NOTE P, A,. A P ( A, P ),,.,. P A. (set) (set), (). (element), (element).,.,. ( A, B, X, Y, P ), { } (),..3 (union) A, B,A B, A B (union),. A B = {x x

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1 2. (set) , (union) (intersection) (power set) (cartesian product set) , 3 2. (length) (epsilon) (star closure) (language) (concatenation) (DFA) DFA DFA DFA (dead state) DFA ϕ : ϕ {} : {a} : a {b} : b {a} {b} () : a+b {a}{b} () : ab {a} () : (a) DFA Rij = P mod (NFA) NFA NFA DFA NFA DFA NFA (-NFA) (RE) 3 6. (regular set) (regular expression).... 4

2 NOTE P, A,. A P ( A, P ),,.,. P A. (set) (set), (). (element), (element).,.,. ( A, B, X, Y, P ), { } (),..3 (union) A, B,A B, A B (union),. A B = {x x A or x B} P A B P = {,, 2, 3, 4, 5, 6},. A = {x x },,.,,,. A B :.4 (intersection) A, B,A B,A B (intersection),. A B = {x x A and x B}.2, P, a,. P A B a P (a, P ) P a A B : 2

3 .5 (power set) A,A,A (power set),. 2 A = {P P A},A = {a, b},a {a, b}, {a}, {b}, ϕ() 4,2 A,. 2 A = {{a, b}, {a}, {b}, ϕ} 2 A, 2 (A ). 2 A,. ϕ..6 (cartesian product set) A,B,A, B (pair),a,b (cartesian product set).. A B = {(a, b) a A, b B} (a, b) a A,b B.,. B A = {(b, a) b B, a A},(a, b) (b, a), (). A = {a, b}, B = {c, d}. A B = {(a, c), (a, d), (b, c), (b, d)} B A = {(c, a), (c, b), (d, a), (d, b)} 2, () A. A = {,,,,, } A = {A, B, C,, X, Y, Z} A,A.,. A = {a} A a, aa, aaa, aaaa,, aa aa, B = {a, b} B a, b, ab, aba, bb, aabb, 2. (length),. abaabba = (epsilon),. = 2.3 (star closure) A, A (star closure). A A = {a} A = {, a, aa, aaa, aaaa, aaaaa, } B = {a, b} B = {, a, b, ab, ba, aaa, aab, } 2,,. a, b 2aa, ab, ba, bb 3aaa, aab, aba, abb, baa, bab,,!! 3

4 2.4 (language) A,A (language). = { a, b, c,, z, A, B,, Z }.,.,. 2.5 (concatenation), (concatenation).,, ω, ω = ω.,,. q. F. F = {q, q 3 } 5,.,.5,. M = {Q, Σ, δ, q, F },. 3. M = {Q, Σ, δ, q, F }. Q = {r, s, t} () Σ = {, } () δ(r, ) = r, δ(r, ) = s, δ(s, ) = r δ(s, ) = t, δ(t, ) = r, δ(t, ) = r ( δ(, )) q = r () 3 (DFA), 5 5 (5-tuple). Q. Q = {q, q, q 2, q 3 }. = {, } δ,. δ( q }{{}, }{{} ) = q }{{}.,,.,. F = {t} (),,. r s t.,... 4

5 3.2 }{{}. r r s s r t t r t,.,,. r t }{{} s, Σ Σ. r Σ = {, },. t. }{{} s r t }{{} r t,.,.,. 3.4 DFA DFA, Σ,. s s Σ L(M) r s t Σ L(M) M 5

6 ,DFA,. q () F () ( )!! 3.5 DFA DFA M, M 2,,.,, 2.. M, M 2 q, r,,.,..,. M, M 2,. () 2. q, r,. q r q 2 r q r 3.6 M, M 2,.,,.,..,, M, M 2,. () M, M 2. M q q 2 M, M 2!! M, M 2.(),,DFA. M 2 q q DFA DFA, DFA.,. r r 3.8 (dead state), (dead state). q dead state 6

7 3.9 DFA DFA,., DFA, DFA. DFA,..., M DFA. 4 (NFA) (NFA), (DFA).NFA DFA,. DFA. NFA.,NFA,.,2. p p p 2, 4. NFA,DFA 5. M = {Q, Σ, δ, q, F },NFA DFA.,, p 2.,p 2. p 2,.,,,M M 2.,DFA, δ, δq Σ Q (, Q ),NFA,.. p δq Σ 2 Q (,Q 2 Q ),,p3.5,.6 p, p ( DFA),., r r r 2,, DFA DFA,. NFA., r,r, r., 7

8 . δ(r, ) = {r, r } 2 Q (2 Q = {ϕ, {r }, {r }, {r 2 }, {r, r } }{{}, {r, r 2 }, {r, r 2 }, {r, r 2 }, {r, r, r 2 }}) 4.2 NFA NFA. r {r } {r, r } r ϕ {r 2 } r 2 {r 2 } ϕ {}. NFA,.,,., r }{{} r }{{} r }{{} (, ) r }{{} r }{{} r 2 }{{} r 2,NFA,,.,,,. 4.3 NFA DFA NFA,DFA,DFA.,.,. NFA DFA (NFA DFA ),,,, NFA NFA DFA.., 2 Q,. r r r 2 (2 Q,,, ),.,., NFA DFA,. DFA NFA (DFA, NFA ),.,,. (NFA DFA) (DFA NFA) NFA = DFA!!,NFA DFA. 8

9 4.4 NFA DFA, r r r 2 NFA DFA.,. r {r } {r, r } r ϕ {r 2 } r 2 {r 2 } ϕ, r, {r }.,. [r ], {}.,,.,,. r r, r r, [r ] [r ] [r, r ],,,. [r ] [r ] [r, r ] [r, r ] r, r,r., r, r, r 2,,. [r ] [r ] [r, r ] [r, r ] [r ] [r, r, r 2 ] [r, r, r 2 ] [r, r 2 ] [r, r, r 2 ] [r, r 2 ] [r, r 2 ] [r, r ], NFA,r 2.,r 2 2.,, r 2. NFA,,,r 2 [r, r, r 2 ], [r, r 2 ].,. [r ] q [r, r ] q [r, r, r 2 ] q 2 [r, r 2 ] q 3, q q q q q q 2 q 2 q 3 q 2 q 3 q 3 q, NFA DFA.,NFA DFA. [r ] [r ] [r, r ] [r, r ] [r ] [r, r, r 2 ] 9

10 5,,,.. case2 q, r,,. q r 5.,.,.,.,.,., q, r.,q, r. 5.2 q, r M q r,.,. q r s t,4 M., q, r. case q, r,,. (q q) (r q),. q r 5.3,.,.,, q, r,q, r.,.

11 q q q 2 q q 2 q 3 q q q 2 q 3 q NFA, q, q 2. 2,. q,.,!! q,,., q q, q 2 q 2,. 5.4,.,.,.,., q q, q q. (),,. q q q q 3 - q q q q q 2 q 3,.. (q ) (q, q 2, q 3 ). q 3,,,. q 2 q q q q 3 - q q q 2 q 3

12 2. (q, q 2 ),(q 2, q ). q.q 2,.,(q, q 2 ). q q q q 3 - q q q 2 q 3 3. (q, q 3 ),(q 2, q ).,(q, q 3 ). q q q q 3 - q q q 2 q 3 4. (q 2, q 3 ) (q, q ). (q 2, q 3 ) (q 3, q 2 ).,. 5.5 NFA (-NFA) NFA, 2 NFA.,. DFA : δ : Q Σ Q, NFA : δ : Q Σ 2 Q, -NFA : δ : Q (Σ ) 2 Q,, ()!! -NFA. M = ({q, q, q 2 }, {,, 2}, δ, q, {q 2 }) }{{}}{{}}{{} Q Σ F 2 q q q 2 q q q q 3 - q q q 2 q 3, Σ,,. M.., q 2 q 3.,. q q [q 2, q 3 ],!!,., =,. NFA,,!! 2. =, = 2,. 4.,,. 2 p3 2.2 : (epsilon) 2

13 5.6,,.,, -NFA, NFA ( )., -NFA,. 2 q q q 2,. 2 q {q } {} {} {q } q {} {q } {} {q 2 } q 2 {} {} {q 2 } {},.,.,..,.,q q, q, q 2,. q {q, q, q 2 } q q 2 2,. 2 q {q, q, q 2 } {q, q 2 } {q 2 } q {} {q, q 2 } {q 2 } q 2 {} {} {q 2 },, NFA. 2,,2 q q 2 q,,2, -NFA, NFA.,. q q q 2 2 NFA NFA,-NFA,NFA, NFA NFA,q., q, q 2,q.. q.,,.,,. 2, NFA = NFA. 6 (RE),Linux ls *.gif, (regular expression:re). 3

14 6. (regular set),.. ϕ() ϕ : ϕ () 2. {}. 3. {a}. (,a Σ) {} : 4. R, S, (a) R S. (b) RS. (c) R, S {a} : a,. 4.,,, {b} : b. () a () b 6.2 (regular expression) (), { }.,.,. (,R, S.) {a} {b} () : a+b a ϕ {} {a} R S RS R ϕ a R + S RS (R) b (a b ) {a}{b} () : ab a 6.3,.,. b (a b ),,,. 4

15 6.3.7 {a} () : (a) a. (() (). 3 ) 6.5. = =, (a ),!! 6.4 DFA R ij,, i j.,.,,, (DFA). 6.5 Rij 3 Rij, i j (path).,rij,.. (,2,3, ) R = {, }, R2 = {}, R3 = {},!! Rij, i j,.,rij i j,. OK 2 R 2 = {}, R 22 = {, }, R 23 = {} R 3 = {}, R 32 = {}, R 33 = {, },Rij,. if i j 2 3 Rij = {x a Σ, δ(q i, a) = q j } if i j Rij = {x a Σ, δ(q i, a) = q j } {} R ij, R 3.,. R ij = R ( ) ij R ( ) i (R ( ) ) R ( ) j 5

16 ,. R ij = R ( ) ij }{{} R ( ) i }{{} (R ( ) ) } {{ } R ( ) j }{{} R R2 R3 R2 = 2 i, (, ) j,,,.,. R 22 R 23 R 3 R 32 R 33. R 2,, a (),R 2 = a!!). 2. R 2 2,,.,. () R 3 = R 2 R3(R 2 33) 2 R3 2,b.,a, b, 2 a.,a, b.,b, 2 a,r2 2 = a ba. Rij 2 (i, j Q). a b a 2 b a 3 b, Rij 2.R2 ij,,,2,.,.,. = 2 R a R2 R3 R2 R22 R23 R3 R32 R33 a ba a ba b ϕ a a b aa aa ba aa ba + b + (,.) 6

17 7,,2., 3. 7.,,. a b q q q }{{} q }{{} a b 7.2,,. q q a q }{{} q a 7.3 2,. =. + =.,,. =,,. 7.4 P mod 3, 2 P 3 P mod 3. 2 P.,P N. [P ] 2 = [N] 2, 2.(2 3 ),2, 2 +.(2 +), P = N, P = 2N, P = 2N + (P : P ) (P : P.),. mod 3 P N 2 P 2N 2 P 2N + 2, 2 P 4,, P mod 3 = P mod 3 = P mod 3 = 2,mod 3 2 P,,P mod 3., mod 3. 3,,2 2. 4,, ( ).,. 7