WORDS 2011 Bounded Parikh Automata Cadilhac , Finkel &McKenzie - - PowerPoint PPT Presentation

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

WORDS 2011 Bounded Parikh Automata

• M. Cadilhac1, A. Finkel2, and P. McKenzie1

1: 2:

September 15th, 2011

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

WORDS 2011 Bounded Parikh Automata

• M. Cadilhac1, A. Finkel2, and P. McKenzie1

1: 2:

September 15th, 2011

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

WORDS 2011 Bounded Parikh Automata

• M. Cadilhac1, A. Finkel2, and P. McKenzie1

1: 2:

September 15th, 2011

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

WORDS 2011 Bounded Parikh Automata

• M. Cadilhac1, A. Finkel2, and P. McKenzie1

1: 2:

September 15th, 2011

1 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Context

A practical problem:

▸ In model checking, properties described by automata

2 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Context

A practical problem:

▸ In model checking, properties described by automata ▸ Efficiency depends, in particular, on determinism

2 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Context

A practical problem:

▸ In model checking, properties described by automata ▸ Efficiency depends, in particular, on determinism ▸ Finite automata offer poor expressiveness

2 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Context

A practical problem:

▸ In model checking, properties described by automata ▸ Efficiency depends, in particular, on determinism ▸ Finite automata offer poor expressiveness ▸ Goal: provide large classes of “deterministic” languages

2 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Context

A practical problem:

▸ In model checking, properties described by automata ▸ Efficiency depends, in particular, on determinism ▸ Finite automata offer poor expressiveness ▸ Goal: provide large classes of “deterministic” languages

Natural models of automata allow for a study of:

▸ Small complexity classes (within NC2)

2 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Context

A practical problem:

▸ In model checking, properties described by automata ▸ Efficiency depends, in particular, on determinism ▸ Finite automata offer poor expressiveness ▸ Goal: provide large classes of “deterministic” languages

Natural models of automata allow for a study of:

▸ Small complexity classes (within NC2) ▸ Word logics (variants of MSO)

2 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Context

A practical problem:

▸ In model checking, properties described by automata ▸ Efficiency depends, in particular, on determinism ▸ Finite automata offer poor expressiveness ▸ Goal: provide large classes of “deterministic” languages

Natural models of automata allow for a study of:

▸ Small complexity classes (within NC2) ▸ Word logics (variants of MSO) ▸ Algebra (pseudovarieties of monoids associated with

language varieties)

2 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Outline

Result and definitions BSL ⊆ DetPA Corollaries and Further Work

3 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Outline

Result and definitions BSL ⊆ DetPA Corollaries and Further Work

3 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

4 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

4 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: semilinear

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

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Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: semilinear

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definitions

▸ Linear set: of the form

E = { ⃗ c0 + ∑m

i=1 ⃗

ci ⋅ ki ∣ ki ∈ N}

5 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: semilinear

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definitions

▸ Linear set: of the form

E = { ⃗ c0 + ∑m

i=1 ⃗

ci ⋅ ki ∣ ki ∈ N} E.g., {(0 0) + (1 2) ⋅ k1} linear

5 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: semilinear

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definitions

▸ Linear set: of the form

E = { ⃗ c0 + ∑m

i=1 ⃗

ci ⋅ ki ∣ ki ∈ N} E.g., {(0 0) + (1 2) ⋅ k1} linear, {2n ∣ n ∈ N} not linear

5 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: semilinear

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definitions

▸ Linear set: of the form

E = { ⃗ c0 + ∑m

i=1 ⃗

ci ⋅ ki ∣ ki ∈ N} E.g., {(0 0) + (1 2) ⋅ k1} linear, {2n ∣ n ∈ N} not linear

▸ Semilinear set: finite union of linear sets (equiv. FO[+])

5 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: semilinear

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definitions

▸ Linear set: of the form

E = { ⃗ c0 + ∑m

i=1 ⃗

ci ⋅ ki ∣ ki ∈ N} E.g., {(0 0) + (1 2) ⋅ k1} linear, {2n ∣ n ∈ N} not linear

▸ Semilinear set: finite union of linear sets (equiv. FO[+])

Why are they natural? One of many reasons: Theorem ([Parikh, 1966]) With Σ = {a1,...,an} and w ∈ Σ∗, let Parikh(w) = (∣w∣a1,...,∣w∣an) ∈ Nn be the Parikh image of w.

5 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: semilinear

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definitions

▸ Linear set: of the form

E = { ⃗ c0 + ∑m

i=1 ⃗

ci ⋅ ki ∣ ki ∈ N} E.g., {(0 0) + (1 2) ⋅ k1} linear, {2n ∣ n ∈ N} not linear

▸ Semilinear set: finite union of linear sets (equiv. FO[+])

Why are they natural? One of many reasons: Theorem ([Parikh, 1966]) With Σ = {a1,...,an} and w ∈ Σ∗, let Parikh(w) = (∣w∣a1,...,∣w∣an) ∈ Nn be the Parikh image of w. L context-free ⇒ Parikh(L) semilinear

5 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4)

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4}

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 +

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 +

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = Parikh(traced run):

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = Parikh(traced run): (0, 0, 0, 0) ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = t1 Parikh(traced run): (1, 0, 0, 0) ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = t1t1 Parikh(traced run): (2, 0, 0, 0) ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = t1t1t2 Parikh(traced run): (2, 1, 0, 0) ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = t1t1t2t3 Parikh(traced run): (2, 1, 1, 0) ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = t1t1t2t3t4 Parikh(traced run): (2, 1, 1, 1) ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = t1t1t2t3t4t4 Parikh(traced run): (2, 1, 1, 2) ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = t1t1t2t3t4t4t4 Parikh(traced run): (2, 1, 1, 3) ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Example: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Example L = {w ⋅ a∣w∣+1 ∣ w ∈ {a,b}∗} ∈ PA a (t1) b (t2) a (t3) a (t4) δ = {t1,t2,t3,t4} C = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k1 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⋅ k2 + ⎛ ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Word: aabaaaa π = t1t1t2t3t4t4t4 Parikh(traced run): (2, 1, 1, 3) ∈ C ▴

6 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

7 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition ([Klaedtke and Rueß, 2003])

▸ Parikh automaton (PA): a pair (A,C) with:

▸ A a finite automaton of transition set δ ▸ C ⊆ N∣δ∣ semilinear 7 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition ([Klaedtke and Rueß, 2003])

▸ Parikh automaton (PA): a pair (A,C) with:

▸ A a finite automaton of transition set δ ▸ C ⊆ N∣δ∣ semilinear

▸ L(A,C) = {Label(π) ∣ π ∈ Runs(A) ∧ Parikh(π) ∈ C}

7 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition ([Klaedtke and Rueß, 2003])

▸ Parikh automaton (PA): a pair (A,C) with:

▸ A a finite automaton of transition set δ ▸ C ⊆ N∣δ∣ semilinear

▸ L(A,C) = {Label(π) ∣ π ∈ Runs(A) ∧ Parikh(π) ∈ C} ▸ Deterministic Parikh automaton (DetPA) if A is

7 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition ([Klaedtke and Rueß, 2003])

▸ Parikh automaton (PA): a pair (A,C) with:

▸ A a finite automaton of transition set δ ▸ C ⊆ N∣δ∣ semilinear

▸ L(A,C) = {Label(π) ∣ π ∈ Runs(A) ∧ Parikh(π) ∈ C} ▸ Deterministic Parikh automaton (DetPA) if A is ▸ L = {w ⋅ a∣w∣+1} ∈ PA but ∉ DetPA

7 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition ([Klaedtke and Rueß, 2003])

▸ Parikh automaton (PA): a pair (A,C) with:

▸ A a finite automaton of transition set δ ▸ C ⊆ N∣δ∣ semilinear

▸ L(A,C) = {Label(π) ∣ π ∈ Runs(A) ∧ Parikh(π) ∈ C} ▸ Deterministic Parikh automaton (DetPA) if A is ▸ L = {w ⋅ a∣w∣+1} ∈ PA but ∉ DetPA, same for PAL

7 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition ([Klaedtke and Rueß, 2003])

▸ Parikh automaton (PA): a pair (A,C) with:

▸ A a finite automaton of transition set δ ▸ C ⊆ N∣δ∣ semilinear

▸ L(A,C) = {Label(π) ∣ π ∈ Runs(A) ∧ Parikh(π) ∈ C} ▸ Deterministic Parikh automaton (DetPA) if A is ▸ L = {w ⋅ a∣w∣+1} ∈ PA but ∉ DetPA, same for PAL ▸ PAL ∉ PA

7 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Why is PA a relevant model?

8 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Why is PA a relevant model?

▸ Nice decidability properties (emptiness)

8 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Why is PA a relevant model?

▸ Nice decidability properties (emptiness) ▸ Nice closure properties for DetPA

8 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Why is PA a relevant model?

▸ Nice decidability properties (emptiness) ▸ Nice closure properties for DetPA ▸ Equivalent to reversal bounded counter machines

(RBCM) of [Ibarra, 1978]

8 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Why is PA a relevant model?

▸ Nice decidability properties (emptiness) ▸ Nice closure properties for DetPA ▸ Equivalent to reversal bounded counter machines

(RBCM) of [Ibarra, 1978]

▸ Equivalent to extended automata over (Zk,+,0) of

[Mitrana and Stiebe, 2001]

8 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Why is PA a relevant model?

▸ Nice decidability properties (emptiness) ▸ Nice closure properties for DetPA ▸ Equivalent to reversal bounded counter machines

(RBCM) of [Ibarra, 1978]

▸ Equivalent to extended automata over (Zk,+,0) of

[Mitrana and Stiebe, 2001]

▸ Related deterministic models used for model checking

8 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Why is PA a relevant model?

▸ Nice decidability properties (emptiness) ▸ Nice closure properties for DetPA ▸ Equivalent to reversal bounded counter machines

(RBCM) of [Ibarra, 1978]

▸ Equivalent to extended automata over (Zk,+,0) of

[Mitrana and Stiebe, 2001]

▸ Related deterministic models used for model checking ▸ Logical characterization (WS1S + cardinalities)

8 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Parikh automata

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Why is PA a relevant model?

▸ Nice decidability properties (emptiness) ▸ Nice closure properties for DetPA ▸ Equivalent to reversal bounded counter machines

(RBCM) of [Ibarra, 1978]

▸ Equivalent to extended automata over (Zk,+,0) of

[Mitrana and Stiebe, 2001]

▸ Related deterministic models used for model checking ▸ Logical characterization (WS1S + cardinalities) ▸ Low complexity (PA ⊊ NL, DetPA ⊊ NC1)

8 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Bounded languages

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

9 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Bounded languages

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition

▸ L bounded: L ⊆ w∗ 1 ⋯w∗ n for some words wi’s

9 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Bounded languages

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition

▸ L bounded: L ⊆ w∗ 1 ⋯w∗ n for some words wi’s ▸ Define Iter ⃗ w(L) = {(i1,...,in) ∣ wi1 1 ⋯win n ∈ L} ⊆ Nn

9 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Bounded languages

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition

▸ L bounded: L ⊆ w∗ 1 ⋯w∗ n for some words wi’s ▸ Define Iter ⃗ w(L) = {(i1,...,in) ∣ wi1 1 ⋯win n ∈ L} ⊆ Nn ▸ BSL = {L ⊆ w∗ 1 ⋯w∗ n ∣ Iter ⃗ w(L) semilinear}

9 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Bounded languages

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition

▸ L bounded: L ⊆ w∗ 1 ⋯w∗ n for some words wi’s ▸ Define Iter ⃗ w(L) = {(i1,...,in) ∣ wi1 1 ⋯win n ∈ L} ⊆ Nn ▸ BSL = {L ⊆ w∗ 1 ⋯w∗ n ∣ Iter ⃗ w(L) semilinear} ▸ {aib2i} ∈ BSL

9 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Bounded languages

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition

▸ L bounded: L ⊆ w∗ 1 ⋯w∗ n for some words wi’s ▸ Define Iter ⃗ w(L) = {(i1,...,in) ∣ wi1 1 ⋯win n ∈ L} ⊆ Nn ▸ BSL = {L ⊆ w∗ 1 ⋯w∗ n ∣ Iter ⃗ w(L) semilinear} ▸ {aib2i} ∈ BSL ▸ Σ∗ not bounded

9 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Bounded languages

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition

▸ L bounded: L ⊆ w∗ 1 ⋯w∗ n for some words wi’s ▸ Define Iter ⃗ w(L) = {(i1,...,in) ∣ wi1 1 ⋯win n ∈ L} ⊆ Nn ▸ BSL = {L ⊆ w∗ 1 ⋯w∗ n ∣ Iter ⃗ w(L) semilinear} ▸ {aib2i} ∈ BSL ▸ Σ∗ not bounded ▸ {a2n} bounded ∉ BSL

9 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

Definition: Bounded languages

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Definition

▸ L bounded: L ⊆ w∗ 1 ⋯w∗ n for some words wi’s ▸ Define Iter ⃗ w(L) = {(i1,...,in) ∣ wi1 1 ⋯win n ∈ L} ⊆ Nn ▸ BSL = {L ⊆ w∗ 1 ⋯w∗ n ∣ Iter ⃗ w(L) semilinear} ▸ {aib2i} ∈ BSL ▸ Σ∗ not bounded ▸ {a2n} bounded ∉ BSL ▸▸ BSL intensively studied, e.g., Ginsburg & Spanier, 60’s

9 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

The big picture

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

10 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

The big picture

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Theorem, restated: Theorem PA ∩ BOUNDED ⊆ BSL ⊆ DetPA ∩ BOUNDED

10 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

The big picture

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Theorem, restated: Theorem PA ∩ BOUNDED ⊆ Parikh(any L in PA) semilinear PA closed under h−1, ∩ BSL ⊆ DetPA ∩ BOUNDED

10 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

The result

The big picture

Theorem Parikh automata and their deterministic variant recognize the same bounded languages: those with a semilinear iteration set

Theorem, restated: Theorem PA ∩ BOUNDED ⊆ Parikh(any L in PA) semilinear PA closed under h−1, ∩ BSL ⊆ Rest of this talk DetPA ∩ BOUNDED

10 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Outline

Result and definitions BSL ⊆ DetPA Corollaries and Further Work

10 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Preliminary

We make use of a related model:

11 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Preliminary

We make use of a related model: Definition

▸ Affine Parikh automaton given by:

▸ A finite automaton ▸ A labelling of the transitions by affine functions ▸ A semilinear set

a [⃗ x ← M.⃗ x + ⃗ v]

11 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Preliminary

We make use of a related model: Definition

▸ Affine Parikh automaton given by:

▸ A finite automaton ▸ A labelling of the transitions by affine functions ▸ A semilinear set

a [⃗ x ← M.⃗ x + ⃗ v]

▸ Its language: accepted words which take ⃗

0 to some ⃗ x in the semilinear set

11 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Preliminary

We make use of a related model: Definition

▸ Affine Parikh automaton given by:

▸ A finite automaton ▸ A labelling of the transitions by affine functions ▸ A semilinear set

a [⃗ x ← M.⃗ x + ⃗ v]

▸ Its language: accepted words which take ⃗

0 to some ⃗ x in the semilinear set

▸ PA: APA in which every M = Identity

11 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Preliminary

We make use of a related model: Definition

▸ Affine Parikh automaton given by:

▸ A finite automaton ▸ A labelling of the transitions by affine functions ▸ A semilinear set

a [⃗ x ← M.⃗ x + ⃗ v]

▸ Its language: accepted words which take ⃗

0 to some ⃗ x in the semilinear set

▸ PA: APA in which every M = Identity ▸ Known facts: PAL, COPY, {a2n} ∈ APA ∖ PA ▸ Open: Dyck ∉ APA, PAL ∉ DetAPA

11 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Outline

Result and definitions BSL ⊆ DetPA BSL ⊆ DetAPA with some property X DetAPA with this property ⊆ DetPA Corollaries and Further Work

11 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

▸ Let L ⊆ w∗ 1 w∗ 2 w∗ 3 ∈ BSL. Then a PA describes L:

A: qf ǫ ǫ w1 w2 w3 and C = Iter ⃗

w(L)

12 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

▸ Let L ⊆ w∗ 1 w∗ 2 w∗ 3 ∈ BSL. Then a PA describes L:

A: qf ǫ ǫ w1 w2 w3 and C = Iter ⃗

w(L) ▸ Let π1,π2 two accepting paths with same label:

Parikh(π1) ∈ C ⇔ Parikh(π2) ∈ C

12 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

▸ Let L ⊆ w∗ 1 w∗ 2 w∗ 3 ∈ BSL. Then a PA describes L:

A: qf ǫ ǫ w1 w2 w3 and C = Iter ⃗

w(L) ▸ Let π1,π2 two accepting paths with same label:

Parikh(π1) ∈ C ⇔ Parikh(π2) ∈ C

▸▸ In SubsetDeterminize(A), when reaching a final state

{qf ,...}, we need only recall the Parikh image of one possible run to qf

12 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

▸ Let L ⊆ w∗ 1 w∗ 2 w∗ 3 ∈ BSL. Then a PA describes L:

A: qf ǫ ǫ w1 w2 w3 and C = Iter ⃗

w(L) ▸ Let π1,π2 two accepting paths with same label:

Parikh(π1) ∈ C ⇔ Parikh(π2) ∈ C

▸▸ In SubsetDeterminize(A), when reaching a final state

{qf ,...}, we need only recall the Parikh image of one possible run to qf

▸▸ In SubsetDeterminize(A), when reaching some state

{q1,...,qk}, we need only recall the Parikh image of

• ne possible run to each qi

12 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• 13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 3

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 ⃗ x = ( ⃗ i ⃗ j) 3

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 ⃗ x = ( Parikh(a path to 1) ⃗ i ⃗ j) 3

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 ⃗ x = ( Parikh(a path to 1) ⃗ i Parikh(a path to 2) ⃗ j) 3

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 ⃗ x = ( Parikh(a path to 1) ⃗ i Parikh(a path to 2) ⃗ j) 3 ⃗ x = ( Parikh(a path to 3) ⃗ i ⃗ 0)

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 ⃗ x = ( Parikh(a path to 1) ⃗ i Parikh(a path to 2) ⃗ j) 3 ⃗ x = ( Parikh(a path to 3) ⃗ i ⃗ 0) a [( ⃗ i ⃗ j) ← ( ⃗ i + Parikh(t1) ⃗ )]

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 ⃗ x = ( Parikh(a path to 1) ⃗ i Parikh(a path to 2) ⃗ j) 3 ⃗ x = ( Parikh(a path to 3) ⃗ i ⃗ 0) a [( ⃗ i ⃗ j) ← ( ⃗ 1 ⃗ ⃗ ⃗ 0) ⋅ ( ⃗ i ⃗ j) + (Parikh(t1) ⃗ )]

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 ⃗ x = ( Parikh(a path to 1) ⃗ i Parikh(a path to 2) ⃗ j) 3 ⃗ x = ( Parikh(a path to 3) ⃗ i ⃗ 0) a [( ⃗ i ⃗ j) ← ( ⃗ i + Parikh(t1) ⃗ )]

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

BSL ⊆ DetAPA with some property X

Thus we do the following:

• 1. Let (A,C) be the PA for L ∈ BSL
• 2. Determinize A by the subset construction
• 3. Associate functions to compute the Parikh image:

A 1 2 3 a (t1) a (t2)

• SubsetDeterminize(A)

1, 2 ⃗ x = ( Parikh(a path to 1) ⃗ i Parikh(a path to 2) ⃗ j) 3 ⃗ x = ( Parikh(a path to 3) ⃗ i ⃗ 0) a [( ⃗ i ⃗ j) ← ( ⃗ j + Parikh(t2) ⃗ )]

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Outline

Result and definitions BSL ⊆ DetPA BSL ⊆ DetAPA with some property X DetAPA with this property ⊆ DetPA Corollaries and Further Work

13 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

DetAPA with the property X ⊆ DetPA

▸ We can assume the DetAPA, of language L, is of the

form: u′

1

u1 [⃗ x ← M1.⃗ x + ⃗ v1] u′

2

u2 [⃗ x ← M2.⃗ x + ⃗ v2] u3 [⃗ x ← M3.⃗ x + ⃗ v3]

14 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

DetAPA with the property X ⊆ DetPA

▸ We can assume the DetAPA, of language L, is of the

form: u′

1

u1 [⃗ x ← M1.⃗ x + ⃗ v1] u′

2

u2 [⃗ x ← M2.⃗ x + ⃗ v2] u3 [⃗ x ← M3.⃗ x + ⃗ v3]

▸ Property X: For all i, there are pi,ki s.t. Mpi i

= Mpi+ki

i

14 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

DetAPA with the property X ⊆ DetPA

▸ We can assume the DetAPA, of language L, is of the

form: u′

1

u1 [⃗ x ← M1.⃗ x + ⃗ v1] u′

2

u2 [⃗ x ← M2.⃗ x + ⃗ v2] u3 [⃗ x ← M3.⃗ x + ⃗ v3]

▸ Property X: For all i, there are pi,ki s.t. Mpi i

= Mpi+ki

i ▸ For any ⃗

a ∈ {pi,...,pi + ki}{1,2,3}, we give a DetPA for L ∩ (ua1

1 )(uk1 1 )∗ ⋅ u′ 1 ⋅ (ua2 2 )(uk2 2 )∗ ⋅ u′ 2 ⋅ (ua3 3 )(uk3 3 )∗

14 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

DetAPA with the property X ⊆ DetPA

▸ We can assume the DetAPA, of language L, is of the

form: u′

1

u1 [⃗ x ← M1.⃗ x + ⃗ v1] u′

2

u2 [⃗ x ← M2.⃗ x + ⃗ v2] u3 [⃗ x ← M3.⃗ x + ⃗ v3]

▸ Property X: For all i, there are pi,ki s.t. Mpi i

= Mpi+ki

i ▸ For any ⃗

a ∈ {pi,...,pi + ki}{1,2,3}, we give a DetPA for L ∩ (ua1

1 )(uk1 1 )∗ ⋅ u′ 1 ⋅ (ua2 2 )(uk2 2 )∗ ⋅ u′ 2 ⋅ (ua3 3 )(uk3 3 )∗ ▸▸ Construction main idea: suppose a word contains t

times uk2

2 . Final value of ⃗

x contains:

14 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

DetAPA with the property X ⊆ DetPA

▸ We can assume the DetAPA, of language L, is of the

form: u′

1

u1 [⃗ x ← M1.⃗ x + ⃗ v1] u′

2

u2 [⃗ x ← M2.⃗ x + ⃗ v2] u3 [⃗ x ← M3.⃗ x + ⃗ v3]

▸ Property X: For all i, there are pi,ki s.t. Mpi i

= Mpi+ki

i ▸ For any ⃗

a ∈ {pi,...,pi + ki}{1,2,3}, we give a DetPA for L ∩ (ua1

1 )(uk1 1 )∗ ⋅ u′ 1 ⋅ (ua2 2 )(uk2 2 )∗ ⋅ u′ 2 ⋅ (ua3 3 )(uk3 3 )∗ ▸▸ Construction main idea: suppose a word contains t

times uk2

2 . Final value of ⃗

x contains: ∑t

i=1 ⋯Mi×k2 2

Ma2

2 ⋯

14 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

DetAPA with the property X ⊆ DetPA

▸ We can assume the DetAPA, of language L, is of the

form: u′

1

u1 [⃗ x ← M1.⃗ x + ⃗ v1] u′

2

u2 [⃗ x ← M2.⃗ x + ⃗ v2] u3 [⃗ x ← M3.⃗ x + ⃗ v3]

▸ Property X: For all i, there are pi,ki s.t. Mpi i

= Mpi+ki

i ▸ For any ⃗

a ∈ {pi,...,pi + ki}{1,2,3}, we give a DetPA for L ∩ (ua1

1 )(uk1 1 )∗ ⋅ u′ 1 ⋅ (ua2 2 )(uk2 2 )∗ ⋅ u′ 2 ⋅ (ua3 3 )(uk3 3 )∗ ▸▸ Construction main idea: suppose a word contains t

times uk2

2 . Final value of ⃗

x contains: ∑t

i=1 ⋯Mi×k2 2

Ma2

2 ⋯ = t × ⋯Ma2 2 ⋯

14 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

DetAPA with the property X ⊆ DetPA

▸ We can assume the DetAPA, of language L, is of the

form: u′

1

u1 [⃗ x ← M1.⃗ x + ⃗ v1] u′

2

u2 [⃗ x ← M2.⃗ x + ⃗ v2] u3 [⃗ x ← M3.⃗ x + ⃗ v3]

▸ Property X: For all i, there are pi,ki s.t. Mpi i

= Mpi+ki

i ▸ For any ⃗

a ∈ {pi,...,pi + ki}{1,2,3}, we give a DetPA for L ∩ (ua1

1 )(uk1 1 )∗ ⋅ u′ 1 ⋅ (ua2 2 )(uk2 2 )∗ ⋅ u′ 2 ⋅ (ua3 3 )(uk3 3 )∗ ▸▸ Construction main idea: suppose a word contains t

times uk2

2 . Final value of ⃗

x contains: ∑t

i=1 ⋯Mi×k2 2

Ma2

2 ⋯ = t × ⋯Ma2 2 ⋯ ▸▸ Contribution of (uk2 2 ) constant

14 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Outline

Result and definitions BSL ⊆ DetPA Corollaries and Further Work

14 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Corollaries and Further Work

Corollaries:

▸ Recall PA = RBCM; moreover DetPA ⊊ DetRBCM, thus

RBCM accepting a bounded language can be determinized

15 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Corollaries and Further Work

Corollaries:

▸ Recall PA = RBCM; moreover DetPA ⊊ DetRBCM, thus

RBCM accepting a bounded language can be determinized

▸ In fact, the resulting DetPA has a special form: union of

flat automata → CQDD [Bouajjani and Habermehl, 1999]; thus CQDD = BSL

15 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Corollaries and Further Work

Corollaries:

▸ Recall PA = RBCM; moreover DetPA ⊊ DetRBCM, thus

RBCM accepting a bounded language can be determinized

▸ In fact, the resulting DetPA has a special form: union of

flat automata → CQDD [Bouajjani and Habermehl, 1999]; thus CQDD = BSL

▸ Properties of BSL: closure under concat, under

h−1 ∩ BSL,

15 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Corollaries and Further Work

Corollaries:

▸ Recall PA = RBCM; moreover DetPA ⊊ DetRBCM, thus

RBCM accepting a bounded language can be determinized

▸ In fact, the resulting DetPA has a special form: union of

flat automata → CQDD [Bouajjani and Habermehl, 1999]; thus CQDD = BSL

▸ Properties of BSL: closure under concat, under

h−1 ∩ BSL, and L ∈ BSL ⇒ (∀⃗ w)[L ⊆ w∗

1 ⋯w∗ n → Iter ⃗ w(L) is SL]

15 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Corollaries and Further Work

Corollaries:

▸ Recall PA = RBCM; moreover DetPA ⊊ DetRBCM, thus

RBCM accepting a bounded language can be determinized

▸ In fact, the resulting DetPA has a special form: union of

flat automata → CQDD [Bouajjani and Habermehl, 1999]; thus CQDD = BSL

▸ Properties of BSL: closure under concat, under

h−1 ∩ BSL, and L ∈ BSL ⇒ (∀⃗ w)[L ⊆ w∗

1 ⋯w∗ n → Iter ⃗ w(L) is SL]

What implications to algebraic/logic characterizations?

15 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Thank you

Result and definitions BSL ⊆ DetPA BSL ⊆ DetAPA with some property X DetAPA with this property ⊆ DetPA Corollaries and Further Work

16 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Bibliography I

Bouajjani, A. and Habermehl, P. (1999). Symbolic reachability analysis of FIFO-channel systems with nonregular sets of configurations. In Theoretical Computer Science. Ibarra, O. H. (1978). Reversal-bounded multicounter machines and their decision problems.

• J. ACM, 25(1):116–133.

Klaedtke, F. and Rueß, H. (2003). Monadic second-order logics with cardinalities. In Proceedings of the 30th International Colloquium on Automata, Languages, and Programming (ICALP 2003), volume 2719 of Lecture Notes in Computer Science, pages 681–696. Springer-Verlag.

17 / 18

Bounded Parikh Automata Cadilhac, Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA

BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA

Corollaries and Further Work Bibliography

Bibliography II

Mitrana, V. and Stiebe, R. (2001). Extended finite automata over groups. Discrete Appl. Math., 108(3):287–300. Parikh, R. J. (1966). On context-free languages. Journal of the ACM, 13(4):570–581.

18 / 18