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The Chomsky-Schützenberger Theoremfor Weighted Automata with Storage
Luisa Herrmann Heiko Vogler
Institute of Theoretical Computer Science
Faculty of Computer Science
Technische Universität Dresden
September 2, 2015
Theorem (CS-Theorem) [Chomsky, Schützenberger 63]
L(G ) = h(Dn ∩ R)
G context-free grammar
h homomorphism
Dn n-Dyck language
R regular language
Alternative CS-Theorem [Harrison 78]
L(G ) = h(g−1(D2) ∩ R)
[Weir 88]:
CS-Theorem for string languages generated by tree-adjoining grammars
[Yoshinaka, Kaji, Seki 10]:
CS-Theorem for multiple context-free languages
[Fratani, Voundy 15]:
CS-Theorem for indexed languages
Luisa Herrmann, Heiko Vogler 2/16
Theorem (CS-Theorem) [Chomsky, Schützenberger 63]
L(G ) = h(Dn ∩ R)
Alternative CS-Theorem [Harrison 78]
L(G ) = h(g−1(D2) ∩ R)
[Weir 88]:
CS-Theorem for string languages generated by tree-adjoining grammars
[Yoshinaka, Kaji, Seki 10]:
CS-Theorem for multiple context-free languages
[Fratani, Voundy 15]:
CS-Theorem for indexed languages
Luisa Herrmann, Heiko Vogler 2/16
Theorem (CS-Theorem) [Chomsky, Schützenberger 63]
L(G ) = h(Dn ∩ R)
Alternative CS-Theorem [Harrison 78]
L(G ) = h(g−1(D2) ∩ R)
[Weir 88]:
CS-Theorem for string languages generated by tree-adjoining grammars
[Yoshinaka, Kaji, Seki 10]:
CS-Theorem for multiple context-free languages
[Fratani, Voundy 15]:
CS-Theorem for indexed languages
Luisa Herrmann, Heiko Vogler 2/16
Theorem (CS-Theorem) [Chomsky, Schützenberger 63]
L(G ) = h(Dn ∩ R)
Alternative CS-Theorem [Harrison 78]
L(G ) = h(g−1(D2) ∩ R)
[Weir 88]:
CS-Theorem for string languages generated by tree-adjoining grammars
[Yoshinaka, Kaji, Seki 10]:
CS-Theorem for multiple context-free languages
[Fratani, Voundy 15]:
CS-Theorem for indexed languages
Luisa Herrmann, Heiko Vogler 2/16
Theorem (CS-Theorem) [Chomsky, Schützenberger 63]
L(G ) = h(Dn ∩ R)
Alternative CS-Theorem [Harrison 78]
L(G ) = h(g−1(D2) ∩ R)
[Weir 88]:
CS-Theorem for string languages generated by tree-adjoining grammars
[Yoshinaka, Kaji, Seki 10]:
CS-Theorem for multiple context-free languages
[Fratani, Voundy 15]:
CS-Theorem for indexed languages
Luisa Herrmann, Heiko Vogler 2/16
weighted language: L : Σ∗ → K for some weight algebra K
(power series)
[Salomaa, Soittola 78]:
CS-Theorem for semiring weighted CF languages
[Droste, Vogler 13]:
CS-Theorem for unital valuation monoid weighted CF languages
[Denkinger 15]:
CS-Theorem for strong bimonoid weighted multiple CF languages
our work:
CS-Theorem for weighted iterated pushdown languages
CS-Theorem for weighted automata with storage
Luisa Herrmann, Heiko Vogler 3/16
weighted language: L : Σ∗ → K for some weight algebra K
(power series)
[Salomaa, Soittola 78]:
CS-Theorem for semiring weighted CF languages
[Droste, Vogler 13]:
CS-Theorem for unital valuation monoid weighted CF languages
[Denkinger 15]:
CS-Theorem for strong bimonoid weighted multiple CF languages
our work:
CS-Theorem for weighted iterated pushdown languages
CS-Theorem for weighted automata with storage
Luisa Herrmann, Heiko Vogler 3/16
weighted language: L : Σ∗ → K for some weight algebra K
(power series)
[Salomaa, Soittola 78]:
CS-Theorem for semiring weighted CF languages
[Droste, Vogler 13]:
CS-Theorem for unital valuation monoid weighted CF languages
[Denkinger 15]:
CS-Theorem for strong bimonoid weighted multiple CF languages
our work:
CS-Theorem for weighted iterated pushdown languages
CS-Theorem for weighted automata with storage
Luisa Herrmann, Heiko Vogler 3/16
weighted language: L : Σ∗ → K for some weight algebra K
(power series)
[Salomaa, Soittola 78]:
CS-Theorem for semiring weighted CF languages
[Droste, Vogler 13]:
CS-Theorem for unital valuation monoid weighted CF languages
[Denkinger 15]:
CS-Theorem for strong bimonoid weighted multiple CF languages
our work:
CS-Theorem for weighted iterated pushdown languages
CS-Theorem for weighted automata with storage
Luisa Herrmann, Heiko Vogler 3/16
weighted language: L : Σ∗ → K for some weight algebra K
(power series)
[Salomaa, Soittola 78]:
CS-Theorem for semiring weighted CF languages
[Droste, Vogler 13]:
CS-Theorem for unital valuation monoid weighted CF languages
[Denkinger 15]:
CS-Theorem for strong bimonoid weighted multiple CF languages
our work:
CS-Theorem for weighted iterated pushdown languages
CS-Theorem for weighted automata with storage
Luisa Herrmann, Heiko Vogler 3/16
weighted language: L : Σ∗ → K for some weight algebra K
(power series)
[Salomaa, Soittola 78]:
CS-Theorem for semiring weighted CF languages
[Droste, Vogler 13]:
CS-Theorem for unital valuation monoid weighted CF languages
[Denkinger 15]:
CS-Theorem for strong bimonoid weighted multiple CF languages
our work:
CS-Theorem for weighted iterated pushdown languages
CS-Theorem for weighted automata with storage
Luisa Herrmann, Heiko Vogler 3/16
Table of Contents
Weights
Storage
Weighted automata with storage
Chomsky-Schützenberger result
Luisa Herrmann, Heiko Vogler 4/16
Unital Valuation Monoid
(K ,+, val, 0, 1) [Droste, Vogler 13]
I (K ,+, 0) commutative monoidI val : K ∗ → K
I val(a) = aI val(. . . 0 . . . ) = 0I val(. . . 1 . . . ) = val(. . . . . . ) ignore “1”sI val(ε) = 1
Examples
1) average
2) strong bimonoid (K ,+, ·, 0, 1)I (K ,+, 0) commutative monoidI (K , ·, 1) monoidI 0 absorbing for ·
e.g., semirings, bounded lattices
Luisa Herrmann, Heiko Vogler Weights 5/16
Unital Valuation Monoid
(K ,+, val, 0, 1) [Droste, Vogler 13]
I (K ,+, 0) commutative monoidI val : K ∗ → K
I val(a) = aI val(. . . 0 . . . ) = 0I val(. . . 1 . . . ) = val(. . . . . . ) ignore “1”sI val(ε) = 1
Examples
1) average
2) strong bimonoid (K ,+, ·, 0, 1)I (K ,+, 0) commutative monoidI (K , ·, 1) monoidI 0 absorbing for ·
e.g., semirings, bounded lattices
Luisa Herrmann, Heiko Vogler Weights 5/16
Unital Valuation Monoid
(K ,+, val, 0, 1) [Droste, Vogler 13]
I (K ,+, 0) commutative monoidI val : K ∗ → K
I val(a) = aI val(. . . 0 . . . ) = 0I val(. . . 1 . . . ) = val(. . . . . . ) ignore “1”sI val(ε) = 1
Examples
1) average
2) strong bimonoid (K ,+, ·, 0, 1)I (K ,+, 0) commutative monoidI (K , ·, 1) monoidI 0 absorbing for ·
e.g., semirings, bounded lattices
Luisa Herrmann, Heiko Vogler Weights 5/16
Storage [Scott 67]
storage type S = (C ,P,F ,C0)
I C set (configurations)
I P set of functions p : C → true, false (predicates)
I F set of partial functions f : C → C (instructions)
I C0 subset of C (initial configurations)
Luisa Herrmann, Heiko Vogler Storage 6/16
Storage [Scott 67]
storage type S = (C ,P,F ,C0)
I C set (configurations)
I P set of functions p : C → true, false (predicates)
I F set of partial functions f : C → C (instructions)
I C0 subset of C (initial configurations)
Luisa Herrmann, Heiko Vogler Storage 6/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
ptrue(c) = true
fid(c) = c
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
bottom
(δ c
)= true
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
δ c...
(top = γ) true if γ = δ
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
δ c...
(top = γ)
test(p)
true if γ = δ
p(c)
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
I F ′ = push(γ, f ) | γ ∈ Γ, f ∈ F ∪ stay(γ) | γ ∈ Γ ∪ pop
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
I F ′ = push(γ, f ) | γ ∈ Γ, f ∈ F ∪ stay(γ) | γ ∈ Γ ∪ pop
δ c...
δ c
γ f (c)
...
push(γ, f )
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
I F ′ = push(γ, f ) | γ ∈ Γ, f ∈ F ∪ stay(γ) | γ ∈ Γ ∪ pop
...
γ c...
δ c stay(γ)
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I let S = (C ,P,F ,C0) storage type [Engelfriet 86/14]
define pushdown of S P(S) = (C ′,P ′,F ′,C ′0) by
I C ′ = (Γ × C)+, C ′0 = (Γ × C0) (no empty pushdown)
I P ′ = bottom ∪ (top = γ) | γ ∈ Γ ∪ test(p) | p ∈ P
I F ′ = push(γ, f ) | γ ∈ Γ, f ∈ F ∪ stay(γ) | γ ∈ Γ ∪ pop
...
δ c1...
δ c1
γ c2
pop
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I P(S) = (C ′,P ′,F ′, I ′,E ′) for given S [Engelfriet 86/14]
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I P(S) = (C ′,P ′,F ′, I ′,E ′) for given S [Engelfriet 86/14]
I iterated pushdown storage [Engelfriet 86/14; Engelfriet, Vogler 86]
I P0 = TRIV
I Pn+1 = P(Pn)
Luisa Herrmann, Heiko Vogler Storage 7/16
Examples S = (C ,P,F ,C0)
I TRIV = (c, ptrue, fid, c)
I P(S) = (C ′,P ′,F ′, I ′,E ′) for given S [Engelfriet 86/14]
I iterated pushdown storage [Engelfriet 86/14; Engelfriet, Vogler 86]
I P0 = TRIV
I Pn+1 = P(Pn)
...
· · ·
· · ·· · ·
Luisa Herrmann, Heiko Vogler Storage 7/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
I Q finite set (states), Qf ⊆ Q (final states),q0 ∈ Q (initial state)
I c0 ∈ C0
I T finite set (transitions)
τ = (q1, x , p, q2, f )
I wt : T → K (weight assignment)
q1, q2 ∈ Q, x ∈ Σ ∪ ε, p ∈ P, f ∈ F
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
I Q finite set (states), Qf ⊆ Q (final states),q0 ∈ Q (initial state)
I c0 ∈ C0
I T finite set (transitions)
τ = (q1, x , p, q2, f )
I wt : T → K (weight assignment)
q1, q2 ∈ Q, x ∈ Σ ∪ ε, p ∈ P, f ∈ F
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
I Q finite set (states), Qf ⊆ Q (final states),q0 ∈ Q (initial state)
I c0 ∈ C0
I T finite set (transitions)
τ = (q1, x , p, q2, f )
I wt : T → K (weight assignment)
q1, q2 ∈ Q, x ∈ Σ ∪ ε, p ∈ P, f ∈ F
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
I Q finite set (states), Qf ⊆ Q (final states),q0 ∈ Q (initial state)
I c0 ∈ C0
I T finite set (transitions)
τ = (q1, x , p, q2, f )
I wt : T → K (weight assignment)
q1, q2 ∈ Q, x ∈ Σ ∪ ε, p ∈ P, f ∈ F
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
I Q finite set (states), Qf ⊆ Q (final states),q0 ∈ Q (initial state)
I c0 ∈ C0
I T finite set (transitions)
τ = (q1, x , p, q2, f )
I wt : T → K (weight assignment)
q1, q2 ∈ Q, x ∈ Σ ∪ ε, p ∈ P, f ∈ F
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
τ = (q1, x , p, q2, f )
x y x y z y
q1
c
computation step with τ
(q1, xw , c) `τ (q2,w , f (c))
if p(c) = trueand f (c) defined
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
τ = (q1, x , p, q2, f )
x y x y z y
q1
c
computation step with τ
(q1, xw , c) `τ (q2,w , f (c))
if p(c) = trueand f (c) defined
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
τ = (q1, x , p, q2, f )
x y x y z y
q1
c
p(c)?
computation step with τ
(q1, xw , c) `τ (q2,w , f (c))
if p(c) = trueand f (c) defined
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
τ = (q1, x , p, q2, f )
x y x y z y
q1
c
computation step with τ
(q1, xw , c) `τ (q2,w , f (c))
if p(c) = trueand f (c) defined
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
τ = (q1, x , p, q2, f )
x y x y z y
q2
c
computation step with τ
(q1, xw , c) `τ (q2,w , f (c))
if p(c) = trueand f (c) defined
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
τ = (q1, x , p, q2, f )
x y x y z y
q2
f (c)
computation step with τ
(q1, xw , c) `τ (q2,w , f (c))
if p(c) = trueand f (c) defined
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with StorageS = (C ,P,F ,C0) storage typeΣ alphabetK unital valuation monoid
A = (Q, Σ, c0, q0,Qf ,T ,wt) (S , Σ,K )-automaton
τ = (q1, x , p, q2, f )
x y x y z y
q2
f (c)
computation step with τ
(q1, xw , c) `τ (q2,w , f (c))
if p(c) = trueand f (c) defined
Luisa Herrmann, Heiko Vogler Weighted automata with storage 8/16
Weighted Automata with Storage
computation:
(q0, x1x2 . . . xn, c0) `τ1 (q1, x2 . . . xn, c1) `τ2 . . . `τn (qf , ε, cn)
Θ(w)
weighted language recognized by A: ||A|| : Σ∗ → K
||A||(w) =∑
θ∈Θ(w)
wt(θ)
wt(τ1 . . . τn) = val(wt(τ1) . . .wt(τn))
REC(S , Σ,K )
w = x1 . . . xn ∈ Σ∗θ = τ1 . . . τn, τi ∈ Tqf ∈ Qf
appropriate finitenesscondition!
Luisa Herrmann, Heiko Vogler Weighted automata with storage 9/16
Weighted Automata with Storage
computation:
(q0, x1x2 . . . xn, c0) `τ1 (q1, x2 . . . xn, c1) `τ2 . . . `τn (qf , ε, cn)
Θ(w)
weighted language recognized by A: ||A|| : Σ∗ → K
||A||(w) =∑
θ∈Θ(w)
wt(θ)
wt(τ1 . . . τn) = val(wt(τ1) . . .wt(τn))
REC(S , Σ,K )
w = x1 . . . xn ∈ Σ∗θ = τ1 . . . τn, τi ∈ Tqf ∈ Qf
appropriate finitenesscondition!
Luisa Herrmann, Heiko Vogler Weighted automata with storage 9/16
Weighted Automata with Storage
computation:
(q0, x1x2 . . . xn, c0) `τ1 (q1, x2 . . . xn, c1) `τ2 . . . `τn (qf , ε, cn) Θ(w)
weighted language recognized by A: ||A|| : Σ∗ → K
||A||(w) =∑
θ∈Θ(w)
wt(θ)
wt(τ1 . . . τn) = val(wt(τ1) . . .wt(τn))
REC(S , Σ,K )
w = x1 . . . xn ∈ Σ∗θ = τ1 . . . τn, τi ∈ Tqf ∈ Qf
appropriate finitenesscondition!
Luisa Herrmann, Heiko Vogler Weighted automata with storage 9/16
Weighted Automata with Storage
computation:
(q0, x1x2 . . . xn, c0) `τ1 (q1, x2 . . . xn, c1) `τ2 . . . `τn (qf , ε, cn) Θ(w)
weighted language recognized by A: ||A|| : Σ∗ → K
||A||(w) =∑
θ∈Θ(w)
wt(θ)
wt(τ1 . . . τn) = val(wt(τ1) . . .wt(τn))
REC(S , Σ,K )
w = x1 . . . xn ∈ Σ∗θ = τ1 . . . τn, τi ∈ Tqf ∈ Qf
appropriate finitenesscondition!
Luisa Herrmann, Heiko Vogler Weighted automata with storage 9/16
Weighted Automata with Storage
computation:
(q0, x1x2 . . . xn, c0) `τ1 (q1, x2 . . . xn, c1) `τ2 . . . `τn (qf , ε, cn) Θ(w)
weighted language recognized by A: ||A|| : Σ∗ → K
||A||(w) =∑
θ∈Θ(w)
wt(θ)
wt(τ1 . . . τn) = val(wt(τ1) . . .wt(τn)) REC(S , Σ,K )
w = x1 . . . xn ∈ Σ∗θ = τ1 . . . τn, τi ∈ Tqf ∈ Qf
appropriate finitenesscondition!
Luisa Herrmann, Heiko Vogler Weighted automata with storage 9/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B)
I REC(P1, Σ,K )
quantitative CF languages over Σ and K[Droste, Vogler 13]
I REC(Pn, Σ,B)
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) TRIV = (c, ptrue, fid, c)
(q0, x1x2 . . . xn, c) `τ1 (q1, x2 . . . xn, c) `τ2 . . . `τn (qn, ε, c)
I REC(P1, Σ,K )
quantitative CF languages over Σ and K[Droste, Vogler 13]
I REC(Pn, Σ,B)
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) TRIV = (c, ptrue, fid, c)
(q0, x1x2 . . . xn, c) `τ1 (q1, x2 . . . xn, c) `τ2 . . . `τn (qn, ε, c)
I REC(P1, Σ,K )
quantitative CF languages over Σ and K[Droste, Vogler 13]
I REC(Pn, Σ,B)
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) recognizable languages
(q0, x1x2 . . . xn, c) `τ1 (q1, x2 . . . xn, c) `τ2 . . . `τn (qn, ε, c)
I REC(P1, Σ,K )
quantitative CF languages over Σ and K[Droste, Vogler 13]
I REC(Pn, Σ,B)
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) recognizable languages
I REC(P1, Σ,K )
quantitative CF languages over Σ and K[Droste, Vogler 13]
I REC(Pn, Σ,B)
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) recognizable languages
I REC(P1, Σ,K )
quantitative CF languages over Σ and K[Droste, Vogler 13]
(q0, x1x2 . . . xn, ) `τ1 (q1, x2 . . . xn, ) `τ2 . . . `τn (qn, ε,... )
I REC(Pn, Σ,B)
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) recognizable languages
I REC(P1, Σ,K ) quantitative CF languages over Σ and K[Droste, Vogler 13]
(q0, x1x2 . . . xn, ) `τ1 (q1, x2 . . . xn, ) `τ2 . . . `τn (qn, ε,... )
I REC(Pn, Σ,B)
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) recognizable languages
I REC(P1, Σ,K ) quantitative CF languages over Σ and K[Droste, Vogler 13]
I REC(Pn, Σ,B)
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) recognizable languages
I REC(P1, Σ,K ) quantitative CF languages over Σ and K[Droste, Vogler 13]
I REC(Pn, Σ,B) iterated pushdown languages[Maslov 76; Engelfriet 83; Damm, Goerdt 86]
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Examples for REC(S , Σ,K )
I REC(TRIV, Σ,B) recognizable languages
I REC(P1, Σ,K ) quantitative CF languages over Σ and K[Droste, Vogler 13]
I REC(Pn, Σ,B) iterated pushdown languages[Maslov 76; Engelfriet 83; Damm, Goerdt 86]
Let L : Σ∗ → K .
L is a weighted iterated pushdown language if
L ∈⋃n
REC(Pn, Σ,K )
Luisa Herrmann, Heiko Vogler Weighted automata with storage 10/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid.
If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,
I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]
such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′
I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
h : ∆→ K [Σ ∪ ε]h(δ) = a.y y ∈ Σ ∪ ε
ε-free (S, ∆,B)-automaton A
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′
I separating weightsI separating storage
I decomposition of GSMI composition of g2 and h
M generalized sequentialmachine (GSM)
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2
h′
I separating weightsI separating storageI decomposition of GSM
I composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 11/16
Separating Storage of (S , ∆,B)-automaton A
(q0, ab, c0) (q1, b, f1(c0)) (q2, ε, f2(f1(c0)))
(q0, a, p1, q1, f1)
p1(c0) = true
(q1, b, p2, q2, f2)
p2(f1(c0)) = true
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 12/16
Separating Storage of (S , ∆,B)-automaton A
(q0, ab, c0) (q1, b, f1(c0)) (q2, ε, f2(f1(c0)))
(q0, a, p1, q1, f1)
p1(c0) = true
(q1, b, p2, q2, f2)
p2(f1(c0)) = true
Pf × Ff
(p1, f1)(p2, f2) ∈ (Pf × Ff )∗
B(Pf × Ff , c0)
q0 q1 q2
(p1, f1) / a (p2, f2) / b
GSMM
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 12/16
Separating Storage of (S , ∆,B)-automaton A
(q0, ab, c0) (q1, b, f1(c0)) (q2, ε, f2(f1(c0)))
(q0, a, p1, q1, f1)
p1(c0) = true
(q1, b, p2, q2, f2)
p2(f1(c0)) = true
Pf × Ff
(p1, f1)(p2, f2) ∈ (Pf × Ff )∗
B(Pf × Ff , c0)
q0 q1 q2
(p1, f1) / a (p2, f2) / b
GSMM
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 12/16
Separating Storage of (S , ∆,B)-automaton A
(q0, ab, c0) (q1, b, f1(c0)) (q2, ε, f2(f1(c0)))
(q0, a, p1, q1, f1)
p1(c0) = true
(q1, b, p2, q2, f2)
p2(f1(c0)) = true
Pf × Ff
(p1, f1)(p2, f2) ∈ (Pf × Ff )∗
B(Pf × Ff , c0)
q0 q1 q2
(p1, f1) / a (p2, f2) / b
GSMM
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 12/16
Theorem (CS-Theorem for weighted automata with storage)
Let S = (C ,P,F ,C0) be a storage type, Σ an alphabet, and K a unitalvaluation monoid. If r : Σ∗ → K is (S, Σ,K)-recognizable, then there are
I a regular language R ⊆ Φ∗,I a finite set Ω ⊆ P × F and a configuration c ∈ C ,
I a letter-to-letter morphism g1 : Φ→ Ω, and
I an alphabetic morphism h′ : Φ→ K [Σ ∪ ε]such that r = h′(g−11 (B(Ω, c)) ∩ R).
r : Σ∗ → K
∆∗ ⊇ L(A)
h
B(Ω, c) ⊆ Ω∗ M
R ⊆ Φ∗g1 g2 h′ I separating weights
I separating storageI decomposition of GSMI composition of g2 and h
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 13/16
Instantiated CS-Theorem for (S , Σ,K )-recognizable languages
I S = Pn: K -weighted n-iterated pushdown languages
I S = P1: quantitative CF languages[Droste, Vogler 13]
I S = SC(TRIV): K -weighted stack automata[Greibach 69]
I S = NS(TRIV): K -weighted nested stack automata[Engelfriet, Vogler 86]
I S = MON(M): K -weighted M-automata for somemonoid M [Kambites 07]
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 14/16
Instantiated CS-Theorem for (S , Σ,K )-recognizable languages
I S = Pn: K -weighted n-iterated pushdown languages
I S = P1: quantitative CF languages[Droste, Vogler 13]
I S = SC(TRIV): K -weighted stack automata[Greibach 69]
I S = NS(TRIV): K -weighted nested stack automata[Engelfriet, Vogler 86]
I S = MON(M): K -weighted M-automata for somemonoid M [Kambites 07]
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 14/16
Instantiated CS-Theorem for (S , Σ,K )-recognizable languages
I S = Pn: K -weighted n-iterated pushdown languages
I S = P1: quantitative CF languages[Droste, Vogler 13]
I S = SC(TRIV): K -weighted stack automata[Greibach 69]
I S = NS(TRIV): K -weighted nested stack automata[Engelfriet, Vogler 86]
I S = MON(M): K -weighted M-automata for somemonoid M [Kambites 07]
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 14/16
Instantiated CS-Theorem for (S , Σ,K )-recognizable languages
I S = Pn: K -weighted n-iterated pushdown languages
I S = P1: quantitative CF languages[Droste, Vogler 13]
I S = SC(TRIV): K -weighted stack automata[Greibach 69]
I S = NS(TRIV): K -weighted nested stack automata[Engelfriet, Vogler 86]
I S = MON(M): K -weighted M-automata for somemonoid M [Kambites 07]
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 14/16
Instantiated CS-Theorem for (S , Σ,K )-recognizable languages
I S = Pn: K -weighted n-iterated pushdown languages
I S = P1: quantitative CF languages[Droste, Vogler 13]
I S = SC(TRIV): K -weighted stack automata[Greibach 69]
I S = NS(TRIV): K -weighted nested stack automata[Engelfriet, Vogler 86]
I S = MON(M): K -weighted M-automata for somemonoid M [Kambites 07]
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 14/16
Instantiated CS-Theorem for (S , Σ,K )-recognizable languages
I S = Pn: K -weighted n-iterated pushdown languages
I S = P1: quantitative CF languages[Droste, Vogler 13]
I S = SC(TRIV): K -weighted stack automata[Greibach 69]
I S = NS(TRIV): K -weighted nested stack automata[Engelfriet, Vogler 86]
I S = MON(M): K -weighted M-automata for somemonoid M [Kambites 07]
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 14/16
References I
N. Chomsky and M.-P. Schützenberger (1963). The algebraic theory ofcontext-free languages. Computer Programming and Formal Systems, pp. 118–161.
T. Denkinger (2015). A Chomsky-Schützenberger representation for weightedmultiplecontext-free languages. FSMNLP, accepted for publication.
M. Droste and H. Vogler (2013). The Chomsky-Schützenberger Theorem forQuantitative Context-Free Languages. DLT, pp. 203–214.
J. Engelfriet (2014). Context-Free Grammars with Storage. CoRR abs/1408.0683
S. Fratani and E.M. Voundy (2015). Dyck-based characterizations of indexedlanguages. published on arXiv http://arxiv.org/abs/1409.6112, March 13, 2015.
S. Greibach (1969). Checking automata and one-way stack languages. Journal ofComputer and System Sciences, 3(2), pp. 196–217.
L. Herrmann and H. Vogler (2015). A Chomsky-Schützenberger Theorem forWeighted Automata with Storage. CAI, accepted for publication.
M. Kambites (2009). Formal languages and groups as memory. Communications inAlgebra, 37(1), pp. 193–208.
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 15/16
References II
A. N. Maslov (1976). Multilevel Stack Automata. Probl. Peredachi Inf.,pp. 55–62.
W. Damm and A. Goerdt (1986). An automata-theoretical characterization of theOI-hierarchy. Inform. Control, 71:1-32.
D. Scott (1967). Some definitional suggestions for automata theory. Journal ofComputer and System Sciences 1.2, pp. 187–212.
A. Salomaa and M. Soittola (1978). Automata-theoretic aspects of formal powerseries. Springer New York.
D. J. Weir (1988). Characterizing Mildly Context-Sensitive Grammar Formalisms.PhD thesis, University of Pennsylvania.
R. Yoshinaka, Y. Kaji, and H. Seki (2010). Chomsky-Schützenberger-typecharacterization of multiple context-free languages. LATA, pp. 596-607.
Luisa Herrmann, Heiko Vogler Chomsky-Schützenberger result 16/16
Separating Weights
“⇒”
B = (Q, Σ, c0, q0,Qf ,T ,wt)
τ = (q, x , p, q′, f ) in T
; A = (Q,T , c0, q0,Qf ,T ′)
τ ′ = (q, τ, p, q′, f ) in T ′
h(τ ′) = wt(τ).x
“⇐”
A = (Q, ∆, c0, q0,Qf ,T )
h : ∆→ K [Σ ∪ ε]
(q, δ, p, q′, f ) in T
h(δ) = a.y
; B = (Q ′, Σ, c0, q′0, ∆×Qf ,T ′,wt)
Q ′ = q′0 ∪ ∆×Q
τ = ((δ′, q), y , p, (δ, q′), f ) in T ′
wt(τ) = a
Luisa Herrmann, Heiko Vogler 17/16
Decomposition of GSM
gsmM
τ = (q, x , q′, y)
; regular grammar R,letter-to-letter morphisms h1, h2
q → τq′ rule in R
h1(τ) = x
h2(τ) = y
Luisa Herrmann, Heiko Vogler 18/16