Non-Self-Embedding Grammars,Constant-Height Pushdown Automata,

and Limited Automata

Bruno Guillon Giovanni Pighizzini Luca Prigioniero

Dipartimento di Informatica, Università degli Studi di Milano

ciaaAugust 1, 2018

0 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

G : S → (S)|SS|ε ( | push ) | pop

Definition (nse [Chomsky 1959])G is self-embedding if for some X ,X ∗=⇒ αXβ with both α, β nonempty.Otherwise, G is non-self-embedding.

Definition (h-pda)An h-height pda is a pdawith stack size ≤ h ∈ N.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

G : S → (S)|SS|ε ( | push ) | pop

Definition (nse [Chomsky 1959])G is self-embedding if for some X ,X ∗=⇒ αXβ with both α, β nonempty.Otherwise, G is non-self-embedding.

Definition (h-pda)An h-height pda is a pdawith stack size ≤ h ∈ N.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

G : S → (S)|SS|ε ( | push ) | pop

Definition (nse [Chomsky 1959])G is self-embedding if for some X ,X ∗=⇒ αXβ with both α, β nonempty.Otherwise, G is non-self-embedding.

Definition (h-pda)An h-height pda is a pdawith stack size ≤ h ∈ N.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

G : S → (S)|SS|ε ( | push ) | pop

Definition (nse [Chomsky 1959])G is self-embedding if for some X ,X ∗=⇒ αXβ with both α, β nonempty.Otherwise, G is non-self-embedding.

Definition (h-pda)An h-height pda is a pdawith stack size ≤ h ∈ N.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

G : S → (S)|SS|ε ( | push ) | pop

Definition (nse [Chomsky 1959])G is self-embedding if for some X ,X ∗=⇒ αXβ with both α, β nonempty.Otherwise, G is non-self-embedding.

Definition (h-pda)An h-height pda is a pdawith stack size ≤ h ∈ N.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Context-free ability: describe recursive structure

Context-free

RegularcfG

reG

pda

fa

d-la [Hibbard’67]

nseG [Chomsky’59] h-pda 1-la [Wechsung,Wagner’86]

. ( ( ) ( ) ) ( ) /

→→→→ ←←→→→→ ←←→→→ ←←←←←←→→→→ ←←→→→ ←←←←yes.

-- -- -- --

Cells can be rewritten only in the first 2 visits!

Definition (Hibbard 1967) For d ∈ N, a d-la is a 1-tape TMallowed to rewrite a cell content only during its first d visits.

1 / 5

Concise representations of regular languages

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Definition: Sizes of models:grammars∑

X→α∈P(2 + |α|)

h-pdapoly in #Q, #∆, h

1-lapoly in #Q, #Γ

fa: poly in #Q

2 / 5

Concise representations of regular languages

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Definition: Sizes of models:grammars∑

X→α∈P(2 + |α|)

h-pdapoly in #Q, #∆, h

1-lapoly in #Q, #Γ

fa: poly in #Q

2 / 5

Concise representations of regular languages

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Definition: Sizes of models:grammars∑

X→α∈P(2 + |α|)

h-pdapoly in #Q, #∆, h

1-lapoly in #Q, #Γ

fa: poly in #Q

2 / 5

Concise representations of regular languages

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Definition: Sizes of models:grammars∑

X→α∈P(2 + |α|)

h-pdapoly in #Q, #∆, h

1-lapoly in #Q, #Γ

fa: poly in #Q

2 / 5

Concise representations of regular languages

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Definition: Sizes of models:grammars∑

X→α∈P(2 + |α|)

h-pdapoly in #Q, #∆, h

1-lapoly in #Q, #Γ

fa: poly in #Q

2 / 5

Concise representations of regular languages

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Definition: Sizes of models:grammars∑

X→α∈P(2 + |α|)

h-pdapoly in #Q, #∆, h

1-lapoly in #Q, #Γ

fa: poly in #Q

2 / 5

Concise representations of regular languages

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Definition: Sizes of models:grammars∑

X→α∈P(2 + |α|)

h-pdapoly in #Q, #∆, h

1-lapoly in #Q, #Γ

fa: poly in #Q

2 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseG

of particular form:CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseG

of particular form:CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseG

of particular form:CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseG

of particular form:CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseG

of particular form:CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseG

of particular form:CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseGof particular form:

CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseGof particular form:CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to h-pda and back

Production graphThere is an edgeX → Y if there is aproduction X → αY β

S A E H I J

B C F G K

I each SCC defines a left- or right-linear grammar[Anselmo, Giammarresi, Varricchio 2002]

I with a polynomial size increase,we can assume that each such SCC-grammar is right-linear

I the resulting grammar can in turn be transformedinto an h-pda of polynomial size (adapting [AGV02])

I Conversely, we can transform each h-pda into a poly-size nseGof particular form:CNF in which each production X → YZ is such that Y > X

3 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

From nseG to 1-la from nseG in CNF withX → YZ =⇒ Y > X

a b a b b a a a b b a

A F B D

A F E C

B E B A A S

A F E D

E C

S

. a b a b b a a a b b a /. a,A, 2 b,B, 2 a,A, 2 b,E , 1 b,B, 2 a,E , 1 a,A, 1 a,A, 2 a,E , 1 b,B, 1 a,S , 0 /

0

1

2

2

2

1

2 1

1

2 1

1

1

1

0

1

0

0

1

0

0

2 2 2 1 2 1 1 2 1 1 0

the rootits left child

C

guessed andstored in

finite control

D

guessed andstored in

finite control

S

guessed andstored in

finite control

C

guessed andstored in

finite control

D

guessed andstored in

finite control

C

D

S

C

F

E

F

A

F

4 / 5

Simulation of 1-la: an exponential gap

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Theorem: Ln = {un | n ∈ N, u ∈ {a, b}n}

I accepted by a 2dfa with O(n) statesI for which a pda or a cfG requires a size exponential in n

Thank you for your attention.5 / 5

Simulation of 1-la: an exponential gap

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Theorem: Ln = {un | n ∈ N, u ∈ {a, b}n}

I accepted by a 2dfa with O(n) statesI for which a pda or a cfG requires a size exponential in n

Thank you for your attention.5 / 5

Simulation of 1-la: an exponential gap

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Theorem: Ln = {un | n ∈ N, u ∈ {a, b}n}

I accepted by a 2dfa with O(n) statesI for which a pda or a cfG requires a size exponential in n

Thank you for your attention.5 / 5

Simulation of 1-la: an exponential gap

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Theorem: Ln = {un | n ∈ N, u ∈ {a, b}n}

I accepted by a 2dfa with O(n) statesI for which a pda or a cfG requires a size exponential in n

Thank you for your attention.5 / 5

Simulation of 1-la: an exponential gap

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Theorem: Ln = {un | n ∈ N, u ∈ {a, b}n}

I accepted by a 2dfa with O(n) statesI for which a pda or a cfG requires a size exponential in n

Thank you for your attention.5 / 5

Simulation of 1-la: an exponential gap

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Theorem: Ln = {un | n ∈ N, u ∈ {a, b}n}

I accepted by a 2dfa with O(n) statesI for which a pda or a cfG requires a size exponential in n

Thank you for your attention.5 / 5

Simulation of 1-la: an exponential gap

Regular

reG

nseG

fa

h-pda 1-la

1dfa 1nfa2nfa2dfaexp

exp exp

exp e

xp

exp expexp exp exp?

??

?

poly

poly

poly

exp

exp expexp

[Pighizzini,Prigioniero’17]

[Geffert,Mereghetti,Palano’10]

[Pighizzini,Pisoni’14]

[Pighizzini,Prigioniero’17]

Theorem: Ln = {un | n ∈ N, u ∈ {a, b}n}

I accepted by a 2dfa with O(n) statesI for which a pda or a cfG requires a size exponential in n

Thank you for your attention.5 / 5