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The first-order theory of linear one-step rewriting is undecidable

Ralf Treinen
1998 Theoretical Computer Science  
The theory of one-step rewriting for a given rewrite system R and signature C is the firstorder theory of the following structure: its universe consists of all C-ground terms, and its only predicate is  ...  We show that there is no algorithm deciding the 3*V*-fragment of this theory for an arbitrary finite, linear and non-erasing term-rewriting system.  ...  I would like to thank the anonymous referee for pointing out an error in the proof of Theorem 4.2 in an earlier version of the paper.  ... 
doi:10.1016/s0304-3975(98)00083-8 fatcat:dsalacxrzzaxpkfiwxx7afajeu

The Existential Fragment of the One-Step Parallel Rewriting Theory [chapter]

Aleksy Schubert
2009 Lecture Notes in Computer Science  
We show that the firstorder theory of → p is undecidable already for formulae with ∃ 7 prefix and left-linear rewrite systems.  ...  Several decidability results exist for the fragment of the theory in which the formulae start with the ∃ prefix only.  ...  The early theorems in [Tis90] imply that the theory is decidable in the positive case (i.e. for formulae with no negation) in the case of left-linear right-ground systems.  ... 
doi:10.1007/978-3-642-02348-4_6 fatcat:jcqj6xfcefc3ffx2kg3xxi3d4y

Problems in rewriting III [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1995 Lecture Notes in Computer Science  
Acknowledgements We thank all the individuals who contributed questions, updates and solutions. Many thanks to Stefan Blom for substantial help producing this list.  ...  Given an arbitrary finite term rewriting system R, is the first order theory of one-step rewriting (~/~) decidable?  ...  The answer is yes [86, Thm. 3.13] . Problem 63 (M. Oyamaguchi). Is confluence of right-ground term-rewriting systems decidable? Compare [87; 33; 34; 88] . Related is [76] . Problem 65 (D.  ... 
doi:10.1007/3-540-59200-8_82 fatcat:in5aspimbrbvngnrhzcm7tfwei

The Reachability Problem over Infinite Graphs [chapter]

Wolfgang Thomas
2009 Lecture Notes in Computer Science  
The problem has a history of 100 years, and it is central for automatic verification of infinitestate systems. Our focus is on graphs that are presented in terms of word or tree rewriting systems.  ...  We survey classical and selected recent work on the reachability problem over finitely presented infinite graphs.  ...  Even the FO(E * )-theory of a ground tree rewriting graph is decidable ( [8] ).  ... 
doi:10.1007/978-3-642-03351-3_2 fatcat:ocehdjg765ejvbzjlb3qshwxve

Open problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1991 Lecture Notes in Computer Science  
Acknowledgements We thank all the individuals mentioned above who contributed questions, F.-J. de Vries for helping prepare this list, and Ron Book, for going out of his way t o m a k e it possible.  ...  A t e r m t is ground reducible" with respect to a rewrite system R if all its ground variable-free instances contain a redex. Ground reducibility is decidable for ordinary rewriting ! R t 0 2 !  ...  What is the complexity of the decision problem for the con uence of ground variable-free term-rewriting systems? Decidability w as shown in 22, 78 ; see also 23 . Problem 13 J.-J. L evy.  ... 
doi:10.1007/3-540-53904-2_120 fatcat:bsnbx2vghrc2tmv6xcywwijbjq

Page 4757 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
Summary: “We show that the termination of ground term-rewriting systems is decidable in polynomial time. This result is extended to ground rational term-rewriting systems.  ...  As a simple consequence, it is decidable in polynomial time whether there exists a simplification ordering which shows that a ground term rewriting system terminates.”  ... 

Undecidability of the first order theory of one-step right ground rewriting [chapter]

Jerzy Marcinkowski
1997 Lecture Notes in Computer Science  
Is the first order theory of one-step rewriting decidable? For an arbitrary finite term rewriting system R, is the first order theory of one-step rewriting (→ R ) decidable?  ...  Decidability would imply the decidability of the first-order theory of encompassment (that is, being an instance of a subterm) [CCD93], as well as several known decidability results in rewriting.  ...  Undecidability of the first order theory of one-step right ground rewriting. In Comon [Com97], pages 241- 253. [NPR97] Joachim Niehren, Manfred Pinkal, and Peter Ruhrberg.  ... 
doi:10.1007/3-540-62950-5_75 fatcat:ptw6ahlbincw3jkewjqbaly7du

Decidability of the confluence of finite ground term rewrite systems and of other related term rewrite systems

Max Dauchet, Thierry Heuillard, Pierre Lescanne, Sophie Tison
1990 Information and Computation  
The aim of this paper is to propose an algorithm to decide the confluence of finite ground term rewrite systems.  ...  It is well known that the confluence is not decidable for general term rewrite systems, but this paper proves it is for ground term rewrite systems following a conjecture made by Huet and Oppen in their  ...  ACKNOWLEDGMENTS The authors thank Jurgen Avenhaus, Wayne Snyder, Sandor Vagvolgyi, Zoltan Fiilop, and the referees for their useful comments.  ... 
doi:10.1016/0890-5401(90)90015-a fatcat:qvkzgcapyzgq7ndqm4e3kmkf2e

Algorithms and reductions for rewriting problems [chapter]

Rakesh M. Verma, Michael Rusinowitch, Denis Lugiez
1998 Lecture Notes in Computer Science  
We also show that there is a family of string rewrite systems for which the word problem is trivially decidable but con uence undecidable, and we show a linear equational theory with decidable word problem  ...  We then give a polynomial-time algorithm for Unique-normal-form property of ground systems for the rst time.  ...  The authors thank the reviewers and the committee for constructive comments.  ... 
doi:10.1007/bfb0052369 fatcat:a4jqeyr33ve7jpkcunin6vdg6q

More problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1993 Lecture Notes in Computer Science  
Is the first order theory of one-step rewriting (-+R) decidable?  ...  Is confluence of ordered rewriting (using the intersection of one step replacement of equals and a reduction ordering that is total on ground terms) decidable when the (existential fragment of the) ordering  ... 
doi:10.1007/3-540-56868-9_39 fatcat:cibthhywyrgmvcsv73tryp3v2e

Decidable matching for convergent systems [chapter]

Nachum Dershowitz, Subrata Mitra, G. Sivakumar
1992 Lecture Notes in Computer Science  
We describe decision procedures for certain classes of semantic matching problems, where the equational theory with respect to which the semantic matching is performed has a convergent rewrite system.  ...  We give c o u n terexamples to show that semantic matching becomes undecidable as it generally is when the conditions we g i v e are weakened.  ...  Acknowledgement We thank Deepak Kapur and the referees for their helpful comments.  ... 
doi:10.1007/3-540-55602-8_194 fatcat:babf6iidprdpbbuahsbqu6fsqq

Page 1592 of Mathematical Reviews Vol. , Issue 92c [page]

1992 Mathematical Reviews  
It is known that ground confluence is undecidable for both left-linear and right-linear monadic systems. We consider partial results on deciding ground confluence of linear monadic systems.”  ...  Using the Peterson-Stickel algorithm we show that confluence is decidable for right-linear monadic term rewriting systems.  ... 

Page 4630 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
First of all it is pointed out that the first-order theory of one-step rewriting of a fixed rewrite system R becomes decidable when it is restricted to a fixed finite quantifier prefix (Proposition 3).  ...  There is no algo- rithm for deciding the IVVV-theory of one-step rewriting for every finite linear canonical rewrite system. Main Theorem. Part B: Strong undecidability.  ... 

Page 3377 of Mathematical Reviews Vol. , Issue 92f [page]

1992 Mathematical Reviews  
Summary: “The aim of this paper is to propose an algorithm to decide the confluence of finite ground term rewrite systems.  ...  It is well known that the confluence is not de- cidable for general term rewrite systems, but this paper proves that it is decidable for ground term rewrite systems, following a con- jecture made by Huet  ... 

Rewriting Modulo SMT and Open System Analysis [chapter]

Camilo Rocha, José Meseguer, César Muñoz
2014 Lecture Notes in Computer Science  
The proposed technique is illustrated with the formal analysis of a real-time system that is beyond the scope of timed-automata methods.  ...  This paper proposes rewriting modulo SMT, a new technique that combines the power of SMT solving, rewriting modulo theories, and model checking.  ...  The first author would like to thank the National Institute of Aerospace for a short visit supported by the Assurance of Flight Critical System's project of NASA's Aviation Safety Program at Langley Research  ... 
doi:10.1007/978-3-319-12904-4_14 fatcat:v4r2osetzjekpmx2bxnrvazf3y
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