39:
752:
if and only if all critical pairs are convergent. Thus, to find out if a term rewriting system is weakly confluent, it suffices to test all critical pairs and see if they are convergent. This makes it possible to find out algorithmically if a term rewriting system is weakly confluent or not, given
167:
The actual definition of a critical pair is slightly more involved as it excludes pairs that can be obtained from critical pairs by substitution and orients the pair based on the overlap. Specifically, for a pair of overlapping rules
272:
219:
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111:(lower row, left and right), respectively. The latter two terms form the critical pair. They can be eventually rewritten to a common term, if the rewrite rule set is
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364:
344:
145:
for which two different applications of a rewrite rule (either the same rule applied differently, or two different rules) yield the terms
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32:
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224:
171:
435:
849:
741:, the critical pair may not converge, so critical pairs are potential sources where confluence will fail.
396:
124:
28:
59:
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have a common reduct and thus the critical pair is convergent. If the term rewriting system is not
822:
785:
499:
When both sides of the critical pair can reduce to the same term, the critical pair is called
503:. Where one side of the critical pair is identical to the other, the critical pair is called
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17:
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105:
90:
73:
67:
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127:
when two rewrite rules overlap to yield two different terms. In more detail, (
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721:
Confluence clearly implies convergent critical pairs: if any critical pair ⟨
635:
As another example, consider the term rewriting system with the single rule
38:
854:
42:
Triangle diagram of a critical pair obtained from two rewrite rules
753:
that one can algorithmically check if two terms converge.
821:. Cambridge, UK: Cambridge University Press. p. 53.
669:
By applying this rule in two different ways to the term
27:
This article is about terms resulting from overlaps in
612:)⟩. Both of these terms can be derived from the term
515:
For example, in the term rewriting system with rules
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765:, an algorithm based on critical pairs to compute a
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96:(lower row, middle) can be rewritten to the term
773:term rewriting system equivalent to a given one
8:
432:that are most general, the critical pair is
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267:{\displaystyle \rho _{1}:l_{1}\to r_{1}}
214:{\displaystyle \rho _{0}:l_{0}\to r_{0}}
141:) is a critical pair if there is a term
37:
809:
748:states that a term rewriting system is
489:{\displaystyle (D\sigma ,r_{0}\sigma )}
632:) by applying a single rewrite rule.
7:
853:, Cambridge University Press, 1998
425:{\displaystyle s\sigma =l_{1}\tau }
25:
750:weakly (a.k.a. locally) confluent
713:)) is a (trivial) critical pair.
366:(that is not a variable) matches
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448:
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333:
329:
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274:, with the overlap being that
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1:
85:. The resulting overlay term
850:Term Rewriting and All That
588:the only critical pair is ⟨
886:
26:
393:under some substitutions
763:Knuth–Bendix completion
309:{\displaystyle l_{0}=D}
819:Term rewriting systems
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50:(upper row, left) and
31:. For other uses, see
29:term rewriting systems
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386:{\displaystyle l_{1}}
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125:term rewriting system
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18:Critical pair (logic)
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746:critical pair lemma
717:Critical pair lemma
316:for some non-empty
786:Weisstein, Eric W.
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870:Rewriting systems
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359:{\displaystyle s}
339:{\displaystyle D}
16:(Redirected from
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689:), we see that (
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789:"Critical Pair"
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729:⟩ arises, then
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828:0-521-39115-6
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121:critical pair
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58:(right). The
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53:
49:
46: →
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40:
34:
33:Critical pair
30:
19:
848:
841:Franz Baader
818:
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792:
745:
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166:
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135:
128:
123:arises in a
120:
118:
107:
101:
97:
92:
86:
82:
77:
70:
60:substitution
55:
51:
47:
43:
771:terminating
163:Definitions
804:References
501:convergent
794:MathWorld
767:confluent
739:confluent
481:σ
462:τ
446:σ
420:τ
404:σ
252:→
230:ρ
199:→
177:ρ
113:confluent
864:Category
757:See also
511:Examples
505:trivial
318:context
68:subterm
64:unifies
825:
74:|
81:with
843:and
823:ISBN
769:and
744:The
733:and
221:and
152:and
100:and
66:the
701:),
600:),
866::
847:,
791:.
725:,
685:),
661:.
657:→
628:),
580:,
576:→
557:)
545:→
538:),
507:.
496:.
159:.
134:,
119:A
98:tσ
62:σ
831:.
797:.
735:b
731:a
727:b
723:a
711:x
709:,
707:x
705:(
703:f
699:x
697:,
695:x
693:(
691:f
687:x
683:x
681:,
679:x
677:(
675:f
673:(
671:f
659:x
654:)
652:y
650:,
648:x
646:(
644:f
630:z
626:y
624:,
622:x
620:(
618:g
616:(
614:f
610:z
608:,
606:x
604:(
602:f
598:z
596:,
594:x
592:(
590:g
578:x
573:)
571:y
569:,
567:x
565:(
563:g
555:z
553:,
551:x
549:(
547:g
542:)
540:z
536:y
534:,
532:x
530:(
528:g
526:(
524:f
484:)
476:0
472:r
468:,
465:]
457:1
453:r
449:[
443:D
440:(
415:1
411:l
407:=
401:s
379:1
375:l
354:s
334:]
330:[
327:D
304:]
301:s
298:[
295:D
292:=
287:0
283:l
260:1
256:r
247:1
243:l
239::
234:1
207:0
203:r
194:0
190:l
186::
181:0
157:2
154:t
150:1
147:t
143:t
139:2
136:t
132:1
129:t
115:.
108:p
104:σ
102:s
93:p
89:σ
87:s
83:l
78:p
71:s
56:r
54:→
52:l
48:t
44:s
35:.
20:)
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