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The Undecidability of the First-Order Theories of One Step Rewriting in Linear Canonical Systems

Sergei Vorobyov
2002 Information and Computation  
We also present a fixed such system with undecidable ∃∀ * -theory of one step rewriting (strong undecidability). This improves over all previously known results of the same kind.  ...  By reduction from the halting problem for Minsky's two-register machines we prove that there is no algorithm capable of deciding the ∃∀∀∀-theory of one step rewriting of an arbitrary finite linear confluent  ...  (4) is true in the first-order theory of one step rewriting generated by R.  ... 
doi:10.1006/inco.2002.3151 fatcat:hkrbegpopnextj4bdl6la35apq

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

2003 Mathematical Reviews  
(S-UPPS-IT:; Uppsala) The undecidability of the first-order theories of one step rewriting in linear canonical systems. (English summary) Inform. and Comput. 175 (2002), no. 2, 182-213.  ...  There exists a finite linear canonical rewrite system for which the 4V*-theory of one-step rewriting is undecidable.  ... 

Page 5456 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
Let C, be the set of all canonical term-rewriting systems.  ...  Usually, one considers simplification orderings, which were first introduced by Higman in 1952, and can be proved well- founded by an argument based on Kruskal’s theorem.  ... 

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?  ...  One might also ask whether the first-order theory of total recursive path orderings is decidable.  ... 
doi:10.1007/3-540-59200-8_82 fatcat:in5aspimbrbvngnrhzcm7tfwei

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

On the unification problem for Cartesian closed categories

Paliath Narendran, Frank Pfenning, Richard Statman
1997 Journal of Symbolic Logic (JSL)  
We show that the unification problem for this theory is undecidable, thus settling an open question.  ...  It also has potential applications to the problem of polymorphic type inference and polymorphic higher-order unification, which in turn is relevant to theorem proving and logic programming.  ...  Both theories have canonical rewriting systems modulo associativity and commutativity (of ×) which can be obtained by orienting the remaining equations left to right. 2.  ... 
doi:10.2307/2275552 fatcat:2ftrbhvubfefddjdo47f6hi7le

Contributions of Ronald V. Book to the theory of string-rewriting systems

Robert McNaughton
1998 Theoretical Computer Science  
Part of our appreciation of the impact that he had on the field of rewriting systems was what these students and post-dots went on to do after they left Santa Barbara.  ...  He was, in effect, the leader of a group that included all or most of these.  ...  in recent rewriting theory.  ... 
doi:10.1016/s0304-3975(98)00053-x fatcat:wgpelyippzbfbo2fkyqzw3yb4y

Page 6185 of Mathematical Reviews Vol. , Issue 91K [page]

1991 Mathematical Reviews  
Many decision problems that are undecidable in general can be solved with various degrees of complexity for various classes of finite canonical string-rewriting systems.  ...  Then we give a simple and practical algorithm, based on solving systems of linear inequalities, for determining whether or not a set of rules can be ordered by a Knuth-Bendix ordering.  ... 

Page 4866 of Mathematical Reviews Vol. , Issue 2004f [page]

2004 Mathematical Reviews  
We also show that there is a family of string rewrite systems for which the word problem is triv- ially decidable but confluence is undecidable, and we show a linear equational theory with decidable word  ...  Depending on the success of the application of a team to the actual state of the system, it is spec- ified which teams are allowed to be applied in the next step.  ... 

On ground-confluence of term rewriting systems

Deepak Kapur, Paliath Narendran, Friedrich Otto
1990 Information and Computation  
On the other hand, if all right-hand sides of rules of a finite Noetherian rewriting system R are either ground terms or proper subterms of the corresponding left-hand sides. then it is decidable whether  ...  This undecidability result holds even when systems involving only unary function symbols and one constant are being considered, or when leftlinear or right-linear monadic systems are being considered.  ...  Book for helpful discussions at an early stage of this work and the referees for useful suggestions.  ... 
doi:10.1016/0890-5401(90)90023-b fatcat:nqqwr2x6gjdt3nxclxfm4rpffq

Completeness of Unfolding for Rewriting Logic Theories

Mari Alpuente, M Baggi, M Falaschi, D Ballis
2010 2010 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing  
More precisely, our methodology identifies the sources of incompleteness and derives a set of rules that are added to the transformed rewrite theory in order to preserve the semantics of the original theory  ...  In this paper, we investigate the semantic properties of a narrowing-based unfolding transformation that is useful to transform rewriting logic theories.  ...  In [2] we presented an implementation of the Code Carrying Theory methodology based of a Fold/Unfold transformation system for rewrite theories.  ... 
doi:10.1109/synasc.2010.32 dblp:conf/synasc/AlpuenteBFB10 fatcat:666tzbrfgnbs5oa365zo7g2qcq

Computing with rewrite systems

Nachum Dershowitz
1985 Information and Control  
We show how, in addition, restricted forms of the Knuth-Bendix "completion" procedure may be used to interpret logic programs written as a set of equivalence-preserving rewrite rules.  ...  Term-rewriting systems, that is, sets of directed equations, provide a paradigm of computation with particularly simple syntax and semantics.  ...  sincerely thank Leo Bachmair, Steve Crocker, Jieh Hsiang, Alan Josephson, Yves Kodratoff, Emmanuel Kounalis, Leo Marcus, David Plaisted, Jean-Luc R6my, and Ehud Shapiro for many stimulating discussions, and the  ... 
doi:10.1016/s0019-9958(85)80003-6 fatcat:apkrzxf6end77lxuy5ho6gnbd4

Equations and Rewrite Rules: A Survey [chapter]

Gérard Huet, Derek C. Oppen
1980 Formal Language Theory  
We restrict ourselves to first-order equations, and do not treat equations which define non-terminating computations or recent work on rewrite rules applied to equational congruence classes.  ...  In addition to covering these topics, we discuss the problem of "solving*' equations (the "unification" problem), the problem of proving termination of sets of rewrite rules, and the decidability and complexity  ...  A Meta-unification Algorithm In this section, we show how we can do unification in theories that admit a canonical term rewriting system.  ... 
doi:10.1016/b978-0-12-115350-2.50017-8 fatcat:sh4rrqoqxvecvdslxyjtr3l5v4

Multiple Congruence Relations, First-Order Theories on Terms, and the Frames of the Applied Pi-Calculus [chapter]

Florent Jacquemard, Étienne Lozes, Ralf Treinen, Jules Villard
2012 Lecture Notes in Computer Science  
We investigate the problem of deciding first-order theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms.  ...  As motivation and application, we show how to translate the model-checking problem of AπL, a spatial equational logic for the applied pi-calculus, to the validity of first-order formulas in term algebras  ...  Let Σ, (E i ) i∈I , (L j ) j∈J be as in Def. 1, and R be a canonical, left- linear rewrite system.  ... 
doi:10.1007/978-3-642-27375-9_10 fatcat:plkghzuw35dnrjaog5m3uteq7e

A Collaborative Framework for Non-Linear Integer Arithmetic Reasoning in Alt-Ergo

Sylvain Conchon, Mohamed Iguernelala, Alain Mebsout
2013 2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing  
The first component is used to handle equalities of linear integer arithmetic and associativity and commutativity properties of non-linear multiplication.  ...  The interval calculus component is used -in addition to standard linear operations over inequalities -to refine bounds of non-linear terms and to inform the SAT solver about judicious case-splits on bounded  ...  In order to integrate can in ground AC completion, we adapt the notion of ground AC-rewriting to cope with canonizers.  ... 
doi:10.1109/synasc.2013.29 dblp:conf/synasc/ConchonIM13 fatcat:57jfxmg7ubfddgjhgsmp7kwx3a
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