mc problemas3
DESCRIPTION
MC_Problemas3.pdfTRANSCRIPT
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f IK@gJ\BC@=h)i:TS`cjF@kEl`HIQD7@NEGPmX=B]Ec@gWYDFBCD'[email protected]=D'IQDFBV@Z@L<
f IK@gJ\BC@=h)i:TS`cjF@kEl`HIQD7@NEGPmX=B]Ec@Z`lp7o\LK`lD'BCWQ@goULQDqJ=D'IQDFBV@Z@L<
r Es@=LYTSb\hg@NTC@%tKIQ`lTSX#WQDuTSD'BShg`HIQvcROTC`HjFXZhg`HIQ`Hhg@=EsoULQD.@=jFDFPYTV@%D'EGEHDFIQJ\LK@:MODL<
wY< r IKjuX\I\TCBC@=B)Ra`]D7RgP[XURa`HeQEHDxLKIK@yJ=BV@Nh)i:TC`Hj'@zEH`lIKD'@NE]PmX=B)Ec@yWYDFBCD'jVn[@{XyLQIK@yJ\BC@=h)i:TS`cjF@{El`HeQBCD|WYDFEjFX=IUTSDF^_TSXgo\LKD.J=DFIKDFBCD2D'EGElD'IQJ=L[@:MOD
LRaLQPKLQD'RaTSX)oULQD
L ⊂ {a, b, c}∗ }�~ DFBC`�t[j'@Q�
u ∈ LRS` } RSX=Ec@NhgDFIUTCDqRS` ~ D'BS`lt[j'@%oULQD u
IQX)jFX=IUTS`HDFIQD.WYXUR]RavHhkemX=EHX\R�e|jFX=IKRSD'jFLYTS` ~ XURF<
u ∈ LRS` } RSX=Ec@NhgDFIUTCDqRS` ~ D'BS`lt[j'@%oULQD u
jFX=IUTS`HDFIQD.WYXUR]RavHhkemX=EHX\R�e5juX\IKRSD'juLQTS` ~ X\R'<
u ∈ LRS` } RSX=Ec@NhgDFIUTCDqRS` ~ D'BS`lt[j'@%oULQD�jFX=IUTS`HDFIQDqLQI5I_�QhgDFBCXZ`lhgPK@=B�WYD�RavHhkemX=EHX\R]j=<
u ∈ LRS` } RSX=Ec@NhgDFIUTCDqRS` ~ D'BS`lt[j'@%oULQD.IQX)jFX=IUTS`HDFIQDqD'EGhg`HRShgXZI_�QhgD'BSXgWYD�RSvlh%e[X\ElXUR�exoULQD
WYD�RSvlh%e[X\ElXUR�j=<�Q< r IKjuX\I\TCBC@=B�LQI|������hg`lIK`lh)@NEsPK@NBV@kD'EGEHDFIQJ\LK@:MOD
(a + b)∗(aa + bb)(a + b)∗
� <?>A@=BC@gjF@\WQ@gLQIQX+WYD�EHX\R�RS`HJ=LQ`HDFIUTSD7R]EHDFIQJ\LK@:MOD7R]BSD'J=LQEc@NBCD'R d DFI[juX=IUTCBC@=B]DFE�@=LYTSb\h)@:TC@ghg`HIQ`Hhg@=E�@\RaX=�jF`H@\WYX d } @kP[@NBSTS`HB0WYDqWY`cjVnQX%@=LYTSb\hg@NTC@khZ`HIQ`Hh)@NE d WQDuTSD'BShg`HIK@NB�Ec@�J=BV@Nh)iNTS`[email protected]\LQEH@=B0oULQD�J=D'IQDFBV@D'E�ElD'IQJ=L[@:MOD=�
a+b+
a(a + b)∗b
�Y<��0X\IKRS`HWYD'BC@�EH@�J=BV@Nh)i:TC`Hj'@�juL } @=R�PQBCX_WQLKjFjF`lX\IQD'R�RaD0PQBCD'RSDFIUTC@=Ik@�jFX=IUTS`HI_LK@=jF`lb\I } WYX=IKWQD0D'E_RSvHhkemX=EHX`HIQ`cju`c@NE�D7R
S�
S → xN |x
N → yM |y
M → zN |z
r RCjuBC`lemD2D'E�WY`H@=J=BV@Nh)@�WYD2TSBV@NI[Ra`cju`HX=IQD7R!P[@NBV@�@Nex������oULQD�@\juD'PYTSD�DFE�EHDFIQJ\LK@:MODL(G)
J=D'IQDu�BC@\WYXZP[X\B
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r IKjFLQDFIUTSBV@�LQIK@�J=BV@Nh)i:TC`Hj'@zBCDFJ=LKEH@=BkPmX=BgEH@�`Hp'oULQ`HDFBVWQ@{@\o\DxJ=D'IQDFBCD�D7RaDxhg`HRShgX�EHDFIKJ=LK@NMODL(G)
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<�Q<��0X\IKRaTSBCLQ`lB�LQI|�2���3hZ`HIQ`Hh)@NEGPK@=BC@%DFEsElD'[email protected]@=WYXgPmX=B]Ec@ZDu^YPQBCD'RS`lb\I�BCDFJ=LKEH@=B
1+01∗
� <��2eQTSDFIKDFB�@=LYTSb\hg@NTC@\R�tKIQ`lTSXUR�WYDFTSDFBChg`lIKvHRaTS`cjuXUR�PK@=BC@gEHX\R2Ra`HJ=LQ`HDFIUTCD'R]EHDFIQJ\LK@:MOD7R�RSX=eQBCD.DFE�@NE�� @NemDuTCX{0, 1}
<
>A@NEc@NeQBV@=R�DFI5Ec@=R]oULQDqDFEsIU�KhZD'BSXgWYD);qD'R�h%�QElTS`HPQElX#WQD�� } DFEGI_�QhgDFBCX)WYD��gD'R�PK@=B'<
{(01)2i | i ≥ 0}
{(02i12i) | i ≥ 0}
Q<��0X\IKRaTSBCLQ`lB�LQI ��LYTCb=h)@:TV@ ��`lIK`�TCX6�2DuTSD'BShg`HIQvcROTC`HjFX oULQD{@=jFDFPYTCDzDFE2ElD'IQJ=L[@:MODzJ\DFIQD'BC@\WYXyP[X\B�Ec@RS`HJ=LQ`HDFIUTSDqJ\BC@=hgiNTS`cjF@K�
S → AB, A → aA, A → c
B → bBb, B → d
Q<���@NBALKIK@qDu^YPQBCD'RS`lb\I%BCDFJ\LQEc@NB�PK@NBV@�EH@�`HIUTSDFBVRSD'jFjF`lb\IZWQD]ElXURAElD'IQJ=L[@:MOD'RA@\RaXYjF`H@\WYX\R�@�EH@\RADu^YPQBCD'RS`HX=IQD7RBCDFJ\LQEc@NBCD'R
(01 + 1)∗0 } (10 + 0)∗<��_D ~ @NEHX=BV@NBVi�oULQD#RSD)juX\IKROTCBSL } @+DFE!@=LYTSb\hg@NTC@xoULQD#@=jFDFPYTV@
Ec@x`lIUTCDFBVRaD7jFju`Hb=I WYD)D'RaTSXURqElD'IQJ=L[@:MOD'R d RSDZhg`HIQ`HhZ`cjuD } d @5PK@=BaTC`lB�WYD'E BCD'RSLQE�TV@=WYX d RaD#jFX=IKRaTSBCL } @+Ec@DF^YPQBSD7Ra`Hb=I+BSD'J=LQEc@NB7<
; �Q<��0X\IKRaTSBCLQ`lB+LQI ��LYTCb=h)@:TV@ ��`lIK`�TCX �2DuTCDFBChZ`HIQ`cROTV@��z`lIQ`Hh)@NE.oULQDy@=juD'PYTSD DFE�EHDFIQJ\LK@:MOD RaX\eQBSD�DFE@=E�� @=e[DFTSX
{a, b, c}WQD]TSXYWQ@\R!@=oULQD'ElEc@=R!PK@NEc@NeKBC@\RAo\LKD ~ D'BS`lt[oULQD'IgRS`HhkLQElTC@=IQD'@=hZD'IUTSD�Ec@=R0Ra`HJ=LQ`HDFIUTCD'R
jFX=IKWQ`HjF`lX\IQD'R
����� @ZPK@=EH@=eQBC@ZjuJ\IUTS`HDFIQDqLQI5I_�QhgDFBCXgPK@NB�WYD a �R
� ��� @ZEHX=IQJ\`�TCLKW5WYDqEH@gPK@=EH@=eQBV@�D'R�LQI5hk�QElTS`HPQEHXgWQD��Q<� ��� @ZPK@=EH@=eQBC@%IQX)juX\IUTS`HDFIQD.Ec@gRaLQe�jF@\WYDFI[@ abc
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{0ibj | i = 2jb
2i = j}
{uu−1 | u ∈ {0, 1}∗, |u| ≤ 1000}
{uu−1 | u ∈ {0, 1}∗, |u| ≥ 1000}
w
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{0i1j2k | i = jb
j = k}
;8wY<��2DuTCDFBChg`lIK@=B�oULQDqElD'IQJ=LK@NMOD'R�RSX=IxBSD'J=LQEc@NBCD'R�XgEl`HeQBCD'R]WYD�jFX=IUTSDF^_TSXgWYD.ElXUR]Ra`HJ=LK`lD'I\TCD'R'�
��� {u0u−1 | u ∈ {0, 1}∗}
� ��� �QhgD'BSXUR�DFI5eQ`HIK@NBC`lXgoULQD�RSD'@NIxhk�KE�TC`lPQEHX\R]WQD �� � >A@NEc@NeQBV@=R�WYD
{0, 1}∗oULQD.IQX)juX\I\TC`lD'IQDFIxEc@)jF@Ne�jF@\WYDFI[@
0110
;7�Q<��2DuTCDFBChg`lIK@=B�@NLYTCb=h)@:TV@=Rqhg`HIQ`lh)@=ElD7R.PK@[email protected]\LK@:MOD7RL(M1) ∪ L(M2)
} L(M1) ∩ L(M2)WQX=IKWYD d
M1 = ({q0, q1, q2, q3}, {a, b, c}, δ1, q0, {q2})WYX\IKWYD
δ1 q0 q1 q2 q3
a q1 q1 q3 q3
b q2 q1 q1 q3
c q3 q3 q0 q3
M2 = ({q0, q1, q2, q3}, {a, b, c}, δ2, q0, {q2})
δ2 q0 q1 q2 q3
a q1 q1 q3 q3
b q1 q2 q2 q3
c q3 q3 q0 q3
; � <��2DuTCDFBChg`lIK@=B�oUL��ZEHDFIQJ\LK@:MOD7R2RSX=I�BSD'J=LQEc@NBCD'R } o\L��%EHDFIQJ\LK@:MOD7R2RSX=I{El`HeQBCD'RqWYDgjuX=IUTCDu^_TSX+DFIUTSBCD%EHX\RRS`HJ=LQ`HDFIUTSD7RF�
@ � �0X=I:MOLKI\TCX)WYD.PK@NEc@NeKBC@\R0RSX=eQBCD�DFEs@=E�� @=e[DFTSX {0, 1}D'I+Ec@=R]oULQD.j'@=WQ@
1 ~ @%PKBSD7juD'WQ`HWYXZPmX=B]LQII_�QhgDFBCXZP[@NB�WYD.juD'BSXURF<
e � �0X=I:MOLKI\TCX {0i12j0i+j |i, j ≥ 0}
j � �0X=I:MOLKI\TCX {0i1j0i∗j |i, j ≥ 0}
;8�Y<���@=WQXZD'EGjuX\I:MOLQIUTSX)BCDFJ\LQEH@=B�BSD'PQBSD7RaD'IUTC@=WQXkPmX=B]Ec@%DF^YPQBSD7Ra`HX=IxBCDFJ=LKEH@=Ba∗b∗ + b∗a∗ d jFX=IKRaTSBCLQ`HB�LKI
@=LYTSb\h)@:TC@ktKIQ`lTSX#WQDuTSD'BShg`HIQvcROTC`HjFXZhg`HIQ`Hhg@=EsoULQDqElX)@\juDFPQTSD=<; �Q< �YD'@NIxEHX\R�EHDFIQJ\LK@:MOD7RF�
� L1 = (01 + 1)∗00
� L2 = 01(01 + 1)∗
�
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jFX=IKRaTSBCLQ`HB�LQI @=LYTSb\hg@NTC@xtKIK`�TCX�WYDFTSDFBChg`lIKvHRaTS`cjuX|hg`HIQ`lh)@=E�oULQD�@\juD'PYTSD#D'E!EHDFIQJ\LK@:MODL1 − L2 d @P[@NBSTS`HB�WQD�@NLYTCb=h)@:TV@=R�oULQD�@=jFDFPYTCDFI
L1} L2
<; � <���@=WK@%LKIK@ZPK@NEc@NeQBV@
u = a1 . . . an ∈ A∗ d RaDqEHEH@=h)@ Per(u)@NEGjFX=I:MOLQIUTCX
{aσ(1), . . . , aσ(n) : σD'R�LQI[@%PmDFBChkLQTC@=jF`lb\IxWQD
{1, . . . , n}}
<��@=WQXZLKIxEHDFIKJ=LK@NMOD
L d RaDqEHEH@=h)@ Per(L) =⋃
u∈L Per(u)<
��@NB]DF^YPQBSD7Ra`HX=IKD'R�BSD'J=LQEc@NBCD'R } @=LYTSb\hg@NTC@\R�hZ`HIQ`Hh)@NEHD'R�PK@=BC@ Per(L)DFI5EHX\R]RS`lJ\LQ`lD'IUTSD'R]j'@=RSX\R'�
@ � L = (00 + 1)∗
e � L = (0 + 1)∗0
j � L = (01)∗
� r R�P[XURa`HeQEHDqoULQD d RS`lD'IKWYX LBSD'J=LQEc@NB d Per(L)
IQXgElX)RSD'@��; Q<���@=WQX\R�ElXUR]@NE�� @NemDuTCX\R
A = {0, 1, 2, 3} } B = {0, 1} } DFE�nQX\hgX=hgX=BSt[RShZX fWYD
A∗D'I
B∗WQ@=WQX
PmX=B7�
f(0) = 00, f(1) = 01, f(2) = 10, f(3) = 11
�YD'@LDFE�juX\I:MOLQIUTSXZWYD�EH@\R!PK@=EH@=eQBV@=R!WQD
B∗D'I�Ec@=R�oULQD�D'EmI_�QhgD'BSXZWYD�RSvHhkemX=EHX\R �%D'R0PK@=B } DFE�WYD
RSvHhkemX=EHX\R.;�IKX�D'R�hk�QElTS`HPQEHX+WYD%�Q<G�0X=IKRaTSBCLQ`HB�LQI{@=LYTSb\hg@NTC@ZtKIK`�TCX+WYDFTSDFBChg`lIK`HRaTC@#oULQDZ@=jFDFPYTCD�DFEEHDFIKJ=LK@NMOD
f−1(L)<
; Q<��2DuTCDFBChg`lIK@=B]LQIK@g@NLQTSb=h)@NTC@%tKIQ`lTSX#WYDFTSD'BShg`HIQvHRaTS`cjuXghg`HIQ`lh)@=EGPK@NBV@ZDFEGD'EGEHDFIQJ\LK@:MOD.RSX=eQBCD2D'Es@NE�� @:�emDuTCX
A = {a, b, c}WQ@=WYXgPmX=B]Ec@ZDu^YPQBCD'RS`lb\I+BCDFJ\LQEH@=B
b(a + b)∗ + cb∗<
w �Q<���@NB]DF^YPQBSD7Ra`HX=IKD'R�BSD'J=LQEc@NBCD'R0PK@=BC@%ElXUR]Ra`HJ=LK`lD'I\TCD'R�EHDFIQJ\LK@:MOD7R�RaX\eQBSDqD'EG@NE�� @NemDuTSXA = {a, b, c} d
>A@NEc@NeQBV@=R�DFI5Ec@=R]oULQD.jF@\WQ@c ~ @gPQBCD'jFD'WY`[email protected]@ a
XgLQI[@b
>A@NEc@NeQBV@=R�WYD.EHX=IQJ\`�TCLKW+`lhgPK@=B>A@NEc@NeQBV@=R�WYD.EHX=IQJ\`�TCLKW+`lhgPK@=B]DFI5EH@\R]oULQDqDFEsRSvlh%e[X\ElX)jFDFIUTSBV@NE�D'R�LKIK@
c
>A@NEc@NeQBV@=R�DFI EH@\R2oULQDZElXUR2WQX\R�PQBC`lhgDFBCX+RSvHhkemX=EHX\R�RaX\Iz`lJ\LK@NEHD'R�@+ElXUR2WYXUR��QE�TC`lhgX5RavHhkemX=EHX\RDFI5X=BVWYD'I+`HI ~ DFBVRSXK�_Ra`GEc@ZPK@NEc@NeKBC@%DFhgPQ`HDFp7@%PmX=B ab d WQDFemDqWQDqTSDFBChg`lI[@NB]PmX=B ba
<wQ;=<��2DuTCDFBChg`lIK@=B�Ra`�Ec@=R�DF^YPQBSD7Ra`HX=IKD'R�BSD'J=LQEc@NBCD'R�RS`lJ\LQ`lD'IUTSD'R�BCDFPKBSD7RaD'I\TV@NI+DFE�hg`HRShgX)ElD'IQJ=L[@:MOD=�
��� (b + (c + a)a∗(b + c))∗(c + a)a∗
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� � b∗(c + a)((b + c)b∗(c + a))∗a∗
� � b∗(c + a)(a∗(b + c)b∗(c + a))∗a∗
�\LKRaTS`lt[jF@=B�EH@ZBCD'RSPQLQD'RaTC@K<w\wY<��0X\IKRaTSBCLQ`lBkLQI @NLYTCb=h)@:TV@xtKIQ`lTSX WYDuTCDFBChZ`HIQ`cROTV@5hg`lIQ`Hh)@NE�oULQD�@=jFDFPYTCD#DFE�juX\I:MOLQIUTSX{WQD#PK@NEc@NeKBC@\R
RSX=eKBSD�DFEs@=E�� @=e[DFTSXA = {0, 1}
oULQD.BCDFPQBCD'RSDFIUTSD'I�I_�QhgDFBCX\R�IQX#WQ` ~ `cRa`HeQEHD'R�PmX=B�WYXUR�IQ`GPmX=B�TSBCD'R'<w=�Q<��2DuTCDFBChg`lIK@=B]LQIK@ZDu^YPQBCD'RS`Hb=I�BCDFJ\LQEc@NB�PK@NBV@%ElXUR]Ra`HJ=LQ`HDFIUTCD'R�EHDFIQJ\LK@:MOD7R�RSX=eKBSDqDFEG@=E�� @=e[DFTSX
{0, 1}�
>A@NEc@NeQBV@=R�DFI5Ec@=R]oULQDqDFEGTSDFBVjuD'B]RavHhkemX=EHXgD'R�LQI �Q<>A@NEc@NeQBV@=R�DFI5Ec@=R]oULQDqDFE�@NIUTSD'P[D'IQE�TC`lhgX)RSvlh%e[X\ElXgD'R�LKI ;\<
�0X\IKRaTSBCLQ`lB�LQI|@=LYTSb\hg@NTC@ktKIQ`lTSX)hg`HIQ`lh)@=EsoULQD.@=juD'PYTSD.Ec@Z`lIUTCDFBVRaD7jFju`Hb=IxWYD�@Nh%e[XUR�ElD'IQJ=LK@NMOD'R'<w � <��0X\IKRaTSBCLQ`lB|@=LYTSb\hg@NTC@\R#tKIK`�TCX\R|hg`HIQ`lh)@=ElD7R5PK@NBV@ EHX\R|RS`lJ\LQ`HDFIUTSD7R+EHDFIKJ=LK@NMOD'R5RaX\eQBCD�D'Ek@NE�� @NemDuTCX
{0, 1}�
��� >A@NEc@NeQBV@=R�oULQD.juX\I\TC`lD'IQDFI|jFX=hgXZRSLQe�jF@\WYDFIK@ZLQI[@gPK@NEc@NeQBV@ZWYDFEGjFX=I:MOLQIUTCX{00, 11}2 <
� � >A@NEc@NeQBV@=R�oULQD.juX\I\TC`lD'IQDFI|jFX=hgXZRSLQe�jF@\WYDFIK@ZLQI[@gPK@NEc@NeQBV@ZWYDFEGjFX=I:MOLQIUTCX{0011, 1100}
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w\�Y< �YD'@ ��D'Es@NLQTSb=h)@NTC@%tKIQ`lTSX)WYDFTSDFBChg`lIK`HRaTC@K�
q0
q2q
1
q4
q3
q5
a
bb
b
b
aa
a
a
a,b
b
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w �Q< r IKjuLQD'IUTSBV@�PK@=BC@�jF@\WQ@�LQIQX.WYD�EHX\R�Ra`HJ=LK`lD'I\TCD'R�EHDFIQJ\LK@:MOD7R�LQIK@�J=BV@Nh)i:TC`Hj'@�WYD�TC`lPmX��2oULQD]EHX�J=DFIKDFBCDXgLQI|@=LYTSb\hg@NTC@ktKIQ`lTSX#oULQDqEHX)BSD7juX=IKX=p'j'@Q�
��� L = {u ∈ {a, b}∗ : uIKXgjFX=IUTS`HDFIKD�Ec@gRaLKemj'@=WYD'IK@ ′abab′}
� � L = {anbmcp : n ≥ 0 } hgElTS`HPQEHX#WYD 3, m ≥ 0, p > 0}
� � L = {(ab)j(cd)i/j ≥ i ≥ 0}
w � < �0X=IKRaTSBCL } D2LQI[@gJ=BV@Nh)i:TC`Hj'@�BCDFJ=LKEH@=B�oULQD.J\DFIQD'BSD�DFE�Ra`HJ=LQ`HDFIUTCD�EHDFIKJ=LK@NMOD=�
L1 = {u ∈ {0, 1}∗ |DFE�I_�QhgD'BSX)WYD
1 �R } DFEsIU�KhZD'BSXgWYD 0 �
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}
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L1 = {u ∈ 0, 1∗ : uIQX)juX\IUTS`HDFIQDqEc@ZRSLQe�jF@\WYDFIK@
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L2 = {0i1j0k : i ≥ 1, k ≥ 0, i`HhZP[@NB
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