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15. Theorietag Automaten und Formale Sprachen Lauterbad, 2829 September 2005 On the Parikh Images of Level-Two Pushdown Automata Wong Karianto karianto@informatik.rwth-aachen.de Lehrstuhl f ur Informatik VII Motivations Correspondence

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  • 15. Theorietag Automaten und Formale Sprachen

Lauterbad, 28–29 September 2005

On the Parikh Images of Level-Two Pushdown Automata

Wong Karianto

karianto@informatik.rwth-aachen.de

Lehrstuhl f¨ ur Informatik VII

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Motivations

Correspondence between automata (and formal language) theory and number theory:

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 2

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Motivations

Correspondence between automata (and formal language) theory and number theory: Automata theory finite automata, pushdown automata Number theory semi-linear sets

✲

(Parikh mapping)

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 2

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Motivations

Correspondence between automata (and formal language) theory and number theory: Automata theory finite automata, pushdown automata Number theory semi-linear sets

✲

(Parikh mapping) higher-order pushdown automata (HOPDA)

???

✲

HOPDA: finite-state automata with a stack of stacks of . . . of stacks

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 2

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SLIDE 5

Motivations

Correspondence between automata (and formal language) theory and number theory: Automata theory finite automata, pushdown automata Number theory semi-linear sets

✲

(Parikh mapping) higher-order pushdown automata (HOPDA)

???

✲

HOPDA: finite-state automata with a stack of stacks of . . . of stacks In this talk: HOPDA of level 2 (2-PDA) Two questions for a class characterizing the Parikh images of 2-PDA’s: Can all sets from this class be generated (via the Parikh mapping)? Does the Parikh image of each 2-PDA belong to this class?

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 2

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Outline

Semi-linear sets and Parikh’s theorem Level 2 pushdown automata Semi-polynomial sets From semi-polynomial sets to 2-PDA’s From 2-PDA’s to semi-polynomial sets? Conclusions

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 3

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Semi-Linear Sets

A ⊆ Nn linear: A = {¯ x0 + k1¯ x1 + . . . + km¯ xm | k1, . . . , km ∈ N} for some ¯ x0, ¯ x1, . . . , ¯ xm

  • ∈ Nn

constant vector periods

❆ ❆ ❑ ✁ ✁ ✁ ✕

Semi-linear set: finite union of linear sets. Example: B := {(x1, x2, x3) ∈ N3 | x1 < x2 < x3} is linear: {(0, 1, 2) + k1(0, 0, 1) + k2(0, 1, 1) + k3(1, 1, 1) | k1, k2, k3 ∈ N} .

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 4

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Semi-Linear Sets

A ⊆ Nn linear: A = {¯ x0 + k1¯ x1 + . . . + km¯ xm | k1, . . . , km ∈ N} for some ¯ x0, ¯ x1, . . . , ¯ xm

  • ∈ Nn

constant vector periods

❆ ❆ ❑ ✁ ✁ ✁ ✕

Semi-linear set: finite union of linear sets. Example: B := {(x1, x2, x3) ∈ N3 | x1 < x2 < x3} is linear: {(0, 1, 2) + k1(0, 0, 1) + k2(0, 1, 1) + k3(1, 1, 1) | k1, k2, k3 ∈ N} . Properties of semi-linear sets: effective closure under Boolean operations [Ginsburg & Spanier] equivalence to Presburger-definable sets [Ginsburg & Spanier]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 4

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Parikh Mapping and Parikh’s Theorem

Σ = {a1, . . . , an} Parikh mapping Φ: Σ∗ → Nn Φ(w) := (|w|a1, . . . , |w|an) . Φ(w): the Parikh image of w Φ(L) := {Φ(w) | w ∈ L} ⊆ Nn: the Parikh image of L

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 5

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Parikh Mapping and Parikh’s Theorem

Σ = {a1, . . . , an} Parikh mapping Φ: Σ∗ → Nn Φ(w) := (|w|a1, . . . , |w|an) . Φ(w): the Parikh image of w Φ(L) := {Φ(w) | w ∈ L} ⊆ Nn: the Parikh image of L Theorem (Parikh (1961)): The Parikh image of any context-free language is effectively semi-linear.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 5

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Higher-Order Pushdown Automata

Finite-state automata augmented with a nested pushdown stack, i.e., a stack of stacks of . . . stacks Level n HOPDA: n-fold nested stacks

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 6

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Higher-Order Pushdown Automata

Finite-state automata augmented with a nested pushdown stack, i.e., a stack of stacks of . . . stacks Level n HOPDA: n-fold nested stacks Background:

◮ Maslov (1976): Formal definition; correspondence with

generalized indexed languages

◮ Damm and Goerdt (1982): automaton characterization of the

OI hierarchy

◮ Engelfriet (1983): correspondence to complexity classes ◮ Carayol and W¨

  • hrle (2003): correspondence to a hierarchy of

infinite graphs introduced by Caucal

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 6

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Level 2 PDA

Stack alphabet Γ with initial symbol ⊥: Level 1 stack (1-stack): [Zm · · · Z1]; Zm is the topmost symbol. Level 2 stack (2-stack): [sr, . . . , s1], where s1, . . . , sr are 1-stacks, and sr is the topmost 1-stack. Empty level 2 stack [[ε]]; initial level 2 stack [[⊥]].

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 7

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Level 2 PDA

Stack alphabet Γ with initial symbol ⊥: Level 1 stack (1-stack): [Zm · · · Z1]; Zm is the topmost symbol. Level 2 stack (2-stack): [sr, . . . , s1], where s1, . . . , sr are 1-stacks, and sr is the topmost 1-stack. Empty level 2 stack [[ε]]; initial level 2 stack [[⊥]]. Instructions on 1-stacks: push and pop Instructions on level 2 stacks:

◮ push and pop on the topmost 1-stack ◮ copy the topmost 1-stack ◮ remove the topmost 1-stack

Access: only to the topmost symbol of the topmost 1-stack !

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 7

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Example: A Non-Semi-Linear Language

Lquad := {akbk2 | k ∈ N} ⊆ {a, b}∗ Take Γ := {⊥, Z, Z2} and process input akbk2 as follows: [[Z2k⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8

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SLIDE 16

Example: A Non-Semi-Linear Language

Lquad := {akbk2 | k ∈ N} ⊆ {a, b}∗ Take Γ := {⊥, Z, Z2} and process input akbk2 as follows: [[Z2Z2k⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8

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SLIDE 17

Example: A Non-Semi-Linear Language

Lquad := {akbk2 | k ∈ N} ⊆ {a, b}∗ Take Γ := {⊥, Z, Z2} and process input akbk2 as follows: [[Z2Z2k⊥], [Z2Z2k⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8

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Example: A Non-Semi-Linear Language

Lquad := {akbk2 | k ∈ N} ⊆ {a, b}∗ Take Γ := {⊥, Z, Z2} and process input akbk2 as follows: [[ZZ2(k−1)⊥], [Z2 Z2k ⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8

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SLIDE 19

Example: A Non-Semi-Linear Language

Lquad := {akbk2 | k ∈ N} ⊆ {a, b}∗ Take Γ := {⊥, Z, Z2} and process input akbk2 as follows: [[Z⊥], [ZZZ⊥], [ZZZZZ⊥], . . . [Z Z2(k−1) ⊥], [Z2 Z2k ⊥]] Number of Z’s above Z2: k−1

i=0 (2i + 1) = k2

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8

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SLIDE 20

Example: A Non-Semi-Linear Language

Lquad := {akbk2 | k ∈ N} ⊆ {a, b}∗ Take Γ := {⊥, Z, Z2} and process input akbk2 as follows: [[Z⊥], [ZZZ⊥], [ZZZZZ⊥], . . . [Z Z2(k−1) ⊥], [Z2 Z2k ⊥]] Number of Z’s above Z2: k−1

i=0 (2i + 1) = k2

Φ(Lquad) = {(x1, x2) ∈ N2 | x2

1 = x2}

= ⇒ not semi-linear (proof by growth rate arguments)

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 8

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Semi-Polynomial Sets

A ⊆ Nn linear: ¯ x ∈ A iff k1, . . . , km ∈ N exist such that ¯ x = ¯ x0 + k1¯ x1 + . . . + km¯ xm.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 9

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Semi-Polynomial Sets

A ⊆ Nn linear: ¯ x ∈ A iff k1, . . . , km ∈ N exist such that ¯ x = ¯ x0 + k1¯ x1 + . . . + km¯ xm. Replace the vectors with their components: ¯ x = (x01, . . . , x0n) + (k1x11, . . . , k1x1n) + . . . + (kmxm1, . . . , kmxmn) and replace linear terms kixij with polynomial ones.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 9

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Semi-Polynomial Sets

A ⊆ Nn linear: ¯ x ∈ A iff k1, . . . , km ∈ N exist such that ¯ x = ¯ x0 + k1¯ x1 + . . . + km¯ xm. Replace the vectors with their components: ¯ x = (x01, . . . , x0n) + (k1x11, . . . , k1x1n) + . . . + (kmxm1, . . . , kmxmn) and replace linear terms kixij with polynomial ones. A ⊆ Nn, n ≥ 1, is called polynomial of degree d ≥ 1 if there exist ¯ x0 (the constant) and (¯ xi,j)1≤i≤m,1≤j≤d (the periods) with A = {¯ x0 + k1¯ x1,1 + k2

1¯

x1,2 + · · · + kd−1

1

¯ x1,d−1 + kd

1 ¯

x1,d + · · · + km¯ xm,1 + k2

m¯

xm,2 + · · · + kd−1

m

¯ xm,d−1 + kd

m¯

xm,d | k1, . . . , km ∈ N} . Semi-polynomial set of degree d: finite union of polynomial sets of degree d

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 9

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From Semi-Polynomial Sets to 2-PDA’s

Aim: show that each semi-polynomial set can be generated (via the Parikh mapping) by some 2-PDA

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 10

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From Semi-Polynomial Sets to 2-PDA’s

Aim: show that each semi-polynomial set can be generated (via the Parikh mapping) by some 2-PDA Idea: extend the idea used for the case of quadratic sets

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 10

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From Semi-Polynomial Sets to 2-PDA’s

Aim: show that each semi-polynomial set can be generated (via the Parikh mapping) by some 2-PDA Idea: extend the idea used for the case of quadratic sets A technical lemma: decomposition of polynomials Lemma: Let k ≥ 0 and e > 0. Then, ke =

k−1

  • i=0
  • ce−1ie−1 + ce−2ie−2 + · · · + c2i2 + c1i + 1
  • ,

(1) where cj := e

j

  • for j = 1, . . . , e − 1.
  • Proof. By induction on k.
  • RWTH Aachen – Wong Karianto

On the Parikh Images of Level-Two Pushdown Automata – p. 10

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The Core Construction

Lemma: Let d ≥ 1 and Γ := {⊥, Z1, . . . , Zd}. Then, for 1 ≤ e ≤ d, k ∈ N, and w ∈ Σ∗, we can construct a 2-PDA A with states p and q which proceeds from configuration (p, [[ZdZ2k⊥], sr, . . . , s1]) to configuration (q, [[ZdZ2k⊥], sr, . . . , s1]) after reading wke.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 11

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The Core Construction

Lemma: Let d ≥ 1 and Γ := {⊥, Z1, . . . , Zd}. Then, for 1 ≤ e ≤ d, k ∈ N, and w ∈ Σ∗, we can construct a 2-PDA A with states p and q which proceeds from configuration (p, [[ZdZ2k⊥], sr, . . . , s1]) to configuration (q, [[ZdZ2k⊥], sr, . . . , s1]) after reading wke.

  • Proof. By induction on d.

For simplicity, let Z := Z1. d = 1: Only subcase e = 1 ⇒ ke = k. [[ZdZ2k⊥], . . . ]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 11

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SLIDE 29

The Core Construction

Lemma: Let d ≥ 1 and Γ := {⊥, Z1, . . . , Zd}. Then, for 1 ≤ e ≤ d, k ∈ N, and w ∈ Σ∗, we can construct a 2-PDA A with states p and q which proceeds from configuration (p, [[ZdZ2k⊥], sr, . . . , s1]) to configuration (q, [[ZdZ2k⊥], sr, . . . , s1]) after reading wke.

  • Proof. By induction on d.

For simplicity, let Z := Z1. d = 1: Only subcase e = 1 ⇒ ke = k. [[ZdZ2k⊥], [ZdZ2k⊥], . . . ]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 11

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SLIDE 30

The Core Construction

Lemma: Let d ≥ 1 and Γ := {⊥, Z1, . . . , Zd}. Then, for 1 ≤ e ≤ d, k ∈ N, and w ∈ Σ∗, we can construct a 2-PDA A with states p and q which proceeds from configuration (p, [[ZdZ2k⊥], sr, . . . , s1]) to configuration (q, [[ZdZ2k⊥], sr, . . . , s1]) after reading wke.

  • Proof. By induction on d.

For simplicity, let Z := Z1. d = 1: Only subcase e = 1 ⇒ ke = k. [[Z2k⊥], [ZdZ2k⊥], . . . ]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 11

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SLIDE 31

The Core Construction

Lemma: Let d ≥ 1 and Γ := {⊥, Z1, . . . , Zd}. Then, for 1 ≤ e ≤ d, k ∈ N, and w ∈ Σ∗, we can construct a 2-PDA A with states p and q which proceeds from configuration (p, [[ZdZ2k⊥], sr, . . . , s1]) to configuration (q, [[ZdZ2k⊥], sr, . . . , s1]) after reading wke.

  • Proof. By induction on d.

For simplicity, let Z := Z1. d = 1: Only subcase e = 1 ⇒ ke = k. d = 2: For e = 1, as before. For e = 2, as with Lquad.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 11

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SLIDE 32

The Core Construction

Lemma: Let d ≥ 1 and Γ := {⊥, Z1, . . . , Zd}. Then, for 1 ≤ e ≤ d, k ∈ N, and w ∈ Σ∗, we can construct a 2-PDA A with states p and q which proceeds from configuration (p, [[ZdZ2k⊥], sr, . . . , s1]) to configuration (q, [[ZdZ2k⊥], sr, . . . , s1]) after reading wke.

  • Proof. By induction on d.

For simplicity, let Z := Z1. d = 1: Only subcase e = 1 ⇒ ke = k. d = 2: For e = 1, as before. For e = 2, as with Lquad. d ≥ 3: For e = 1, 2, as before. For e = 3, . . . , d, apply Decomposition Lemma: ke =

k−1

  • i=0
  • ce−1ie−1 + ce−2ie−2 + · · · + c2i2 + c1i + 1
  • ,

where cj := e

j

  • , for j = 1, . . . , e − 1.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 11

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SLIDE 33

[[Zd Z2k ⊥], . . . ] .

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 12

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SLIDE 34

[[Ze−1 Z2(k−1) ⊥], [Zd Z2k ⊥], . . . ] .

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 12

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SLIDE 35

[[Ze−1⊥], [Ze−1ZZ⊥], [Ze−1ZZZZ⊥], . . . [Ze−1 Z2(k−1) ⊥], [Zd Z2k ⊥], . . . ] .

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 12

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SLIDE 36

[[Ze−1⊥], [Ze−1ZZ⊥], [Ze−1ZZZZ⊥], . . . [Ze−1 Z2(k−1) ⊥], [Zd Z2k ⊥], . . . ] . For each i = 0, . . . , k − 1, starting from 1-stack [Ze−1Z2i⊥], process (ce−1ie−1 + ce−2ie−2 + · · · + c2i2 + c1i + 1) . successive words w, using the induction hypothesis. The procedure ends if Zd appears, having processed

k−1

  • i=0
  • ce−1ie−1 + ce−2ie−2 + · · · + c2i2 + c1i + 1
  • ,

i.e. (ke)-many successive words w.

  • RWTH Aachen – Wong Karianto

On the Parikh Images of Level-Two Pushdown Automata – p. 12

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SLIDE 37

Main Theorem

Theorem: Every semi-polynomial set is the Parikh image of a 2-PDA-recognizable language.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 13

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SLIDE 38

Main Theorem

Theorem: Every semi-polynomial set is the Parikh image of a 2-PDA-recognizable language.

  • Proof. W

.l.o.g., consider only polynomial sets (closure under union). Let A ⊆ Nn be a polynomial set of degree d, given by ¯ x0 and ¯ xi,j (1 ≤ i ≤ m, 1 ≤ j ≤ d). Define Σ := {a1, . . . , an} and assign to ¯ x0 and ¯ xi,j words w0 and wi,j in a∗

1 · · · a∗ n.

Construct 2-PDA A with L(A) = {w0 wk1

1,1wk2

1

1,2 · · · wkd−1

1

1,d−1wkd

1

1,d

· · · wkm

m,1wk2

m

m,2 · · · wkd−1

m

m,d−1wkd

m

m,d | k1, . . . , km ∈ N} .

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 13

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SLIDE 39

Main Theorem (continued)

Read w0 (without using the stack). For i = 1, . . . , m:

◮ Guess ki by pushing (2ki)-many Z’s followed by a Zd, resulting

in [[ZdZ2ki⊥]].

◮ For j = 1, . . . , d: By previous lemma, process (kj

i )-many

successive words wi,j.

◮ After having processed

wki

i,1wk2

i

i,2 · · · w kd−1

i

i,d−1wkd

i

i,d ,

remove ki from the stack and proceed with next i. The procedure ends after we have processed w0 wk1

1,1wk2

1

1,2 · · · wkd−1

1

1,d−1wkd

1

1,d · · · wkm m,1wk2

m

m,2 · · · wkd−1

m

m,d−1wkd

m

m,d .

  • RWTH Aachen – Wong Karianto

On the Parikh Images of Level-Two Pushdown Automata – p. 14

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SLIDE 40

From 2-PDA to Semi-Polynomial Sets? (1)

Lexp := {axb2x | x ∈ N} is 2-PDA-recognizable (Carayol and W¨

  • hrle):

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 15

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SLIDE 41

From 2-PDA to Semi-Polynomial Sets? (1)

Lexp := {axb2x | x ∈ N} is 2-PDA-recognizable (Carayol and W¨

  • hrle):

Take Γ := {⊥, #, Z, 0, 1} and process input akb2k as follows: [[Zk⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 15

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SLIDE 42

From 2-PDA to Semi-Polynomial Sets? (1)

Lexp := {axb2x | x ∈ N} is 2-PDA-recognizable (Carayol and W¨

  • hrle):

Take Γ := {⊥, #, Z, 0, 1} and process input akb2k as follows: [[#Zk⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 15

slide-43
SLIDE 43

From 2-PDA to Semi-Polynomial Sets? (1)

Lexp := {axb2x | x ∈ N} is 2-PDA-recognizable (Carayol and W¨

  • hrle):

Take Γ := {⊥, #, Z, 0, 1} and process input akb2k as follows: [[#Zk⊥], [#Zk⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 15

slide-44
SLIDE 44

From 2-PDA to Semi-Polynomial Sets? (1)

Lexp := {axb2x | x ∈ N} is 2-PDA-recognizable (Carayol and W¨

  • hrle):

Take Γ := {⊥, #, Z, 0, 1} and process input akb2k as follows: [[0Zk−1⊥], [# Zk ⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 15

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SLIDE 45

From 2-PDA to Semi-Polynomial Sets? (1)

Lexp := {axb2x | x ∈ N} is 2-PDA-recognizable (Carayol and W¨

  • hrle):

Take Γ := {⊥, #, Z, 0, 1} and process input akb2k as follows: [[0⊥], [0Z⊥], [0ZZ⊥], . . . [0 Zk−1 ⊥], [# Zk ⊥]] Implement a binary counting using top symbols of 1-stacks as bits.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 15

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SLIDE 46

From 2-PDA to Semi-Polynomial Sets? (1)

Lexp := {axb2x | x ∈ N} is 2-PDA-recognizable (Carayol and W¨

  • hrle):

Take Γ := {⊥, #, Z, 0, 1} and process input akb2k as follows: [[0⊥], [0Z⊥], [0ZZ⊥], . . . [0 Zk−1 ⊥], [# Zk ⊥]] Implement a binary counting using top symbols of 1-stacks as bits. However, Φ(Lexp) = {(x, 2x) | x ∈ N} is not semi-polynomial ! (proof by growth rate arguments).

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 15

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SLIDE 47

From 2-PDA to Semi-Polynomial Sets? (2)

Lprod := {axbycxy | x, y ∈ N} is 2-PDA-recognizable:

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 16

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SLIDE 48

From 2-PDA to Semi-Polynomial Sets? (2)

Lprod := {axbycxy | x, y ∈ N} is 2-PDA-recognizable: Take Γ := {⊥, Z} and process input axbycxy as follows: [[Zx⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 16

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SLIDE 49

From 2-PDA to Semi-Polynomial Sets? (2)

Lprod := {axbycxy | x, y ∈ N} is 2-PDA-recognizable: Take Γ := {⊥, Z} and process input axbycxy as follows: [[Zx⊥], [Zx⊥]]

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 16

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SLIDE 50

From 2-PDA to Semi-Polynomial Sets? (2)

Lprod := {axbycxy | x, y ∈ N} is 2-PDA-recognizable: Take Γ := {⊥, Z} and process input axbycxy as follows: [[Zx⊥], . . . [Zx⊥], [Zx⊥]]                y-times Number of Z’s: xy

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 16

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SLIDE 51

From 2-PDA to Semi-Polynomial Sets? (2)

Lprod := {axbycxy | x, y ∈ N} is 2-PDA-recognizable: Take Γ := {⊥, Z} and process input axbycxy as follows: [[Zx⊥], . . . [Zx⊥], [Zx⊥]]                y-times Number of Z’s: xy However, Φ(Lprod) = {(x, y, xy) | x, y ∈ N} is not semi-polynomial ! (the proof involves a number-theoretical analysis)

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 16

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SLIDE 52

Summary and Further Prospects

Summary: Semi-Polynomial sets cannot capture the Parikh images of 2-PDA’s. Ingredients needed for a characterization:

◮ polynomials ◮ product relations ◮ exponential relations

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 17

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SLIDE 53

Summary and Further Prospects

Summary: Semi-Polynomial sets cannot capture the Parikh images of 2-PDA’s. Ingredients needed for a characterization:

◮ polynomials ◮ product relations ◮ exponential relations

Future work: Extending semi-polynomial sets to capture the Parikh images of 2-PDA’s (n-PDA’s). Restricting 2-PDA’s such that only semi-polynomial sets are generated.

RWTH Aachen – Wong Karianto On the Parikh Images of Level-Two Pushdown Automata – p. 17