ask_2012 (Systems Analysis)
38
ΑΣΚΗΣΕΙΣ ΓΙΑ ΚΑΝΟΝΙΚΕΣ ΓΛWΣΣΕΣ ΚΑΙ ΠΕΠΕΡΑΣΜΕΝΑ ΑΥΤΟΜΑΤΑ (1) L = a(a + b) * . Απ.: Λέξεις που αρχίζουν mε a. a a b (2) L =(a + b) * ab. Απ.: Λέξεις που τελειώνουν σε ab. a b b a a b (3) L =(a + b) * bb(a + b) * a. Απ.: Λέξεις mε δυο διαδοχικά b, δηλαδή, mε το bb, οι οποίες τελειώνουν σε a. b a a b b a a b (4) L =(a + ab) * . Απ.: Λέξεις χωρίς καθόλου διαδοχικά b, δηλαδή, χωρίς το bb, οι οποίες αρχίζουν mε a. a a b a 1
description
Algorithm and Systems Analysis
Transcript of ask_2012 (Systems Analysis)
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(1) L = a(a+ b).
pi.: pi a.
a
a
b
(2) L = (a+ b)ab.
pi.: pi ab.
a
b
b
a
a
b
(3) L = (a+ b)bb(a+ b)a.
pi.: b, , bb, pi a.
b
a
a
b
b
a
a
b
(4) L = (a+ ab).
pi.: b, , bb, pi a.
a
a
b
a
1
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(5) L = (+ b)(a+ ab).
pi.: b, , bb.
a b
a
b
a
(6) L = (a+ bab).
pi.: pi b.
b
b
a a
(7) pi a a.
pi.: L = a(a+ b) + (a+ b)a.
a b
a
b
b a
a
b
2
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(8) pi a a, .
pi.: L = a(a+ b)b+ b(a+ b)a.
a b
a b ab
a
b
b
a
(9) pi .
pi.: L = a(a+ b)a+ b(a+ b)b+ a+ b.
a b
a b ab
a
b
b
a
(10) = {a, b, c}, pi a, pi b ( pi) c, , cc.
pi.: L = a(a+ c)b(a+ c)b(a+ c)cc.
a b b
c
c
a
a
a+ c a+ c a c
3
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(11) = {a, b, c}, pi b pi pi c.
pi.: L =((a+ c)bc+
)(a+ c).
b
c
b
a
a+ c c
(12) pi , , pi bab bb. ( babb bab bb, , ,, 2 , pi babb ...)
pi.:
a aa
b
a
b
b
a a
a
b
a
(13) pi .
pi.: L =((a+ b)(a+ b)
)= (aa+ ab+ ba+ bb).
a+ b
a+ b
(14) pi pi .
pi.: L = (a+ b)((a+ b)(a+ b)
)= (a+ b)(aa+ ab+ ba+ bb).
a+ b
a+ b
4
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(15) = {a, b, c}, 3.
pi.: L = (a+ b+ c)(a+ b+ c)(a+ b+ c) = (a+ b+ c)3.
a+ b+ c a+ b+ c a+ b+ c
(16) = {a, b, c}, 3.
pi.: L = + (a+ b+ c) + (a+ b+ c)2.
a+ b+ c a+ b+ c
(17) = {a, b, c}, 3.pi.: L = (a+ b+ c)3(a+ b+ c)+.
a+ b+ c a+ b+ c a+ b+ c a+ b+ c
a+ b+ c
(18) pi pi ab .
pi.: L = (a + b)ab(a+ b).
a b
b a a+ b
(19) pi pi ab .
pi.: L = (a + b)ab(a + b)ab(a+ b).
a b a b
b a b a a+ b
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(20) a, , aa.
pi.: L = (b+ ab)(+ a).
a
b
b
(21) a , , pi pi aa .
pi.: L = (b+ ab)aa(b+ ab).
a
b
b
a
b
a
b
(22) a , , pi pi aa .
pi.: L = (b+ ab)aa(a+ b+(ab)aa
)(a+ b).
a
b
b
a
a
b
b
a
b
a
a+ b
6
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(23) a b, , pi pi aa bb.
pi.: L = (a+ b)aa(a+ b)bb(a+ b) + (a+ b)bb(a+ b)aa(a+ b).
a
b
b
a
a
ba
b
a
b
a
b
a
b
a+ b
(24) pi pi ab ba.
pi.: L = (bab+a+ aba+b)(a+ b).
a
b
b
a
a
b
a
b
b
a
b
a
a
b
a+ b
(25) = {a, b, c}, a, , aa.
pi.: L = (b+ c+ ab+ ac)(+ a).
a
b+ c
b+ c
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(26) = {a, b, c}, a, , pi pi aa.
pi.: L =(b+ c+ a(b+ c)
)aa(a+ b+ c).
a
b+ c
b+ c
a
a+ b+ c
(27) pi pi pi b, , pi pi bb.
pi.: L =(a(ba)a
))(abb+ bb(a+ b)
)((a+ b)(a+ b)
).
b
ab
a+ b
a+ b
a a
b
a
b
a+ b
a+ b
(28) = {a, b, c}, pi pi a.
pi.: L =((b+ c)(b+ c)
)a(b+ c)
((b+ c)(b+ c)
)+ (b+ c)
((b+ c)(b+ c)
)a((b+ c)(b+ c)
).
a
b+ c
b+ c
b+ c b+ c
a
b+ c
b+ c
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(29) pi a, , pi aaa.
pi.: L = b(a+ b) + ab(a+ b) + aab(a+ b) = (+ a+ aa)b(a+ b).
a a
bb
b
a+ b
(30) a, , aaa.
pi.: L = (b+ ab+ aab).
a
b
a
b
b
(31) aba.
pi.: L =(abb+a
)+ a + b + ab + ba.
b
a a b
b b
a
a b
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(32) pi a (pi na) pi 3.
pi.: L = b(ababab).
a
a
a
bb
b
(33) nb < na 2.
pi.: L = L0 + L1, pi L0 pi pi na = 2 nb = 0 1 L1 pina = 1 nb = 0. , : L0 = aa+ baa+ aba+ aab L1 = a.
a a
b
a a
b b
(34) = {a, b, c}, nb + nc = 3.
pi.: L = L0 + L1 + L2 + L3, pi L0 pi pi nb = 3 nc = 0, L1 pi nb = 2 nc = 1, L2 pi nb = 1 nc = 2 L3 pi nb = 0 nc = 3., : L0 = abababa, L1 = ababaca + abacaba + acababa, L2 =acacaba + acabaca + abacaca L3 = acacaca.
c
c
b+ c
b+ c
b+ cc
b
b
ba
a a
a a
a a
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(35) pi a b pi pi .
pi.: L = (b+ ab+ ba+ aba).
a b
a
b
a
b
b
(36) pi 4 pipi :
q0 q1
q2q3
a
a
bb
a
a
b b
pi.:
q0 na = nb = . q1 na = pi nb = . q2 na = pi nb = pi. q3 na = nb = pi.
pipi, pi pi- . .., pipi q3:
q0 q1
q2q3
a
a
bb
a
a
b b
q1 pipi :
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q0 q2 q3
ab
ba
a
a
b
b
aa bb
pi pi q2 pipi :
q0 q3
b+ ab(bb)a
b+ a(bb)ba
aa+ ab(bb)ba a(bb)a
, pi, ( na = pi nb =pi) :
L =(aa+ab(bb)ba
)(b+ab(bb)a
)((a(bb)a
)(b+a(bb)ba
)(aa+ab(bb)ba
)(b+ab(bb)a
)).
(37) pi 3 pipi :
q0
q1
q2
a
b
a
b
a
b
pi.:
q0 na nb = 0 mod 3. q1 na nb = 1 mod 3 nb na = 2 mod 3. q2 na nb = 2 mod 3 nb na = 1 mod 3.
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(38) pi 4 pipi :
q0 q1
q2q3
a
b
ab
a
b
a b
pi.:
q0 na nb = 0 mod 4. q1 na nb = 1 mod 4 nb na = 3 mod 4. q2 |na nb| = 2 mod 4. q3 na nb = 3 mod 4 nb na = 1 mod 4.
(39) pi 9 pipi :
q0 q1 q2
q3 q4 q5
q6 q7 q8
b b
b
b
b b
a a a
a a a
b
b
a a a
b
pi.:
q0 na = 0 mod 3 nb = 0 mod 3. q1 na = 0 mod 3 nb = 1 mod 3. q2 na = 0 mod 3 nb = 2 mod 3. q3 na = 1 mod 3 nb = 0 mod 3. q4 na = 1 mod 3 nb = 1 mod 3. q5 na = 1 mod 3 nb = 2 mod 3. q6 na = 2 mod 3 nb = 0 mod 3. q7 na = 2 mod 3 nb = 1 mod 3. q8 na = 2 mod 3 nb = 2 mod 3.
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(40) , (i) ( {a, b}), pi na = 1 mod 3 nb = 2 mod 3.
pi.: pipi pi- . pi pi pi pi pi.
(0) pi :
q0 q1 q2 q3 q4 q5 q6 q7 q8q0 b a
q1 b a
q2 b a
q3 b a
q4 b a
q5 b a
q6 a b
q7 a b
q8 a b
(1) pi q8:
q0 q1 q2 q3 q4 q5 q6 q7 q8q0 b a
q1 b a
q2 b a
q3 b a
q4 b a
q5 aa b ab
q6 a b
q7 a ba bb
q8
(2) pi q7:
q0 q1 q2 q3 q4 q5 q6 q7 q8q0 b a
q1 b a
q2 b a
q3 b a
q4 aa aba b abb
q5 aa b ab
q6 a ba bba bbb
q7q8
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(3) pi q6:
q0 q1 q2 q3 q4 q5 q6 q7 q8q0 b a
q1 b a
q2 b a
q3 a(bbb)a a(bbb)ba a(bbb)bba b
q4 abb(bbb)a aa+ abb(bbb)ba aba+ abb(bbb)bba b
q5 ab(bbb)a ab(bbb)ba aa+ ab(bbb)bba b
q6q7q8
(4) pi q4:
q0 q1 q2 q3 q4 q5 q6 q7 q8q0 b a
q1 aabb(bbb)a aaa+aabb(bbb)ba b+ aaba+
aabb(bbb)bbaq2 b a
q3a(bbb)a+babb(bbb)a
a(bbb)ba+ baa+babb(bbb)ba
a(bbb)bba+baba+
babb(bbb)bbaq4q5 ab(bbb)
a ab(bbb)ba aa+ ab(bbb)bba bq6q7q8
(5) pi q3:
q0 q1 q2 q5
q0 aa(bbb)a+ababb(bbb)a b+ aa(bbb)
ba+ abaa+ababb(bbb)ba
aa(bbb)bba+ ababa+ababb(bbb)bba
q1 aabb(bbb)a aaa+ aabb(bbb)ba b+aaba+aabb(bbb)bba
q2 b a
q5ab(bbb)a+ ba(bbb)a+
bbabb(bbb)aab(bbb)ba+ba(bbb)ba+bbaa+ bbabb(bbb)ba
aa+ ab(bbb)bba+ba(bbb)bba+ bbaba+
bbabb(bbb)bba
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(6) pi q2:
q0 q1 q5
q0
aa(bbb)a+ ababb(bbb)a+aa(bbb)bbab+ ababab+
ababb(bbb)bbab
b+ aa(bbb)ba+ abaa+ababb(bbb)ba
aa(bbb)bbaa+ ababaa+ababb(bbb)bbaa
q1aabb(bbb)a+ bb+ aabab+
aabb(bbb)bbab aaa+ aabb(bbb)ba ba+ aabaa+ aabb(bbb)bbaa
q5
ab(bbb)a+ ba(bbb)a+bbabb(bbb)a+ aab+
ab(bbb)bbab+ ba(bbb)bbab+bbabab+ bbabb(bbb)bbab
ab(bbb)ba+ ba(bbb)ba+bbaa+ bbabb(bbb)ba
aaa+ ab(bbb)bbaa+ba(bbb)bbaa+ bbabaa+
bbabb(bbb)bbaa
(7) pi q1:
q0 q5
q0
(b+ aa(bbb)ba+ abaa+
ababb(bbb)ba)(aa(bbb)a+ ababb(bbb)a+
aa(bbb)bbab+ ababab+ababb(bbb)bbab
)(aabb(bbb)a+ bb+ aabab+
aabb(bbb)bbab)
(b+ aa(bbb)ba+ abaa+
ababb(bbb)ba)(aa(bbb)a+ ababb(bbb)a+
aa(bbb)bbab+ ababab+ababb(bbb)bbab
)(aa(bbb)bbaa+ ababaa+
ababb(bbb)bbaa)
q5
(ab(bbb)ba+ ba(bbb)ba+ bbaa+
bbabb(bbb)ba)(aa(bbb)a+ ababb(bbb)a+
aa(bbb)bbab+ ababab+ababb(bbb)bbab
)(aabb(bbb)a+ bb+ aabab+
aabb(bbb)bbab)
(ab(bbb)ba+ ba(bbb)ba+ bbaa+
bbabb(bbb)ba)(aa(bbb)a+ ababb(bbb)a+
aa(bbb)bbab+ ababab+ababb(bbb)bbab
)(aa(bbb)bbaa+ ababaa+
ababb(bbb)bbaa)
, pi, pipi pipi (pi Aij pi 2 2):
q0 q5
A12
A21
A11 A22
pi, :
L = (A11)A12
((A22)
A21(A11)A12).
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(41) :
(i) na 1 nb = 2.(ii) na 2 nb 2.(iii) na 4 nb 2.
pi. (i):
q0 q1
q2 q3
q4 q5
a
a
a
b b
b b
a
a
a
pi. (ii):
q0 q1 q2
q3 q4 q5
q6 q7 q8
a a
a a
a a
b b b
b b b
b
b
b
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pi. (iii):
q0 q1 q2 q3 q4
q5 q6 q7 q8 q9
q10 q11 q12 q13 q14
a a a a
a a a a
a a a a
b b b b b
b b b b b
a a a a a
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pi :
(1) S aSa | aBa,B bB | b.pi.: S
n= anSan m= an(amBam)an k= an+mbkBan+m an+mbk+1an+m. ,
L = {aibjai: i 0, j > 0}.(2) S aSdd |A,A bAc | bc.
pi.: Sn
= anS(dd)n anA(dd)n m= an(bmAcm)(dd)n anbmbccm(dd)n =anbm+1cm+1(dd)n. , L = {aibjcjd2i: i 0, j > 0}.
(3) S aSb | aSbb | .pi.: S
n= anSbn m= an(amS(bb)m)bn an+mbn+2m. , L = {aibj: 0
i j 2i}.(4) S abScB | ,B bB | b.
pi.: Sn
= (ab)nS(cB)n (ab)n(cB)n m= (ab)n(c(bmB))n (ab)n(cbm+1)n. , L = {(ab)i(cbj)i: i 0, j > 0}.
pi :
(5) L = {aibi: i 0}.pi.: S aSb | .
(6) L = {(ab)i: i 0}.pi.: S abS | .
(7) L = {aibi+1: i 0}.pi.: S aSb | b. S Bb,B aBb | pi.
(8) L = {aib2i: i 0}.pi.: S aSbb | .
(9) L = {aibj: i > 0, j 0}.pi.: S AB,A aA | a,B Bb | .
(10) L = {ai+2bi: i 1}.pi.: S aSb | aaab.
(11) L = {aibi3: i 3}.pi.: S aSb | aaa.
(12) L = {aibj: j > i 0}.pi.: S aSb |Sb | b.
(13) L = {aibj: i 6= j}.pi.: S S> |S aS>b | aS> | a, S< aS
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(17) L = {aibj: 2i j 3i}.pi.: S aSbb | aSbbb | .
(18) L = {a2ib3j: i, j 0}.pi.: S AB,A aaA | ,B Bbbb | .
(19) L = {aibjck: i = j j k}.pi.: S S1C |AS2, S1 aS1b | , C Cc | ,A aA | , S2 bS2c |S2c | .
(20) L = {aibjck: i = j j 6= k}.pi.: S S1C |AS3, S1 aS1b | , C Cc | ,A aA | , S3 S2 |S4,S2 bS2c |S2c | c, S4 bS4c | bS4 | b.
(21) L = {aibjck: i 6= j j 6= k}.pi.: S S1C |AS3, S1 S5 |S6, S5 aS5b | aS5 | a, S6 aS6b |S6b | b, S3 S2 |S4,S2 bS2c |S2c | c, S4 bS4c | bS4 | b.
(22) L = {aibjck: k = i+ j}.pi.: S aSc |S1, S1 bS1c | .
(23) L = {aibjck: k 6= i+ j}.pi.: S S1 |S2, S1 aS1c |S3 | c, S3 bS3c |S3c | c,S2 AS4, A aA | , S4 aS4c |S5 | a, S5 bS5 | b.
(24) L = {aibjck: i+ 2j = k}.pi.: S aSc |S1 | , S1 bS1cc | .
(25) L = {aibjck: k = |i j|}.pi.: S S1 |S2S3, S1 aS1c |S2 | , S2 aS2b | , S3 bS3c | .
(26) L = {aibick: i 0, k 3}.pi.: S S1C, S1 aS1b | , C Cc | ccc.
(27) (i) L = {w (a+ b): |w| = }.(ii) L = {w (a+ b): |w| = pi}.pi.: pipi ( ):
S T
a+ b
a+ b
(i) S : S aT | bT | , T aS | bS.(ii) T : S aT | bT, T aS | bS | .
(28) (i) L = {w (a+ b): |w| mod 3 = 0}.(ii) L = {w (a+ b): |w| mod 3 > 0}.(iii) L = {w (a+ b): |w| mod 3 6= |w| mod 2}.(iv) L = {w (a+ b): |w| mod 3 |w| mod 2}.pi.: pipi ( ):
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US Ta+ b
a+ ba+ b
(i) S : S (a+b)T | , T (a+b)U,U (a+b)S( S (a+ b)3S | ).
(ii) T U : S (a + b)T, T (a +b)U | , U (a+ b)S | ( S (a+ b)T | (a+ b)2U, T (a+ b)3T | , U (a+ b)3U | ).
(iii) , |w| mod 3 = |w| mod 2 |w| mod 6 = 0, pi |w|mod 3 6= |w| mod 2 |w| mod 6 = 1, 2, 3, 4 5. pi, pipi :
S
T U
V
X W
a+ b
a+ b
a+ b
a+ b
a+ b
a+ b
, : S (a + b)T, T (a + b)U | , U (a + b)V | , V (a +b)W | ,W (a+ b)X | ,X (a+ b)S | .
(iv) , |w| mod 3 |w| mod 2 |w| mod 6 6= 3, pi |w|mod 3 |w| mod 2 |w| mod 6 = 0, 1, 2 4, 5, 6. pi, pipi :
S
T U
V
X W
a+ b
a+ b
a+ b
a+ b
a+ b
a+ b
, : S (a+ b)T | , T (a+ b)U | , U (a+ b)V | , V (a+ b)W,W (a+ b)X | ,X (a+ b)S | .
(29) (i) L = {w (a+ b): na(w) = }.(ii) L = {w (a+ b): na(w) = pi}.(iii) L = {w (a+ b): na(w) = nb(w) = }.
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(iv) L = {w (a+ b): na(w) = nb(w) = pi}.pi.: pipi ( ):
S T
UV
a
a
bb
a
a
b b
(i) S V : S aT | bV | , T aS | bU, U aV | bT, V aU | bS | . : pi na(w) na(ww) , , , na
((ww)
), pi pi na(bw) na(wb) , pi
() : S AA | ,A AAA | bA |Ab | a.(ii) T U : S aT | bV, T
aS | bU | , U aV | bT | , V aU | bS. : pi na(w) pi na(www) pi, , -, na
(w(ww)
)pi, pi pi na(bw) na(wb) pi,
pi () : S AAA,A AAA | bA |Ab | a.(iii) S : S aT | bV | , T aS | bU, U
aV | bT, V aU | bS.(iv) V : S aT | bV, T aS | bU, U
aV | bT, V aU | bS | .
(30) L = {w (a+ b+ c): abc pi w}.pi.: pipi ( ):
U
S T
a
c a
b
b
b+ c
a
S, T U : S aU | bS | cS | , T aU | bS | , U aU | bT | cS | .
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(31) L = {w (a+ b): na(w) = 1}.pi.: pipi :
S Ta
b b
T : S aT | bS, T bT | .(32) L = {w (a+ b): na(w) 1}.
pi.: pi pi , pi pipi T(pi a T ): S aT | bS, T aT | bT | .
(33) L = {w (a+ b): na(w) = 2}.pi.: pipi :
S T Ua a
b b b
U : S aT | bS, T aU | bT, U bU | .(34) L = {w (a+ b): na(w) 2}.
pi.: pi pi , pi pipi U(pi a U): S aT | bS, T aU | bT, U aU | bU | .
(35) L = {w (a+ b): na(w) 3}.pi.: pipi :
S T U Va a a
b b b b
S, T, U V : S aT | bS | , T aU | bT | , U aV | bU | , V bV | .
(36) L = {w (a+ b): na(w) 3}.pi.: pipi :
S T U Va a a
b b b a+ b
S, T, U V : S aT | bS, T aU | bT, U aV | bU, V aV | bV | .
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(37) L = {w (a+ b): na(w) = nb(w)}.pi.: pi w L w1w2 L, w1, w2 L, awb L bwa L, : S SS | aSb | bSa | .
(38) L = {w (a+ b): na(w) = nb(w) + 1}.pi.: S0 pi pi a b(pi pi ), : S S0aS0, S0 S0S0 | aS0b | bS0a | .
(39) L = {w (a+ b): na(w) > nb(w)}.pi.: S0 pi , :S S0AS0, S0 S0S0 | aS0b | bS0a | ,A aA | a.
(40) L = {w (a+ b): nb(w) na(w)}.pi.: , pi pi, : S SS | aSb | bSa | bS | .
(41) L = {w (a+ b): na(w) = 2nb(w)}.pi.: pi , pi , aab, aba baa, : S SS | aSaSb | aSbSa | bSaSa | .
(42) L = {w (a+ b): na(w) = 2nb(w) + 1}.pi.: S2 pi pi a pi pi b (pi pi ), : S S2aS2, S2 S2S2 | aS2aS2b | aS2bS2a | bS2aS2a | .
(43) L = {w (a+ b): na(w) = 2nb(w) 1}.pi.: , pi na(w) = 2nb(w)1 na(w)1 = 2(nb(w)1), pi , w L, a b pi pi pi w, , pipi , pi a pi pi b. pipi, a b, , ab ba. pi,pi pi , pi a pi pi b, pi ab ba, L. pi a pi pi b :S SS | aSaSb | aSbSa | bSaSa | . , S ab ba, pi () ab ba, S .pi, pipi pi pi pi a pi pi b S . pi S SS pi pipi : pi.., S aSaSb S Saab | aSab | aaSb | aabS ... pi, L :
S Saab | aSab | aaSb | aabS |Saba | aSba | abSa | abaS |Sbaa | bSaa | baSa | baaS |ab | ba.
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(44) L = {w (a+ b): |na(w) nb(w)| = 1}.pi.: pi pi, : S S0aS0 |S0bS0, S0 S0S0 | aS0b | bS0a | .
(45) L = {w (a+ b): w = wR}.pi.: , : S aSa | bSb | a | b | .
(46) L = {wwR: w (a+ b)+}.pi.: , : S aSa | bSb | aa | bb.
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pi :
(1) L = {aibi: i 0}.
| , | ,
a | , a | , b | ,
(2) L = {aib2i: i 0}.
| , | ,
a | , a | , b | ,
(3) L = {a2ibi: i 0}.
| , | ,
a | , a | , a | ,
b | ,
(4) L = {a2ib3i: i 1}.
| , | ,
a | , a | , a | ,
b | ,
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(5) L = {aibj: i 6= j, i, j 0}.
a | ,
| , | ,
| ,
| ,
| , b | ,
| ,
a | , a | , a | , b | ,
a | , a | ,
b | , b | ,
(6) L = {aibj: 0 i j 2i}.
| , | ,
a | , a | , a | , a | , b | ,
(7) L = {aibi: i 6= 0 mod 3}.
a | , a | ,
a | ,
a | ,
b | ,
b | , b | ,
| ,
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:
b | 0, b | 1,
| ,
a | , 0a | 0, 10a | 1, 21a | 2, 02
b | 0, b | 1, b | 2,
(8) L = {ab(ab)ib(ba)i: i 0}.
a | , b | , b | , b | , | ,
a | , b | , b | , b | , a | ,
(9) L = {anw : n 0, w (a+ b) |w| n}.
| , | ,
a | , a | , a+ b+ | ,
(10) L = {w (a+ b): |w| }.
| ,
a+ b | , a+ b | ,
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(11) L = {w (a+ b): |w| pi}.
| , | ,
a+ b | , a+ b | ,
(12) L = {w1aw2 (a+ b): |w1| |w2| }.
a | ,
a+ b | , a+ b | ,
a+ b | , a+ b | ,
| ,
(13) L = {aibjci: i, j 0}.
| , | , | ,
a | , a | , b | , c | ,
(14) L = {aibjck: i = j + k}.
| , | , | ,
a | , a | , b | , c | ,
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(15) L = {aibjck: j = i+ k}.
| , | , | ,
a | , a | ,
b | , b | , b | , c | ,
(16) L = {aibjck: k = i+ j}.
| , | , | ,
a | , a | ,
b | , b | , c | ,
(17) L = {a3bici: i 0}.
a | , a | , a | , | , | ,
a | , a | , c | ,
(18) L = {aibjck: i = j j = k}.
| ,
| , | ,
| ,
| ,
| , b | ,
| ,
a | , a | , b | , c | ,
a | , b | , b | ,
c | ,
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(19) L = {aibjck: j < i j < k}.
| ,
| ,
| , | ,
| ,
| ,
| ,
a | ,
a | , a | , b | ,
c | ,
b | , b | ,
c | ,
(20) L = {aibici: i 0}.
| , ; , | , ; , | 1, ; 2,
a | 1, 11; 2, 22a | 1, 11; 2, 22 b | 1, ; 2, 2 c | 1, 1; 2,
(21) L = {aibicidi: i 0}.
| , ; , | , ; , | , ; , | 1, ; 2,
a | 1, 11; 2, 2a | 1, 11; 2, 2
b | 1, ; 2, 22b | 1, ; 2, 22
c | 1, 11; 2, c | 1, 11; 2, d | 1, ; 2, 2
(22) L = {aibjcjdi: i, j 0}.
| , ; , | , ; , | , ; , | 1, ; 2,
a | 1, 11; 2, 2a | 1, 11; 2, 2
b | 1, 1; 2, 22b | 1, 1; 2, 22 c | 1, 1; 2, d | 1, ; 2, 2
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(23) L = {a3ibjcid2j: i, j 0}.
| , ; , | , ; , | , ; , | 1, ; 2,
a3 | 1, 11; 2, 2a3 | 1, 11; 2, 2
b | 1, 1; 2, 222b | 1, 1; 2, 222 c | 1, ; 2, 2 d | 1, 1; 2,
(24) L = {w (a+ b)?: na(w) = nb(w)}.
| ,
a | , aa | a, aaa | b,
b | , bb | b, bbb | a,
(25) L = {w (a+ b)?: na(w) = nb(w) + 1}.
na = nb na = nba | , | ,
(26) L = {w (a+ b)?: na(w) nb(w) + 1}.
na = nb na = nb | , a | , | , | , | ,
a | , a | , a | ,
(27) L = {w (a+ b)?: na(w) < nb(w)}.
na = nb na = nb | , | , | , | , | ,
b | , b | , b | ,
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(28) L = {w (a+ b)?: na(w) = 2nb(w)}.
a | , aa | a, aa
a | , a | a, a
a | b, b | , bbb | b, bbbb | a,
b | , bbb | b, bbbb | a,
a | b, b | , bbb | b, bbbb | a,
a | , a
| ,
| ,
| ,
(29) L = {w (a+ b)?: na(w) = 2nb(w) + 1}.
na = 2nb
na = 2nb
na = 2nb
na = 2nb | ,
a | ,
| ,
a | ,
| ,
| ,
a | ,
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(30) L = {w (a+ b)?: na(w) > 2nb(w)}.
a | , aa | a, aa
a | , a | a, a
a | b, b | , bbb | b, bbbb | a,
b | , bbb | b, bbbb | a,
a | b, b | , bbb | b, bbbb | a,
a | , a | a, | ,
| a, | ,
| a, | ,
| ,
| a,
():
na = 2nb | a, | ,
| ,
| a,
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(31) L = {w (a+ b)?: na(w) = 3nb(w)}.
E1
E2
E3
E4
E5
E4
E6
| ,
| ,
| ,
pi
E1 : a | , aa | a, aa
E2 : a | , a | a, a
E3 : a | , a | a, a | ,
E4 : b | , bbbb | b, bbbbb | a, a | b,
E5 : b | , bbbb | b, bbbbb | a,
E6 : a | , a
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(32) L = {w (a+ b)?: 2na(w) nb(w) 3na(w)}.
E1
E2
E3
E4
E5
E4
E6
| ,
| ,
| ,
pi
E1 : b | , bb | b, bb
E2 : b | , b | b, b
E3 : b | , b | b, b | ,
E6 : b | , b
E4 : b | , bbba | , aaaab | b, bbbba | a, aaaaaa | b, b | a,
E5 : a | , aaaa | , aaaaa | a, aaaaa | a, aaaaa
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(33) L = {w (a+ b): |na(w) nb(w)| = 2}.
a | , aa | a, aaa | b,
b | , bb | b, bbb | a,
a | , aa | a, aaa | b,
b | , bb | b, bbb | a,
a | , aa | a, aaa | b,
b | , bb | b, bbb | a,
a | ,
b | ,
a | ,
b | ,
| ,
(34) L = {w (a+ b+ c)?: na(w) + nb(w) = nc(w)}.
| ,
a | , a | , a | , b | , b | , b | ,
c | , c | , c | ,
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(35) L = {wwR: w (a+ b)?}.
| , | ,
a | , aa | a, aa
b | , bb | b, bb
a | a,
b | b,
(36) L = {wcwR: w (a+ b)?}.
c | , | ,
a | , aa | a, aa
b | , bb | b, bb
a | a,
b | b,
(37) L = {w (a+ b)?: w = wR}.pi.: , w = wR w = zzR w = zzR, pi z (a+ b) pi = a b. pi:
| ,
| , | ,
a | , aa | a, aa
b | , bb | b, bb
a | a,
b | b,
38
