The document discusses various concepts related to automata theory, including deterministic finite automata (DFA), non-deterministic finite automata (NFA), and their transitions and relationships. It outlines mathematical notations and rules governing the behavior of these automata, including state transitions based on input sequences. Additionally, the document addresses the inclusion and properties of languages accepted by these automated systems.

1. 1
:2
version 5
04.11.06
DFA
KMP
DFA∑ },,,,{ 0 FqQ δ
Q
∑
δ
∑→× QQ:δ
0q
F
Q={“on”,”off”}
∑= }"_{" pressedkey
"")"_","("
"")"_","("
onpressedkeyoff
offpressedkeyon
=
=
δ
δ
""0 offq =
""offF =
DFA

2. 2






∈∈=
∈=
=
∑∑
∑
+
awawqwaq
aaqaq
qq
,)),,(
~
(),(
~
),(),(
~
),(
~
δδδ
δδ
εδ
LDFA
}),(
~
|{ 0 FwqwL ∈= δ
"
},01|{ *
∑∈== yxyxwwL
•
3
00
321
}1,0{
},,{
qF
qq
qqqQ
=
=
=
=
∑
),( aqδ
0q2q0q
3q2q2q
3q3q3q
)(3 nSn3q

3. 3
)(2 nSn2q
n-1
)(0 nSn0q
n-1
n
n=0
•
nkkS ,,2,1,0)(3 =∀)(2 kS)(0 kS
)1(3 +nS)1(2 +nS)1(0 +nS
•3qn+1
3q
3q
)(3 nS3q
),( aqδn2q)(2 nS
nn+12q3qn+1
),( aqδ
•2qn+1
n
•0qn+1
nn=0
L
∑=
=
}1,0{
}3|{ bydividablenumberbinaryiswwL

4. 4

5. 5
LDFA
LW
}|{
}|{
}|{
}|{
*
LwWandworWwandLwwWL
LwwL
WwandLwwWL
WworLwwWL
∉∈∉∈=⊕
∈=
∈∈=∩
∈∈=∪
∑
LWWL ∪
‫ו‬
¡
WL ∩
¢ ‫ג‬‫ש‬‫ת‬ ‫ו‬ ‫י‬ ‫ר‬ ‫א‬ ‫ל‬ ‫ו‬ ‫ג‬ ‫ר‬ ‫ת‬ ‫ו‬ ‫פ‬
.
LL
‫ת‬ ‫י‬ ‫ר‬ ‫א‬ ‫ל‬ ‫ו‬ ‫ג‬ ‫ר‬ ¢ ‫ג‬
.
LWWL ⊕
‫ת‬ ‫י‬ ‫ר‬ ‫א‬ ‫ל‬ ‫ו‬ ‫ג‬ ‫ר‬ ‫ה‬ ‫פ‬ ‫ש‬
.
KMP
LALA
L
A
A

6. 6
)(4 nSn4q
)(3 nSn3q
)(2 nSn2q
)(1 nSn1q
w
w|w|+1
KMP
wy
wywy
w
ywy
O((|y|-|w|)|w|)
KMP
•:preprocessing
w
O(|w|)

7. 7
•y
O(|y|)
•O(|w|+|y|)
•
KMP
CLRS
KMP
NFA
NFADFA
DFANFA
NFA
- NFA
DFA
NFA
NFANFA∑ },,,,{ 0 FqQ δ
Q
∑
0q
F
δ
∑ →× )(: QPQ εδ U
.

8. 8
∑ →× )(: QPQ εδ U
0q1q,0q0q
2qΘ1q
ΘΘ2q
NFA
NFA
NFA
),,,...,,,( 12110 +kkk qrqrqrqi
iq1+iqir
wkrrw ...1=
LNFA
}|{ FinendwhichwforpathlegalexistwL =
DFA-NFA
NFA
}'1'0|{ stwoexactlyorsofnumberevenancontainswwL =

9. 9
NFA
}110|{ substringthecontainnotdoeswwL =
NFA
}010000|{ substringasoreithercontainswwL =
DFANFADFA
NFA"" DFANFA ⊆
DFANFA
DFANFA
DFANFA

10. 10
LNFA
NFADFADFA
DFANFA
L
DFADFA
LWWL ∪
‫ת‬ ‫י‬ ‫ר‬ ‫א‬ ‫ל‬ ‫ו‬ ‫ג‬ ‫ר‬ ‫ה‬ ‫פ‬ ‫ש‬
.
NFA
LL
‫ת‬ ‫י‬ ‫ר‬ ‫א‬ ‫ל‬ ‫ו‬ ‫ג‬ ‫ר‬ ¢ ‫ג‬
.
LWWL ⊕
‫ת‬ ‫י‬ ‫ר‬ ‫א‬ ‫ל‬ ‫ו‬ ‫ג‬ ‫ר‬ ‫ה‬ ‫פ‬ ‫ש‬
.
}|{
......if 121121
LwwL
aaaawthenaaaaw
RR
kk
R
kk
∈=
== −−
LR
L
‫ת‬ ‫י‬ ‫ר‬ ‫א‬ ‫ל‬ ‫ו‬ ‫ג‬ ‫ר‬ ¢ ‫ג‬
.
}|{* WlandLwlwWL ∈∈=
LWWL*
‫ת‬ ‫י‬ ‫ר‬ ‫א‬ ‫ל‬ ‫ו‬ ‫ג‬ ‫ר‬ ‫ה‬ ‫פ‬ ‫ש‬
.
NFA
NFA