Formal denition of rewriting

Given a set of objects A, an (abstract) rewriting system is a relation A A. R ⊆ ×

Example : Abstract Rewriting A = the set of nite sequences over , , . {◦ • •}

◦ • →1 • ◦ Rewriting system  = 2 R  • • → • •  • ◦ →3 ◦ •  

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Closure notions

 An -rewrite sequence has the form R s = s0 s1 ... sn = t (for n 0). →R →R →R ≥  s + t is the transitive closure of . → →R  = Ris the reexive closure. Example of -rewrite sequence R →R  ∗ is the reexive transitive closure of . →R →R

 is the symmetrique closure. ↔R  ∗ is the reexive, symetrique and transitive closure. ↔R

3 4 ◦ • • ◦ • • ◦ • • ↓

• ◦ • ◦ • • ◦ • • More basic vocabulary ↓  A term t is -reducible i there exists s s.t. t s. • ◦ • • ◦ • ◦ • • R →R  A term t is in -normal form i it is no -reducible. ↓ R R  A term s is a -normal form of t i t s and s is in R →∗ • ◦ • • ◦ • ◦ • • -normal form. R R ↓ • • ◦ • ◦ • ◦ • • ↓∗ • • • ◦ ◦ ◦ • • • 5 6

Same meaning for equivalent terms Dierent meaning for equivalent terms

Given again is Church-Rosser i f(x, x) c R → =  a b R  u ∗ v  →  f(x, b) d ↔R →  ∗R ∗R we can compute from the same term f(a, a) two dierent results c ↘ ↙ and d. s

7 8 Conuence notions Conuence diagrams  is conuent i R A diagram like : t ∗ u →R

t 1 u ↓∗R ↓∗R R v ∗ s →R R2 R3 v 4 s  is locally conuent i R R reads : t u →R for all t, u, v such that t 1 u and t 2 v, R R ↓R ↓∗R there exist such that and . s u 3 s v 4 s v ∗ s R R →R

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 is strongly conuent i R t u →R

↓R ↓∗R v = s →R Equivalent notions  has the diamond property i R Theorem : is Church-Rosser i is conuent. R R t u →R

↓R ↓R v s →R This is a particular case of strongly conuence.

11 12 Not equivalent notions Termination notions The following system [Curry] :  The element s is -weakly normalising (WN) i s has at least R c a one normal form. →  c d  The element s is - strongly normalising (SN) i there is no = → R  innite sequence i every -reduction R  d c s = s0 s1 ...  →R →R R  → sequence starting at is nite. We note . d b s s SN  → ∈ S   The system is weakly normalising (WN) i every element is  R is locally conuent but not conuent : WN.  The system terminates or is strongly normalising (SN) or R noetherien or well-founded (WF) i every element is SN. a c ∗ b s ← →

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Weak vs strong normalisation Convergent Systems

Dénition : The system is convergent i it is conuent and f(a) c R = → strongly normalising. R  f(x) f(a)  → The system is weakly normalising but not strongly normalising : Remarque : f(b) f(a) c  If is conuent, then every element has at most a normal form. R → →  If is convergent, then every element has one and only one R normal form. In this case, we use the functional notation (t) to f(b) f(a) f(a) ... R → → denote the -normal form of t. R

15 16 Conuence from local conuence Important remark

Lemma : (Newmann) Let be a SN system. Then is The following (innite) system on natural numbers : R R locally conuent i is conuent. 2.n 2.n + 1 R → Proof. (By Huet) By well-founded induction on s SN.  2.n a ∈ =  → R   2.m + 1 2.m + 2 s t′ ∗ t  → → → t′ < s 2.m + 1 b LC  → ↓ u <↓∗ s ↓∗  ′  u′ ∗ v I.H. is locally conuent but not conuent : a 0 b → ↓∗ ← →∗ I.H. In fact it is not SN ↓∗ ↓∗ ↓∗ u ∗ p ∗ p′ → → 0 1 2 3 ... → → → →

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