Complexity of Parikhs Theorem Anthony Widjaja Lin Yale-NUS College, - - PowerPoint PPT Presentation

ACTS 2015 – 1 / 43

Complexity of Parikh’s Theorem

Anthony Widjaja Lin Yale-NUS College, Singapore

Introduction

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Classical automata theory

w = abaabba T = a b a a b

Introduction

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Classical automata theory

w = abaabba T = a b a a b

Parikh (1961) suggested to remove ordering

P(w) = (4, 3) P(T) = (3, 2)

Introduction

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Classical automata theory

w = abaabba T = a b a a b

Parikh (1961) suggested to remove ordering

P(w) = (4, 3) P(T) = (3, 2)

Q: What’s the expressive power of standard automata models “modulo Parikh mapping P” (i.e. treated as sets of vectors)?

Parikh’s Theorem

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Parikh’s Theorem: Parikh images of regular and context-free languages are effectively semilinear.

Parikh’s Theorem

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Parikh’s Theorem: Parikh images of regular and context-free languages are effectively semilinear. Semilinear sets = Presburger-definable subsets of Nk.

Parikh’s Theorem Almost Everywhere

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• Verification of concurrent systems
• Bounded context-switch analysis [Esparza-Ganty’11, Hague-Lin’12]
• Asynchronous programs [Ganty-Majumdar’10]
• Message-passing programs [Abdulla-Atig-Cederberg’13]
• Verification of (restrictions of) counter machines:
• Reversal-bounded verification [Ibarra’79]
• Flat counter machines [Fribourg-Olsen’97, Comon-Jurski’98]
• Flattable counter machines [Bardin-Finkel-Leroux-Schnoebelen’05,

Leroux-Sutre’05]

• Path logics over graph databases [Barcelo-Libkin-Lin-Woods’12]
• Cryptographic Analysis of C programs [Verma et al.’06]

Complexity of Parikh’s Theorem

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Descriptional Complexity: How succinct are the different automata models for representing semilinear sets?

Complexity of Parikh’s Theorem

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Descriptional Complexity: How succinct are the different automata models for representing semilinear sets? i.e. the size of the smallest semilinear sets for NFA, CFG, ...?

Complexity of Parikh’s Theorem

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Descriptional Complexity: How succinct are the different automata models for representing semilinear sets? i.e. the size of the smallest semilinear sets for NFA, CFG, ...? Computational complexity: Can we compute semilinear sets for Parikh images

• f these models efficiently?

Parikh’s Theorem (more precisely)

Intro. Intro. Intro. Complexity Parikh’s Theorem (more precisely) Parikh’s Theorem Semilinearity Parikh’s Theorem CFG Case NFA Case What about a fixed alphabet size? Normal Form Theorem for Semilinear Sets Normal Form Theorem for Parikh Images of NFA Parikh images of extensions of NFA Conclusion

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Parikh Images

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• Parikh Image P(L) of a language L over Σ = {a1, . . . , ak} is a subset of Nk

Parikh Images

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• Parikh Image P(L) of a language L over Σ = {a1, . . . , ak} is a subset of Nk
• L = {anbn : n ∈ N}

Parikh Images

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• Parikh Image P(L) of a language L over Σ = {a1, . . . , ak} is a subset of Nk
• L = {anbn : n ∈ N}
• P(L) = {(0, 0), (1, 1), (2, 2), (3, 3), . . .}

Parikh Images

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• Parikh Image P(L) of a language L over Σ = {a1, . . . , ak} is a subset of Nk
• L = {anbn : n ∈ N}
• P(L) = {(0, 0), (1, 1), (2, 2), (3, 3), . . .}
• Note: L′ = (ab)∗ and P(L) = P(L′).

Semilinear Sets

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• A linear set (over Nk) is a set of the form

L(v0; {v1, . . . , vm}) :=

• v0 +

m

• i=1

aivi : a1, . . . , am ∈ N

• for some offset v0 ∈ Nk and periods v1, . . . , vm ∈ Nk and m ∈ N.

Semilinear Sets

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• A linear set (over Nk) is a set of the form

L(v0; {v1, . . . , vm}) :=

• v0 +

m

• i=1

aivi : a1, . . . , am ∈ N

• for some offset v0 ∈ Nk and periods v1, . . . , vm ∈ Nk and m ∈ N.
• Example: {(i, i) : i ∈ N} = L((0, 0); {(1, 1)})

Semilinear Sets

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• A linear set (over Nk) is a set of the form

L(v0; {v1, . . . , vm}) :=

• v0 +

m

• i=1

aivi : a1, . . . , am ∈ N

• for some offset v0 ∈ Nk and periods v1, . . . , vm ∈ Nk and m ∈ N.
• Example: {(i, i) : i ∈ N} = L((0, 0); {(1, 1)})
• A semilinear set over Nk is a finite union of linear sets.

Complexity of Parikh’s Theorem

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Theorem (Parikh): Parikh images of context-free and regular languages are effectively semilinear.

• Descriptional complexity: size of smallest description
• Computational complexity: how efficient to compute

Complexity of Parikh’s Theorem

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Theorem (Parikh): Parikh images of context-free and regular languages are effectively semilinear.

• Descriptional complexity: size of smallest description
• Computational complexity: how efficient to compute

Parikh’s original proof gives exponential complexity bound with linearly many periods for each linear set!

CFG Case

Intro. Intro. Intro. Complexity Parikh’s Theorem (more precisely) CFG Case CFG upper CFG upper CFG lower NFA Case What about a fixed alphabet size? Normal Form Theorem for Semilinear Sets Normal Form Theorem for Parikh Images of NFA Parikh images of extensions of NFA Conclusion

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Esparza’s Theorem

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S → A

(1)

A → aBb

(2)

B → aAb

(3)

B → ε

(4)

Esparza’s Theorem

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S → A

(1)

A → aBb

(2)

B → aAb

(3)

B → ε

(4) (Esparza’97): a sufficient and necessary condition for a multiset X ⊆ {1, . . . , 4} to be realisable based on (C1) flow condition, and (C2) connectivity condition.

Esparza’s Theorem

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S → A

(1)

A → aBb

(2)

B → aAb

(3)

B → ε

(4) (Esparza’97): a sufficient and necessary condition for a multiset X ⊆ {1, . . . , 4} to be realisable based on (C1) flow condition, and (C2) connectivity condition. The following are not realisable:

• {12, 2, 4} — need at least two S

(C1)

Esparza’s Theorem

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S → A

(1)

A → aBb

(2)

B → aAb

(3)

B → ε

(4) (Esparza’97): a sufficient and necessary condition for a multiset X ⊆ {1, . . . , 4} to be realisable based on (C1) flow condition, and (C2) connectivity condition. The following are not realisable:

• {12, 2, 4} — need at least two S

(C1)

• {1, 2, 37, 4} — need at least seven B

(C1)

Esparza’s Theorem

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S → A

(1)

A → aBb

(2)

B → aAb

(3)

B → ε

(4) (Esparza’97): a sufficient and necessary condition for a multiset X ⊆ {1, . . . , 4} to be realisable based on (C1) flow condition, and (C2) connectivity condition. The following are not realisable:

• {12, 2, 4} — need at least two S

(C1)

• {1, 2, 37, 4} — need at least seven B

(C1)

• {1, 3, 4} — 3 cannot be fired without 2

(C2)

Esparza’s Theorem

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• Flow and connectivity conditions are expressible as an exponential sized

semilinear set (Esparza’97)

Esparza’s Theorem

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• Flow and connectivity conditions are expressible as an exponential sized

semilinear set (Esparza’97)

• Flow and connectivity conditions are expressible as a linear sized existential

Presburger formula (Verma et al.’05)

Esparza’s Theorem

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• Flow and connectivity conditions are expressible as an exponential sized

semilinear set (Esparza’97)

• Flow and connectivity conditions are expressible as a linear sized existential

Presburger formula (Verma et al.’05) n.b. checking existential Presburger formulas are NP-complete: can use fast SMT solvers.

Esparza’s Theorem

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• Flow and connectivity conditions are expressible as an exponential sized

semilinear set (Esparza’97)

• Flow and connectivity conditions are expressible as a linear sized existential

Presburger formula (Verma et al.’05) n.b. checking existential Presburger formulas are NP-complete: can use fast SMT solvers. Note: This has been successfully used in many applications in infinite-state verification.

Lower bound for semilinear sets for CFGs

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Proposition: There is an infinite sequence {Gn}n∈N of CFGs over Σ = {a} s.t. P(L(Gn)) must have at least 2Ω(|Gn|) linear sets.

Lower bound for semilinear sets for CFGs

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Proposition: There is an infinite sequence {Gn}n∈N of CFGs over Σ = {a} s.t. P(L(Gn)) must have at least 2Ω(|Gn|) linear sets. The CFG Gn generates {aj : j ∈ [0, 2n − 1]}:

Lower bound for semilinear sets for CFGs

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Proposition: There is an infinite sequence {Gn}n∈N of CFGs over Σ = {a} s.t. P(L(Gn)) must have at least 2Ω(|Gn|) linear sets. The CFG Gn generates {aj : j ∈ [0, 2n − 1]}: (2n linear sets!!)

Lower bound for semilinear sets for CFGs

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Proposition: There is an infinite sequence {Gn}n∈N of CFGs over Σ = {a} s.t. P(L(Gn)) must have at least 2Ω(|Gn|) linear sets. The CFG Gn generates {aj : j ∈ [0, 2n − 1]}: (2n linear sets!!)

S → A0 . . . An−1 Ai → ε

for each 0 ≤ i < n

Ai → Bi

for each 0 ≤ i < n

Bi → Bi−1Bi−1

for each 0 < i < n

B0 → a

Lower bound for semilinear sets for CFGs

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Proposition: There is an infinite sequence {Gn}n∈N of CFGs over Σ = {a} s.t. P(L(Gn)) must have at least 2Ω(|Gn|) linear sets. The CFG Gn generates {aj : j ∈ [0, 2n − 1]}: (2n linear sets!!)

S → A0 . . . An−1 Ai → ε

for each 0 ≤ i < n

Ai → Bi

for each 0 ≤ i < n

Bi → Bi−1Bi−1

for each 0 < i < n

B0 → a

n.b. this kind of encoding for CFG is from (Stockmeyer-Meyer’73)

NFA Case

Intro. Intro. Intro. Complexity Parikh’s Theorem (more precisely) CFG Case NFA Case NFA case What about a fixed alphabet size? Normal Form Theorem for Semilinear Sets Normal Form Theorem for Parikh Images of NFA Parikh images of extensions of NFA Conclusion

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Exponential lower bound for DFAs

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Proposition: There is an infinite sequence {An}n∈N of DFAs s.t. P(L(An)) must have at least 2Ω(|An|) linear sets.

Exponential lower bound for DFAs

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Proposition: There is an infinite sequence {An}n∈N of DFAs s.t. P(L(An)) must have at least 2Ω(|An|) linear sets.

An is over Σn := {a1, . . . , an+1}: Σn · Σn · · · · · Σn

• n copies

Exponential lower bound for DFAs

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Proposition: There is an infinite sequence {An}n∈N of DFAs s.t. P(L(An)) must have at least 2Ω(|An|) linear sets.

An is over Σn := {a1, . . . , an+1}: Σn · Σn · · · · · Σn

• n copies

P(L(An)) contains each (r1, . . . , rn+1) s.t. n+1

i=1 ri = n.

Exponential lower bound for DFAs

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Proposition: There is an infinite sequence {An}n∈N of DFAs s.t. P(L(An)) must have at least 2Ω(|An|) linear sets.

An is over Σn := {a1, . . . , an+1}: Σn · Σn · · · · · Σn

• n copies

P(L(An)) contains each (r1, . . . , rn+1) s.t. n+1

i=1 ri = n. There are

2n

n

• ≥ 22n−1

√n

• f these.

Exponential lower bound for DFAs

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Proposition: There is an infinite sequence {An}n∈N of DFAs s.t. P(L(An)) must have at least 2Ω(|An|) linear sets.

An is over Σn := {a1, . . . , an+1}: Σn · Σn · · · · · Σn

• n copies

P(L(An)) contains each (r1, . . . , rn+1) s.t. n+1

i=1 ri = n. There are

2n

n

• ≥ 22n−1

√n

• f these.

Note: Σn grows with n

What about a fixed alphabet size?

Intro. Intro. Intro. Complexity Parikh’s Theorem (more precisely) CFG Case NFA Case What about a fixed alphabet size? Chrobak-Martinez Kopczynski-Lin Proof outline Normal Form Theorem for Semilinear Sets Normal Form Theorem for Parikh Images of NFA Parikh images of extensions of NFA Conclusion

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Unary alphabet case: Chrobak-Martinez Theorem

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Let us assume that the alphabet is unary, i.e., Σ = {a}.

Unary alphabet case: Chrobak-Martinez Theorem

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Let us assume that the alphabet is unary, i.e., Σ = {a}. Theorem (Chrobak-Martinez): Descriptional and computational complexity of Parikh Images of unary regular languages are polynomial.

Unary alphabet case: Chrobak-Martinez Theorem

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Let us assume that the alphabet is unary, i.e., Σ = {a}. Theorem (Chrobak-Martinez): Descriptional and computational complexity of Parikh Images of unary regular languages are polynomial. Note:

• quadratically many union of arithmetic progressions with periods of linear size

suffice.

Chrobak-Martinez Theorem in Action

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Chrobak-Martinez Theorem in Action

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Parikh image of L(A) is 8 + 4N + 3N

Chrobak-Martinez Theorem in Action

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Parikh image of L(A) is 8 + 4N + 3N which is equal to

(8 + 4N) ∪ (11 + 4N) ∪ (14 + 4N) ∪ (17 + 4N)

Generalised Chrobak-Martinez Theorem

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Let Σ := {a1, . . . , ak} for fixed k ∈ Z>0.

Generalised Chrobak-Martinez Theorem

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Let Σ := {a1, . . . , ak} for fixed k ∈ Z>0. Theorem (Kopczynski & Lin’10): Descriptional and computational complexity of Parikh Images of NFAs are polynomial.

• union of polynomially many linear sets with at most k polynomially-bounded

periods

• Complexities are exponential in k
• Generalizes Chrobak-Martinez Theorem (case k = 1).

Outline of proof

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• Normal-Form Theorem for Semilinear sets.
• Normal-Form Theorem for Parikh images of NFA

Normal Form Theorem for Semilinear Sets

Intro. Intro. Intro. Complexity Parikh’s Theorem (more precisely) CFG Case NFA Case What about a fixed alphabet size? Normal Form Theorem for Semilinear Sets Real cones Real cone example Caratheodory’s thm Our theorem Intuition Intuition Higher dim. Normal Form Theorem for Parikh Images of NFA Parikh images of extensions of NFA

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Digression to Convex Geometry: Real Cones

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Given V = {v1, . . . , vn} ⊆ Rk, define the real cone over V : cone(V ) := {Σn

i=1aivi : ai ∈ R≥0}.

Note: akin to definition of vector subspace of Rk “spanned” by some set V (but ...)

Examples

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0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Real cone of (1, 1, 0.5), (1, 0.5, 1), (0.5, 1, 1)

Examples

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• 1

1

• 1
• 0.5

0.5 1 0.2 0.4 0.6 0.8 1

Real cone over {1, −1} × {1, −1} × {1}

Caratheodory’s theorem

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Theorem: Given a finite V ⊆ Rk of rank d ≤ k, we have cone(V ) =

• V ′⊆V,|V ′|=V ′=d

cone(V ′).

• Cones over Rk can be decomposed into smaller subcones with ≤ k vertices

Caratheodory’s theorem

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Theorem: Given a finite V ⊆ Rk of rank d ≤ k, we have cone(V ) =

• V ′⊆V,|V ′|=V ′=d

cone(V ′).

• Cones over Rk can be decomposed into smaller subcones with ≤ k vertices
• Note: if k is fixed, there are only polynomially many such subcones.

Caratheodory’s theorem (example)

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• 1

1

• 1
• 0.5

0.5 1 0.2 0.4 0.6 0.8 1

Real cone over {1, −1} × {1, −1} × {1}

Caratheodory’s theorem (example)

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• 1
• 0.5

0.5 1

• 1

1 0.2 0.4 0.6 0.8 1

Real cone over (1, 1, 1), (1, −1, 1), (−1, 1, 1)

Caratheodory’s theorem (example)

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• 1
• 0.5

0.5 1

• 1

1 0.2 0.4 0.6 0.8 1

Real cone over (−1, −1, 1), (1, −1, 1), (−1, 1, 1)

Caratheodory-like theorem for linear sets

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• Question: Does the integer version of Caratheodory’s theorem hold?

Caratheodory-like theorem for linear sets

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• Question: Does the integer version of Caratheodory’s theorem hold?
• Unfortunately, no!

Caratheodory-like theorem for linear sets

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• Question: Does the integer version of Caratheodory’s theorem hold?
• Unfortunately, no!
• Fortunately, it does if you allow nonzero offsets :)

Caratheodory-like theorem for linear sets

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Theorem: Fix k ∈ N. Given a finite V ⊆ Zk of rank d ≤ k, we can compute in pseudopolynomial time linear sets L(w1; S1), . . . , L(ws; Ss) s.t. each Si ⊆ V and |Si| = d

L(0; V ) =

s

• i=1

L(wi; Si).

Each number in wi is polynomially large (in unary)

Caratheodory-like theorem for linear sets

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Theorem: Fix k ∈ N. Given a finite V ⊆ Zk of rank d ≤ k, we can compute in pseudopolynomial time linear sets L(w1; S1), . . . , L(ws; Ss) s.t. each Si ⊆ V and |Si| = d

L(0; V ) =

s

• i=1

L(wi; Si).

Each number in wi is polynomially large (in unary) When V ⊆ Nk, each wi ∈ Nk

Caratheodory-like theorem for linear sets

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Theorem: Fix k ∈ N. Given a finite V ⊆ Zk of rank d ≤ k, we can compute in pseudopolynomial time linear sets L(w1; S1), . . . , L(ws; Ss) s.t. each Si ⊆ V and |Si| = d

L(0; V ) =

s

• i=1

L(wi; Si).

Each number in wi is polynomially large (in unary) When V ⊆ Nk, each wi ∈ Nk The same holds when the initial offset is v = 0

Caratheodory-like theorem for linear sets

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Theorem: Fix k ∈ N. Given a finite V ⊆ Zk of rank d ≤ k, we can compute in pseudopolynomial time linear sets L(w1; S1), . . . , L(ws; Ss) s.t. each Si ⊆ V and |Si| = d

L(0; V ) =

s

• i=1

L(wi; Si).

Each number in wi is polynomially large (in unary) When V ⊆ Nk, each wi ∈ Nk The same holds when the initial offset is v = 0

Caratheodory-like theorem for linear sets

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Theorem: Fix k ∈ N. Given a finite V ⊆ Zk of rank d ≤ k, we can compute in pseudopolynomial time linear sets L(w1; S1), . . . , L(ws; Ss) s.t. each Si ⊆ V and |Si| = d

L(0; V ) =

s

• i=1

L(wi; Si).

Use Caratheodory’s theorem

Caratheodory-like theorem for linear sets

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Theorem: Fix k ∈ N. Given a finite V ⊆ Zk of rank d ≤ k, we can compute in pseudopolynomial time linear sets L(w1; S1), . . . , L(ws; Ss) s.t. each Si ⊆ V and |Si| = d

L(0; V ) =

s

• i=1

L(wi; Si).

Use Caratheodory’s theorem Use bounds from integer programming (Papadimitriou’83) for estimating the biggest entries in wi’s

Caratheodory-like theorem for linear sets

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Theorem: Fix k ∈ N. Given a finite V ⊆ Zk of rank d ≤ k, we can compute in pseudopolynomial time linear sets L(w1; S1), . . . , L(ws; Ss) s.t. each Si ⊆ V and |Si| = d

L(0; V ) =

s

• i=1

L(wi; Si).

Use Caratheodory’s theorem Use bounds from integer programming (Papadimitriou’83) for estimating the biggest entries in wi’s Use dynamic programming to compute all wi’s

Intuition for dimension 2

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Intuition for dimension 2

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Intuition for dimension 2

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b b b b b b b b b b b b b b b b b b

Intuition for dimension 2

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b b b b b b b b b b b b b b b b b b

Intuition for dimension 2

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b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

Intuition for dimension 2

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Key observations:

• Each parallelogram has a quadratically many integer points

Intuition for dimension 2

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Key observations:

• Each parallelogram has a quadratically many integer points
• There is a natural ordering on these parallelograms:
• The parallelogram above or to the right of a parallelogram is larger.

Intuition for dimension 2

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Key observations:

• Each parallelogram has a quadratically many integer points
• There is a natural ordering on these parallelograms:
• The parallelogram above or to the right of a parallelogram is larger.
• Once a point “appears”, it stays in the larger parallelograms.

Intuition for dimension 2

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Key observations:

• Each parallelogram has a quadratically many integer points
• There is a natural ordering on these parallelograms:
• The parallelogram above or to the right of a parallelogram is larger.
• Once a point “appears”, it stays in the larger parallelograms.
• So, only need to keep track of minimal representatives M, i.e.,

L(0; {(3, 1), (2, 3), (2, 2)}) = M + (3, 1)N + (2, 3)N

where:

M = {(2, 2), (4, 4), (6, 6), (8, 8), (10, 10), (12, 12)}

Intuition for dimension 2

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Key observations:

• Each parallelogram has a quadratically many integer points
• There is a natural ordering on these parallelograms:
• The parallelogram above or to the right of a parallelogram is larger.
• Once a point “appears”, it stays in the larger parallelograms.
• So, only need to keep track of minimal representatives M, i.e.,

L(0; {(3, 1), (2, 3), (2, 2)}) = M + (3, 1)N + (2, 3)N

where:

M = {(2, 2), (4, 4), (6, 6), (8, 8), (10, 10), (12, 12)}

• |M| and all numbers in M are “not big”.

Case of dimension > 2

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• Need to use linear sets with different periods (unlike d = 2)
• Replace finding two outermost vectors with Caratheodory’s
• The rest are similar:
• Dynamic programming,
• Bounds from integer programming

Normal Form Theorem for Parikh Images of NFA

Intro. Intro. Intro. Complexity Parikh’s Theorem (more precisely) CFG Case NFA Case What about a fixed alphabet size? Normal Form Theorem for Semilinear Sets Normal Form Theorem for Parikh Images of NFA Proof idea

• Comp. complex.

Parikh images of extensions of NFA Conclusion

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Statement of the Theorem

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Let Σ := {a1, . . . , ak} for fixed k ∈ Z>0. Theorem (Kopczynski & Lin, LICS 2010): Descriptional and computational complexity of Parikh Images of NFAs are polynomial.

• poly many union of linear sets with at most k periods with poly-bounded

numbers

• Complexities are exponential in k .
• Generalizes Chrobak-Martinez Theorem (case k = 1).

Proof idea: Path types

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Notation: Q = {q0, . . . , qn−1} with initial state q0 and final state qn−1.

Proof idea: Path types

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Notation: Q = {q0, . . . , qn−1} with initial state q0 and final state qn−1. Given a path π and simple cycles C1, . . . , Cm meeting with π:

Cm π p q C1 C2

Proof idea: Path types

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Notation: Q = {q0, . . . , qn−1} with initial state q0 and final state qn−1. Given a path π and simple cycles C1, . . . , Cm meeting with π:

Cm π p q C1 C2

The path type Tπ of π is the linear set L(P(π); {P(Ci)}m

i=1).

Proof idea: Path types

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Notation: Q = {q0, . . . , qn−1} with initial state q0 and final state qn−1. Given a path π and simple cycles C1, . . . , Cm meeting with π:

Cm π p q C1 C2

The path type Tπ of π is the linear set L(P(π); {P(Ci)}m

i=1).

There are at most nkO(1) many periods.

Proof idea

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Characterization of Parikh images of L(A) Lemma: P(L(A)) =

π Tπ, where π ranges over paths from initial to final

state of length at most O(n2).

Proof idea

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Characterization of Parikh images of L(A) Lemma: P(L(A)) =

π Tπ, where π ranges over paths from initial to final

state of length at most O(n2). Problem: There are 2(n+k)O(1) such π.

Proof idea

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Characterization of Parikh images of L(A) Lemma: P(L(A)) =

π Tπ, where π ranges over paths from initial to final

state of length at most O(n2). Problem: There are 2(n+k)O(1) such π. BUT: can apply Caratheodory-like theorem on each Tπ:

Proof idea

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Characterization of Parikh images of L(A) Lemma: P(L(A)) =

π Tπ, where π ranges over paths from initial to final

state of length at most O(n2). Problem: There are 2(n+k)O(1) such π. BUT: can apply Caratheodory-like theorem on each Tπ: Corollary: P(L(A)) coincides with a union of polynomially many linear sets with polynomially large offsets and at most k polynomially large periods.

Computational complexity

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Theorem: P(L(A)) coincides with a union of polynomially many linear sets with polynomially large offsets and at most k polynomially large periods. A naive implementation takes exponential time. Can improve to PTIME using dynamic programming!

Parikh images of extensions of NFA

Intro. Intro. Intro. Complexity Parikh’s Theorem (more precisely) CFG Case NFA Case What about a fixed alphabet size? Normal Form Theorem for Semilinear Sets Normal Form Theorem for Parikh Images of NFA Parikh images of extensions of NFA LG LG RBCA Conclusion

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Linear grammars

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S → A

(5)

A → aBb

(6)

B → aAb

(7)

B → ε

(8)

Linear grammars

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S → A

(5)

A → aBb

(6)

B → aAb

(7)

B → ε

(8) The r.h.s. of a rule has at most 1 variable.

Linear grammars

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S → A

(5)

A → aBb

(6)

B → aAb

(7)

B → ε

(8) The r.h.s. of a rule has at most 1 variable. Proposition: Generalised Chrobak-Martinez Theorem extends to Linear Grammars.

Linear grammars

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S → A

(5)

A → aBb

(6)

B → aAb

(7)

B → ε

(8) The r.h.s. of a rule has at most 1 variable. Proposition: Generalised Chrobak-Martinez Theorem extends to Linear Grammars. Proof: Linear grammars are essentially NFA ...

CFG of fixed “dimensions”

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A1 A1 a A2 c A1 a

Theorem (Esparza-Ganty-Kiefer-Luttenberger’11): Generalised Chrobak-Martinez Theorem extends to CFG of fixed dimensions. Dimensions measure how many times “doubling tricks” possibly get used in a CFG.

CFG of fixed “dimensions”

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A1 A1 a A2 c A1 a

Theorem (Esparza-Ganty-Kiefer-Luttenberger’11): Generalised Chrobak-Martinez Theorem extends to CFG of fixed dimensions. Dimensions measure how many times “doubling tricks” possibly get used in a CFG. Linear grammars have dimension 0.

CFG of fixed “dimensions”

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A1 A1 a A2 c A1 a

Theorem (Esparza-Ganty-Kiefer-Luttenberger’11): Generalised Chrobak-Martinez Theorem extends to CFG of fixed dimensions. Dimensions measure how many times “doubling tricks” possibly get used in a CFG. Linear grammars have dimension 0. Proof: Compute an equivalent NFA of polynomial size and use Generalised Chrobak-Martinez Theorem.

Reversal-bounded counter automata

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Minsky’s counter automata

Control X3 X2 X1 Counters

Reversal-bounded counter automata

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Time R R R This variable has 3 reversals

Reversal-bounded counter automata

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Time R R R This variable has 3 reversals Restricted problem: examine paths with r ∈ N reversals for all variables

Reversal-bounded counter automata

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Time R R R This variable has 3 reversals Restricted problem: examine paths with r ∈ N reversals for all variables No finite bounds on length of 0-reversal-bounded paths!

Reversal-bounded counter automata

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Theorem: Generalised Chrobak-Martinez Theorem extends to reversal-bounded counter automata with a fixed number of reversals and a fixed number of counters.

Reversal-bounded counter automata

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Theorem: Generalised Chrobak-Martinez Theorem extends to reversal-bounded counter automata with a fixed number of reversals and a fixed number of counters. Proof idea: follow Ibarra’s proof, but use Generalised Chrobak-Martinez for Parikh images of NFA.

Reversal-bounded counter automata

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Theorem: Generalised Chrobak-Martinez Theorem extends to reversal-bounded counter automata with a fixed number of reversals and a fixed number of counters. Proof idea: follow Ibarra’s proof, but use Generalised Chrobak-Martinez for Parikh images of NFA. Note: This result can be used to derive optimal complexity for reversal-bounded verification counter automata.

Reversal-bounded counter automata

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Theorem: Generalised Chrobak-Martinez Theorem extends to reversal-bounded counter automata with a fixed number of reversals and a fixed number of counters. Proof idea: follow Ibarra’s proof, but use Generalised Chrobak-Martinez for Parikh images of NFA. Note: This result can be used to derive optimal complexity for reversal-bounded verification counter automata. Theorem (Hague-Lin’11): This does not work if # reversals/# counters are non-fixed, but you can compute an existential Presburger formula in polynomial time (even if a pushdown stack is added).

Alternative notion of Complexity

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Study the complexity of decision problems (e.g. membership, universality, ...)

• ver different models.

Alternative notion of Complexity

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Study the complexity of decision problems (e.g. membership, universality, ...)

• ver different models.

There is also a stark difference, e.g., for emptiness:

• NFA: P (f.a.), NP (u.a.) [Kopczynski-Lin’10]
• CFG: NP (f.a.), NP (u.a.) [Hyunh’83]

Alternative notion of Complexity

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Study the complexity of decision problems (e.g. membership, universality, ...)

• ver different models.

There is also a stark difference, e.g., for emptiness:

• NFA: P (f.a.), NP (u.a.) [Kopczynski-Lin’10]
• CFG: NP (f.a.), NP (u.a.) [Hyunh’83]

Can be proven by first constructing Parikh images (as semilinear set representation or existential Presburger formulas).

Conclusion

Intro. Intro. Intro. Complexity Parikh’s Theorem (more precisely) CFG Case NFA Case What about a fixed alphabet size? Normal Form Theorem for Semilinear Sets Normal Form Theorem for Parikh Images of NFA Parikh images of extensions of NFA Conclusion

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Challenges

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• Does generalised Chrobak-Martinez Theorem extend to one-counter

automata?

Challenges

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• Does generalised Chrobak-Martinez Theorem extend to one-counter

automata?

• How do we compare the descriptional complexity of alternating finite-state

automata vs. CFG?

Challenges

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• Does generalised Chrobak-Martinez Theorem extend to one-counter

automata?

• How do we compare the descriptional complexity of alternating finite-state

automata vs. CFG?

• Study succinctness hierarchy in Parikh’s Theorem.

Challenges

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• Does generalised Chrobak-Martinez Theorem extend to one-counter

automata?

• How do we compare the descriptional complexity of alternating finite-state

automata vs. CFG?

• Study succinctness hierarchy in Parikh’s Theorem.

THANKS!!