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Shallow confluence of conditional term rewriting systems

Claus-Peter Wirth
2009 Journal of symbolic computation  
Recursion can be conveniently modeled with left-linear positive/ negative-conditional term rewriting systems, provided that nontermination, non-trivial critical overlaps, non-right-stability, nonnormality  ...  For such systems we present novel sufficient criteria for shallow confluence and arrive at the first decidable confluence criterion admitting nontrivial critical overlaps.  ...  Note 2 to Definition 22(5) (Fulfilledness of Condition Literals in QuodLibet) The condition literals in QuodLibet (cf.  ... 
doi:10.1016/j.jsc.2008.05.005 fatcat:zjp4ce43onbfxbt3sv5g5wndoq

Confluence of conditional rewrite systems [chapter]

Nachum Dershowitz, Mitsuhiro Okada, G. Sivakumar
1988 Lecture Notes in Computer Science  
We review known sufficient conditions for confluence, and also prove two new positive results for systems having critical pairs and arbitrarily big terms in conditions.  ...  In this paper, we settle negatively the question whether "joinabihty of critical pairs" is, in general, sufficient for confluence of terminating conditional systems.  ...  A conditional rewrite system is shallow-joinable if each of its critical pairs is.  ... 
doi:10.1007/3-540-19242-5_3 fatcat:2njs65kbzbgm5j2tmbqstea5wq

Deciding confluence of certain term rewriting systems in polynomial time

Guillem Godoy, Ashish Tiwari, Rakesh Verma
2004 Annals of Pure and Applied Logic  
The same approach also shows the decidability of confluence for shallow and linear term rewriting systems.  ...  We present a characterization of confluence for term rewriting systems, which is then refined for special classes of rewriting systems.  ...  Acknowledgements The research of the first author was partially supported by the Spanish CICYT project MAVERISH ref. TIC2001-2476-C03-01.  ... 
doi:10.1016/j.apal.2004.04.005 fatcat:bbegyqguzzdwpdf6dls4k5odo4

Classes of term rewrite systems with polynomial confluence problems

Guillem Godoy, Robert Nieuwenhuis, Ashish Tiwari
2004 ACM Transactions on Computational Logic  
The confluence property of ground (i.e., variable-free) term rewrite systems (TRS) is well-known to be decidable.  ...  In [CGN01]) we gave the first polynomial-time algorithm for deciding the confluence of ground TRS.  ...  Confluence is, of course, a well-studied property for different classes of rewrite systems.  ... 
doi:10.1145/976706.976712 fatcat:v7ikpqerkzb2ndswxwt47wr4ji

On the confluence of lambda-calculus with conditional rewriting

Frédéric Blanqui, Claude Kirchner, Colin Riba
2010 Theoretical Computer Science  
In this paper, we investigate the confluence of \lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic.  ...  Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules.  ...  We show the shallow confluence of β-reduction with βconditional rewriting for these systems, hence extending Theorem 3.12. Our results are summarized in Fig. 1 .  ... 
doi:10.1016/j.tcs.2009.07.058 fatcat:dmgbkp3yszffzcxoz37w2ucqny

Confluence of Shallow Right-Linear Rewrite Systems [chapter]

Guillem Godoy, Ashish Tiwari
2005 Lecture Notes in Computer Science  
We show that confluence of shallow and right-linear term rewriting systems is decidable.  ...  The new decidability result is obtained by building upon our prior work for the class of ground term rewriting systems and shallow linear term rewriting systems.  ...  Confluence of shallow right-linear rewrite systems is decidable. Proof.  ... 
doi:10.1007/11538363_37 fatcat:jg3gfegjjnf35c2dz5hkygphsi

Weakly-non-overlapping non-collapsing shallow term rewriting systems are confluent

Masahiko Sakai, Mizuhito Ogawa
2010 Information Processing Letters  
This paper shows that weakly-non-overlapping, non-collapsing and shallow term rewriting systems are confluent, which is a new sufficient condition on confluence for non-left-linear systems.  ...  A term rewriting system (TRS) is a set R of rewriting rules.  ...  Introduction Confluence, which guarantees the uniqueness of a computation, is an important property for term rewriting systems (TRSs).  ... 
doi:10.1016/j.ipl.2010.06.015 fatcat:wuggmrnmprhltlmhvqna2q6hpq

New Undecidability Results for Properties of Term Rewrite Systems

Rakesh Verma
2012 Electronical Notes in Theoretical Computer Science  
This paper is on several basic properties of term rewrite systems: reachability, joinability, uniqueness of normal forms, unique normalization, confluence, and existence of normal forms, for subclasses  ...  Reachability, joinability and confluence were shown to be decidable for the class of linear, shallow rewrite systems [7], for right-ground systems [8] , and for shallow right-linear systems [6] .  ...  Confluence. Instance: Rewrite system R. Question: Does R have the UN = /UN → /Confluence property? Existence of normal form. Instance: Rewrite system R and term s.  ... 
doi:10.1016/j.entcs.2012.11.012 fatcat:m3oxyvzbnvfovhvxwma25ddub4

The Confluence Property for Petri Nets and its Applications

Ioana Leahu, Ferucio Tiplea
2006 2006 Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing  
Thus, in Section 4 we prove that the confluence problem for Petri nets is reducible to the confluence problem for 2-linear-shallow term rewriting systems.  ...  There is also a very close relationship between confluence for Petri nets and confluence for term rewriting systems.  ...  Confluence of 2-Shallow Term Rewriting Systems Confluence is a widely studied property for term rewriting systems.  ... 
doi:10.1109/synasc.2006.71 dblp:conf/synasc/LeahuT06 fatcat:i2ybzhuxuvgajlgrpvwdl3f73q

Reachability and confluence are undecidable for flat term rewriting systems

Florent Jacquemard
2003 Information Processing Letters  
Ground reachability, ground joinability and confluence are shown undecidable for flat term rewriting systems, i.e. systems in which all left and right members of rule have depth at most one.  ...  Acknowledgements The author wishes to thank Ralf Treinen who has carefully read and commented a preliminary version of this paper, and Harald Ganzinger for his useful suggestions.  ...  Introduction The confluence of a term rewriting systems (TRS) guarantees that every term has at most one normal form.  ... 
doi:10.1016/s0020-0190(03)00310-7 fatcat:jfp55qtfjrgfzini23b4oocquu

A rationale for conditional equational programming

Nachum Dershowitz, Mitsuhiro Okada
1990 Theoretical Computer Science  
Some termination issues are also considered, including the case of rewriting with higher-order terms.  ...  We discuss some aspects of the theory of conditional rewriting, and the reasons underlying certain choices in designing a language based on them.  ...  Let R be a terminating left-linear, conditional rewrite normal system all critical pairs of which are shallow-joinable.  ... 
doi:10.1016/0304-3975(90)90064-o fatcat:rojkwf4kx5avrk7kcs23gkyogi

On the confluence of lambda-calculus with conditional rewriting [article]

Frédéric Blanqui, Claude Kirchner (INRIA Lorraine - LORIA), Colin Riba
2006 arXiv   pre-print
In this paper, we investigate the confluence of lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic.  ...  Second, we go beyond the algebraic framework and get new confluence results using an extended notion of orthogonality that takes advantage of the conditional part of rewrite rules.  ...  In this paper, we study the confluence property of the combination of βreduction with a confluent conditional rewrite system.  ... 
arXiv:cs/0609002v1 fatcat:dtw6coxnj5dgxgxedpn3u5bulq

Deciding Fundamental Properties of Right-(Ground or Variable) Rewrite Systems by Rewrite Closure [chapter]

Guillem Godoy, Ashish Tiwari
2004 Lecture Notes in Computer Science  
Right-(ground or variable) rewrite systems (RGV systems for short) are term rewrite systems where all right hand sides of rules are restricted to be either ground or a variable.  ...  The rewrite closure is used to obtain decidability of the reachability, joinability, termination, and confluence properties of the RGV system R.  ...  Deciding Confluence Theorem 5. The confluence of RGV rewrite systems is decidable. Proof.  ... 
doi:10.1007/978-3-540-25984-8_5 fatcat:kimodwpbo5f5bd2qpqzollxeyq

Non-E-Overlapping, Weakly Shallow, and Non-Collapsing TRSs are Confluent [chapter]

Masahiko Sakai, Michio Oyamaguchi, Mizuhito Ogawa
2015 Lecture Notes in Computer Science  
A term is weakly shallow if each defined function symbol occurs either at the root or in the ground subterms, and a term rewriting system is weakly shallow if both sides of a rewrite rule are weakly shallow  ...  This paper proves that non-E-overlapping, weakly-shallow, and noncollapsing term rewriting systems are confluent by extending reduction graph techniques in our previous work [SO10] with towers of expansions  ...  A further step will be to relax the weakly shallow to the almost weakly shallow condition, which allows at most one occurrence of a defined function symbol in each path from the root to a variable.  ... 
doi:10.1007/978-3-319-21401-6_7 fatcat:zumkwput2rcmdde67jvgfytjfu

On the Confluence of λ-Calculus with Conditional Rewriting [chapter]

Frédéric Blanqui, Claude Kirchner, Colin Riba
2006 Lecture Notes in Computer Science  
In this paper, we investigate the confluence of λ-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic.  ...  Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules.  ...  In this paper, we study the confluence property of the combination of βreduction with a confluent conditional rewrite system.  ... 
doi:10.1007/11690634_26 fatcat:o3tkznadkndfvg4ibpvohdrg6y
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