## Simple Termination is Difficult (1993)

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Venue: | Applicable Algebra in Engineering, Communication and Computing |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Middeldorp93simpletermination,

author = {Aart Middeldorp and Bernhard Gramlich},

title = {Simple Termination is Difficult},

booktitle = {Applicable Algebra in Engineering, Communication and Computing},

year = {1993},

pages = {228--242},

publisher = {Springer}

}

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### Abstract

A terminating term rewriting system is called simply terminating if its termination can be shown by means of a simplification ordering, an ordering with the property that a term is always bigger than its proper subterms. Almost all methods for proving termination yield, when applicable, simple termination. We show that simple termination is an undecidable property, even for one-rule systems. This contradicts a result by Jouannaud and Kirchner. The proof is based on the ingenious construction of Dauchet who showed the undecidability of termination for one-rule systems. Our results may be summarized as follows: being simply terminating, (non-)selfembedding, and (non-)looping are undecidable properties of orthogonal, variable preserving, one-rule constructor systems. 1. Introduction It is well-known that termination is an undecidable property of term rewriting systems. This result was obtained by Huet and Lankford [9] in 1978. They showed that every Turing machine can be coded as a strin...

### Citations

751 | Rewrite systems
- Dershowitz, Jouannaud
- 1992
(Show Context)
Citation Context ...ing the powerful distribution elimination technique of Zantema [19]. 2. Simple Termination We start with a brief introduction to term rewriting. Term rewriting is surveyed in Dershowitz and Jouannaud =-=[5]-=- and Klop [11]. 2 A signature is a set F of function symbols. Associated with every f 2 F is a natural number denoting its arity. Function symbols of arity 0 are called constants. Let T (F ; V) be the... |

565 | Term Rewriting Systems
- Klop
- 1992
(Show Context)
Citation Context ...ful distribution elimination technique of Zantema [19]. 2. Simple Termination We start with a brief introduction to term rewriting. Term rewriting is surveyed in Dershowitz and Jouannaud [5] and Klop =-=[11]-=-. 2 A signature is a set F of function symbols. Associated with every f 2 F is a natural number denoting its arity. Function symbols of arity 0 are called constants. Let T (F ; V) be the set of all te... |

453 | Termination of rewriting
- Dershowitz
- 1985
(Show Context)
Citation Context ...stem is equivalent to the uniform halting problem for the originating Turing machine. The number of rules in their construction depends on the number of Turing machine instructions. Later, Dershowitz =-=[3]-=- showed that every Turing machine can be simulated by means of a two-rule term rewriting system. This result was improved by Dauchet [2], who showed that termination remains undecidable even if we res... |

74 |
On the uniform halting problem for term rewriting systems
- Huet, Lankford
- 1978
(Show Context)
Citation Context ...ariable preserving, one-rule constructor systems. 1. Introduction It is well-known that termination is an undecidable property of term rewriting systems. This result was obtained by Huet and Lankford =-=[9]-=- in 1978. They showed that every Turing machine can be coded as a string rewriting system---a term rewriting system with only unary function symbols---such that termination of the resulting string rew... |

49 | Generalized sufficient conditions for modular termination of rewriting. Applicable Algebra
- Gramlich
- 1994
(Show Context)
Citation Context ...imple termination, however. As a matter of fact, it is known that negative results for the class of non-self-embedding systems do not always carry over to the class of simply terminating systems, see =-=[7]-=-. In this paper we show the undecidability of simple termination for one-rule term rewriting systems. This contradicts a result of Jouannaud and Kirchner [10]. The undecidability proof is based on the... |

47 |
Modularity of simple termination of term rewriting systems
- Kurihara, Ohuchi
- 1990
(Show Context)
Citation Context ... The proof is not difficult. This lemma appeared for the first time in Zantema [19], although it is implicit in many earlier works on termination, see Dershowitz [3] for a survey. Kurihara and Ohuchi =-=[12, 13] proved th-=-e related equivalence "a TRS (F ; R) is simplifying () the transitive closure of the rewrite relation associated to the TRS (F ; R) [ Emb (F) is irreflexive". The above lemma facilitates an ... |

27 |
The Undecidability of the Turing Machine Immortality Problem
- Hooper
- 1966
(Show Context)
Citation Context ...whether M halts for all its configurations.sA proof of this statement can be found in Caron [1], where a reduction to Post's Correspondence Problem is given. Caron ascribes the above result to Hooper =-=[8]-=-, but she obtained it independently. Moreover, [8] is very hard to read---there is for instance no notion of LBA---and it is not clear at all whether we may assume the simple definition of LBA given a... |

21 |
Simulation of Turing machines by a regular rewrite rule
- Dauchet
- 1992
(Show Context)
Citation Context ...on the number of Turing machine instructions. Later, Dershowitz [3] showed that every Turing machine can be simulated by means of a two-rule term rewriting system. This result was improved by Dauchet =-=[2]-=-, who showed that termination remains undecidable even if we restrict our attention to one-rule term rewriting systems that are orthogonal and variable preserving. His skillful construction will be ex... |

13 | Termination of term rewriting by interpretation
- ZANTEMA
- 1993
(Show Context)
Citation Context ...type removal. In Section 5 we prove the equivalence RM is terminating () RM is simply terminating for all linear bounded automata M by using the powerful distribution elimination technique of Zantema =-=[19]-=-. 2. Simple Termination We start with a brief introduction to term rewriting. Term rewriting is surveyed in Dershowitz and Jouannaud [5] and Klop [11]. 2 A signature is a set F of function symbols. As... |

12 |
The undecidability of self-embedding for term rewriting systems
- Plaisted
- 1985
(Show Context)
Citation Context ...ld, when applicable, simple termination. Simple termination is closely related to the non-self-embedding property, since every simply terminating term rewriting system is non-self-embedding. Plaisted =-=[16]-=- showed that the non-self-embedding property is undecidable. From this result we cannot infer the undecidability of simple termination, however. As a matter of fact, it is known that negative results ... |

11 |
bounded automata and rewrite systems: Influence of initial configurations on decision properties
- Linear
- 1991
(Show Context)
Citation Context ...trict our attention to one-rule term rewriting systems that are orthogonal and variable preserving. His skillful construction will be explained in detail later in this paper. On the other hand, Caron =-=[1]-=- recently showed that termination is an undecidable property of length-preserving string rewriting systems---systems in which the left-hand side and the right-hand side of each A preliminary version o... |

10 |
Termination und konfluenz von semi-Thue-systemen mit nur einer regel
- Kurth
- 1990
(Show Context)
Citation Context ... and oe = fx 7! f(a; b; b)g.) We would like to conclude this section with mentioning a (famous) open problem: the decidability of termination for one-rule SRSs. Partial results were obtained by Kurth =-=[14]-=-. 4 He showed that termination is decidable in case the number of function symbols in the righthand side of the single rewrite rule does not exceed six. Deciding the termination of one-rule non-length... |

9 | A structural analysis of modular termination of term rewriting systems
- Gramlich
- 1991
(Show Context)
Citation Context ...imple termination, however. As a matter of fact, it is known that negative results for the class of non-self-embedding systems do not always carry over to the class of simply terminating systems, see =-=[6]-=-. In this paper we show the undecidability of simple termination for one-rule term rewriting systems. This contradicts a result of Jouannaud and Kirchner [9]. The undecidability proof is based on the ... |

5 |
Type Removal in Term Rewriting
- Zantema
- 1992
(Show Context)
Citation Context ...omewhat simpler construction. We show that the equivalence M halts for all configurations () RM is terminating is easily obtained for all linear bounded automata M by using a recent result of Zantema =-=[18]-=- on type removal. In Section 5 we prove the equivalence RM is terminating () RM is simply terminating for all linear bounded automata M by using the powerful distribution elimination technique of Zant... |

4 | A Note on Simple Termination of Infinite Term Rewriting Systems, report nr
- Ohlebusch
- 1992
(Show Context)
Citation Context ...(F ; R) with F or R finite is simply terminating, as a consequence of Kruskal's Tree Theo3 rem. There exists (infinite) simplifying and terminating TRSs that are not simply terminating, see Ohlebusch =-=[15]-=-. This does not concern us too much as we will deal with decidability issues in the sequel, in which one considers only finite (both with respect to signature and set of rewrite rules) TRSs. Next we p... |

1 |
Corrigendum: Termination of Rewriting, JSC
- Dershowitz
- 1987
(Show Context)
Citation Context ...text C, and a substitution such that t ! + R C[toe]. In the literature various different definitions of the property of TRSs (or rewrite sequences) to be (non-)looping are given (cf. e.g. [16], [17], =-=[4]-=-). The one given above is the most general. One easily shows that (1) every finite non-self-embedding TRS is terminating, (2) every looping TRS is non-terminating, and (3) every simply terminating TRS... |

1 |
Construction d'un Plus Petit Ordre de Simplification
- Jouannaud, Kirchner
- 1984
(Show Context)
Citation Context ...e class of simply terminating systems, see [7]. In this paper we show the undecidability of simple termination for one-rule term rewriting systems. This contradicts a result of Jouannaud and Kirchner =-=[10]-=-. The undecidability proof is based on the ingenious construction of Dauchet. He showed in [2] that with every Turing machine M one can associate a term rewriting system RM consisting of a single rewr... |

1 |
Modularity of Simple Termination of Term Rewriting
- Kurihara, Ohuchi
- 1992
(Show Context)
Citation Context ... The proof is not difficult. This lemma appeared for the first time in Zantema [19], although it is implicit in many earlier works on termination, see Dershowitz [3] for a survey. Kurihara and Ohuchi =-=[12, 13] proved th-=-e related equivalence "a TRS (F ; R) is simplifying () the transitive closure of the rewrite relation associated to the TRS (F ; R) [ Emb (F) is irreflexive". The above lemma facilitates an ... |

1 |
Detecting Looping Simplifications
- Jr
- 1987
(Show Context)
Citation Context ... a context C, and a substitution such that t ! + R C[toe]. In the literature various different definitions of the property of TRSs (or rewrite sequences) to be (non-)looping are given (cf. e.g. [16], =-=[17]-=-, [4]). The one given above is the most general. One easily shows that (1) every finite non-self-embedding TRS is terminating, (2) every looping TRS is non-terminating, and (3) every simply terminatin... |