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