Abstract Rewriting

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Delia KESNER IRIF, CNRS et Universit ´e Paris [email protected] www.irif.fr/˜kesner

Abstract Rewriting

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Formal definition of rewriting

Given a set ofobjectsA, an (abstract) rewriting system is arelationR ⊆A×A.

Example

LetAbe the set of finite sequences over{◦,•,•}. LetRbe the rewriting system given by the three rules{◦•7→•◦,•• 7→••,•◦ 7→ ◦•}. We consider the rewriting predicate t→u(which reads“trewrites tou”), obtained by applying one of the rules ofRto some subsequence oft. Thus for example,

◦•• ◦ •• ◦ ••

↓

• ◦ • ◦ •• ◦ ••

↓

• ◦ •• ◦ • ◦ ••

↓

• ◦ •• ◦ • ◦ ••

↓^∗

• • • ◦ ◦ ◦ • • •

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Closure notions

s→⁺_Rtis the transitive closure of→_R_:

s→⁺_Rtiffs=s0→R s1→R . . .→R sn=tforn≥1.

→⁼_Ris the reflexive closure: s→⁼_Rtiffs=tors→R t.

→^∗_Ris the reflexive transitive closure of→R:

s→^∗_Rtiffs=s0→R s1→R . . .→R sn=tforn≥0.

→ⁿ_Rmeansn-reductions steps:

s→ⁿ_Rtiffs=s0→R s1→R . . .→R sn=tforn≥0.

The objects=s0→R s1→R . . .→R sn=t(forn≥0) is called anR-rewrite sequence.

↔Ris the symmetrique closure:

s↔Rtiffs→R tort→R s.

↔^∗_Ris the reflexive, symetrique and transitive closure:

s↔^∗_Rtiffs=s0↔Rs1↔R. . .↔R sn=tforn≥0.

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More basic vocabulary

A termtisR-reducibleiff there existsss.t.t→R s. A termtisinR-normal formifftis noR-reducible.

A termsisaR-normal form oftifft→^∗_R sandsis inR-normal form.

The symbol7→is used to denote rewriting rules, while→denotes the rewriting predicate.

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Different meaning for equivalent terms

Given

R=











f(x,x) 7→ c

a 7→ b

f(x,b) 7→ d

we can compute from the same termf(a,a)two different normal-formscandd.

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Same meaning for equivalent terms

RisChurch-Rosser (CR)iff for allu,vsuch thatu↔^∗_Rv, there existsssuch that u→^∗_Rsandv→^∗_R s. Graphically,

u ↔

^∗_R

v

&

∗R ∗R

.

s

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Confluence diagrams

A diagram like:

t R₁ u

R₂ R₃

v R₄ s

reads:

for allt,u,vsuch thattR₁uandtR₂v, there existssuch thatuR₃sandvR₄ s.

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Confluence notions

Risconfluent (C)iff

t →^∗_R u

↓∗R ↓∗R

v →^∗_R s

Rislocally confluent (LC)iff

t →_R u

↓R ↓∗R

v →^∗_R s

Risstrongly confluent (SC)iff

t →R u

↓R ↓∗R

v →⁼_R s

Rhas thediamond property (DP)iff

t →R u

↓R ↓R

v →R s

This is a particular case of strongly confluence.

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Equivalent notions

Theorem

RisChurch-RosseriffRisconfluent.

Proof.

On blackboard.

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Sufficient conditions

Theorem

Rhas thediamond propertyimpliesRisconfluent.

Proof.

On blackboard.

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Non-equivalent notions

The following system [Curry]:

R=











c 7→ a

c 7→ d

d 7→ c

d 7→ b

islocally confluentbut notconfluent:

a←c→^∗b

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Termination notions

TheelementsisR-weakly normalising (WN)iffshas at least one normal form.

TheelementsisR- strongly normalising (SN)iff there is no infinite sequence

s=s0→R s1→R . . .iff everyR-reduction sequence starting atsis finite. We note

s∈S NS.

ThesystemRis weakly normalising (WN)iff every element is WN.

ThesystemRterminatesor isstrongly normalising (SN)ornoetherienor well-founded (WF)iff every elementsis SN.

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Weak vs strong normalisation

R=

( f(a) 7→ c f(x) 7→ f(a) The system is weakly normalising but not strongly normalising:

f(b)→ f(a)→c

f(b)→ f(a)→ f(a). . .

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Convergent Systems

Definition

ThesystemRisconvergentiff it is confluent and strongly normalising.

Remark

IfRis confluent, then every element hasat most a normal form.

IfRis convergent, then every element hasone and only one normal form. In this case, we use thefunctionalnotationR(t)to denote theR-normal form oft.

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Confluence from local confluence

Lemma

(Newmann)LetRbe aSNsystem. ThenRis locally confluent iffRis confluent.

Proof.

(By Huet) By well-founded induction ons∈S N.

s → t1 →^∗ tn

↓ LC ↓∗ ↓∗

u1 →^∗ v t1<s ↓∗

↓_∗ u1<s ↓_∗ ↓_∗

um →^∗ w →^∗ p

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Important remark

The following (infinite) system on natural numbers:

R=











2.n 7→ 2.n+1

2.n 7→ a

2.m+1 7→ 2.m+2 2.m+1 7→ b islocally confluentbut notconfluent:a←0→^∗b In fact it is not SN

0→ 1→ 2→ 3→ . . .