Post on 01-Feb-2021
The Chomsky-Schützenberger Theorem for
Quantitative Context-Free Languages
Manfred Droste, University of Leipzig, GermanyHeiko Vogler, TU Dresden, Germany
DLT 2013
context-free CF
1/23
G : ρ1 : S → AS A
ρ2 : S → b
ρ3 : A→ a
2/23
G : ρ1 : S → AS A
ρ2 : S → b
ρ3 : A→ a
S → [1ρ1A]1ρ1 [
2ρ1S ]
2ρ1 [
3ρ1A]
3ρ1
S → [ρ2 ]ρ2A→ [ρ3 ]ρ3
2/23
G : ρ1 : S → AS A
ρ2 : S → b
ρ3 : A→ a
S → [1ρ1A]1ρ1 [
2ρ1S ]
2ρ1 [
3ρ1A]
3ρ1
S → [ρ2 ]ρ2A→ [ρ3 ]ρ3
3/23
G : ρ1 : S → AS A
ρ2 : S → b
ρ3 : A→ a
S → [1ρ1A]1ρ1 [
2ρ1S ]
2ρ1 [
3ρ1A]
3ρ1
S → [ρ2 ]ρ2A→ [ρ3 ]ρ3
[1ρ1 [ρ3 ]ρ3 ]1ρ1 [
2ρ1 [ρ2 ]ρ2 ]
2ρ1 [
3ρ1 [ρ3 ]ρ3 ]
3ρ1
3/23
G : ρ1 : S → AS A
ρ2 : S → b
ρ3 : A→ a
S → [1ρ1A]1ρ1 [
2ρ1S ]
2ρ1 [
3ρ1A]
3ρ1
S → [ρ2 ]ρ2A→ [ρ3 ]ρ3
[1ρ1 [ρ3 ]ρ3 ]1ρ1 [
2ρ1 [ρ2 ]ρ2 ]
2ρ1 [
3ρ1 [ρ3 ]ρ3 ]
3ρ1 ∈ D({[
1ρ1 , [
2ρ1 , [
3ρ1 , [ρ2 , [ρ3}︸ ︷︷ ︸Y
)
Dyck-language
3/23
G : ρ1 : S → AS A
ρ2 : S → b
ρ3 : A→ a
S → [1ρ1A]1ρ1 [
2ρ1S ]
2ρ1 [
3ρ1A]
3ρ1
S → [ρ2 ]ρ2A→ [ρ3 ]ρ3
[1ρ1 [ρ3 ]ρ3 ]1ρ1 [
2ρ1 [ρ2 ]ρ2 ]
2ρ1 [
3ρ1 [ρ3 ]ρ3 ]
3ρ1 ∈ D({[
1ρ1 , [
2ρ1 , [
3ρ1 , [ρ2 , [ρ3}︸ ︷︷ ︸Y
)
Dyck-language
[ρ3 ]ρ3 [2ρ1 [ρ3 ]ρ3 ]
2ρ1 [
1ρ1 [ρ3 ]ρ3 ]
1ρ1 ∈ D(Y )
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G : ρ1 : S → AS A
ρ2 : S → b
ρ3 : A→ a
S → [1ρ1A]1ρ1 [
2ρ1S ]
2ρ1 [
3ρ1A]
3ρ1
S → [ρ2 ]ρ2A→ [ρ3 ]ρ3
[1ρ1 [ρ3 ]ρ3 ]1ρ1 [
2ρ1 [ρ2 ]ρ2 ]
2ρ1 [
3ρ1 [ρ3 ]ρ3 ]
3ρ1 ∈ D({[
1ρ1 , [
2ρ1 , [
3ρ1 , [ρ2 , [ρ3}︸ ︷︷ ︸Y
)
Dyck-language
[ρ3 ]ρ3 [2ρ1 [ρ3 ]ρ3 ]
2ρ1 [
1ρ1 [ρ3 ]ρ3 ]
1ρ1 ∈ D(Y )
check local properties: D(Y ) ∩ R
3/23
G : ρ1 : S → AS A
ρ2 : S → b
ρ3 : A→ a
S → [1ρ1A]1ρ1 [
2ρ1S ]
2ρ1 [
3ρ1A]
3ρ1
S → [ρ2 ]ρ2A→ [ρ3 ]ρ3
[1ρ1 [ρ3 ]ρ3 ]1ρ1 [
2ρ1 [ρ2 ]ρ2 ]
2ρ1 [
3ρ1 [ρ3 ]ρ3 ]
3ρ1 ∈ D({[
1ρ1 , [
2ρ1 , [
3ρ1 , [ρ2 , [ρ3}︸ ︷︷ ︸Y
)
Dyck-language
[ρ3 ]ρ3 [2ρ1 [ρ3 ]ρ3 ]
2ρ1 [
1ρ1 [ρ3 ]ρ3 ]
1ρ1 ∈ D(Y )
check local properties: D(Y ) ∩ R
alphabetic morphism:
h : Y ∪ Y → Σ ∪ {ε} h(y) =
b if y = [ρ2a if y = [ρ3ε otherwise
3/23
Theorem [Chomsky, Schützenberger 63]
Let L ⊆ Σ∗.
If L = L(G ) for some CF grammar G ,
then there are
I an alphabet Y ,
I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → Σ ∪ {ε}
such that L = h′(D(Y ) ∩ R).
Moreover, for every w ∈ L(G ):
|DG (w)| = |{u ∈ D(Y ) ∩ R | h′(u) = w}|
(degree of ambiguity)
4/23
Theorem [Chomsky, Schützenberger 63]
Let L ⊆ Σ∗.
If L = L(G ) for some CF grammar G ,
then there are
I an alphabet Y ,
I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → Σ ∪ {ε}
such that L = h′(D(Y ) ∩ R).
Moreover, for every w ∈ L(G ):
|DG (w)| = |{u ∈ D(Y ) ∩ R | h′(u) = w}|
(degree of ambiguity)
4/23
Theorem [Chomsky, Schützenberger 63]
Let L ⊆ Σ∗.
If L = L(G ) for some CF grammar G ,
then there are
I an alphabet Y ,
I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → Σ ∪ {ε}
such that L = h′(D(Y ) ∩ R).
Moreover, for every w ∈ L(G ):
|DG (w)| = |{u ∈ D(Y ) ∩ R | h′(u) = w}|
(degree of ambiguity)
L : Σ∗ → N; w 7→ |DG (w)|
4/23
weighted languages (formal power series) L : Σ∗ → K
(K ,+, ·, 0, 1) semiringI (K ,+, 0) commutative monoid
I (K , ·, 1) monoidI · distributes over +I 0 is absorbing for ·, i.e., a · 0 = 0 · a = 0
Examples:
1. natural numbers (N,+, ·, 0, 1)2. Boolean semiring ({true, false},∨,∧, false, true)3. distributive bounded lattices (L,∨,∧, 0, 1)4. fields
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weighted CF languages over semiring (K ,+, ·, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation: wt(ρ1 . . . ρn) = wt(ρ1) · . . . · wt(ρn)
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of
a weighted CF grammar
G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
6/23
weighted CF languages over semiring (K ,+, ·, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation: wt(ρ1 . . . ρn) = wt(ρ1) · . . . · wt(ρn)
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of
a weighted CF grammar
G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
6/23
weighted CF languages over semiring (K ,+, ·, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation: wt(ρ1 . . . ρn) = wt(ρ1) · . . . · wt(ρn)
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of
a weighted CF grammar
G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
6/23
weighted CF languages over semiring (K ,+, ·, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation: wt(ρ1 . . . ρn) = wt(ρ1) · . . . · wt(ρn)
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
6/23
weighted CF languages over semiring (K ,+, ·, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation: wt(ρ1 . . . ρn) = wt(ρ1) · . . . · wt(ρn)
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
6/23
Theorem [Chomsky, Schützenberger 63] (revisited)
Let L : Σ∗ → N and (N,+, ·, 0, 1) natural number semiring.
If L = L(G ) for weighted CF grammar G with wt(p) = 1for every p ∈ P,
then there are
I an alphabet Y ,
I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → Σ ∪ {ε}
such that L = h′(char(D(Y ) ∩ R))
7/23
Theorem [Salomaa, Soittola 78]
Let L : Σ∗ → K and (K ,+, ·, 0, 1) commutative semiring.
The following are equivalent:
1. L is weighted CF language
2. there areI an alphabet Y ,I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → KΣ∪{ε}
such that L = h′(char(D(Y ) ∩ R))
8/23
quantitative languages L : Σ∗ → K “average”[Chatterjee, Doyen, Henzinger 10]
(K ,+, val, 0) valuation monoid [Droste, Meinecke 10]
I (K ,+, 0) commutative monoid
I val : K+ → K , val(a) = a, val(. . . 0 . . .) = 0
9/23
quantitative languages L : Σ∗ → K “average”[Chatterjee, Doyen, Henzinger 10]
(K ,+, val, 0) valuation monoid [Droste, Meinecke 10]
I (K ,+, 0) commutative monoid
I val : K+ → K , val(a) = a, val(. . . 0 . . .) = 0
Examples:
1. (R ∪ {−∞}, sup, avg,−∞) avg(a1 . . . an) = 1n ·∑n
i=1 ai
2. strong bimonoid (K ,+, ·, 0, 1)I (K ,+, 0) commutative monoidI (K , ·, 1) monoidI 0 absorbing for ·
e.g., semirings, bounded lattices
9/23
quantitative languages L : Σ∗ → K “average”[Chatterjee, Doyen, Henzinger 10]
(K ,+, val, 0) valuation monoid [Droste, Meinecke 10]
I (K ,+, 0) commutative monoid
I val : K+ → K , val(a) = a, val(. . . 0 . . .) = 0
Examples:
1. (R ∪ {−∞}, sup, avg,−∞) avg(a1 . . . an) = 1n ·∑n
i=1 ai
2. strong bimonoid (K ,+, ·, 0, 1)I (K ,+, 0) commutative monoidI (K , ·, 1) monoidI 0 absorbing for ·
e.g., semirings, bounded lattices
9/23
quantitative languages L : Σ∗ → K “average”[Chatterjee, Doyen, Henzinger 10]
(K ,+, val, 0
, 1
)
unital
valuation monoid
[Droste, Vogler 13]
I (K ,+, 0) commutative monoid
I val : K+ → K , val(a) = a, val(. . . 0 . . .) = 0
I val(. . . 1 . . .) = val(. . . . . .) ignore “1”s
I val(ε) = 1
10/23
quantitative languages L : Σ∗ → K “average”[Chatterjee, Doyen, Henzinger 10]
(K ,+, val, 0, 1) unital valuation monoid [Droste, Vogler 13]
I (K ,+, 0) commutative monoid
I val : K ∗ → K , val(a) = a, val(. . . 0 . . .) = 0I val(. . . 1 . . .) = val(. . . . . .) ignore “1”s
I val(ε) = 1
10/23
quantitative languages L : Σ∗ → K “average”[Chatterjee, Doyen, Henzinger 10]
(K ,+, val, 0, 1) unital valuation monoid [Droste, Vogler 13]
I (K ,+, 0) commutative monoid
I val : K ∗ → K , val(a) = a, val(. . . 0 . . .) = 0I val(. . . 1 . . .) = val(. . . . . .) ignore “1”s
I val(ε) = 1
Example:
(R ∪ {−∞}, sup, avg,−∞) avg(a1 . . . an) = 1n ·∑n
i=1 ai
(R ∪ {−∞,+∞}, sup, avg,−∞,+∞)
avg(. . .+∞ . . .) = avg(. . . . . .)
10/23
quantitative languages L : Σ∗ → K “average”[Chatterjee, Doyen, Henzinger 10]
(K ,+, val, 0, 1) unital valuation monoid [Droste, Vogler 13]
I (K ,+, 0) commutative monoid
I val : K ∗ → K , val(a) = a, val(. . . 0 . . .) = 0I val(. . . 1 . . .) = val(. . . . . .) ignore “1”s
I val(ε) = 1
10/23
quantitative languages L : Σ∗ → K “average”[Chatterjee, Doyen, Henzinger 10]
(K ,+, val, 0, 1) unital valuation monoid [Droste, Vogler 13]
I (K ,+, 0) commutative monoid
I val : K ∗ → K , val(a) = a, val(. . . 0 . . .) = 0I val(. . . 1 . . .) = val(. . . . . .) ignore “1”s
I val(ε) = 1
In general, val is
I not commutative
I not associative
I not distributive over +
10/23
weighted CF languages over semiring (K ,+, ·, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation:wt(ρ1 . . . ρn) = wt(ρ1) · . . . · wt(ρn)
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
11/23
quantitative CF languages over unital val. monoid (K ,+, val, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation:wt(ρ1 . . . ρn) = wt(ρ1) · . . . · wt(ρn)
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
12/23
quantitative CF languages over unital val. monoid (K ,+, val, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation:wt(ρ1 . . . ρn) = wt(ρ1) · . . . · wt(ρn)
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
13/23
quantitative CF languages over unital val. monoid (K ,+, val, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation:wt(ρ1 . . . ρn) = val(wt(ρ1) . . .wt(ρn))
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
14/23
quantitative CF languages over unital val. monoid (K ,+, val, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation:wt(ρ1 . . . ρn) = val(wt(ρ1) . . .wt(ρn))
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I weighted language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
15/23
quantitative CF languages over unital val. monoid (K ,+, val, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation:wt(ρ1 . . . ρn) = val(wt(ρ1) . . .wt(ρn))
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I quantitative language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a weighted CF language:∃ weighted CF grammar G over semiring K : L = ||G ||.
16/23
quantitative CF languages over unital val. monoid (K ,+, val, 0, 1):
I CF grammar with weights G = (N,Σ,Z ,P,wt)wt : P → K
I weight of a derivation:wt(ρ1 . . . ρn) = val(wt(ρ1) . . .wt(ρn))
I weighted CF grammar:I CF grammar with weights andI K is complete or DG (w) is finite for every w ∈ Σ∗
I quantitative language of a weighted CF grammar G :
||G || : Σ∗ → K , w 7→∑
d∈DG (w) wt(d)
I L : Σ∗ → K is a quantitative CF language:∃ weighted CF gr. G over unital val. monoid K : L = ||G ||.
17/23
... in a very similar way define:
weighted pushdown automaton over unital valuation monoid
18/23
... in a very similar way define:
weighted pushdown automaton over unital valuation monoid
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. L is quantitative behaviour of a weighted PDA.
18/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
19/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
19/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
h′ : ∆∗ → KΣ∗
h′(δ1 . . . δn) = val(a1 . . . an) . x1 . . . xn if h(δi ) = ai .xi
19/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
h′ : ∆∗ → KΣ∗
h′(δ1 . . . δn) = val(a1 . . . an) . x1 . . . xn if h(δi ) = ai .xi
L ⊆ ∆∗h′(L) =
∑v∈L h
′(v)
19/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
h′ : ∆∗ → KΣ∗
h′(δ1 . . . δn) = val(a1 . . . an) . x1 . . . xn if h(δi ) = ai .xi
L ⊆ ∆∗h′(L) =
∑v∈L h
′(v) if (h′(v) | v ∈ L) locally finiteor K complete.
19/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
20/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
3. there areI an alphabet Y ,I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → KΣ∪{ε}
such that L = h′(D(Y ) ∩ R)
21/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
3. there areI an alphabet Y ,I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → KΣ∪{ε}
such that L = h′(D(Y ) ∩ R)
21/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
2. there areI an alphabet ∆,I an unambiguous CF grammar G over ∆, andI an alphabetic morphism h : ∆→ KΣ∪{ε}
such that L = h′(L(G ))
3. there areI an alphabet Y ,I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → KΣ∪{ε}
such that L = h′(D(Y ) ∩ R)
thanks
21/23
22/23
Theorem [Droste, Vogler 13]
Let L : Σ∗ → K and (K ,+, val, 0, 1) unital val. monoid.The following are equivalent:
1. L is quantitative CF language
3. there areI an alphabet Y ,I a recognizable language R over Y ∪ Y , andI an alphabetic morphism h : Y ∪ Y → KΣ∪{ε}
such that L = h′(D(Y ) ∩ R)4. there are
I an alphabet Y ,I a deterministically recognizable series r ∈ K (Y∪Y )∗ , andI an alphabetic morphism h : Y ∪ Y → Σ ∪ {ε}
such that L = h′(D(Y ) ∩ r)
(D(Y ) ∩ r) ∈ K (Y∪Y )∗ w 7→{
(r ,w) if w ∈ D(Y )0 otherwise
23/23