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

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. ...

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

2009
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/978-3-642-02348-4_6
fatcat:jcqj6xfcefc3ffx2kg3xxi3d4y
*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*. ...##
###
Problems in rewriting III
[chapter]

1995
*
Lecture Notes in Computer Science
*

Acknowledgements We thank all

doi:10.1007/3-540-59200-8_82
fatcat:in5aspimbrbvngnrhzcm7tfwei
*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. ...##
###
The Reachability Problem over Infinite Graphs
[chapter]

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] ). ...

##
###
Open problems in rewriting
[chapter]

1991
*
Lecture Notes in Computer Science
*

Acknowledgements We thank all

doi:10.1007/3-540-53904-2_120
fatcat:bsnbx2vghrc2tmv6xcywwijbjq
*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. ...##
###
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]

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. ...

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

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. ...

##
###
Algorithms and reductions for rewriting problems
[chapter]

1998
*
Lecture Notes in Computer Science
*

We also show that there

doi:10.1007/bfb0052369
fatcat:a4jqeyr33ve7jpkcunin6vdg6q
*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. ...##
###
More problems in rewriting
[chapter]

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 ...

##
###
Decidable matching for convergent systems
[chapter]

1992
*
Lecture Notes in Computer Science
*

We describe decision procedures for certain classes

doi:10.1007/3-540-55602-8_194
fatcat:babf6iidprdpbbuahsbqu6fsqq
*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. ...##
###
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]

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 ...

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