SLIDE 1 Viliam Geffert Zuzana Bednárová
- Dept. Computer Science, P.J.Šafárik Univerzity
Košice, Slovakia
SLIDE 2 A = ( Q,Σ,Γ,H,qI,F ) Q
Σ
Γ
qI
F
H
(handling input and pushdown separately)
Nondeterministic pushdown automata
input tape x1 x2 x3 xn
q
. . . pushdown
SLIDE 3 A = ( Q,Σ,Γ,H,qI,F,h ) h ≥ 0
with an additional limit on the height
Constant height NPDA
x1 x2 x3 xn
q
. . . pushdown
h
TOP
input tape
SLIDE 4 input tape
TOP
pushdown overflow: if adding on TOP results in pushdown overflow, ABORT
Constant height NPDA
x1 x2 x3 xn
q
. . . pushdown
h
SLIDE 5 ₵ ₵ ₵ ₵ ₵ ₵
. . .
y1
R
q
X I x1
. . . pushdown state
input tape
TOP
h
Two Way PDA
. . .
├ ┤ yr
R
y2
R
x2 xs $ right
push pop
SLIDE 6 ₵ ₵ ₵ ₵ ₵ ₵
. . .
y1
R
q
X I
. . .
x1
. . . pushdown state
input tape
TOP
h
Two Way PDA
. . .
├ ┤ yr
R
y2
R
x2 xs $ left
push pop
SLIDE 7 Descriptional complexity measure:
- the number of finite control states |Q|
- the size of the pushdown alphabet |Γ |
- the height of the pushdown store h
Descriptional complexity
SLIDE 8 lin
poly det. PDA complement exp nondet. det. PDA intersection, det. PDA union 2exp nondet. PDA det. PDA Known facts about h-PDAs
SLIDE 9 lin
poly det. PDA complement exp nondet. det. PDA intersection, det. PDA union 2exp nondet. PDA det. PDA Known facts about h-PDAs
- exp. . . 2exp nondet. PDA complement
2 det. PDA 1 det. PDA 2 nondet. PDA 1 nondet. PDA
?
Open
SLIDE 10
- exp. . . 2exp nondet. PDA complement
2 det. PDA 1 det. PDA 2 nondet. PDA 1 nondet. PDA lin
poly det. PDA complement exp nondet. det. PDA intersection, det. PDA union 2exp nondet. PDA det. PDA Known facts about h-PDAs
?
SLIDE 11 For each constant height NPDA
NPDA NFA
Lem.:
. . . xi
xn Z1
q
Zj
. . .
x1
. . .
h
pushdown
state
input tape
TOP
SLIDE 12 For each constant height NPDA there exists an equivalent NFA with | Q’ | ≤ |Q | . |Γ ≤ h|
NPDA NFA
Lem.:
. . . xi
xn Z1
q
Zj
. . .
x1
. . .
q’ϵ Q ’ Q ’ Q x Γ ≤ h h
pushdown
state
input tape
TOP
SLIDE 13 For each constant height NPDA there exists an equivalent NFA with | Q’ | ≤ |Q | . |Γ ≤ h|
Lem.:
2NPDA 2NFA
. . . xi
xn Z1
q
Zj
. . .
x1
. . .
q’ϵ Q ’ h
pushdown
state
input tape
TOP
Q ’ Q x Γ ≤ h
SLIDE 14 For each constant height 2NPDA there exists an equivalent 1NFA with | Q’’ | ≤ 4| Q | . |Γ ≤ h |
2NPDA 2NFA 1NFA
Thm.:
Q’’ 4 Q x Γ ≤ h
. . . xi
xn Z1
q
Zj
. . .
x1
. . .
q’ϵ Q ’ h
pushdown
state
input tape
TOP
Q ’ Q x Γ ≤ h
SLIDE 15 For each constant height 2NPDA there exists an equivalent 1NFA with | Q’’ | ≤ 4| Q | . |Γ ≤ h |
2NPDA 1NPDA – upper bound
Thm.:
Q’’ 4 Q x Γ ≤ h
. . . xi
xn Z1
q
Zj
. . .
x1
. . .
q’ϵ Q ’ h
pushdown
state
input tape
TOP
Q ’ Q x Γ ≤ h
SLIDE 16
For any constant height NPDA A = ( Q, Σ, Γ, H, qI, F, h ) there exists a constant height NPDA B accepting L(B) = L(A)c, with at most
2| Q | . | Γ ≤ h | states
Thm.:
Complement – upper bound
SLIDE 17 NPDA A
QA , ΓA , hA L
Complement – upper bound
SLIDE 18 NPDA A NFA A’
QA’ QA x ΓA
≤ hA
QA , ΓA , hA L
Complement – upper bound
SLIDE 19 double exponential NPDA A NFA A’ DFA A’’
QA’ QA x ΓA
≤ hA
QA , ΓA , hA
QA’’ 2QA x ΓA≤hA
L
Complement – upper bound
SLIDE 20 NPDA A NFA A’ DFA A’’ DFA B
QA’ QA x ΓA
≤ hA
QA , ΓA , hA
QA’’ 2QA x ΓA≤hA
L LC
Complement – upper bound
Accept Reject
SLIDE 21 NPDA A NFA A’ NPDA B DFA A’’ DFA B
Accept Reject
QA’ QA x ΓA
≤ hA
QA , ΓA , hA
QA’’ 2QA x ΓA≤hA
hB = 0 L LC
Complement – upper bound
SLIDE 22
Lower bounds - triangular lemma
Lem.:
For each finite sets A, B: |A|, |B| ≥ 2 and each C A x B: |C| ≥ |A|+|B|-1
A x B
C A B
SLIDE 23 Lower bounds - triangular lemma
Lem.:
For each finite sets A, B: |A|, |B| ≥ 2 and each C A x B: |C| ≥ |A|+|B|-1 there exist elements a1 a2 , b1 b2
A x B
a1
C
a2 b1 b2
A B
SLIDE 24 Lower bounds - triangular lemma
Lem.:
For each finite sets A, B: |A|, |B| ≥ 2 and each C A x B: |C| ≥ |A|+|B|-1 there exist elements a1 a2 , b1 b2 such that
[a1,b1], [a1, b2], [a2, b1] ∈ C A x B
a1
C
a2 b1 b2
[a1,b1] [a2,b1] [a1,b2]
A B
SLIDE 25 For a fixed alphabet and n > 0, a string ϕ is well formed, if: xi , yj n Witness language
₵ ₵ ₵ ₵ ₵ ₵
. . .
y1
R
x1
. . . yr
R
y2
R
x2 xs
$ ϕ =
SLIDE 26 LC consists of: I s t s
- ill formed strings ϕ , plus
- well formed strings ϕ
Witness language ∃ xi = yj
₵ ₵ ₵ ₵ ₵ ₵
. . .
y1
R
x1
. . . yr
R
y2
R
x2 xs $
ϕ =
SLIDE 27 LC consists of:
ill formed strings ϕ
Witness language
₵ ₵ ₵ ₵ ₵ ₵
. . .
y1
R
x1
. . . yr
R
y2
R
x2 xs $
ϕ = ∃ xi = yj
∀
SLIDE 28 For each and n > 0, the language L
I n p u t p u s h d
n
h
₵
. . . y1
R
x1
$ Witness language – upper bound
x2
₵
xs
₵ ₵ y2
R
₵ . . . ₵
yr
R
Thm.:
SLIDE 29 For each and n > 0, the language L can be accepted by NPDA with linear cost.
e r a p m
l
d
I n p u t p u s h d
n
h
s c a n s c a n
|Q| = n + O(| |) h = n Γ =
₵
. . . y1
R
x1
$
x2
₵
xs
₵ ₵ y2
R
₵ . . . ₵
yr
R
s c a n Thm.:
Witness language – upper bound
SLIDE 30 For each alphabet Σ and n>3, each constant height NPDA accepting L C must have
- either the number of states |Q|,
- or pushdown height h
above
2|Σ|n-O(1)
Witness language – lower bound
Thm.:
SLIDE 31
Witness language – lower bound By contradiction if |Q|, h < 2|Σ|n-O(1) we fool A: We construct δ ∈ L C such that A accepts δ
SLIDE 32
Witness language – lower bound We define sets of blocks:
U W xi< 1/2 |Σ|n xi ≥ 1/2 |Σ|n
X = Σn It is easy to see: | X X |=|U|+|W| |X|= | Σ|n |U|= 1/2 | Σ|n |W|= 1/2 | Σ|n
SLIDE 33
Witness language – lower bound Now define sets of strings: = {x1₵ … xs₵: x1 < … < xs , x1, … xs ∈ X } X0 U0 W0 = {u1₵ … us₵: u1 < … < us , u1, …us ∈ U } = {w1₵ … ws₵: w1 < … < ws , w1, …ws ∈ W }
SLIDE 34
Witness language – lower bound
For x1 < x2 < . . . < xs , xi ∊ X U0 ∋ x1 ₵ x2 ₵ . . . xi ₵ xi+1 ₵ xi+2 ₵. . . xs ₵ ∊ W0
u w
X0 = U0 .W0
SLIDE 35
Witness language – lower bound
|X0 | = 2 | Σ |n |U0 | = 2 1/2 | Σ |n |W0 | = 2 1/2 | Σ |n For x1 < x2 < . . . < xs , xi ∊ X U0 ∋ x1 ₵ x2 ₵ . . . xi ₵ xi+1 ₵ xi+2 ₵. . . xs ₵ ∊ W0
u w
X0 = U0 .W0
SLIDE 36 Witness language – lower bound u = u1 ₵ u2 ₵ . . . ui ₵ . . . us ₵ uCR = u1
R ₵ u2 R ₵ . . . ui R ₵ . . . ur R ₵
such that
ui ∈ u iff ui
R ∉ uCR
intuitively: U U – { ui ∈ u } = { uj : uj
R ∈ uCR }
SLIDE 37
Witness language – lower bound
w h
$ u CR
wCR u
uw$uCRwCR LC
SLIDE 38
w $ u CR
Witness language – lower bound ql
h yl wCR u
SLIDE 39
Witness language – lower bound ql
xl w uw h
$
yl u CR wCR u CASE 1
SLIDE 40
Witness language – lower bound ql
xl w uw h
$
yl u CR wCR u CASE 2
SLIDE 41
Witness language – lower bound ql
xl w uw h
$
yl u CR wCR xk qk u CASE 1
SLIDE 42
Witness language – lower bound for each uw ∈ X0 we have some [ yl
l , ql l , xl l , yk , q k ]
X0
SLIDE 43
Witness language – lower bound Pigeonhole argument
X1= strings uw ∈ X0 sharing
the same [ yl
l , ql l , xl l , yk , qk ]
X1 X0
SLIDE 44 X1 Witness language – lower bound
u1w1, u1w2, u2w1 ∈ X1
share the same [ yl
l , ql l , xl l , yk , qk ]
X1 ⊆ U0 .W0 |X1| ≥ |U0|+|W0|-1 Lemma
u1w1
u1w2
u2w1
X0
+
w1 , w2 ∈ W0 w1 ≠ w2 u1 , u2 ∈ U0 u1 ≠ u2
SLIDE 45 X1 Witness language – lower bound
u1w1, u1w2, u2w1 ∈ X1
share the same [ yl
l , ql l , xl l , yk , qk ]
X1 ⊆ U0 .W0 |X1| ≥ |U0|+|W0|-1 Lemma
u1w1
u1w2
u2w1
X0
+
w1 , w2 ∈ W0 w1 ≠ w2 u1 , u2 ∈ U0 u1 ≠ u2
SLIDE 46 Witness language – lower bound ql
xl w1 u’ h
$
yl u1
CR
w1
CR
xk qk u1 CASE 1
SLIDE 47 Witness language – lower bound ql
xl h u’’
$
yl u2
CR
xk qk u2 CASE 1 w1 w1
CR
SLIDE 48 Witness language – lower bound
u1 , u2 ∈ U0 u1 ≠ u2
∃ui : ui ∈ u1 & ui
∉ u2
SLIDE 49 Witness language – lower bound
u1 , u2 ∈ U0 u1 ≠ u2
∃ui : ui ∈ u1 & ui
∉ u2
∃ui : ui
∉ u1
C & ui
∈ u2
C
SLIDE 50 Witness language – lower bound
u1 , u2 ∈ U0 u1 ≠ u2
∃ui : ui ∈ u1 & ui
∉ u2
∃ui : ui
∉ u1
C & ui
∈ u2
C
u1 w1 $ u2
CR w1 CR ∉ LC
SLIDE 51 Witness language – lower bound ql
xl u’ h
$
yl u2
CR
xk qk u1 CASE 1 w1 w1
CR
SLIDE 52 Witness language – lower bound ql
xl u’ h
$
yl u2
CR
xk qk u1 CASE 1 w1 w1
CR
u1 w1 $ u2
CR w1 CR ∉ LC
SLIDE 53 ql
yl
Witness language – lower bound
xl u’ h
$
u2
CR
xk qk u1 CASE 1 w1 w1
CR
u1 w1 $ u2
CR w1 CR ∉ LC
SLIDE 54 For L , we have NPDA with n + O(|Σ|) states n pushdown height |Σ| pushdown symbols Summing up: each NPDA for LC requires at least
2|Σ|n-O(1)
states pushdown height
On the other hand,
SLIDE 55
Lem.:
Witness language by two-way machine For each Σ and n≥1, language LC can be accepted by constant height 2DPDA using
n + 6|Σ|+21 states n + 1
pushdown height
|Σ|+1
pushdown symbols
SLIDE 56 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
X I
. . . pushdown state
TOP
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input?
Witness language by two-way machine
SLIDE 57 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
X I
. . . pushdown state
TOP
├ ┤
$
1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input? (Exactly one “$” ?)
Witness language by two-way machine
SLIDE 58 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input? X I
pushdown
TOP
$ Witness language by two-way machine
SLIDE 59 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q0
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input? (All blocks of equal length?) X I
pushdown
TOP
Witness language by two-way machine
SLIDE 60 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q1
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input? X I
pushdown
TOP
Witness language by two-way machine
SLIDE 61 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q2
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input? X I
pushdown
TOP
Witness language by two-way machine
SLIDE 62 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q3
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input? X I
pushdown
TOP
Witness language by two-way machine
SLIDE 63 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q4
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1
0 1 0 0
Well formed input? X I
pushdown
TOP
Witness language by two-way machine
SLIDE 64 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q0
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input? X I
pushdown
TOP
. . .
4
Witness language by two-way machine
SLIDE 65 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q0
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input? X I
pushdown
TOP
Witness language by two-way machine
SLIDE 66 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Well formed input?
. . .
X I
pushdown
TOP
4
Witness language by two-way machine
SLIDE 67 ₵ ₵ ₵ ₵ ₵ ₵ ₵
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
Well formed input?
. . .
4 X I
pushdown
TOP
YES
n + O(1) states Witness language by two-way machine
SLIDE 68 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0
1 0 1 1
0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Idea
pushdown
TOP
1 1 1 X I
Z
? ∃ Z : Witness language by two-way machine xi = Z = yj
R
SLIDE 69 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0
1 0 1 1
0 0 0 1 1 0 1 0
1 1 0 1
0 1 0 0 Idea
pushdown
TOP
1 1 1 X I
Z
? ∃ Z : Witness language by two-way machine xi = Z = yj
R
SLIDE 70 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
4
TOP
X I Initialize PD memory
Witness language by two-way machine
SLIDE 71 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
4 X I Initialize PD memory
Witness language by two-way machine
SLIDE 72 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $
1 0 0
0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
4 X I Initialize PD memory
Witness language by two-way machine
SLIDE 73 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
4 X I Initialize PD memory
Witness language by two-way machine
SLIDE 74 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
4 X I Initialize PD memory
Z
Witness language by two-way machine
SLIDE 75 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Initialize PD memory X I
pushdown
TOP
O(1) states
4
Z
Witness language by two-way machine
SLIDE 76 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
X I
Z
= Z ? Witness language by two-way machine
SLIDE 77 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
= Z ? Witness language by two-way machine
SLIDE 78 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
├ ┤ $
1 0 0
0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
Difference
X I Left Part
Z
= Z ? Witness language by two-way machine
SLIDE 79 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
Witness language by two-way machine
SLIDE 80 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
Witness language by two-way machine
SLIDE 81 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
= Z ? Witness language by two-way machine
SLIDE 82 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
Difference
X I Left Part
Z
= Z ? Witness language by two-way machine
SLIDE 83 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0
0 1 1 0
1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
Witness language by two-way machine
SLIDE 84 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
Witness language by two-way machine
SLIDE 85 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
├ ┤ $ 1 0 0 0 1 1 0
1
0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
Difference
X I Left Part
Z
= Z ? Witness language by two-way machine
SLIDE 86 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
Witness language by two-way machine
SLIDE 87 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤
$
1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
Witness language by two-way machine
SLIDE 88 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0
pushdown
TOP
X I Left Part
Z
Witness language by two-way machine
. . .
SLIDE 89 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
1
X I
Witness language by two-way machine
Z := Z + 1
SLIDE 90 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
1 X I
Z
= Z ? Witness language by two-way machine
SLIDE 91 X I
₵ ₵ ₵ ₵ ₵ ₵ ₵
q
├ ┤ $
1 0 0
0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
1
YES
Z
= Z ? Witness language by two-way machine
SLIDE 92 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
1 X I
Z
Z= ? Witness language by two-way machine
SLIDE 93 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
1 X I
Z
Z= ? Witness language by two-way machine
SLIDE 94 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
X I
1
Difference
1
Z
Z= ? Witness language by two-way machine
SLIDE 95 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
X I 1
Z
Witness language by two-way machine
SLIDE 96 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
X I 1
Z
Witness language by two-way machine
SLIDE 97 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
1 X I
Z
Z= ? Witness language by two-way machine
SLIDE 98 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
1 X I
Z
Z= ? Witness language by two-way machine
SLIDE 99 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
1 X I
Z
Z= ? Witness language by two-way machine
SLIDE 100 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤
$
1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
1 X I
No match Z
Witness language by two-way machine
SLIDE 101 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
X I 1
Z
Witness language by two-way machine
SLIDE 102 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
1 X I
. . .
Z := Z + 1
Witness language by two-way machine
SLIDE 103 X I
₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
1 1 1
. . .
Z
= Z ? Witness language by two-way machine
SLIDE 104 X I
₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
1 1 1
Z
= Z ? Witness language by two-way machine
SLIDE 105 X I
₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Left Part
pushdown
TOP
1 1 1
= Z
Z
Witness language by two-way machine
1 0 1 1
SLIDE 106 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
1 1 1 X I
Z
Z= ? Witness language by two-way machine
1 0 1 1
SLIDE 107 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 Right Part
pushdown
TOP
1 1 1 X I
Z
Z= ? Witness language by two-way machine
1 0 1 1
SLIDE 108 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
state
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 Right Part
pushdown
TOP
1 1 1 X I
Z =
q
REJECT
Z
Witness language by two-way machine
1 0 1 1 1 1 0 1
SLIDE 109 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0
pushdown
TOP
1 1 1 1 X I 1 0 1 1
q Z
ACCEPT ∃ Z: X = Z & Z = Y Witness language by two-way machine
SLIDE 110 ₵ ₵ ₵ ₵ ₵ ₵ ₵
q
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 Right Part
pushdown
TOP
1 1 1 1 X I 1 0 1 1
q Z
ACCEPT ∃ Z: X = Z & Z = Y
Left Part
3|Σ|+ O(1) states 2|Σ|+ O(1) states
Witness language by two-way machine
SLIDE 111 X I
q
state
PD Memory
pushdown
TOP
1 1 1
₵ ₵ ₵ ₵ ₵ ₵ ₵
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1 1
Witness language by two-way machine
? Z := Z + 1
SLIDE 112 X I
q
state
PD Memory
pushdown
1 1
TOP
₵ ₵ ₵ ₵ ₵ ₵ ₵
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1
Witness language by two-way machine
SLIDE 113 X I
q
state
PD Memory
pushdown
1
TOP
₵ ₵ ₵ ₵ ₵ ₵ ₵
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1
Witness language by two-way machine
SLIDE 114 X I
q
state
PD Memory
pushdown
1
TOP
₵ ₵ ₵ ₵ ₵ ₵ ₵
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1
Witness language by two-way machine
SLIDE 115 X I
q
state
PD Memory
pushdown
1 1
TOP
₵ ₵ ₵ ₵ ₵ ₵ ₵
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1
Witness language by two-way machine
SLIDE 116 X I
q
state
PD Memory
pushdown
1 1
TOP
₵ ₵ ₵ ₵ ₵ ₵ ₵
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1
Witness language by two-way machine
SLIDE 117 X I
q
state
PD Memory
pushdown
TOP
1 1
Z:=Z+1
₵ ₵ ₵ ₵ ₵ ₵ ₵
├ ┤ $ 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1
|Σ|+ O(1) states
Witness language by two-way machine
SLIDE 118
PD Memory Right Part Left Part Initialize PD memory Well formed input?
n + 6|Σ|+ O(1)
TOTAL
Witness language by two-way machine
n + +|Σ|+ O(1) n + 3|Σ|+ O(1) n + 2|Σ|+ O(1) n + 3|Σ| +O(1) n + 3|Σ|+ O(1)
SLIDE 119 For LC , we have 2DPDA with n + O(|Σ|) states n + 1 pushdown height |Σ|+1 pushdown symbols Summing up: each 1NPDA for LC requires at least
2|Σ|n-O(1)
states pushdown height
On the other hand,
SLIDE 120 The following operations require double-exp blow up: Summary 1 DPDA 1 NPDA 2 NPDA 1 NPDA 2 DPDA 1 DPDA 1 NPDA complement
new new
SLIDE 121 The following operations require double-exp blow up: Summary 1 DPDA 1 NPDA 2 NPDA 1 NPDA 2 DPDA 1 DPDA 1 NPDA complement
new new
SLIDE 122 The following operations require double-exp blow up: Summary 1 DPDA 1 NPDA 2 NPDA 1 NPDA 2 DPDA 1 DPDA 1 NPDA complement
new new
SLIDE 123
The following operations require double-exp blow up: Summary 1 DPDA 1 NPDA 2 NPDA 1 NPDA 2 DPDA 1 DPDA 1 NPDA complement Other operations with 2NPDA, DPDA (union, intersection, complement, homomorphism, …) OPEN
SLIDE 124
