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version 4
20.11.06
Mihill-Nerode
PDA
}#|#{ 01
wwwL ==
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
L
}#|#{ 01
wwwL ==
LnwLwnn ∈= 10
yi=0
010
10)0(0
0 >∉=
∈==−
−−
bLzxy
Lzxyw
nbn
nabnba
L
Lw
}#|#{ 01
wwwL ≥=
LnwLwnn ∈= 10
yi=2
01010
10)0(0
22 >∉==
∈==+−−++
−−
bLzxy
Lzxyw
nbnnbanba
nbanba
L
}0,|{}1,0|{ ≥∪≥≥= jicbikcbaLjikki
LL
}0|{ 2 ≥= nbaaLnn
}|{ jikcbaLkji +>=
}#2|#},{{ * b
x
a
xbaxL <∈=
}|}1,0{{ *palindromeisxxL ∈=
LnwLwnnnn ∈= 0110
y
i=0
00110
0||||0110)0(0
0 >∉=
>==∈==−
−−
bLzxy
byaxLzxyw
nnnbn
nnnbanba
L
}}1,0{|{ *∈= wwwL
}|}1,0{{ *
palindromenotisxxL ∈=
}|1{ primeispLp=
}|1{ numbernaturalofsquareispLp=
LnwLwn ∈=
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{ 3NttpL
p ∈==
}:2|1{ NtpLtp ∈==
}210{ nmnL =
Nerode-Mihill
-
xyyx L~Lyx ∈,Lyx ∉,
xyyx L≡yuxuu L~:*∑∈∀ x
yL
}1{ *=L
LLNnnn ∈⇔∈∈∀ 11111:
}|1{ evenpLp=
LandLnfor nn ∈=∉== 111111111111:2
TL
ab⇔aTb
Lvww ,...,1jLi wwji ≠≠∀ :LDFA
v
∞=vL
LTL
DFA L
T
L
}|01{ Ν∈= pLpp
}1{ n
na =
ijLandLijii ∉∈ 0101
}}1,0{|{ *∈= wwwL
}01{ n
na =
ijLandLiijii ∉∈ 01010011
}|0{ 2 Ν∈= nLn
}0{ 2n
na =i
jLandLijiii
∉∈=+ 22222 00001
}|0{2
Ν∈= nLn
}0{2
n
na =i
jjiLandLijiii <∀∉∈= +++ ,00000 12)1(12 222
222 )1(12 +<++< jijj,K
},|010{ Ν∈= +mnL
nmnm
∑= ),,,,{ 0 FqQM δ
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)(MLφ=∑ )(\ 1
*ML
)()()(\)( 2121 MLMLMLML ⊆⇔= φ
)()()(\)()(\)( 211221 MLMLMLMandLMLML =⇔== φϑ
||21 ... wwwww =w12|| ... wwww w
R =
L}:{ LwwL RR ∈=
a MLNFA
L
b MLNFARL
MLDFA
suffix(L)
∑ ∑∈∈=∃∈= *},,:*{)( yLxywxwLsuffix
NFA L={w: w has two different consecutive characters (over the alphabet{0,1})}
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LPDA
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PDA∑ Γ },,,,,{ 0 FqQ δ
Q
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