Parikh Image of Pushdown Automata Elena Guti errez and Pierre Ganty - - PowerPoint PPT Presentation

Parikh Image of Pushdown Automata

Elena Guti´ errez and Pierre Ganty

Pushdown Automata Context-free Grammars

Introduction

(PDAs) (CFGs) P G

L(P) = L(G)

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Context-free Languages (CFLs)

Pushdown Automata Context-free Grammars (PDAs) (CFGs)

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P G Context-free Languages

Introduction

(CFLs)

PDA2CFG Pushdown Automata Context-free Grammars (PDAs) (CFGs)

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P G Context-free Languages

Introduction

(CFLs)

PDAs and CFGs

q

X a b Z Pushdown Automata Context-free Grammar

S ⇒ aSa ⇒ abSba ⇒ . . . ⇒ abaaba

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PDAs and CFGs

q

X a b Z Pushdown Automata Context-free Grammar

S ⇒ aSa ⇒ abSba ⇒ . . . ⇒ abaaba

PDA2CFG

PDA PDA2CFG CFG n states p s.s.

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PDAs and CFGs

q

X a b Z Pushdown Automata Context-free Grammar

S ⇒ aSa ⇒ abSba ⇒ . . . ⇒ abaaba

PDA2CFG

PDA PDA2CFG V = {[q X q′] | q, q′ ∈ Q, X ∈ Γ} CFG n states p s.s.

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PDAs and CFGs

q

X a b Z Pushdown Automata Context-free Grammar

S ⇒ aSa ⇒ abSba ⇒ . . . ⇒ abaaba

PDA2CFG

PDA PDA2CFG V = {[q X q′] | q, q′ ∈ Q, X ∈ Γ} CFG n states p s.s.

|V | = n2p + 1 3

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PDA2CFG PDAs CFGs P G CFLs

Introduction

4 Goldstine et. al.(1982): PDA2CFG is optimal

PDA2CFG PDAs CFGs P G CFLs

Introduction

Introduction

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Introduction

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Introduction

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PDA2CFG is also optimal∗ in the unary case

Family P(n,k) n states p = 2n + k + 4 stack symbols Σ = {a}

Thm: There is a family of unary PDAs with n states and p stack symbols for which every equivalent CFG has Ω(n2(p − 2n − 4)) variables.

Lower bound 7

Set of actions of P(n,k):

(q0, a, S) ֒ → (q0, Xk r0) (qi, a, Xj) ֒ → (qi, Xj−1 rm si Xj−1 rm) ∀ i, m ∈ {0, . . . , n − 1}, ∀ j ∈ {1, . . . , k}, (qj, a, si) ֒ → (qi, ε) ∀i, j ∈ {0, . . . , n − 1}, (qi, a, ri) ֒ → (qi, ε) ∀i ∈ {0, . . . , n − 1}, (qi, a, X0) ֒ → (qi, Xk⋆) ∀i ∈ {0, . . . , n − 1}, (qi, a, X0) ֒ → (qi+1, Xk$) ∀i ∈ {0, . . . , n − 2}, (qi, a, ⋆) ֒ → (qi−1, ε) ∀i ∈ {1, . . . , n − 1}, (q0, a, $) ֒ → (qn−1, ε) (qn−1, a, X0) ֒ → (qn−1, ε)

PDA2CFG is also optimal∗ in the unary case

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Properties of P(n,k): P has only one accepting run L(P) = {aℓ} with ℓ ≥ 2n2k

PDA2CFG is also optimal∗ in the unary case

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PDA2CFG is also optimal∗ in the unary case

Thm: There is a family of unary PDAs with n states and p stack symbols for which every equivalent CFG has Ω(n2(p − 2n − 4)) variables.

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Proof: Find G s.t.: L(G) = L(P) = {aℓ} with ℓ ≥ 2n2k.

PDA2CFG is also optimal∗ in the unary case

Thm: There is a family of unary PDAs with n states and p stack symbols for which every equivalent CFG has Ω(n2(p − 2n − 4)) variables.

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Proof: Find G s.t.: L(G) = L(P) = {aℓ} with ℓ ≥ 2n2k. [Charikar et. al., 2005]: The smallest CFG that generates exactly one word of length ℓ has Ω(log(ℓ)) variables.

PDA2CFG is also optimal∗ in the unary case

Thm: There is a family of unary PDAs with n states and p stack symbols for which every equivalent CFG has Ω(n2(p − 2n − 4)) variables.

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Proof: Find G s.t.: L(G) = L(P) = {aℓ} with ℓ ≥ 2n2k. [Charikar et. al., 2005]: The smallest CFG that generates exactly one word of length ℓ has Ω(log(ℓ)) variables. Then G has Ω(log(2n2k)) = Ω(n2k) variables.

PDA2CFG is also optimal∗ in the unary case

Thm: There is a family of unary PDAs with n states and p stack symbols for which every equivalent CFG has Ω(n2(p − 2n − 4)) variables.

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Proof: Find G s.t.: L(G) = L(P) = {aℓ} with ℓ ≥ 2n2k. [Charikar et. al., 2005]: The smallest CFG that generates exactly one word of length ℓ has Ω(log(ℓ)) variables. Then G has Ω(log(2n2k)) = Ω(n2k) variables. As k = p − 2n − 4, G has Ω(n2(p − 2n − 4)) variables.

PDA2CFG is also optimal∗ in the unary case

Thm: There is a family of unary PDAs with n states and p stack symbols for which every equivalent CFG has Ω(n2(p − 2n − 4)) variables.

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PDA2CFG is also optimal∗ in the unary case

P(n, k) Equivalent CFG Lower bound Upper bound Ω(n2k) O(n2(k + n)) 11

PDA2CFG is also optimal∗ in the unary case

P(n, k) Equivalent CFG Lower bound Upper bound Ω(n2k) O(n2(k + n))

Asymptotically tight if n ≤ Ck with C > 0

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PDA2CFG is also optimal∗ in the unary case

PDA2CFG is optimal |Σ| > 1

|Σ| = 1

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PDA2CFG is also optimal∗ in the unary case

PDA2CFG is optimal |Σ| > 1

|Σ| = 1

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PDA2CFG PDAs CFGs P G CFLs

PDA2CFG PDAs CFGs P G CFLs

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{abb, ab} {bab, ba}

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P G PDAs CFGs CFLs

Parikh equivalence

Parikh-equivalent words Parikh-equivalent languages

{abb, ab}

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abb bab {bab, ba}

Parikh equivalence

Parikh-equivalent words Parikh-equivalent languages

{abb, ab}

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abb bab {bab, ba}

Parikh equivalence

Parikh-equivalent words Parikh-equivalent languages 16

abb ≈ bab {abb, ab} {bab, ba}

Parikh equivalence

Parikh-equivalent words Parikh-equivalent languages 16

abb ≈ bab {abb, ab} {bab, ba}

Idea: Find F such that: For all L ∈ F : every CFG G with L(G) ≈ L needs Ω(n2 p) variables

PDA2CFG

PDAs CFGs P G CFLs

PDA2CFG for Parikh equivalence

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Idea: Find F such that: For all L ∈ F : every CFG G with L(G) ≈ L needs Ω(n2 p) variables

PDA2CFG

PDAs CFGs P G CFLs

PDA2CFG for Parikh equivalence

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{abb, ab} {abb, ab}

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{abb, ab} {abb, ab} L = L′ ⇒ L ≈ L′

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{abb, ab} {abb, ab} L = L′ ⇒ L ≈ L′

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{abb, ab} {abb, ab} L = L′ ⇒ L ≈ L′ ⇐

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If |Σ| = 1 :

aaa

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If |Σ| = 1 :

aaa aaa

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If |Σ| = 1 :

aaa aaa {aaa, aa} {aaa, aa}

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If |Σ| = 1 :

aaa aaa {aaa, aa} {aaa, aa}

If |Σ| = 1 :

L = L′ ⇐ ⇒ L ≈ L′

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|Σ| = 1 Idea: Find F For all L ∈ F : every CFG G with L(G) ≈ L needs Ω(n2 p) variables

with |Σ| = 1

such that:

PDA2CFG

PDAs CFGs P G CFLs

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|Σ| = 1 Idea: Find F For all L ∈ F : every CFG G with L(G) ≈ L needs Ω(n2 p) variables

with |Σ| = 1

such that:

PDA2CFG

PDAs CFGs P G CFLs P(n, k) is unary

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PDA2CFG is optimal∗ for Parikh equivalence

PDA2CFG is optimal |Σ| > 1

|Σ| = 1

Parikh equivalence 21

PDAs

P

(FSAs)

F

2-step procedure for Parikh-equivalent FSA

Thm: Every CFL is Parikh-equivalent to some regular language

CFLs 22

Finite State Automata

P F Regular Languages

2-step procedure for Parikh-equivalent FSA

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PDAs FSAs Thm: Every CFL is Parikh-equivalent to some regular language

Upper bound PDA

Parikh-equivalent

FSA

2-step procedure

2-step procedure for Parikh-equivalent FSA

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Upper bound PDA

Parikh-equivalent

FSA

PDA2CFG procedure Equivalent

CFG

[Esparza et. al., 2011]

Procedure 2-step procedure

n states p s.s. O(n2p) variables

2-step procedure for Parikh-equivalent FSA 2-step procedure for Parikh-equivalent FSA 2-step procedure for Parikh-equivalent FSA

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Upper bound PDA

Parikh-equivalent

FSA

PDA2CFG procedure Equivalent

CFG

[Esparza et. al., 2011]

Procedure 2-step procedure n

variables O(4n) states

2-step procedure for Parikh-equivalent FSA

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Upper bound PDA

Parikh-equivalent

FSA

PDA2CFG procedure Equivalent

CFG

[Esparza et. al., 2011]

Procedure 2-step procedure

O(4n2p) states n states p s.s.

Thm: Given a PDA with n states and p s.s., there is a Parikh-equivalent FSA with O(4n2p) states.

2-step procedure for Parikh-equivalent FSA 2-step procedure for Parikh-equivalent FSA

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Lower bound Using the family P(n, k) L(P) = {aℓ} with ℓ ≥ 2n2k q0 q1 qℓ a a a

Thm: There is a family of unary PDAs with n states and p stack symbols for which every equivalent FSA needs at least 2n2(p−2n−4) + 1 states.

2-step procedure for Parikh-equivalent FSA

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P(n, k) Parikh-equivalent FSA Lower bound Upper bound Ω(2n2k) O(4n2(k+2n+4))

Asymptotically tight if n ≤ Ck with C > 0

2-step procedure for Parikh-equivalent FSA

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Conclusions

PDA2CFG is also optimal in the unary case PDA2CFG is optimal for Parikh-equivalence

PDA2CFG-based procedure for Parikh-equivalent FSA is close to optimal

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Conclusions

PDA2CFG is also optimal in the unary case PDA2CFG is optimal for Parikh-equivalence

PDA2CFG-based procedure for Parikh-equivalent FSA is close to optimal

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Thank you!