Strategy
Dénition : A reduction strategy (one-steps or multi-step) for a
rewriting system R is a function IF : T (X , Σ) 7→ T (X , Σ) s.t.
1. IF (t) = t if t is in R-normal form.
2. t →+ IF (t) otherwise.
Strategies
IF is normalizing i for every WN term t there is no innite
sequence t → IF (t) → IF (IF (t)) → IF (IF (IF (t))) → . . .
In what follows, we will focus only on orthogonal systems.
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Classication
Strategies without history
Innermost
Leftmost-innermost
Parallel-innermost
Leftmost-outermost (standard)
Parallel-outermost
Complete
Strategies with history
Complete development
Standard
Strategies without history
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Innermost Strategy(s)
Leftmost-innermost strategy
The innermost strategy rewrites ONE redex of the set of all the
innermost redexes.
The leftmost-innermost strategy rewrites the leftmost redex of the
set of the innermost redexes.
Example :
Example :

 f (a, x) → x
R
 g(a, x) → a
f (b, x) → b
g(b, x) → x
f (f (a, f (a, b)), g(f (a, b), g(b, a)))
f (f (a, f (a, b)), g(f (a, b), g(b, a)))


f (a, x)




 f (b, x)
R

 g(a, x)




g(b, x)
→ x
→ b
→ a
→ x
f (f (a, f (a, b)), g(f (a, b), g(b, a)))
f (f (a, f (a, b)), g(f (a, b), g(b, a)))
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Parallel-innermost strategy
The parallel-innermost strategy rewrites simultaneously ALL the
innermost redexes.
Remark : There is only one "leftmost-innermost" or
"parallel-innermost" strategy but many "innermost" strategies.
f (f (a, f (a, b)), g(f (a, b), g(b, a)))
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Leftmost-outermost strategy
Parallel-outermost strategy
The leftmost-outermost strategy rewrites the leftmost redex of all
the set of outermost redexes.
The parallel-outermost strategy rewrites simultaneously the set of
ALL the outermost redexes.
f (f (a, f (a, b)), g(f (a, b), g(b, a)))
f (f (a, f (a, b)), g(f (a, b), g(b, a)))
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Complete strategy (denition)
Let R be an orthogonal system. Dene IF (t) = u i t ≫c u, where
Complete strategy
s in R-normal form
The complete strategy rewrites simultaneously all the redexes.
s ≫c s
f (f (a, f (a, b)), g(f (a, b), g(b, a)))
(reexivity)
l → r ∈ R and σ ≫c δ
s1 ≫c t1 . . . . . . sn ≫c tn and s is not a redex
s = f (s1 , . . . , sn ) ≫c f (t1 , . . . , tn )
(head)
σ(l) ≫c δ(r)
(context)
and σ ≫c δ i dom(σ) = dom(δ) and σx ≫c δx ∀x ∈ dom(σ).
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The leftmost-innermost strategy is not normalizing
Example : Let R be the following system :
Are these strategies normalizing ?
R



f (x, b) → d


a



 c
→ b
→ c
f (c, a) → f (c, a) → . . .
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The parallel-innermost strategy is not normalizing
Example : Let R be the following system :
R
In general...



f (x, b) → d


a



 c
Theorem : Let R an orthogonal system. Then the innermost
→ b
strategy is normalizing for R i R is strongly normalizing.
→ c
f (c, a) → f (c, b) → . . .
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Making the leftmost-outermost strategy normalizing
The leftmost-outermost strategy is not normalizing
Example :
Dénition : A term is left normal i all the function symbols



f (x, b) → d


R
a
→ b



 c
→ c
appear before the variables. A system R is left normal i for every
rule l → r ∈ R the term l is left normal.
f (c, a) → f (c, a) → . . .
Theorem : The leftmost-outermost strategy is normalizing for all
Example :
The term f (x, b) is not left normal, the term f (b, x) is left normal.
the orthogonal systems which are left normal.
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Complete Strategy
Parallel Outermost
The complete strategy is normalizing for orthogonal systems.
The parallel-outermost strategy is normalizing for orthogonal
systems.
A (not trivial) proof can be found in the TERESE book.
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Residual theory
Strategies with history
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Accidents
Descendants

 f (x) → h(x, x)
i
 g(a) →ii a
f (g(a)) →i
R1 : I(x) → x
h(g(a), g(a))
Where does the reduction R1 take place in I(I(x)) → I(x) ?
↓ii
h(a, g(a))
↓ii
f (a)
↓ii
→i
h(a, a)
R2 : a → a
Where does the reduction R2 take place in f (a, a) → f (a, a) ?
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Solution
We need to specify the rule which is used, and the position in
which reduction takes place.
Descendants
f (g(a)) →ii,1 f (a)
Let s and s′ be two subterms of u, i.e. u = t[s]p and u = v[s′ ]p′ .
Consider the reduction ρ : u →p′ u′ (i.e. s′ is a redex).
f (g(a)) →i,ϵ h(g(a), g(a)) →ii,1 h(a, g(a))
I(I(x)) →R1 ,ϵ I(x)
f (a, a) →R2 ,1 f (a, a)
The operation p/ρ species the descendants in u′ of the subterm s
at position p of u after the reduction of the redex s′ at position p′
of u. Thus, the descendants are given by a set of positions of u′ .
I(I(x)) →R1 ,1 I(x)
f (a, a) →R2 ,2 f (a, a)
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Example
Formally,
If p and p′ are disjoint : p/ρ = {p} so that s appears inside u′ .
If p and p′ are the same : p/ρ = ∅ so that the subterm s is
erased in u′ .
If p′ > p (s′ is a strict subterm of s) : p/ρ = {p}, s′ is an
argument of s, so that s appears in u′ with a dierent argument.
If p > p′ (s is a strict subterm of s′ ) : p/ρ = the set of all the
positions of s in u′ since s appears n ≥ 0 times in u′ .
All the other redexes in u′ are created.
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

a






b


R
c





f (x, y)




g(x, h(y))
→1 b
→2 c
→3 d
→4 g(y, y)
→5 h(y)
t : g(f (|{z}
c , h(|{z}
a )), h(|{z}
b )) →4 g(g(h(|{z}
a ), h(|{z}
a )), h(|{z}
b )) : t′
|
{z
}
|
{z
}
|
{z
}
|
{z
}
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Descendants and residuals
Remark : Let R an orthogonal system. Descendants of a redex
are always redexes.
The redex a in t is duplicated twice in t′ .
The redex b in t appears once in t′ .
The redex c in t is erased in t′ .
The redex f (c, h(a)) in t has no descendant in t′ .
The redex g(f (c, h(a)), h(b)) in t has a descendant
g(g(h(a), h(a)), h(b)) in t′ .
The redex g(h(a), h(a)) in t′ is created.
Dénition : A redex which is descendant of a redex is a residual,
otherwise it is a created redex.
Remark : Let ρ : t →
u and σ : t → u. Then p/ρ and p/σ are
not necessarily equal. Example : ρ : I(I(x)) →R1 ,ϵ I(x) and
σ : I(I(x)) →R1 ,1 I(x) but ϵ/ρ = ∅ and ϵ/σ = ϵ.
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Residual by a reduction sequence
Dénition : Let s be a subterm of t at position p. The set of
residuals of p w.r.t. the reduction sequence ρ : t →∗ t′ , written
p/∗ ρ, is given by :
Residual of a set of redexes
Let S a set of redexes of t. The set of residuals of S w.r.t. the
reduction ρ : t →s′ t′ is given by
S/ρ =
∪
p/∗ ϵ
p/∗ ρ
= {s}
= s/ρ, if |ρ| = 1
p/∗ δ0 ρ0 = (p/δ0 )/∗ ρ0
s/ρ
s∈S
where /∗ is extended to sets as follows :
S/∗ ρ =
∪
s/∗ ρ
s∈S
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Developments
Complete Developments
A development of a term t is a reduction sequence where no
created redex can be reduced, that is, only residual of the set of
redex of t can be reduced. Formally,
Dénition : Let S
development ρ : t
Dénition :
Let S be the set of all the redexes of t. A development of a term t
is a reduction sequence :
ρ : t →s0 t0 →s1 t1 →s2 t2 . . .
be the set of all the redexes of t. A
is complete i S/∗ ρ = ∅.
→ ∗ t′
Theorem : Let R be an orthogonal system. Then,
The complete development strategy is well dened
(computation of complete developments terminates).
The complete development strategy is normalizing.
such that for every si we have si ∈ S/∗ s0 . . . si−1 .
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Relation with the underlined Reduction
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Example : An example.


a






b


R
c





f (x, y)




g(x, h(y))
Let us consider a new signature
Σ = Σ ∪ {f | f ∈ Σ}
We construct a new rewriting system
R = {f (l1 , . . . , ln ) → r | f (l1 , . . . , ln ) → r ∈ R}
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→1 b
→2 c
→3 d
→4 g(y, y)
→5 h(y)
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R


a






b


Properties of the underlined reduction
→1 b
1. If R is orthogonal, R is also orthogonal (and thus conuent).
→2 c
2. R is strongly normalizing.
c
→3 d





f (x, y)
→4 g(y, y)




g(x, h(y)) →5 h(y)
3. Every development ρ : t →∗ t′ can be projected into an
R-reduction sequence of the same length.
g(f (c, h(a)), h(b)) →4
Corollary : Let R be an orthogonal system and let t be a term.
g(g(h(a), h(a)), h(b)) →5
Then, every development of t is nite.
h(b) →2
Remark : The complete strategy and the complete development
h(c)
strategy are the same.
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