Limited Automata and Context-Free Languages - Università degli
Limited Automata and Context-Free Languages Giovanni Pighizzini
Andrea Pisoni
Dipartimento di Informatica Università degli Studi di Milano, Italy
NCMA 2013 Umeå, Sweden August 13–14, 2013
The Chomsky Hierarchy
type 0
1-tape Turing Machines
type 1
Linear Bounded Automata Pushdown Automata Finite Automata
type 2
type 3
Limited Automata [Hibbard’67] One-tape Turing machines with restricted rewritings
Definition Fixed an integer d ≥ 1, a d -limited automaton is I
a one-tape Turing machine
I
which is allowed to rewrite the content of each tape cell only in the first d visits
Limited Automata [Hibbard’67] One-tape Turing machines with restricted rewritings
Definition Fixed an integer d ≥ 1, a d -limited automaton is I
a one-tape Turing machine
I
which is allowed to rewrite the content of each tape cell only in the first d visits
Limited Automata [Hibbard’67] One-tape Turing machines with restricted rewritings
Definition Fixed an integer d ≥ 1, a d -limited automaton is I
a one-tape Turing machine
I
which is allowed to rewrite the content of each tape cell only in the first d visits
Computational power I
For each d ≥ 2, d -limited automata characterize context-free languages [Hibbard’67]
I
1-limited automata characterize regular languages [Wagner&Wechsung’86]
The Chomsky Hierarchy
type 0
1-tape Turing Machines
type 1
Linear Bounded Automata Pushdown Automata Finite Automata
type 2
type 3
The Chomsky Hierarchy
type 0
1-tape Turing Machines
type 1
Linear Bounded Automata 2-Limited Automata Finite Automata
type 2
type 3
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
I
Determinism vs Nondeterminism
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
2-LAs → PDAs Exponential gap
I
Determinism vs Nondeterminism
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
2-LAs → PDAs Exponential gap
I
Determinism vs Nondeterminism
PDAs → 2-LAs Polynomial upper bound
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
2-LAs → PDAs Exponential gap
I
Determinism vs Nondeterminism
PDAs → 2-LAs Polynomial upper bound
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
2-LAs → PDAs Exponential gap
I
PDAs → 2-LAs Polynomial upper bound
Determinism vs Nondeterminism
Deterministic Context-Free Languages ≡ Deterministic 2-LAs
Example: Balanced Parentheses B
(
)
(
(
(
)
)
)
C
6
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
−→ (i) (ii) (iii) (iv) (v)
(
)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
−→ (i) (ii) (iii) (iv) (v)
(
)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
−→ (i) (ii) (iii) (iv) (v)
(
#
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
(
# 6
(i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
( 6
(i) (ii) (iii) (iv) (v)
#
(
(
(
)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # 6
(i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # 6
(i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
#
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
(
# 6
(i) (ii) (iii) (iv) (v)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
( 6
(i) (ii) (iii) (iv) (v)
#
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
# # 6
(i) (ii) (iii) (iv) (v)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
# #
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
# #
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
# #
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
# # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
# # # 6
(i) (ii) (iii) (iv) (v)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
# # # 6
(i) (ii) (iii) (iv) (v)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
# # # 6
(i) (ii) (iii) (iv) (v)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
( 6
(i) (ii) (iii) (iv) (v)
# # #
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # 6
(i) (ii) (iii) (iv) (v)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6 ←−
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
( 6
(i) (ii) (iii) (iv) (v)
# # # # # C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
←− 6 (i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6 ←−
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
yes!
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject Each cell is rewritten only in the first 2 visits!
Simulation of 2-Limited Automata by Pushdown Automata
Problem How much it costs, in the description size, the simulation of 2-LAs by PDAs?
This work Exponential cost!
Simulation of 2-Limited Automata by Pushdown Automata
Problem How much it costs, in the description size, the simulation of 2-LAs by PDAs?
This work Exponential cost!
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} w
w q
q p p
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} −→
j
w
q
p
(q, −1, p, −1) ∈ τw
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} ←−
w
q
j
p
(q, −1, p, −1) ∈ τw
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} w
w
p p
←−
j
q
q
(q, −1, p, −1) ∈ τw
(q, +1, p, −1) ∈ τw
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} w
←−
w q
q
j
p p
(q, −1, p, −1) ∈ τw
(q, +1, p, −1) ∈ τw
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B a b c d e f g h i . . .C 6
... 2-LA
q0
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B X b c d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B X B c d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B c d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B c d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y Z f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y Z F g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y Z E F g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y Z E F g h i . . .C 6
... 2-LA
q
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEFn Y X q τAB PDA
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Yn Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Yn X q τAB PDA
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B Xn Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y Xn q τ PDA AB
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τABn PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
δ(q, g ) 3 (p, Z , +1) move to the right
⇓
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
δ(q, g ) 3 (p, Z , +1) move to the right
⇓ B A B X Y E F Z h i . . .C 6
2-LA
p
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
δ(q, g ) 3 (p, Z , +1)
normal mode
move to the right
push and direct simulation
⇓
⇓
B A B X Y E F Z h i . . .C 6
2-LA
p
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
δ(q, g ) 3 (p, Z , +1)
normal mode
move to the right
push and direct simulation
⇓
⇓
B A B X Y E F Zn h i . . .C 6
2-LA
p
. a b c d e f g h i . .Zn 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
δ(p, h) 3 (r , H, −1) move to the left
⇓
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
δ(p, h) 3 (r , H, −1) move to the left
⇓
B A B X Y E F Z H i . . .C 6
2-LA
r
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
δ(p, h) 3 (r , H, −1) move to the left
⇓
B A B X Y E F Z H i . . .C 6
2-LA
r
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
back mode
⇓ . a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
δ(p, h) 3 (r , H, −1) move to the left
⇓
B A B X Y E F Z Hn i . . .C 6
2-LA
r
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
back mode
⇓ . a b c d e f g h i . .Z 6 τEF Y X r , τHn τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
δ(r , Z ) 3 (q, G , −1) move to the left
⇓
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
δ(r , Z ) 3 (q, G , −1) move to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
q
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
δ(r , Z ) 3 (q, G , −1) move to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
q
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
back mode
⇓ a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
δ(r , Z ) 3 (q, G , −1) move to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
q
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
back mode
⇓ a b c d e f g h i ... 6 - τEF Y X q, τGHn τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
(q, +1, s, −1) ∈ τEF exit to the left
⇓
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
(q, +1, s, −1) ∈ τEF exit to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
s
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
(q, +1, s, −1) ∈ τEF exit to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
s
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
back mode
⇓ a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
(q, +1, s, −1) ∈ τEF exit to the left
⇓
B A B X Y E F G H i . . .C
6
2-LA
s
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
back mode
⇓ a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1) move to the right
⇓
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1) move to the right
⇓ B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1)
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
move to the right
back mode
⇓
⇓
B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1)
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
move to the right
back mode
⇓
⇓
B A B X D E F G H i . . .C
6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1)
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
move to the right
back mode
⇓
⇓
B A B X Dn E F G H i . . .C 6
2-LA
p
a b c d e f g h i ... 6 - τDn X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
(p, −1, r , +1) ∈ τE ···H exit to the right
⇓
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
(p, −1, r , +1) ∈ τE ···H exit to the right
⇓ B A B X D E F G H i . . .C 6 X X
2-LA
r
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
(p, −1, r , +1) ∈ τE ···H
resume normal mode
exit to the right
move to the right
⇓
⇓
B A B X D E F G H i . . .C 6 X X
2-LA
r
a b c d e f g h i ... 6 -τD···H X r τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
(p, −1, r , +1) ∈ τE ···H
resume normal mode
exit to the right
move to the right
⇓
⇓
B A B X D E F G H i . . .C
6
2-LA
r
a b c d e f g h i ... 6 -τD···H
X r τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
(p, −1, r , +1) ∈ τE ···H
resume normal mode
exit to the right
move to the right
⇓
⇓
B A B Xn D E F G H i . . .C 6
2-LA
r
a b c d e f g h i ... 6 -τD···H Xn r τ PDA AB
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
(p, −1, r , +1) ∈ τE ···H
resume normal mode
exit to the right
move to the right
⇓
⇓
B A B X D E F G H i . . .C 6
2-LA
r
a b c d e f g h i ... 6 -τD···H X r τABn PDA
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Determinism vs nondeterminism I
Determinism is preserved by the simulation provided that the input of the PDA is right end-marked
I
Double exponential size for the simulation of D2-LAs by DPDAs
I
Conjecture: this cost cannot be reduced
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Determinism vs nondeterminism I
Determinism is preserved by the simulation provided that the input of the PDA is right end-marked
I
Double exponential size for the simulation of D2-LAs by DPDAs
I
Conjecture: this cost cannot be reduced
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Determinism vs nondeterminism I
Determinism is preserved by the simulation provided that the input of the PDA is right end-marked
I
Double exponential size for the simulation of D2-LAs by DPDAs
I
Conjecture: this cost cannot be reduced
Simulation of Pushdown Automata by 2-Limited Automata
PDAs → 2-LAs Polynomial cost! (in the description size)
Simulation of Pushdown Automata by 2-Limited Automata
PDAs → 2-LAs Polynomial cost!
DPDAs → D2-LAs Polynomial cost! (in the description size)
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Z Y X q PDA X
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Z Y X q PDA X
B X X Y f g h i . . .C 6
2-LA
q, Z
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Zm Y X q PDA X
B X X Y f g h i . . .C 6
2-LA
q, Zm
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Z Y X q PDA X
B X X Y f g h i . . .C
6
2-LA
q, Z
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Z Y X q PDA X
B X X Y f g h i . . .C 6
2-LA
q, Z
Normal form for (D)PDAs: I
at each step, the stack height increases at most by 1
I
-moves cannot push on the stack
Each (D)PDA can be simulated by an equivalent (D)2-LA of polynomial size
Determinism vs Nondeterminism in Limited Automata Corollary of the simulations Deterministic 2-LAs ≡ Deterministic Context-Free Languages On the other hand, the language L = {an bn c | n ≥ 0} ∪ {an b2n d | n ≥ 0} is accepted by a deterministic 3-LA, but it is not a DCFL Infinite hierarchy [Hibbard’67] For each d ≥ 2 there is a language which is accepted by a deterministic d -limited automaton and that cannot be accepted by any deterministic (d − 1)-limited automaton
Determinism vs Nondeterminism in Limited Automata Corollary of the simulations Deterministic 2-LAs ≡ Deterministic Context-Free Languages On the other hand, the language L = {an bn c | n ≥ 0} ∪ {an b2n d | n ≥ 0} is accepted by a deterministic 3-LA, but it is not a DCFL Infinite hierarchy [Hibbard’67] For each d ≥ 2 there is a language which is accepted by a deterministic d -limited automaton and that cannot be accepted by any deterministic (d − 1)-limited automaton
Determinism vs Nondeterminism in Limited Automata Corollary of the simulations Deterministic 2-LAs ≡ Deterministic Context-Free Languages On the other hand, the language L = {an bn c | n ≥ 0} ∪ {an b2n d | n ≥ 0} is accepted by a deterministic 3-LA, but it is not a DCFL Infinite hierarchy [Hibbard’67] For each d ≥ 2 there is a language which is accepted by a deterministic d -limited automaton and that cannot be accepted by any deterministic (d − 1)-limited automaton
Futher Investigations
I
Descriptional complexity aspects for d > 2 We conjecture that for d > 2 the size gap from d -limited automata to PDAs remains exponential
I
Descriptional complexity aspects in the unary case Unary context-free language are regular [Ginbsurg&Rice’62]
n
Ex: Ln = (a2 )∗
2-LA DPDA minimal DFA minimal 2NFA
size O(n) O(n) 2n 2n
Futher Investigations
I
Descriptional complexity aspects for d > 2 We conjecture that for d > 2 the size gap from d -limited automata to PDAs remains exponential
I
Descriptional complexity aspects in the unary case Unary context-free language are regular [Ginbsurg&Rice’62]
n
Ex: Ln = (a2 )∗
2-LA DPDA minimal DFA minimal 2NFA
size O(n) O(n) 2n 2n
Futher Investigations
I
Descriptional complexity aspects for d > 2 We conjecture that for d > 2 the size gap from d -limited automata to PDAs remains exponential
I
Descriptional complexity aspects in the unary case Unary context-free language are regular [Ginbsurg&Rice’62]
n
Ex: Ln = (a2 )∗
2-LA DPDA minimal DFA minimal 2NFA
size O(n) O(n) 2n 2n
Futher Investigations
I
Descriptional complexity aspects for d > 2 We conjecture that for d > 2 the size gap from d -limited automata to PDAs remains exponential
I
Descriptional complexity aspects in the unary case Unary context-free language are regular [Ginbsurg&Rice’62]
n
Ex: Ln = (a2 )∗
2-LA DPDA minimal DFA minimal 2NFA
size O(n) O(n) 2n 2n
Thank you for your attention!
Andrea Pisoni
Dipartimento di Informatica Università degli Studi di Milano, Italy
NCMA 2013 Umeå, Sweden August 13–14, 2013
The Chomsky Hierarchy
type 0
1-tape Turing Machines
type 1
Linear Bounded Automata Pushdown Automata Finite Automata
type 2
type 3
Limited Automata [Hibbard’67] One-tape Turing machines with restricted rewritings
Definition Fixed an integer d ≥ 1, a d -limited automaton is I
a one-tape Turing machine
I
which is allowed to rewrite the content of each tape cell only in the first d visits
Limited Automata [Hibbard’67] One-tape Turing machines with restricted rewritings
Definition Fixed an integer d ≥ 1, a d -limited automaton is I
a one-tape Turing machine
I
which is allowed to rewrite the content of each tape cell only in the first d visits
Limited Automata [Hibbard’67] One-tape Turing machines with restricted rewritings
Definition Fixed an integer d ≥ 1, a d -limited automaton is I
a one-tape Turing machine
I
which is allowed to rewrite the content of each tape cell only in the first d visits
Computational power I
For each d ≥ 2, d -limited automata characterize context-free languages [Hibbard’67]
I
1-limited automata characterize regular languages [Wagner&Wechsung’86]
The Chomsky Hierarchy
type 0
1-tape Turing Machines
type 1
Linear Bounded Automata Pushdown Automata Finite Automata
type 2
type 3
The Chomsky Hierarchy
type 0
1-tape Turing Machines
type 1
Linear Bounded Automata 2-Limited Automata Finite Automata
type 2
type 3
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
I
Determinism vs Nondeterminism
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
2-LAs → PDAs Exponential gap
I
Determinism vs Nondeterminism
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
2-LAs → PDAs Exponential gap
I
Determinism vs Nondeterminism
PDAs → 2-LAs Polynomial upper bound
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
2-LAs → PDAs Exponential gap
I
Determinism vs Nondeterminism
PDAs → 2-LAs Polynomial upper bound
Our Contributions
I
2-Limited Automata ≡ Pushdown Automata: descriptional complexity point of view
2-LAs → PDAs Exponential gap
I
PDAs → 2-LAs Polynomial upper bound
Determinism vs Nondeterminism
Deterministic Context-Free Languages ≡ Deterministic 2-LAs
Example: Balanced Parentheses B
(
)
(
(
(
)
)
)
C
6
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
−→ (i) (ii) (iii) (iv) (v)
(
)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
−→ (i) (ii) (iii) (iv) (v)
(
)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
−→ (i) (ii) (iii) (iv) (v)
(
#
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
(
# 6
(i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B
( 6
(i) (ii) (iii) (iv) (v)
#
(
(
(
)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # 6
(i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # 6
(i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
)
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
(
#
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
(
# 6
(i) (ii) (iii) (iv) (v)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
( 6
(i) (ii) (iii) (iv) (v)
#
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
# # 6
(i) (ii) (iii) (iv) (v)
)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
# #
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
# #
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
# #
)
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
(
# # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
# # # 6
(i) (ii) (iii) (iv) (v)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
# # # 6
(i) (ii) (iii) (iv) (v)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
(
# # # 6
(i) (ii) (iii) (iv) (v)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
( 6
(i) (ii) (iii) (iv) (v)
# # #
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # 6
(i) (ii) (iii) (iv) (v)
)
C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
−→ (i) (ii) (iii) (iv) (v)
(
# # # #
)
C
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6 ←−
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
(
# # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # #
( 6
(i) (ii) (iii) (iv) (v)
# # # # # C
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
−→ (i) (ii) (iii) (iv) (v)
6
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Example: Balanced Parentheses B # # # # # # # # C
←− 6 (i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6 ←−
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
←−
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C 6
(i) (ii) (iii) (iv) (v)
yes!
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject
Example: Balanced Parentheses B # # # # # # # # C
(i) (ii) (iii) (iv) (v)
Move to the right to search a closed parenthesis Rewrite it by # Move to the left to search an open parenthesis Rewrite it by # Repeat from the beginning
Special cases: (i’) If in (i) the right end of the tape is reached then scan all the tape and accept iff all tape cells contain # (iii’) If in (iii) the left end of the tape is reached then reject Each cell is rewritten only in the first 2 visits!
Simulation of 2-Limited Automata by Pushdown Automata
Problem How much it costs, in the description size, the simulation of 2-LAs by PDAs?
This work Exponential cost!
Simulation of 2-Limited Automata by Pushdown Automata
Problem How much it costs, in the description size, the simulation of 2-LAs by PDAs?
This work Exponential cost!
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} w
w q
q p p
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} −→
j
w
q
p
(q, −1, p, −1) ∈ τw
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} ←−
w
q
j
p
(q, −1, p, −1) ∈ τw
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} w
w
p p
←−
j
q
q
(q, −1, p, −1) ∈ τw
(q, +1, p, −1) ∈ τw
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Transition Tables of 2-LAs I I
Fixed a 2-limited automaton Transition table τw
w is a “frozen” string
τw ⊆ Q × {−1, +1} × Q × {−1, +1} w
←−
w q
q
j
p p
(q, −1, p, −1) ∈ τw
(q, +1, p, −1) ∈ τw
(q, d 0 , p, d 00 ) ∈ τw iff M on a tape segment containing w has a computation path: entering the segment in q from d 0 exiting the segment in p from d 00 left = −1, right = +1
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B a b c d e f g h i . . .C 6
... 2-LA
q0
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B X b c d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B X B c d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B c d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B c d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X d e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y e f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y Z f g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y Z F g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y Z E F g h i . . .C 6
... 2-LA
...
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
Some computation steps... B A B X Y Z E F g h i . . .C 6
... 2-LA
q
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEFn Y X q τAB PDA
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Yn Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Yn X q τAB PDA
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B Xn Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y Xn q τ PDA AB
Simulation of 2-LAs by PDAs Initial configuration B a b c d e f g h i . . .C 6
2-LA
q0
a b c d e f g h i ... 6
PDA
q0 -
After some steps... B A B X Y Z E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τABn PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
δ(q, g ) 3 (p, Z , +1) move to the right
⇓
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
δ(q, g ) 3 (p, Z , +1) move to the right
⇓ B A B X Y E F Z h i . . .C 6
2-LA
p
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
δ(q, g ) 3 (p, Z , +1)
normal mode
move to the right
push and direct simulation
⇓
⇓
B A B X Y E F Z h i . . .C 6
2-LA
p
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F g h i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q τAB PDA
δ(q, g ) 3 (p, Z , +1)
normal mode
move to the right
push and direct simulation
⇓
⇓
B A B X Y E F Zn h i . . .C 6
2-LA
p
. a b c d e f g h i . .Zn 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
δ(p, h) 3 (r , H, −1) move to the left
⇓
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
δ(p, h) 3 (r , H, −1) move to the left
⇓
B A B X Y E F Z H i . . .C 6
2-LA
r
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
δ(p, h) 3 (r , H, −1) move to the left
⇓
B A B X Y E F Z H i . . .C 6
2-LA
r
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
back mode
⇓ . a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z h i . . .C 6
2-LA
p
δ(p, h) 3 (r , H, −1) move to the left
⇓
B A B X Y E F Z Hn i . . .C 6
2-LA
r
. a b c d e f g h i . .Z 6 τEF Y X p τAB PDA
back mode
⇓ . a b c d e f g h i . .Z 6 τEF Y X r , τHn τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
δ(r , Z ) 3 (q, G , −1) move to the left
⇓
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
δ(r , Z ) 3 (q, G , −1) move to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
q
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
δ(r , Z ) 3 (q, G , −1) move to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
q
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
back mode
⇓ a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F Z H i . . .C 6
2-LA
r
δ(r , Z ) 3 (q, G , −1) move to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
q
. a b c d e f g h i . .Z 6 τEF Y X r , τH τAB PDA
back mode
⇓ a b c d e f g h i ... 6 - τEF Y X q, τGHn τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
(q, +1, s, −1) ∈ τEF exit to the left
⇓
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
(q, +1, s, −1) ∈ τEF exit to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
s
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
(q, +1, s, −1) ∈ τEF exit to the left
⇓
B A B X Y E F G H i . . .C 6
2-LA
s
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
back mode
⇓ a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
q
(q, +1, s, −1) ∈ τEF exit to the left
⇓
B A B X Y E F G H i . . .C
6
2-LA
s
a b c d e f g h i ... 6 - τEF Y X q, τGH τAB PDA
back mode
⇓ a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1) move to the right
⇓
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1) move to the right
⇓ B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1)
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
move to the right
back mode
⇓
⇓
B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1)
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
move to the right
back mode
⇓
⇓
B A B X D E F G H i . . .C
6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X Y E F G H i . . .C 6
2-LA
s
δ(s, Y ) 3 (p, D, +1)
a b c d e f g h i ... 6 - Y X s, τE ···H τAB PDA
move to the right
back mode
⇓
⇓
B A B X Dn E F G H i . . .C 6
2-LA
p
a b c d e f g h i ... 6 - τDn X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
(p, −1, r , +1) ∈ τE ···H exit to the right
⇓
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
(p, −1, r , +1) ∈ τE ···H exit to the right
⇓ B A B X D E F G H i . . .C 6 X X
2-LA
r
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
(p, −1, r , +1) ∈ τE ···H
resume normal mode
exit to the right
move to the right
⇓
⇓
B A B X D E F G H i . . .C 6 X X
2-LA
r
a b c d e f g h i ... 6 -τD···H X r τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
(p, −1, r , +1) ∈ τE ···H
resume normal mode
exit to the right
move to the right
⇓
⇓
B A B X D E F G H i . . .C
6
2-LA
r
a b c d e f g h i ... 6 -τD···H
X r τAB PDA
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
(p, −1, r , +1) ∈ τE ···H
resume normal mode
exit to the right
move to the right
⇓
⇓
B A B Xn D E F G H i . . .C 6
2-LA
r
a b c d e f g h i ... 6 -τD···H Xn r τ PDA AB
Simulation of 2-LAs by PDAs B A B X D E F G H i . . .C X X6
2-LA
p
a b c d e f g h i ... 6 - τD X p, τE ···H τAB PDA
(p, −1, r , +1) ∈ τE ···H
resume normal mode
exit to the right
move to the right
⇓
⇓
B A B X D E F G H i . . .C 6
2-LA
r
a b c d e f g h i ... 6 -τD···H X r τABn PDA
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Determinism vs nondeterminism I
Determinism is preserved by the simulation provided that the input of the PDA is right end-marked
I
Double exponential size for the simulation of D2-LAs by DPDAs
I
Conjecture: this cost cannot be reduced
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Determinism vs nondeterminism I
Determinism is preserved by the simulation provided that the input of the PDA is right end-marked
I
Double exponential size for the simulation of D2-LAs by DPDAs
I
Conjecture: this cost cannot be reduced
Simulation of 2-LAs by PDAs Summing up...
Cost of the simulation I In the resulting PDA transition tables are used for states pushdown alphabet I
Exponential upper bound for the size of the resulting PDA
I
Optimal
Determinism vs nondeterminism I
Determinism is preserved by the simulation provided that the input of the PDA is right end-marked
I
Double exponential size for the simulation of D2-LAs by DPDAs
I
Conjecture: this cost cannot be reduced
Simulation of Pushdown Automata by 2-Limited Automata
PDAs → 2-LAs Polynomial cost! (in the description size)
Simulation of Pushdown Automata by 2-Limited Automata
PDAs → 2-LAs Polynomial cost!
DPDAs → D2-LAs Polynomial cost! (in the description size)
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Z Y X q PDA X
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Z Y X q PDA X
B X X Y f g h i . . .C 6
2-LA
q, Z
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Zm Y X q PDA X
B X X Y f g h i . . .C 6
2-LA
q, Zm
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Z Y X q PDA X
B X X Y f g h i . . .C
6
2-LA
q, Z
Simulation of PDAs by 2-LAs a b c d e f g h i ... 6 - Z Y X q PDA X
B X X Y f g h i . . .C 6
2-LA
q, Z
Normal form for (D)PDAs: I
at each step, the stack height increases at most by 1
I
-moves cannot push on the stack
Each (D)PDA can be simulated by an equivalent (D)2-LA of polynomial size
Determinism vs Nondeterminism in Limited Automata Corollary of the simulations Deterministic 2-LAs ≡ Deterministic Context-Free Languages On the other hand, the language L = {an bn c | n ≥ 0} ∪ {an b2n d | n ≥ 0} is accepted by a deterministic 3-LA, but it is not a DCFL Infinite hierarchy [Hibbard’67] For each d ≥ 2 there is a language which is accepted by a deterministic d -limited automaton and that cannot be accepted by any deterministic (d − 1)-limited automaton
Determinism vs Nondeterminism in Limited Automata Corollary of the simulations Deterministic 2-LAs ≡ Deterministic Context-Free Languages On the other hand, the language L = {an bn c | n ≥ 0} ∪ {an b2n d | n ≥ 0} is accepted by a deterministic 3-LA, but it is not a DCFL Infinite hierarchy [Hibbard’67] For each d ≥ 2 there is a language which is accepted by a deterministic d -limited automaton and that cannot be accepted by any deterministic (d − 1)-limited automaton
Determinism vs Nondeterminism in Limited Automata Corollary of the simulations Deterministic 2-LAs ≡ Deterministic Context-Free Languages On the other hand, the language L = {an bn c | n ≥ 0} ∪ {an b2n d | n ≥ 0} is accepted by a deterministic 3-LA, but it is not a DCFL Infinite hierarchy [Hibbard’67] For each d ≥ 2 there is a language which is accepted by a deterministic d -limited automaton and that cannot be accepted by any deterministic (d − 1)-limited automaton
Futher Investigations
I
Descriptional complexity aspects for d > 2 We conjecture that for d > 2 the size gap from d -limited automata to PDAs remains exponential
I
Descriptional complexity aspects in the unary case Unary context-free language are regular [Ginbsurg&Rice’62]
n
Ex: Ln = (a2 )∗
2-LA DPDA minimal DFA minimal 2NFA
size O(n) O(n) 2n 2n
Futher Investigations
I
Descriptional complexity aspects for d > 2 We conjecture that for d > 2 the size gap from d -limited automata to PDAs remains exponential
I
Descriptional complexity aspects in the unary case Unary context-free language are regular [Ginbsurg&Rice’62]
n
Ex: Ln = (a2 )∗
2-LA DPDA minimal DFA minimal 2NFA
size O(n) O(n) 2n 2n
Futher Investigations
I
Descriptional complexity aspects for d > 2 We conjecture that for d > 2 the size gap from d -limited automata to PDAs remains exponential
I
Descriptional complexity aspects in the unary case Unary context-free language are regular [Ginbsurg&Rice’62]
n
Ex: Ln = (a2 )∗
2-LA DPDA minimal DFA minimal 2NFA
size O(n) O(n) 2n 2n
Futher Investigations
I
Descriptional complexity aspects for d > 2 We conjecture that for d > 2 the size gap from d -limited automata to PDAs remains exponential
I
Descriptional complexity aspects in the unary case Unary context-free language are regular [Ginbsurg&Rice’62]
n
Ex: Ln = (a2 )∗
2-LA DPDA minimal DFA minimal 2NFA
size O(n) O(n) 2n 2n
Thank you for your attention!
