 2 Σ 0 1 Π 0 2 Π 1 1 Π 0 2 single term Σ 0 1 Σ 0 1 Π 0 2 Π 0 2 Σ 0 1 Σ 0 1 Π 1 1 − The contributions of this paper are encircled. The non-encircled uniform properties have been studied in [7] and =-=[12]-=-. For the complexity of these properties for single terms we refer to [3], an extended version of the present paper. We deepen the study of [12] and find surprisingly that weak ground confluence is ha... |

1 |
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Citation Context ... symbols in R and all symbols in S are unmarked. We now sketch how this proof also implies Π1 1-completeness of the property SN∞ in infinitary rewriting, for its definition and basic observations see =-=[9]-=-. Since in Theorem 6.5 we proved Π1 1-hardness even for the case where R and S coincide, we conclude that SN(Stop/S) isΠ1 1-complete. This property SN(Stop/S) states that every infinite S-reduction co... |