מודלים חישוביים - תרגול 4 שפות לא רגולריות למת הניפוח Regular languages Pumping lemma lemat Nimrod Myhill

1. e
:4
version 4
20.11.06
Mihill-Nerode
PDA
}#|#{ 01
ww
wL ==
LDFAnDFA
n+1
TTij(i<j)
Tij
Ti
ij
T
L
LnwL
x,y,z
0:)3
0||)2
||)1
∪∈∀∈
>
≤
=
+
NiLzxy
y
nxy
xyzw
i

2. L
}#|#{ 01
ww
wL ==
LnwLw nn
∈= 10
yi=0
010
10)0(0
0
>∉=
∈==
−
−−
bLzxy
Lzxyw
nbn
nabnba
L
Lw
}#|#{ 01
ww
wL ≥=
LnwLw nn
∈= 10
yi=2
01010
10)0(0
22
>∉==
∈==
+−−++
−−
bLzxy
Lzxyw
nbnnbanba
nbanba
L
}0,|{}1,0|{ ≥∪≥≥= jicbikcbaL jikki
LL
}0|{ 2
≥= nbaaL nn
}|{ jikcbaL kji
+>=
}#2|#},{{ * b
x
a
xbaxL <∈=
}|}1,0{{ *
palindromeisxxL ∈=
LnwLw nnnn
∈= 0110
y
i=0
00110
0||||0110)0(0
0
>∉=
>==∈==
−
−−
bLzxy
byaxLzxyw
nnnbn
nnnbanba
L
}}1,0{|{ *
∈= wwwL

3. }|}1,0{{ *
palindromenotisxxL ∈=
}|1{ primeispL p
=
}|1{ numbernaturalofsquareispL p
=
LnwLw n
∈=
2
0
yi=2
12)1(00
0||||0)0(0
222222 22
2
++=+<+<∉==
>=≥=∈==
+−−++
−−
nnnbnnLzxy
bynaxLzxyw
bnabnba
abnba
L
}|||}9,..2,1,0{{ *
piofdigitswfirsttheiswthewL ∈=
}:|1{ 3
NttpL p
∈==
}:2|1{ NtpL tp
∈==
}210{ nmn
L =
Nerode-Mihill
-
xyyx L~Lyx ∈,Lyx ∉,
xyyx L≡yuxuu L~:*
∑∈∀x
yL
}1{ *
=L
LLNn nn
∈⇔∈∈∀ 11111:
}|1{ evenpL p
=
LandLnfor nn
∈=∉== 111111111111:2
TL
ab⇔aTb

4. Lvww ,...,1jLi wwji ≠≠∀ :LDFA
v
∞=vL
LTL
DFAL
T
L
}|01{ Ν∈= pL pp
}1{ n
na =
ijLandL ijii
∉∈ 0101
}}1,0{|{ *
∈= wwwL
}01{ n
na =
ijLandL iijii
∉∈ 01010011
}|0{ 2
Ν∈= nL
n
}0{ 2n
na =i
jLandL
ijiii
∉∈=
+
22222
0000
1
}|0{
2
Ν∈= nL n
}0{
2
n
na =i
jjiLandL ijiii
<∀∉∈= +++
,00000 12)1(12 222
222
)1(12 +<++< jijj,
K
},|010{ Ν∈= +
mnL nmnm
∑= ),,,,{ 0 FqQM δ

5. w||||),(,*
QwMLww >∈∈∑L(M)
M
∑= ),,,,{ 0 FqQM δ∑+= |||| Qlm
DFA
))(( c
mlO
21,MM
θ=)( 1ML)( 1ML
∑= *
1)(ML
)()( 21 MLML ⊆
)()( 21 MLML =
φ=)( 1ML
θ=)( 1ML
BFS
∑= *
1)(ML
φ=∑ )( 1
*
ML
)()()()( 2121 MLMLMLML ⊆⇔= φ
)()()()()()( 211221 MLMLMLMandLMLML =⇔== φϑ
||21 ... wwwww =w12|| ... wwww w
R
=
L}:{ LwwL RR
∈=
aMLNFA
L
bMLNFA
R
L

7. MLDFA
suffix(L)
∑ ∑∈∈=∃∈= *},,:*{)( yLxywxwLsuffix
NFA
L={w: w has two different consecutive characters (over the alphabet{0,1})}

8. PDA
PDA
PDA
PDA
LPDA
PDA
PDA∑ Γ },,,,,{ 0 FqQ δ
Q
∑
Γ
)(),,(: *
∑ Γ×→∪Γ∪ QPQ εεδ