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