,-.&#/()(&)012#.&%34()5()6#)678%34( 5(0)$’15%9#&0

(1)

!"#$%&'()*+

,-.&#/()(&)012#.&%34()5()6#)678%34(

5(0)$'15%9#&0

+: ;()6#.8#8( L

⁺

!"#$%&$'()*(+,$-(,./0+1&,$0&1$+#-2,&#3&1$4)+$#()1$+#52,&11&#56 #()1$%&*(#1$%7/8(,%$20/,9+,$0/$1:#5/;&

%&$#(5,&$0/#9)&$&5$%2"#+,$0&$0/#9/9&$-(,.&0$%&$0/$0(9+4)&$%&1$',2%+3/51< =/#1$0/$3+#4)+>.&$0&?(#$@''<AB

&5$1)+*/#51C6 #()1$/*(#1$%2"#+$0&$0/#9/9&L4)+$3(#1+15&$&#$%&1$-(,.)0&1$',('(1+5+(##&00&1$3(.'(12&1

%&$'D,/1&1$1+.'0&1$&5$%&$3(##&35&),1$',('(1+5+(##&01< E()1$20/,9+11(#1$./+#5&#/#5$#(5,&$/0'D/8&5$&5 /%('5(#1$)#&$#()*&00&$%2"#+5+(#6 '0)1$0/,9&6 %&$3&$4)7&15$)#&$-(,.)0&$8+&#F-(,.2& G

<1=.%&%7. >?: !"#$%&#'()*+,*$#-.#./L⁺+(*$#*$0.12,(*+(3*%45+16#)3*60-313)(*(-*$(3*31.-(3*3,17#-)3 8 9: +(3*31.-(3*$0.12,(3 8

;#< $(3*60--(6)(,43*=¬. . .>*;= -(?%#3 ><@ =. . .∧ · · ·>*;= (A ><@ =. . .∨ · · ·>*;*= 0, ><@ =. . .→ · · ·>*;= 31?#$043<

() * =. . .↔ · · ·>*;= 331: >< B

;'< $(3*2,#-)1C6#)(,43*=∀x(. . . x· · ·)>*;= %0,4*)0,Ax><*()*=∃x(. . . x· · ·)>*;*= 1$*D*#*#,*E01-3*,Fx>< B

;6< $( * 31.-( * + "1+(-)1)G 8 =. . .!· · ·>*;= (3) * 1+(-)12,( * H >< B

;+< +(3*7#41#'$(3*%0,4*+(3*1-+171+,3 8 =xi>*%0,4*)0,Ai∈NB I: +(3*31.-(3*-0-?$0.12,(3 8

;#< +(3*31.-(3*%0,4*$(3*4($#)10-3 8 =Ri>*%0,4*)0,Ai∈@B

;'< +(3*31.-(3*%0,4*$(3*J0-6)10-3 8 =fi>*%0,4*)0,Ai∈AB

;6< +(3*60-3)#-)(3*%0,4*+(3*1-+171+,3 8 =ci>*%0,4*)0,Ai∈BB K: +(3*31.-(3*#,L1$1#14(3 8 %#4(-)&M3(3@ N14.,$(3

E()1$/''&0(#1$0&1$1+9#&1$H∀I$&5$H∃I$%&1= 2,#-)1C6#)(,43 >6 0&1$&;',&11+(#1$%&$0/$.J.&$-(,.&$4)&$H∀xI

&5$H∃xI$%&1= 2,#-)1C6#)10-3*+(3*7#41#'$(3 >&5$0&1$'D,/1&1$%&$0/$-(,.&$H∀x(. . . x· · ·)I$&5$H∃x(. . . x· · ·)I

%&1= %&4#3(3*2,#-)1C5(3 >< E()1$23,+*(#1$H!I$'(),$0&$1+9#&$%&$0/$,&0/5+(#$%7+%&#5+52$%/#1$0/$0/#9/9&$(8K&5 '(),$0&$%+15+#9)&,$%&$H=I$4)+$,&',21&#5&$0/$,&0/5+(#$%7+%&#5+52$%/#1$0&$.25/0/#9/9&< E()1$/8,29&(#1 H¬(. . . !. . .)I$'/,$H. . . (!. . . I$&5$/%('5(#1$0&1$.J.&1$3(#*&#5+(#1$%723(#(.+&$%&$'/,&#5D>1&1$4)&

'(),$0/$0(9+4)&$',('(1+5+(##&00&<

L&1$-(#35+(#16 3(..&$#()1$07/*(#1$*)$M$0/$'<NAO%25&,.+#&#5$)#$(8K&5$1),$0/$8/1&$%&$0&),1$/,9).&#51< L/

,2-2,&#3&$%&1$&;',&11+(#1$3(..&$H 0&$'>,&$%&$P&/# I$&15$/0(,1$25/80+&$%&$./#+>,&$)#+*(4)& G &00&$1&,/$0/

*/0&),$'(),$0/$-(#35+(#$H 0&$'>,&$%&xI$/''0+4)2&$M$P&/#< L&1$-(#35+(#1$%(#5$#()1$'/,0(#1$&#L⁺1(#5$%&1

(2)

*CD *+: ,-.&#/()(&)012#.&%34()5()6#)678%34()5(0)$'15%9#&0

-(#35+(#1$%7+#%+*+%)16 37&15FMF%+,&$%&1$,&0/5+(#1$&#5,&$%&1$202.&#51$%&$#(5,&$%(./+#&$%&$4)/#5+"3/5+(#<

=/#1$0&1$0/#9)&1$#/5),&00&16 %&$5&00&1$-(#35+(#1$1(#5$&;',+.2&1$'/,$%&1$&;',&11+(#1$3(..&$H 0&$'>,&$%&

<<< I6 H 0&$0+*,&$',2-2,2$%&$<<< I6 H 07+#52,J5$3(..)#$%&$<<<&5· · · I$&5$H 0&$3&#5,&$%)$3&,30&$+#13,+5$%/#1$0&

5,+/#90&$%252,.+#2$'/,$0&1$'(+#51$<<<6· · · &5. . .I< Q0$17/9+5$%&$R ,>90&1$%&$3(#15,)35+(#$S '(),$0&1$5&,.&1 1+#9)0+&,1$3(.'0&;&1< T(#5,/+,&.&#5$M$8&/)3()'$%&$-(#35+(#1$&;',+.2&1$%/#1$0&1$0/#9)&1$#/5),&00&16

#()1$1)''(1(#1$4)&$#(1$-(#35+(#1$%7+#%+*+%)$1(+&#5$3&$4)&$07(#$/''&00&= (-)1M4(E(-) * +5C-1(3 >6 37&15FMF

%+,&$4)7&00&1$(#5$)#&$*/0&),$'(),)0,A/,9).&#5<

E(5(#1$4)&L⁺3(#5+&#5L6 %)$.(+#1$1+$#()1$#()1$+./9+#(#1$5()5&1$0&1$'D,/1&1$1+.'0&1$%&L3(.'(12&1

%7)#$',2%+3/5$@,&0/5+(#$)#/+,&C$&5$%7)#&$3(#15/#5&$+#%+*+%)&00&$HF aI G 3&3+$*&)5$%+,&$4)7)#&$-(,.)0&

8+&#F-(,.2&$%&L1&,/$/)11+$)#&$-(,.)0&$8+&#F-(,.2&$%&L⁺<^N

L&1$1+9#&1$#(#F0(9+4)&1$%&$#(5,&$/0'D/8&5$K()&#5$0&$,U0&$%&1$'D,/1&1$',+.+5+*&1$%&$0/$0(9+4)&$',('(1+F 5+(##&00& G 3(..&$HpI6 HqI$&53< @()$Hp1I6 Hp2I6 Hp3I$&53<C 1(#5$%&1$/8,2*+/5+(#1$/,8+5,/+,&1$'(),$0&1 'D,/1&1$%&$#(5,&$0/#9/9&$(8K&5$4)&$#()1$3(#1+%2,(#1$3(..&$0(9+4)&.&#5$1+.'0&6 0&1$1+9#&1$HRI6 HfI

&5$HcI$@()$HR1I6 HR2I6 HR3I$V6 Hf1I6 Hf2I6 Hf3I6 VHc1I6 Hc2I6 Hc3I$VC$#()1$1&,*&#5$'(),$/8,29&,

#7+.'(,5&$4)&0$1+9#&$%&$,&0/5+(#6 %&$-(#35+(#$&5$#7+.'(,5&$4)&00&$3(#15/#5&< T(..&$%/#1$0&$3/1$%&

0/$0(9+4)&$',('(1+5+(##&00&6 (#$'&)5$5()K(),1$17+./9+#&,$3&1$1+9#&1$,&.'0/321$'/,$%&1$&;',&11+(#1$%) -,/#?/+1$@%)$5:'&$1:#5/;+4)&$3(,,&1'(#%/#5C< W(),$0&1$',2%+3/51$)#/+,&16 #()1$)5+0+1(#1$1()*&#5$HFI6 HGI6 HHI$&53<

L&1$&#1&.80&1@,A &5 B 1(#5$%&1$&#1&.80&1$%7+#%+3&1$4)+$#()1$1&,*&#5$M$%+15+#9)&,$0&1$%+X2,&#51 1:.8(0&1$'(),$0&1$,&0/5+(#16 0&1$-(#35+(#1$&5$0&1$+#%+*+%)1< E()1$'(),,+(#1$'/,$&;&.'0&$2#).&,$0&1 1+9#&1$%&$,&0/5+(#1$'/,$0&1$#(.8,&11,4,7,10,14&53<6 0&1$1+9#&1$%&$-(#35+(#1$'/,2,5,8,11,15&53< &5 0&1$3(#15/#5&1$+#%+*+%)&00&1$'/,3,6,9,12,16&53< !)$0+&)$%&$Hx1I6 Hx2I6 Hx3I6 #()1$23,+*(#1$'/,-(+1 HxI6 HyI6 HzI$'(),$0&1$*/,+/80&1< E()1$/8,29&(#1$07&#1&.80&$%&$5()5&1$0&1$*/,+/80&1$%&$0/$0/#9)&L⁺'/, HQ]g(L⁺)I<

L&1$,&0/5+(#1$&5$0&1$-(#35+(#1$'&)*&#5$J5,&$)#/+,&16 8+#/+,&16 5&,5+/+,&1$&5$/+#1+$%&$1)+5&< Y00&1$'&)*&#5 ',&#%,&$)#6 %&);6 5,(+1$()$'0)1+&),1$/,9).&#5$'(),$-(,.&,$)#&$'D,/1&6 37&15FMF%+,&$/*(+,$)#&$/%+3+52

%&$N6 Z6 [$&53< T7&15$'(),4)(+$#()1$/''&0(#1$)#&= $#-.,/ >L⁺)#$/0'D/8&5$\$+#30)/#5$)#$3D(+;$',23+1

%&$1+9#&1$#(#F0(9+4)&1$\$/*&3$)#$&#1&.80&B%7+#%+3&1$'(),$0&1$3(#15/#5&1$&5$%&);$-(#35+(#1λ:@→ N\ {0}&5µ :A → N\ {0}4)+$%25&,.+#&#5$07/%+3+52$%&$#(1$1+9#&1$%&$,&0/5+(#1$&5$%&$-(#35+(#1<^Z]+

F&15$)#$',2%+3/5$@)#&$,&0/5+(#$)#/+,&C6 '/,$&;&.'0&6λ(F) = 1^ 1+R&15$)#&$,&0/5+(#$8+#/+,&6λ(R) = 2^ 1++&15$)#&$-(#35+(#$8+#/+,&6 /0(,1µ(+) = 2&53< _#&$0/#9)&)L⁺,B, λ, µ*'(),$0/$0(9+4)&$%&1 ',2%+3/51$&15$%(#3$3(.'(12&$%&$3(##&35&),16 %&$4)/#5+"3/5&),16 %)$1+9#&$%7+%&#5+526 %7)#&$+#"#+52$%&

*/,+/80&1$+#%+*+%)&00&16 %7)#$3&,5/+#$#(.8,&$%&$,&0/5+(#1$&5$%&$-(#35+(#1$/:/#5$%&1$/%+3+521$1'23+"4)&1

@%25&,.+#2&1$'/,λ&5µC$&5$%7)#$3&,5/+#$#(.8,&$%&$3(#15/#5&1$+#%+*+%)&00&1<

E()1$'()*(#1$./+#5&#/#5$%2"#+,$3&$4)7&15$)#)(4E/@1+#9)0+&,C$%&$#(5,&$0/#9)&L⁺G

<1=.%&%7. >>@`&,.&1C: !(3*)(4E(3*+",-(*$#-.,/)L⁺,B, λ, µ*30-)*+5C-13*%#4*$(3*6$#,3(3*456,4317(3 8 9: O0,)( * 7#41#'$( * =xi>*;i∈N< * (3) * ,- * )(4E(:

I: O0,)( * 60-3)#-)( * =ci>*;i∈N< * (3) * ,- * )(4E(:

K: P1*=t1>@ =t2>@ Q @ =tµ(j)>*30-)*+(3*)(4E(3(j∈A)@ #$043*=fj(t1, t2, . . . , tµ(j))> * (3) * ,- * )(4E(:

R: S1(- * + "#,)4( * -"(3) * ,- * )(4E(:

N< _#&$/)5,&$',(*+1+(#$&15$#23&11/+,& G 0&1$3(##&35&),1$',('(1+5+(##&016 %2"#+1$%/#1$0&$0/#9/9&$%&$0/$0(9+4)&$',('(1+5+(##&00&

3(..&$,&0+/#5$%&1$'D,/1&16 ,&0+&#5$%/#1$0/$0(9+4)&$%&1$',2%+3/51$%&1$'D,/1&1$4)+$'&)*&#5$J5,&$3(.'0>5&10,()*&,5&1< T&3+$*&)5

%+,&$4)&$0&1$3(##&35&),1$%/#1L1(#5$5()K(),1$%)$5:'&,/,,@3-< '<NaB%/#1$0&$3D<a<bC6 8+&#$4)&$3&);$%&L⁺1(+&#5$/.8+9)c1

&#5,&,/,,&5(,/E)/(,/E)(,/E)<

Z< E()1$&;0)(#1$)#&$/%+3+52$%&0< !05&,#/5+*&.&#56 #()1$/),+(#1$')$%2"#+,$0&1$3(#15/#5&1$+#%+*+%)&00&1$3(..&$%&1$-(#35+(#1

%7/%+3+520<

(3)

*+:* ;()6#.8#8(L⁺ *CF

W)+14)7&00&$)5+0+1&$0&$',2%+3/5$M$%2"#+,$@H V &15$)#$5&,.& IC6 3&55&$%2"#+5+(#$&15$,23),1+*&6 3(..&$0&$1(#5 /)11+$0&1$%2"#+5+(#1$%&$-(,.)0&$/5(.+4)&$&5$%&$-(,.)0&$4)&$#()1$%(##&,(#1$'/,$0/$1)+5&<^[L&1$5&,.&1 1(#5$0&1$&;',&11+(#1$%&$#(5,&$0/#9/9&$4)+$#()1$1&,*&#5$'(),$%21+9#&,$0&1$(8K&51$%)$%(./+#&$%&$%+13(),1<

E()1$%2"#+11(#1$29/0&.&#5$0&1$-(,.)0&1$/5(.+4)&16 4)+$K()&#5$0&$,U0&$%&1$'D,/1&1$1+.'0&1$%&$0/$0(9+4)&

',('(1+5+(##&00& G

<1=.%&%7. >G@d(,.)0&1$/5(.+4)&1C: φ(3) * ,-( * J04E,$( * #)0E12,( * +( * $# * $#-.,/)L⁺,B, λ, µ*31*()*3(,$(?

E(-)*31*,-*+(3*3,17#-)3*(3)*$(*6#3 8

9: φ(3) * +( * $# * J04E( * =t1!t2>*%0,4*+(,L*)(4E(3*=t1>*()*=t2> B

I: φ(3) * +( * $# * J04E( * =Ri(t1, t2, . . . , t_λ(i))>*%0,4*+(3*)(4E(3*=t1>@ =t2>@ Q @ =t_λ(i)>*(Ai∈@:

!:/#5$M$#(5,&$%+1'(1+5+(#$0724)+*/0&#5$%&$'D,/1&1$1+.'0&16 #()1$'()*(#1$3(#15,)+,&$0&1$-(,.)0&1$3(.F '0&;&16 &#$+.+5/#5$0/$',(32%),&$'(),$0&1$3(##&35&),1$%&$0/$0(9+4)&$',('(1+5+(##&00& G

<1=.%&%7. >D@d(,.)0&1C: φ(3) * ,-( * J04E,$( * +( * $# * $#-.,/)L⁺,B, λ, µ*31*()*3(,$(E(-)*31*,-*+(3*3,17#-)3 () * $( * 6#3 8

9: φ(3) * ,-( * J04E,$( * #)0E12,/ B

I: φ(3) * +( * $# * J04E/"(¬ψ)#%0,4 * ,-( * J04E,$/ψB

K: φ(3) * +( * $# * J04E/"(ψ∧χ)#@"(ψ∨χ)#@"(ψ→χ)#0,"(ψ↔χ)#@ %0,4*+(3*J04E,$(3ψ(AχB R: φ(3) * +( * $# * J04E/"∀xi(ψ)#0,"∃xi(ψ)#%0,4 * ,-( * J04E,$/ψ() * ,-( * 7#41#'$( * =xi>@i∈N:

E()1$23(#(.+1(#1$%&1$'/,&#5D>1&1$'/,$0&1$.J.&1$3(#*&#5+(#1$4)&$%/#1$0&$3/1$%&$0/$0(9+4)&$',(F '(1+5+(##&00&< Y#$'0)16 #()1$23,+*(#1$H∀x, y, z(. . .)I$/)$0+&)$%&$H∀x(. . .∀y (. . . ∀z(. . .). . .). . .)I$&5 H∃x, y, z(. . .)I$/)$0+&)$%&$H∃x(. . . ∃y(. . .∃z(. . .). . .). . .)I< Q0$&15$M$,&./,4)&,$4)&$#()1$3(#1+%2,(#1 3(..&$8+&#F-(,.2&1$%&1$4)/#5+"3/5+(#1$*+%&1$\$%&1$'D,/1&1$3(#5&#/#5$%&1$4)/#5+"3/5&),1$4)+$'(,5&#5 1),$%&1$'D,/1&1$&#5+>,&1$3(..&$H∀x(F a)I$&5$H∃y∃x(F x)I<

T(..&$#()1$07/*(#1$*)$%/#1$0&$3D<a<[6 +0$&15$3,)3+/0$'(),$0/$12./#5+4)&$%&$0/$0(9+4)&$%&1$',2%+3/51

%&$%+15+#9)&,$&#5,&$0&1$'D,/1&1$3(.'0>5&1$4)+$1(#5$*,/+&1$()$-/)11&1$&5$%&1$'D,/1&1$()*&,5&1$4)+$1(#5

*,/+&1$()$-/)11&1+/3&,5/+#1$(8K&51< T&55&$%+15+#35+(#$#()1$(80+9&$%&$%+,&$4)/#%$)#&$*/,+/80&$/$)#&

(33),,&#3&$H 0+8,& I$%/#1$)#&$-(,.)0&< E()1$/%('5(#1$%(#3$0/$%2"#+5+(#$1)+*/#5&6 4)+$#()1$-(),#+5$)#&

./#+>,&$'0)1$,+9(),&)1&$%&$'/,0&,$%&$3&$4)&$#()1$/*(#1$/''&026 %&$./#+>,&$.25/'D(,+4)&6 )#$H 5,() I

%/#1$)#&$'D,/1&$()*&,5& G

<1=.%&%7. >F@e33),,&#3&$0+8,&C: P1φ(3) * ,-( * J04E,$( * () * =xi> * ,-( * 7#41#'$(@ -0,3*+130-3*2,(*=xi>*#

,-( * 066,44(-6( * $1'4( * +#-3φ31*()*3(,$(E(-)*31*,-(*+(3*60-+1)10-3*3,17#-)(3*(3)*4(E%$1/ 8 9: φ(3) * ,-( * J04E,$( * #)0E12,( * () * 60-)1(-) * =xi> B

I: φ#*$#*J04E/"¬ψ#() * =xi>*#*,-( *066,44(-6( *$1'4(*+#-3ψB

K: φ#*$#*J04E/"ψ∧χ#@"ψ∨χ#@"ψ →χ#0,"ψ↔χ#() * =xi> * # * ,-( * 066,44(-6( * $1'4( * 301) * +#-3ψ@

301) * +#-3χB

[< T&3+$1+9#+"&$4)&$#(5,&$%2"#+5+(#$/$0/$-(,.&$1)+*/#5& G

@+C `()5&$*/,+/80&$&15$)#$5&,.&<

@++C `()5&$3(#15/#5&$&15$)#$5&,.&<

@+++C `()5$#(.$'(),$0/$*/0&),$%7)#&$-(#35+(#$'(),$)#$,2-2,&#5$%7)#&$()$%&$'0)1+&),1$&;',&11+(#1$%&1$3/529(,+&1$N$&5$Z$&15$)#

5&,.&<

@+*C `()5$#(.$'(),$0/$*/0&),$%7)#&$-(#35+(#$'(),$)#$,2-2,&#5$%7)#&$()$%&$'0)1+&),1$&;',&11+(#1$%&1$3/529(,+&1$N6 Z$&5$[$&15$)#

5&,.&<

@*C `()5$#(.$'(),$0/$*/0&),$%7)#&$-(#35+(#$'(),$)#$,2-2,&#5$%7)#&$()$%&$'0)1+&),1$&;',&11+(#1$%&1$3/529(,+&1$N6 Z6 [$&5$O$&15$)#

5&,.&<

@*+C V

L/$,23),1+(#$%/#1$0/$30/)1&$@[C$%&$0/$%2"#+5+(#$%(##2&$#()1$'&,.&5$%7/8,29&,$0&1$3(#%+5+(#1$@[C6 @OC6 @fC$&53< %&$3&55&$%2"#+5+(#

+52,/5+*&$@37&15FMF%+,&$#(#F,23),1+*&C6 4)+$/$0&$%2-/)5$@3(#1+%2,/80& gC %7J5,&$%7)#&$0(#9)&),$+#"#+&<

(4)

*CH *+: ,-.&#/()(&)012#.&%34()5()6#)678%34()5(0)$'15%9#&0

R: φ#*$#*J04E/"∀xj(ψ)#0,"∃xj(ψ)#@i(=j() * =xi>*#*,-(*066,44(-6(*$1'4(*+#-3ψ:

W/,$0/$%&,#+>,&$30/)1&6 0&1$4)/#5+"3/5&),1$,2%)+1&#5$0&$#(.8,&$%7(33),,&#3&1$0+8,&1$%&1$*/,+/80&1$4)7+01 4)/#5+"&#5$%/#1$0&),$'(,52&<

_#&$*/,+/80&$/$)#&$(33),,&#3&$0+8,&$%/#1$)#&$-(,.)0&$1+$&00&$#7:$&15$'/1$'/,5()5$9()*&,#2&$'/,$)#

4)/#5+"3/5&),$3(,,&1'(#%/#5< E()1$%+1(#1$4)&$0&1$(33),,&#3&1$%&$*/,+/80&1$4)+$#&$1(#5$'/1$0+8,&1$1(#5 H 0+2&1 I$\$0+2&1$'/,$)#$4)/#5+"3/5&),$%/#1$0/$'(,52&$%)4)&0$&00&1$1&$5,()*&#5<^OT(#1+%2,(#1$'/,$&;&.'0&

0/$-(,.)0&$1)+*/#5&$%/#1$0&$0/#9/9&$%&$07/,+5D.25+4)& G

∀x(x≤0→x≤y)∧ ∃z(0≤z∧z≤x∧x(!0∧z(!x)

=/#1$3&55&$-(,.)0&6 0/$*/,+/80&$HxI$/$M$0/$-(+1$%&1$(33),,&#3&1$0+8,&1$&5$%&1$(33),,&#3&1$0+2&16 0/$*/,+/80&

HyI$#7/$4)&$%&1$(33),,&#3&1$0+8,&1$&5$HzI$#7/$4)&$%&1$(33),,&#3&1$0+2&1<

_#&$&;',&11+(#$&15$)#&$'D,/1&$()*&,5&$1+$&5$1&)0&.&#5$1+$&00&$3(#5+&#5$/)$.(+#1$)#&$(33),,&#3&$0+8,&

%7)#&$*/,+/80&< W/,$3(#124)&#56 #()1$'()*(#1$%2"#+, G

<1=.%&%7. >H@WD,/1&1C: T-( * %&4#3( * (3) * ,-( * J04E,$( * 2,1 * -( * 60-)1(-) * #,6,-( * 066,44(-6( * $1'4( * + ",-/

N#41#'$(:

E()1$/8,29&(#1$'/,$HφI6 HψI$&53< %&1$#(.1$'(),$%&1$'D,/1&1$/,8+5,/+,&1$&5$'/,$H"φ(x)#I6 H"ψ(x, y)#I

%&1$#(.1$'(),$%&1$'D,/1&1$()*&,5&1$3(#5&#/#5$%&1$(33),,&#3&1$0+8,&1$%&1$*/,+/80&1$HxI$&5$%&1$*/,+/80&1 HxI$&5$HyI$,&1'&35+*&.&#5<^fL&1$,21)05/51$%&$0/$1)815+5)5+(#$%&1$3(#15/#5&1$+#%+*+%)&00&1$HaI$&5$HbI '(),$0&1$*/,+/80&1$1(#5$%21+9#21$'/,$H"φ(a)#I$&5$H"ψ(a, b)#I$,&1'&35+*&.&#5<

T(..&$0&1$3(##&35&),16 0&1$4)/#5+"3/5&),1$(#5$/)11+$)#&%04)5/\$0/$'D,/1&$()*&,5&$4)+$&15$9()*&,#2&

'/,$0&$4)/#5+"3/5&),< L/$%+15+#35+(#$&#5,&

@+C W&,1(##&$#7&15$D&),&);$&5$'&,1(##&$#7&15$D(##J5&<

@++C W&,1(##&$#7&15$D&),&);$&5$D(##J5&<

&15$4)&$0&$',&.+&,$4)/#5+"3/5&),$)#+*&,1&0$%/#1$0/$',&.+>,&$'D,/1&$#&$9()*&,#&$4)&$0/$'D,/1&$()*&,5&

1+.'0&$Hx&15$D&),&); I$8+&#$4)&$3&0)+$%/#1$0/$%&);+>.&$'D,/1&$0&$'D,/1&$()*&,5&$3(.'0&;&$Hx&15 D&),&);$&5x&15$D(##J5& I G

@+^!C ∀x¬(x&15$D&),&);) ∧ ∀x¬(x&15$D(##J5&)

@++^!C ∀x¬(x&15$D&),&); ∧ xD(##J5&)

W/,$0/$',&.+>,&$'D,/1&6 K&$-/+1$%&);$/11&,5+(#1 G 4)&$'&,1(##&$#7&15$D&),&);$&5$4)&$'&,1(##&$#7&15$D(#F

#J5&$\$17+0$:$/$4)&04)&$3D(1&$4)+$&15$D&),&);$()$17+0$:$/$4)&04)&$3D(1&$4)+$&15$D(##J5&$K7/+$29/0&.&#5 5(,5< W/,$0/$%&);+>.&$'D,/1&6 3&'&#%/#56 K&$-/+1$)#&$/11&,5+(#$8&/)3()'$'0)1$-/+80& G K&$%+1$4)&$'&,1(##&

#7&15$D&),&);$&5$D(##J5&(- * EUE( * )(E%3\$4)&$5()1$3&$4)+$1(#5$D&),&);$#&$1(#5$'/1$D(##J5&1$&5$4)&

5()1$3&$4)+$1(#5$D(##J5&1$#&$1(#5$'/1$D&),&);< E()1$%2"#+11(#1$0/$'(,52&$%7)#$4)/#5+"3/5&), G

<1=.%&%7. >C@W(,52&C: P1φ(3) * ,-( * J04E,$( * 2,1 * 60-)1(-) * ,-( * 2,#-)1C6#)10- * + ",-( * 7#41#'$( * =x>*;0, '1(- * =∀x>*0,*'1(-*=∃x><@ -0,3*#%%($0-3*= %04)5( * +( * $# * 7#41#'$( * =x>*+#-3φ>*$#*%$,3*%()1)(*J04E,$(*2,1*3,1A

$# * 2,#-)1C6#)10- * +( * =x>:

O< h(+3+$4)&04)&1$&;&.'0&1< L/$*/,+/80&$HxI$/$)#&$(33),,&#3&$0+8,&$%/#1$5()5&1$0&1$&;',&11+(#1$1)+*/#5&1 G HxI6 HF xI6 HF x∧F yI6 H∃y(Rxy)I6 Hf(x)I$&5$H∀y∃z(F x→Ryz)I< Y00&$/$/)11+$)#&$(33),,&#3&$0+8,&$%/#1$H∃x(F x)∧GxI$@0/$%&);+>.&C

&5$%/#1$H∀y(Rxy)→ ∃x(F x)I$@0/$',&.+>,&C<

f< E()1$/*(#1$8&1(+#$%&1$%&.+F3,(3D&51$%&$i)+#&$')+14)&$0&1$&;',&11+(#1$H H V &15$5,+15& I@xC I6 H H. . ./+.&· · ·I@x, yC I$#&

1(#5$'/1$%&1$-(,.)0&1$8+&#F-(,.2&1< L&1$&;',&11+(#1$Hx&15$5,+15& I6 Hx/+.&yI$@()6 M$5+5,&$24)+*/0&#56 H V &15$5,+15& I$&5$H. . . /+.&· · ·IC6 '/,$3(#5,&6 4)&$07(#$(85+&#5$&#$1)815+5)/#5$H V &15$5,+15& I$'(),φ(. . .)&5$H. . ./+.&· · ·I$'(),ψ(. . . ,· · ·)6 1(#5$%&1 'D,/1&1$()*&,5&1$8+&#F-(,.2&1<

(5)

*+:I ;#)012#.&%34()5()6#)678%34()5(0)$'15%9#&0 *CC

=/#1$0&1$5,(+1$-(,.)0&1$1)+*/#5&16 #()1$/*(#1$%&);$(33),,&#3&1$%7)#$4)/#5+"3/5&),$)#+*&,1&0 G

J* ∀x(P x)→(∀x(Qxy)∨Rx) JI ∀x(P x→(∀x(Qxy)∨Rx)) J? ∀x(P x→ ∀x(Qxy∨Rx))

=/#1$07+.'0+3/5+(#J*6 0/$'(,52&$%&$',&.+&,$4)/#5+"3/5&),$)#+*&,1&0$4)+$0+&$0/$*/,+/80&$HxI$&15$HP xI6

%/#1$0/$4)/#5+"3/5+(#$)#+*&,1&00&JI6 37&15$HP x→(∀x(Qxy)∨Rx)I$&5$%/#1J?37&15$HP x→ ∀x(Qxy∨ Rx)I< L/$'(,52&$%&$0/$%&);+>.&$4)/#5+"3/5+(#$%&$HxI6 %/#1J*&5$%/#1JI6 &15$HQxyI$&5$HQxy∨RxI

%/#1J?< L/$*/,+/80&$HxI$/$)#&$1&)0&$(33),,&#3&$0+8,&$%/#1J*@0/$5,(+1+>.&C$&5$#7/$'/1$%7(33),,&#3&

0+8,&$%/#1JI()J?< L/$*/,+/80&$HyI6 3&'&#%/#56 /$%&1$(33),,&#3&1$0+8,&1$%/#1$3D/3)#&$%&1$5,(+1 'D,/1&1$3(#1+%2,2&1<

*+:I ;#)012#.&%34()5()6#)678%34()5(0)$'15%9#&0

!',>1$/*(+,$%2"#+6 %&$./#+>,&$,+9(),&)1&6 3&$4)7&15$)#&$0/#9)&$'(),$0/$0(9+4)&$%&1$',2%+3/516 #()1$/00(#1 ./+#5&#/#5$.(#5,&,$3(..&#5$+0$-/)51-)(4%45)(4)#&$5&00&$0/#9)&6 &#$%(##/#5$)#&$12./#5+4)&$'(), 0/$0(9+4)&$%&1$',2%+3/51< _#&$+#5&,',25/5+(#$%7)#&$-(,.)0&$%)$0/#9/9&$%&$0/$0(9+4)&$',('(1+5+(##&00&

3(#1+15&$&#$07/55,+8)5+(#$%&$*/0&),1$%&$*2,+52$M$5()5&1$0&1$'D,/1&1$1+.'0&1$4)&$3&55&$-(,.)0&$3(#5+&#5 ^ )#&$5&00&$/55,+8)5+(#$3(,,&1'(#%$M$)#&$0+9#&$%&$0/$5/80&$%&$*2,+52$'(),$0&$3(##&35&),$',+#3+'/0$%&$0/

-(,.)0&$&#$4)&15+(#< j/0D&),&)1&.&#56 0/$12./#5+4)&$%&$0/$0(9+4)&$%&1$',2%+3/51$&15$3(.'0+4)2&$'/,

%&);$-/35&),16 /81&#51$/)$3/1$%&$0/$0(9+4)&$',('(1+5+(##&00& G

N< 0/$',21&#3&$%&1$1+9#&1$#(#F0(9+4)&1$1)8F1&#5&#5+&01$%/#1$#(5,&$/0'D/8&5$%&$8/1& ^^b Z< 0/$',21&#3&$%&1$*/,+/80&1$&56 &#$3(#124)&#3&6 %&1$'D,/1&1$()*&,5&1<

L/$',&.+>,&$%+k3)052$1+9#+"&$4)&$#()1$#&$'()*(#1$'/1$1+.'0&.&#5$/55,+8)&,$%&1$*/0&),1$%&$*2,+52 M$%&1$'D,/1&1$3(.'0>5&16 ./+1$4)&$#()1$1(..&1$(80+921$%7+#5&,',25&,$0&1$3(#15/#5&16 0&1$1+9#&1$%&

,&0/5+(#1$&5$0&1$1+9#&1$%&$-(#35+(#1< T7&15$'(),4)(+$#()1$+#5,(%)+1(#1$0/$#(5+(#$%7)#&$H 15,)35),& I<

L/$%&);+>.&$%+k3)052$&#5,/l#&$4)&$.J.&$)#&$+#5&,',25/5+(#$%)$*(3/8)0/+,&$#(#F0(9+4)&$#&$1)k,/

'/1$3(..&$+#5&,',25/5+(#$%&$#(5,&$0/#9)&6 ')+14)&$#()1$%&*(#1$+#5&,',25&,$29/0&.&#5$0&1$'D,/1&1 ()*&,5&1 G #()1$/*(#1$8&1(+#$%&$0/$#(5+(#$%7)#&$H /11+9#/5+(#$%&$*/0&),1 I$/);$*/,+/80&1$3(#5&#)&1$%/#1 )#&$-(,.)0&<

L/$%2"#+5+(#$1)+*/#5&$%25&,.+#&$4)&00&1$1&,(#5$0&1$*/0&),1$%&1$5&,.&1$&5$%&1$',2%+3/51$%&$#(5,&$0/#9/9&<

W(),$0&1$3(#15/#5&1$&5$0&1$*/,+/80&16 #()1$%&*(#1$";&,$)#$&#1&.80&$%7(8K&5$3(..&$)#+*&,1$%&$%+13(),1$\

3&5$&#1&.80&$3(#5+&#5$5()1$0&1$+#%+*+%)1$%(#5$#()1$*()0(#1$'/,0&,$M$07/+%&$%)$0/#9/9&$&#$4)&15+(#<^A!);

1+9#&1$%&$,&0/5+(#1$3(,,&1'(#%,(#5$%&1$,&0/5+(#1$M$07+#52,+&),$%&$3&5$&#1&.80&$&5$/);$1+9#&1$%&$-(#35+(#1

%&1$-(#35+(#1$%&$3&5$&#1&.80&$%/#1$3&5$&#1&.80&<^m

b< _#&$&;',&11+(#$8+&#F-(,.2&$&15$/''&02&$H 1)8F1&#5&#5+&00& I$1+$&00&$#&$3(#5+&#5$'/1$%&$'D,/1&$&#5+>,&<

A< E()1$,&*+&#%,(#1$M$0/$4)&15+(#$%&1$)#+*&,1$%&$%+13(),1$*+%&1$'0)1$5/,%$@3-< 0&$3D<NN<fM$0/$'<ZZ[<

m< h(+3+$4)&04)&1$&;&.'0&1$%&$15,)35),&1 G

N< 07&#1&.80&$%&$5()1$0&1$J5,&1$D)./+#16 /*&3$0/$,&0/5+(#$%7/.(),6 0/$-(#35+(#$&;',+.2&$'/,$H 0/$.>,&$%&$VI$&5$%&);$3(#15/#5&1 +#%+*+%)&00&16 H ]/. I$'(),$]/.$&5$H j/,+& I$'(),$j/,+& ^

Z< 07&#1&.80&$%&$5()1$0&1$/#+./);6 /*&3$0&1$,&0/5+(#1$&;',+.2&1$'/,$Hx&15$)#$n/#9(),() I$&5$Hx/%(,&yI6 /)3)#&$-(#35+(#

&5$%&1$3(#15/#5&1$+#%+*+%)&00&1$Hc1I6 Hc2I6 V6 HcnI$'(),$5()1$0&1$n/#9(),()1 ^

[< 07&#1&.80&$%&$5()1$0&1$#(.8,&1$#/5),&016 /*&3$0/$,&0/5+(#$&;',+.2&$'/,$Hx≤yI6 0&1$-(#35+(#1$%&$1()15,/35+(#6 /%%+5+(#

&5$.)05+'0+3/5+(#$&5$%&);$3(#15/#5&1$+#%+*+%)&00&1$'(),$0&1$#(.8,&1$B$&5$N<

(6)

I++ *+: ,-.&#/()(&)012#.&%34()5()6#)678%34()5(0)$'15%9#&0

<1=.%&%7. G+@]5,)35),&1C: P01A)L⁺,B, λ, µ*,-( * $#-.,( * +( * $# * $0.12,( * +(3 * %45+16#)3:T-( * 3)4,6),4/A

%0,4L⁺60-313)( * (F 8

9: ,- * (-3(E'$( * -0-?71+/|A|@ #%%($5 * = $",-17(43*+(*+1360,43 >*0,*$(*= +0E#1-( * +( * 2,#-)1C6#)10F>*+/AB I: ,-(*1-)(4%45)#)10-*+(*)0,3*$(3*31.-(3*+(*4($#)10-3 8 ,-( * J0-6)10- * 2,1 * #))41',( * H * )0,Ai∈@,-( * 4($#)10F

RÂ_i 3,4|A|#7(6λ(i)%$#6(3*#4.,E(-)#$(3@ 6"(3)?H?+14( * ,- * 30,3?(-3(E'$/RÂ_i ⊂ |A|^λ(i):â

K: ,-(*1-)(4%45)#)10-*+(*)0,3*$(3*31.-(3*+(*J0-6)10-3 8 ,-( * J0-6)10- * 2,1 * #))41',( * H * )0,Aj∈A,-( * J0-6)10F f_j^A3,4|A|#7(6µ(j)%$#6(3*#4.,E(-)#$(3@ 6"(3)?H?+14( * ,-( * J0-6)10Ff_j^A:|A|^µ(j)→ |A|:

R: ,-( * 1-)(4%45)#)10- * +( * )0,)(3 * $(3 * 60-3)#-)(3@ 2,1*#))41',(*H*)0,Ak∈B,- * 5$5E(-) * CL/cÂ_k +/|A|: E()1$)5+0+1(#1$%&1$1)'&,13,+'51$'(),$,&#%,&$30/+,$4)&$HRÂI6 HfÂI$&5$HcÂI$%21+9#&#56 ,&1'&35+*&.&#56 )#&$,&0/5+(#6 )#&$-(#35+(#$&5$)#$202.&#5$%&$07)#+*&,1$%&$%+13(),1$4)+$#()1$1&,*&#5$%7+#5&,',25&,$0&$1+9#&

%&$,&0/5+(#$HRI6 0&$1+9#&$%&$-(#35+(#$HfI$&5$0/$3(#15/#5&$HcI$%/#1$0/$15,)35),&A<^NBE()1$/''&0(#1 )#&$5&00&$15,)35),&$)#&= 1-)(4%45)#)10F >%&$#(5,&$0/#9)&$&5$#()1$%+1(#1$4)&$#()1$H +#5&,',25(#1 I$0&1 -(,.)0&1$%&$3&55&$0/#9)&$%/#1$0/$15,)35),&$&#$4)&15+(#<

W,&#(#1$)#&$'D,/1&$%&$#(5,&$0/#9/9&$(8K&56 '/,$&;&.'0&$HR(a, b)I< E()1$1/*(#16 25/#5$%(##2&1$#(1 3(#*&#5+(#1$5:'(9,/'D+4)&16 4)&$HRI$,&',21&#5&$)#&$,&0/5+(#$&5$4)&$HaI$&5$HbI$1(#5$%&1$3(#15/#5&1 +#%+*+%)&00&16 ,&',21&#5/#5$%&1$(8K&51$%&$#(5,&$%(./+#&< _#&$+#5&,',25/5+(#$%&$3&1$1+9#&1$3(#1+15&,/$&#

)#&$/11+9#/5+(#$%7)#&$,&0/5+(#$',23+1&$&5$%&1$,2-2,&#51$'/,5+3)0+&,1$M$HRI6 HaI$&5$HbI< E()1$'()*(#16 '/,

&;&.'0&6 +#5&,',25&,$3&55&$'D,/1&$%/#1$07&#1&.80&$%&$5()5&1$0&1$'&,1(##&16 /11+9#&,$0/$,&0/5+(#$%7/.(), M$HRI6 j/,+&$M$HaI$&5$P&/#$M$HbI< L/$'D,/1&$1&,/$%(#3$*,/+&$17+0$&15$0&$3/1$&5$1&)0&.&#5$17+0$&15$0&$3/1$4)&

j/,+&$/+.&$P&/#<

E(5(#1$4)&$0/$%2"#+5+(#$%7)#&$15,)35),&$#&$#()1$%+5$,+&#$1),$07+#5&,',25/5+(#$%&1$'D,/1&1$()*&,5&1<

]+$#()1$',&#(#1$HR(a, x)I$3(..&$&;&.'0&6 #()1$1/*(#1$1&)0&.&#5$4)7)#$(8K&5$1/5+1-/+5$3&55&$'D,/1&

()*&,5&$1+$&5$1&)0&.&#5$1+$3&5$(8K&5$&15$/+.2$'/,$j/,+&< j/+1$3(..&#5$'()*(#1F#()1$,&#%,&$3&55&

,&0/5+(#$&;',+.2&$'/,$H. . .&15$*,/+$%&· · · I$'0)1$',23+1& o =/#1$0&$3(#5&;5&$%7)#&$15,)35),&$%(##2&6

#()1$'()*(#1$/11+9#&,$%&1$*/0&),1$/);$(33),,&#3&1$0+8,&1$%&$#(1$*/,+/80&1 G

<1=.%&%7. G*@!11+9#/5+(#1$%&$*/0&),1C: P01AL⁺,-(*$#-.,(*+(*$#*$0.12,(*+(3*%45+16#)3*(AA,-( * 3)4,6?

),4( * %0,4L⁺: T-( * #331.-#)10- * +( * 7#$(,43 * %0,4L⁺ (3) * ,-( * J0-6)10Fh2,1 * #331.-( * H * )0,)( * 7#41#'$/xi

;i∈N< * (L#6)(E(-) * ,- * 5$5E(-) * +( * $",-17(43 * +( * +1360,43 8 h:Q]g(L⁺)→ |A|:

_#&$/11+9#/5+(#$%&$*/0&),16 &#$%7/)5,&1$5&,.&16 &15$)#&$+#5&,',25/5+(#$R 3(#%+5+(##&00&$S %&1$*/,+/80&16 H R 3(#%+5+(##&00&$S I$'/,3&$4)7&00&$#&$0&1$+#5&,',>5&$'/1$/)$#+*&/)$%7)#&$15,)35),&$@)#&$'(11+8+0+52$0(9+4)&C6 ./+1$M$07+#52,+&),$%7)#&$15,)35),&< L7/11+9#/5+(#$%&$*/0&),1$#()1$'&,.&56 %/#1$3&,5/+#&1$3(#%+5+(#16 %&

5,/+5&,$0&1$'D,/1&1$()*&,5&1$3(..&$%&1$'D,/1&1$3(.'0>5&1<^NN L/$1+9#+"3/5+(#$%7)#&$'D,/1&$()*&,5&6 5&00&$4)&$Hx/+.&$j/,+& I6 %2'&#%,/$%&$07/11+9#/5+(#$%&$*/0&),1$%/#1$0&$1&#1$4)7)#&$/11+9#/5+(#$%&$]/.$M

a< H|A|^λ(i)I$%21+9#&$)#$',(%)+5$3/,521+&#6 37&15FMF%+,&$07&#1&.80&$%&$5()5&1$0&1$124)&#3&16 Mλ(i).&.8,&16 %7202.&#51$%&

|A|G

|A|^λ(i) := |A| × |A| ×. . .× |A|

| {z }

λ(i)-(+1

NB< !05&,#/5+*&.&#56 #()1$/),+(#1$')$%+,&$4)7)#&$15,)35),&A&15$)#&$124)&#3&$%&$4)/5,&$202.&#516A='|A|, h_@, h_A, h_B(6 3(#1+15/#5$%7)#$&#1&.80&|A|6 %7)#&$-(#35+(#$4)+$/11+9#&$M$5()5$1+9#&$%&$,&0/5+(#$1(#$+#5&,',25/5+(#h_@:Ri)→R^A_i 6 )#&$/)5,&

-(#35+(#$4)+$/11+9#&$M$5()5$1+9#&$%&$-(#35+(#$1(#$+#5&,',25/5+(#h_A:fJ)→f_j^A&5$)#&$-(#35+(#$4)+$/11+9#&$M$5()5&$3(#15/#5&$)#

202.&#5$%&$07)#+*&,1$%&$%+13(),1 Gh_B:c_k)→c^A_k∈|A|<

NN< T&3+$#7&15$'/1$5()5$M$-/+5$&;/35 G +0$1&,/+5$'0)1$/%24)/5$%&$%+,&$4)7)#&$/11+9#/5+(#$%&$*/0&),1$#()1$'&,.&5$%&$%2"#+,$)#&

#(5+(#$92#2,/0&6 3&00&$%&$0/$1/5+1-/35+(#6 4)+$17/''0+4)&$29/0&.&#5$/);$'D,/1&1$()*&,5&1$&5$3(.'0>5&1$&5$4)+$#()1$'&,.&5$%&$%2"#+, 0/$*2,+52$%7)#&$'D,/1&$3(.'0>5&$3(..&$3/1$0+.+5&<

(7)

*+:I ;#)012#.&%34()5()6#)678%34()5(0)$'15%9#&0 I+*

HxI$#()1$%(##&,/$)#&$'D,/1&$*,/+&6 ./+1$)#&$/11+9#/5+(#$%&$d,2%2,+4)&$M$HxI$)#&$'D,/1&$-/)11&< W(), 2*+5&,$07/.8+9)p526 +0$&15$29/0&.&#5$+.'(,5/#5$4)7)#&$/11+9#/5+(#$%&$*/0&),1$1(+5$)#&$-(#35+(# G 5()5&

(33),,&#3&$%7)#&$*/,+/80&$%(+5$J5,&$/11+9#2&$M$)#$1&)0$(8K&5<

_#&$15,)35),&$&5$)#&$/11+9#/5+(#$%&$*/0&),16 ',+1&1$&#1&.80&6 %25&,.+#&#5$%&$4)&01$(8K&51$#()1$'/,0(#1 M$07/+%&$%&$#(1$&;',&11+(#1$%)$0/#9/9&$(8K&5 G 0/$15,)35),&$&#$4)&15+(#$#()1$-(),#+5$0&1$,2-2,&#51$%&1 3(#15/#5&1$+#%+*+%)&00&1< L&1$-(#35+(#1$25/#5$29/0&.&#5$+#5&,',252&1$%/#1$0/$15,)35),&6 #()1$'()*(#1

%(#3$%25&,.+#&,$1/#1$)#+*(4)&$0/$,2-2,&#3&$()$0/$%21+9#/5+(#$%&$5()1$#(1$5&,.&1$1+#9)0+&,1 G

<1=.%&%7. GI@=21+9#/5+(#$%&$5&,.&1C: P01AL⁺,-( * $#-.,( * +( * $# * $0.12,( * +(3 * %45+16#)3@A,-( * 3)4,6),4/

%0,4L⁺(Ah:Q]g(L⁺)→ |A|,-( * #331.-#)10- * +( * 7#$(,43:!#*+531.-#)10Fh(t)+",- * )(4E( * =t>*+/L⁺30,3 6())( * #331.-#)10- * +( * 7#$(,43 * (3) * +5C-1( * 60EE( * 3,1A 8

9: 31*=t> * (3) * ,-( * 7#41#'$(@h(t)(3Ah(t)B I: 31*=t> * (3) * ,-( * 60-3)#-)( * =ck>@h(t)(3Ac^A_k B

K: 31*=t> * (3) * ,- * )(4E( * +( * $# * J04E( * =fj(t1, . . . tµ(j))>@ #$043h(t)(3Af_j^A(h(t1), . . . , h(t_µ(j))):

_#&$+#5&,',25/5+(#$%25&,.+#&$M$4)&01$(8K&51$1&$,2->,&#5$#(1$5&,.&1$1+#9)0+&,1$&5$4)&01$(8K&51$1(#5$,&F ',21&#521$'/,$#(1$*/,+/80&1 G 37&15$/+#1+$4)7)#&$/11+9#/5+(#$%&$*/0&),1h&5$)#&$15,)35),&A%25&,.+#&#5 0/$-(#35+(#h4)+$/11(3+&$M$5()1$#(1$5&,.&1$1+#9)0+&,1$&5$#(1$*/,+/80&1$0&),1$%21+9#/5+(#1$@%/#1$)#&

15,)35),&$&5$1()1$)#&$/11+9#/5+(#$%&$*/0&),1C<

T(..&$#()1$/*(#1$*)$M$0/$'<NAA%&$0/$135<a<[6 #()1$'()*(#1$%(##&,$%&1$3(#%+5+(#1$%&$1/5+1-/35+(#

'(),$%&1$',2%+3/51$0(9+4)&.&#5$3(.'0&;&16 /+#1+$&;'0+4)/#5$0&),1$1+9#+"3/5+(#1$'/,$0/$1+9#+"3/5+(#$%&

0&),1$'/,5+&1<

!)$0+&)$%&$%+,&$4)&$0&1$'D,/1&1$()*&,5&1$#&$1(#5$#+$*,/+&1$#+$-/)11&1$./+1$*,/+&1$()$-/)11&1+/3&,5/+#1 (8K&516 #()1$'()*(#1$./+#5&#/#5$%+,&$4)7&00&1$1(#5$*,/+&1$()$-/)11&130,3 * ,-( * #331.-#)10- * +( * 7#$(,43/);

*/,+/80&1$%(#5$&00&1$3(#5+&##&#5$%&1$(33),,&#3&1$0+8,&1< T&55&$#(5+(#$%&$*2,+52F1()1F)#&F/11+9#/5+(#F

%&F$*/0&),1$&15$0/$#(5+(#$30&-$%&$0/$12./#5+4)&$%&$0/$0(9+4)&$%&1$',2%+3/51 G

<1=.%&%7. G?@h2,+52$1()1$)#$/11+9#/5+(#$%&$*/0&),1C: P01AL⁺,-( * $#-.,( * +( * $# * $0.12,( * +(3 * %45+16#)3@

A,-( * 3)4,6),4( * %0,4L⁺(Ah:Q]g(L⁺) → |A|,-( * #331.-#)10- * +( * 7#$(,43: V0,3*+130-3*2,",-(*J04E,$/

φ+/L⁺(3) * 74#1( * 30,3 * $"#331.-#)10- * +( * 7#$(,43h0, * 2,( * $"#331.-#)10- * +( * 7#$(,43h3#)13J#1) * $# * J04E,$/φ

;#'45.G 8 =A |qhφ><*31*()*3(,$(E(-)*31*,-(*+(3*60-+1)10-3*3,17#-)(3*(3)*4(E%$1/ 8 ,* φ# * $# * EUE( * J04E( * 2,( * =t1!t2> (A h(t1) =h(t2)

,I φ# * $# * EUE( * J04E( * 2,( * =Ri(t1, . . . , tλ(i))> (A R^A_i (h(t1), . . . , h(tλ(i)))

,? φ(3) * +( * $# * J04E/"¬ψ# (A A (|qhψ

,> φ(3) * +( * $# * J04E/"ψ∧χ# (A A |qhψ(AA |qhχ ,G φ(3) * +( * $# * J04E/"ψ∨χ# (A 301AA |qhψ301AA |qhχ ,D φ(3) * +( * $# * J04E/"ψ→χ# (A 301AA (|qhψ301AA |qhχ ,F φ(3) * +( * $# * J04E/"ψ↔χ# (A A |qhψ31*()*3(,$(E(-)*31A |qhχ ,H φ(3) * +( * $# * J04E/"∀xi(ψ)# (A A |q_h(^xia)ψ%0,4*)0,3*$(3a∈|A|

,C φ(3) * +( * $# * J04E/"∃xi(ψ)# (A A |q_h(^xia)ψ%0,4*#,*E01-3*,Fa∈|A|

E()1$/K()5(#1$0&$1+9#&$%&$07/11+9#/5+(#$%&$*/0&),1$/)$1+9#&$%&$3(#124)&#3&$0(9+4)&$&5$23,+*(#1$H|qhI '(),$+#%+4)&,$4)7+0$#&$17/9+5$'/16 3(..&$%/#1$0&$3/1$%&$0/$0(9+4)&$',('(1+5+(##&00&6 %7)#&$,&0/5+(#$&#5,&

)#$&#1&.80&$%&$'D,/1&1$&5$)#&$'D,/1&6 ./+1$%7)#&$,&0/5+(#$5&,#/+,&$&#5,&$)#&$15,)35),&6 )#&$/11+9#/5+(#

%&$*/0&),1$&5$)#&$'D,/1&<

T&1$3(#%+5+(#1$%&$*2,+52F1()1F)#&F/11+9#/5+(#$1(#5$%+14)(5/5+(#&00&1 G 3&$4)+$&15$/k,.2$%/#1$0&$0/#9/9&

(8K&5$&15$1+.'0&.&#5$R 1,&F5,/%)+5S 1$%/#1$0&$.25/0/#9/9&< L&1$3(#%+5+(#1$%&$*2,+52$%+14)(5/5+(#&00&1

(8)

I+I *+: ,-.&#/()(&)012#.&%34()5()6#)678%34()5(0)$'15%9#&0

0&1$'0)1$-/.&)1&1$1(#5$3&);$4)+$(#5$0/$-(,.&$%)$13D2./K%&$`/,1n+ G^NZ H L/$#&+9&$&15$80/#3D& I$&15$*,/+&< ⇐⇒ L/$#&+9&$&15$80/#3D&<

T7&15$K)15&.&#5$0/$5,+*+/0+52$%&$3&55&$3(#%+5+(#$%&$*2,+52$'(),$H L/$#&+9&$&15$80/#3D& I$4)+$#()1$/11),&

%&$1/$3(,,&35+(#< ]+$#()1$/8,29&(#1$H 0/$#&+9& I$'/,$HaI$&5$H <<<&15$80/#3 I$'/,$HF xI6 0/$30/)1&,I%&$0/

%2"#+5+(#$%&$0/$1/5+1-/35+(#$#()1$%(##& G

HF aI$&15$*,/+&$%/#1A ⇐⇒ a^A∈F^A

E()1$%+1(#1$4)7)#&$'D,/1&$()*&,5&$&153#)13J#13#'$( * +#-3 * ,-( * 3)4,6),4/17+0$:$/$)#&$/11+9#/5+(#$%&$*/F 0&),1$4)+$0/$1/5+1-/+5< Y00&$&153#)13J#13#'$/17+0$:$/$)#&$15,)35),&$%/#1$0/4)&00&$&00&$&15$1/5+1-/+1/80&< _#

&#1&.80&$%&$'D,/1&1Σ&15$1/5+1-/+1/80&$1+$5()1$1&1$.&.8,&1φ∈Σ0&$1(#5< T(..&$/*/#56 %+,&$4)&

Σ&15$1/5+1-/+1/80&$,&*+&#5$M$%+,&$4)7+0$&15$0(9+4)&.&#5$'(11+80&$4)&$5()5&1$0&1$'D,/1&1$%/#1Σ1(+&#5

*,/+&1$&#1&.80&16 4)7&00&1$%23,+*&#5$)#&$'(11+8+0+52$0(9+4)& G )#&$'(11+8+0+52$0(9+4)&6 %/#1$0/$0(9+4)&

%&1$',2%+3/516 3(,,&1'(#%,/$%(#3$M$)#&$15,)35),&$&5$)#&$/11+9#/5+(#$%&$*/0&),1<

,*&5,I%+1&#5$4)&$0&1$'D,/1&1$4)+$/k,.&#5$)#&$+%&#5+52$()$4)&$3&,5/+#&1$3D(1&1$1&$5,()*&#5$%/#1 )#&$3&,5/+#&$,&0/5+(#$1(#5$*,/+&1$1()1$)#&$/11+9#/5+(#$%&$*/0&),16 1+$&5$1&)0&.&#5$1+$0&1$3D(1&1$%+5&1 +%&#5+4)&1$1(#5$,2&00&.&#5$5,/+52&1$3(..&$25/#5$+%&#5+4)&1$'/,$3&55&$/11+9#/5+(#$%&$*/0&),1$&5$0&1 3D(1&1$%(#5$(#$%+5$4)7&00&1$1&$5,()*&#5$%/#1$)#&$3&,5/+#&$,&0/5+(#Ri1&$5,()*&#5$,2&00&.&#5$%/#1$3&55&

,&0/5+(#$%7/',>1$07/11+9#/5+(#$%&$*/0&),1$&#$4)&15+(#< Y#$8,&-6 &00&1$%+1&#5$4)&$0&1$'D,/1&1$/5(.+4)&1 1(#5$*,/+&1$1+$&00&1$,&',21&#5&#5$0&1$3D(1&1$3(..&#5$&00&1$1(#5$%7/',>1$07/11+9#/5+(#$&#$4)&15+(#<

L&1$3(#%+5+(#1,?M,F,2'>5&#5$0&1$30/)1&1$4)&$#()1$/*(#1$%(##2&1$'(),$0/$12./#5+4)&$%&1$3(##&3F 5&),1$',('(1+5+(##&016 1/)-$4)7&00&1$1(#5$./+#5&#/#5$/''0+4)2&1$29/0&.&#5$M$%&1$3(##&35&),1$4)+$,&0+&#5

%&1$'D,/1&1$()*&,5&1<^N[

L&1$%&);$30/)1&1$3(#3&,#/#5$0&1$4)/#5+"3/5&),1$1(#5$'0)1$%+k3+0&1$M$3(.',&#%,&< Y00&1$)5+0+1&#5$)#&

#(5+(#$4)&$#()1$#7/*(#1$'/1$&#3(,&$+#5,(%)+5&6 3&00&$%7)#&$H /11+9#/5+(#$*/,+2&$M$0/$'0/3&$HxI I< T&$4)&

#()1$*()0(#1$%+,&6 &#$15+')0/#5$0/$3(#%+5+(#,C6 37&15$4)7)#&$-(,.)0&$4)/#5+"2&$&;+15&#5+&00&.&#5$&15

*,/+&$%/#1A1()1h1+$&5$1&)0&.&#5$17+0$:$/$)#$(8K&5$%/#1$07)#+*&,1$%)$%+13(),1$%(#5$&15$*,/+&$0/$'D,/1&

()*&,5&$',232%2&$'/,$0&$4)/#5+"3/5&),$&;+15&#5+&0< E()1$#&$1/*(#1$'/16 3&'&#%/#56 1+$07/11+9#/5+(#h&#

4)&15+(#$/11+9#&$3&5$(8K&5$M$07(33),,&#3&$0+8,&$%&$0/$*/,+/80&$%/#1$0/$'D,/1&$()*&,5&< E()1$/*(#1$%(#3 8&1(+#$%&$3D/#9&,$07/11+9#/5+(#$&#$4)&15+(#6 '(),$,&#%,&$1r,$4)7&00&$/11+9#&$0&$8(#$(8K&5<

<1=.%&%7. G>@!11+9#/5+(#1$*/,+2&1C: P01AL⁺,-( * $#-.,( * +( * $# * $0.12,( * +(3 * %45+16#)3@A,-( * 3)4,6),4/

%0,4L⁺@h : Q]g(L+) → |A|,-( * #331.-#)10- * +( * 7#$(,43 * () * =xi> * ,-( * 7#41#'$( * +/L⁺: V0,3*+5C-1330-3

NZ< L/$3(#%+5+(#$4)7)#$',2%+3/5$%&$*2,+52$%(+5$,&#%,&$*,/+1$5()5&1$0&1$+#15/#3&1$%)$13D2./K6 3D&sL6 K()&$0&$,U0&$%7)#$3,+5>,&

%7/%24)/5+(#$'(),$)#&$%2"#+5+(#$%&$*2,+52< ]&0(#$0)+6 5()5$',2%+3/5$HF xI$4)+$#&$1/5+1-/+5$'/1$0/$3(#%+5+(#$1)+*/#5&$#&$.2,+5&$'/1 0&$#(.$%7)#$',2%+3/5$%&$*2,+52$@3-< '<fZC G

H L/$#&+9&$&15$80/#3D& I$&15F ⇐⇒ L/$#&+9&$&15$80/#3D&<

L/$',21&#3&$%&1$'D,/1&1$/)5(F,2-2,&#5+&00&16 3&'&#%/#56 #()1$+#5&,%+5$%&$',&#%,&$1+.'0&.&#5$0/$5(5/0+52$%&$5()5&1$3&1$24)+*/F 0&#3&1$3(..&$%2"#+5+(#$%&$0/$*2,+52 ^ &#$1)815+5)/#5$H 3&55&$'D,/1&$&15$-/)11& I$%/#1$0&$13D2./K6 #()1$(85&#(#1$)#&$3(#5,/%+35+(#

T7&15$'(),$&;30),&$3&55&$-(,.&$%7/)5(F,2-2,&#5+/0+52$4)&$0&$R %25(),$R M$5,/*&,1$0/$1:#5/;&$&15$#23&11/+,&<

N[< T(..&$#()1$0&1$/*(#1$%2"#+1$0&1$3(##&35&),1$',('(1+5+(##&016 0&1$3(##&35&),1$,&0+/#5$%&1$'D,/1&1$()*&,5&1$#&$1(#5$'/1

%&1$3(##&35&),1$',('(1+5+(##&01< ]5,+35&.&#5$'/,0/#56 +0$#7&15$%(#3$'/1$*,/+$4)&$0&1$-(,.)0&1$8+&#F-(,.2&1$%&$0/$0(9+4)&$',('(F 1+5+(##&00&$1(#5$/)11+$%&1$-(,.)0&1$8+&#F-(,.2&1$%)$3/03)0$%&1$',2%+3/51< E()1$/),+(#1$')6 3&'&#%/#56 %2"#+,$0&1$3(##&35&),1

%>1$0&$%28)5$&5$'(),$%&1$'D,/1&1$3(.'0>5&1$&5$'(),$%&1$'D,/1&1$()*&,5&1$\$%/#1$0/$0(9+4)&$',('(1+5+(##&00&6 #()1$#7/),+(#1$%(#3 5,/+52$4)&$%7)#$%&$3&1$3/1<

(9)

*+:? ;#)M#6%5%&1 I+?

$"#331.-#)10- * 7#415( * H * $# * %$#6( * =xi>*W*#%%($5(*=h!xi

a

"

>*W*60EE(*3,1A 8

h

#xi

a

$

(xj) := % h(xj) i(=j

a i=j

h!xi

a

"

&15$)#&$-(#35+(#$4)+$/11+9#&$M$5()5&1$0&1$*/,+/80&1$1/)-$HxiI$0&$.J.&$+#%+*+%)$4)&$0&),$/11+9#&h&5 /11+9#&aM$HxiI$\$37&15$)#&$.(%+"3/5+(#$0(3/0&$%&$07/11+9#/5+(#hM$0/$'0/3&$HxiI$4)+$0/$-(,3&$%7/11+9#&, aM$HxiI<

,H%+56 &#$3(#124)&#3&6 4)7)#&$-(,.)0&$)#+*&,1&00&.&#5$4)/#5+"2&$&15$*,/+&$1()1$)#&$/11+9#/5+(#$%&

*/0&),1$1+$&5$1&)0&.&#5$1+$0/$'D,/1&$()*&,5&$&15$*,/+&$1()1$5()5&$.(%+"3/5+(#$%&$3&55&$/11+9#/5+(#$&5

%(#3$1()1$5()5&$/11+9#/5+(#$%7)#$202.&#5$%)$%(./+#&$M$0/$*/,+/80&$&#$4)&15+(#< _#&$4)/#5+"3/5+(#

)#+*&,1&00&$&15$*,/+&$)#+4)&.&#5$/)$3/1$(t$0/$'D,/1&$()*&,5&6 9()*&,#2&$'/,$0&$4)/#5+"3/5&),$)#+*&,1&06

&15$1/5+1-/+5&$&#$5&#/#5$3(.'5&$%&$5()5&1$0&1$*/0&),1$'(11+80&1$%&$1&1$*/,+/80&1<,C6 %&$07/)5,&$3U526 %+5 4)7)#&$'D,/1&$&;+15&#5+&00&$&15$*,/+&$1()1$)#&$/11+9#/5+(#$%&$*/0&),1$17+0$:$/$&5$1&)0&.&#5$17+0$:$/6 %/#1 07)#+*&,1$%&$%+13(),1$&#$4)&15+(#6 /)$.(+#1$)#$+#%+*+%)$4)+$'&)5$J5,&$/11+9#2$3(..&$*/0&),$M$0/$*/,+/80&

/:/#5$)#&$(33),,&#3&$0+8,&$%/#1$0/$'D,/1&$()*&,5&$&#$4)&15+(#<

*+:? ;#)M#6%5%&1

E()1$/*(#1$*)$4)&$0/$%2"#+5+(#$%&$0/$#(5+(#$%&$*2,+52$1()1$)#$/11+9#/5+(#$%&$*/0&),1$3(#1+15&$&#$5,(+1 '/,5+&1 G

N< )#&$',&.+>,&$'/,5+&$4)+$/k,.&$0/$*2,+52$%&1$-(,.)0&1$/5(.+4)&1$%&$0/$-(,.&$Ht1 ! t2I$&5 HRi(t1, . . . , tλ(i))I$1+$07/11+9#/5+(#$&#$4)&15+(#$/55,+8)&$/)$5&,.&1$0&1$.J.&1$%21+9#/5+(#1$@',&F .+&,$3/1C$&5$/)$1+9#&$%&$,&0/5+(#$)#&$,&0/5+(#$&#5,&5&#)&$&#5,&$0&1$%21+9#/5+(#1$%&1$5&,.&1$@%&);+>.&

3/1C ^

Z< )#&$%&);+>.&$'/,5+&$4)+$%(##&$%&1$%2"#+5+(#1$,23),1+*&1$'(),$0&1$-(,.)0&1$3(#5&#/#5$%&1$3(##&3F 5&),1$',('(1+5+(##&01 ^

[< )#&$5,(+1+>.&$'/,5+&$4)+$5,/+5&$%&1$4)/#5+"3/5&),1$&5$)5+0+1&$0/$#(5+(#$%7/11+9#/5+(#$%&$*/0&),1

*/,+2&$M$)#&$'0/3& G )#&$'D,/1&$)#+*&,1&00&.&#5$4)/#5+"2&$&15$*,/+&$1()1$)#&$/11+9#/5+(#$1+$&5 1&)0&.&#5$1+$&00&$&15$*,/+&$1()1$5()5&$/11+9#/5+(#$*/,+2&$M$0/$'0/3&$%&$0/$*/,+/80&$)#+*&,1&00&.&#5 4)/#5+"2& ^ )#&$'D,/1&$&;+15&#5+&00&.&#5$4)/#5+"2&$&15$*,/+&$1()1$)#&$/11+9#/5+(#$1+$&5$1&)0&.&#5 1+$&00&$&15$*,/+&$1()1$/)$.(+#1$)#&$5&00&$/11+9#/5+(#$*/,+2&<

_#&$-(,.)0&$&15$*,/+&$1()1$)#&$/11+9#/5+(#$%&$*/0&),1$1+$&5$1&)0&.&#5$1+$07/11+9#/5+(#$%&1$*/0&),1$M$1&1

*/,+/80&1$3(,,&1'(#%$M$0/$./#+>,&$%(#5$3&1$*/,+/80&1$1(#5$,&0+2&1$%/#1$0/$',('(1+5+(#< Y#$&;/.+#/#5 0&1$30/)1&1,*M,C6 #()1$,&./,4)(#1$4)&$0&),1$,21)05/51$'(),$3&,5/+#&1$'D,/1&1$#&$%2'&#%&#5$'/1$%&

07/11+9#/5+(#h&#$4)&15+(#< ]+$#()1$3D(+1+11(#16 '/,$&;&.'0&6 0/$'D,/1&$H∀x(0≤x)I$&5$#()1$072*/0)(#1

%/#1$0/$15,)35),&$%&1$#(.8,&1$#/5),&01$@&5$/*&3$07+#5&,',25/5+(#$%&$H≤I$'/,$0/$,&0/5+(#$%&$#7J5,&$'/1$'0)1 9,/#%$4)&C6 1&0(#,H0&$,21)05/5$&15$4)7&00&$&15$*,/+&$1()1$5()5&1$0&1$/11+9#/5+(#1< L/$,/+1(#$'(),$3&0/$&15 1+.'0& G +0$#7:$/$,+&#$M$/11+9#&,$M$3&55&$',('(1+5+(#6 ')+14)7&00&$#&$3(#5+&#5$'/1$%7(33),,&#3&$0+8,&$%7)#&

*/,+/80&$@0/$1&)0&$*/,+/80&$4)7&00&$3(#5+&#56 HxI6 /$)#&$1&)0&$(33),,&#3&$&5$3&00&F0M$&15$9()*&,#2&$'/,$0&

4)/#5+"3/5&),$)#+*&,1&0C<

T&55&$(81&,*/5+(#$1&$92#2,/0+1& G +0$&15$*,/+$%&$5()5&1$0&1%&4#3(34)&$0&),1$*/0&),1$%&$*2,+52$#&$%2'&#%&#5 '/1$%7)#&$/11+9#/5+(#$%&$*/0&),1$'/,5+3)0+>,&< _#&$'D,/1&$#&$3(#5+&#5$'/1$%7(33),,&#3&$0+8,&$%7)#&$*/F ,+/80&6 %(#3$+0$#7:$/$,+&#$M$/11+9#&,$\$1+$&00&$&15$*,/+&6 &00&$&15$*,/+&$1()1$5()5&1$0&1$/11+9#/5+(#16 1+$&00&

&15$-/)11&6 &00&$#7&15$*,/+&$1()1$/)3)#&< E()1$(85&#(#1$/+#1+$0/$#(5+(#$%&$*2,+52$@#&$17/''0+4)/#5$4)7M

(10)

I+> *+: ,-.&#/()(&)012#.&%34()5()6#)678%34()5(0)$'15%9#&0

%&1$'D,/1&1$3(.'0>5&1C$3(..&$3/1$0+.+5&$%&$0/$#(5+(#$%&$*2,+52F1()1F)#&F/11+9#/5+(#$@4)+$17/''0+4)&

29/0&.&#5$M$%&1$'D,/1&1$()*&,5&1C G

<1=.%&%7. GG@h2,+52$%/#1$)#&$15,)35),&C: P01AL⁺,-( * $#-.,( * +( * $# * $0.12,( * +(3 * %45+16#)3 * (AA,-/

3)4,6),4( * %0,4L⁺: V0,3*+130-3*2,",-(*J04E,$/φ+/L⁺(3) * 74#1( * +#-3 * $# * 3)4,6),4/A31 * () * 3(,$(E(-) * 31φ (3)*74#1(*30,3*)0,)(3*$(3*#331.-#)10-3*+(*7#$(,43*%0,4L⁺8

A |qφ :⇐⇒ %0,4*)0,Ah:Q]g(L⁺)→ |A| : A |qhφ P1φ(3) * 74#1( * +#-3 * ,-( * 3)4,6),4/A@ -0,3*#%%($0-3A,- * = E0+M$/ >*+/φ:

T&55&$%2"#+5+(#$%&$*2,+52F%/#1F)#&F15,)35),&$3(..&$*2,+52F1()1F5()5&1F0&1F/11+9#/5+(#1$&15$&11&#5+&0F 0&.&#5$3&00&$%&`/,1n+@Na[[C<

L/$#(5+(#$%&$*2,+52F%/#1F)#&F15,)35),&$&15$)#&$92#2,/0+1/5+(#$%&$0/$#(5+(#$(,%+#/+,&$%&$*2,+526 4)+$5,/+5&

0&$.(#%&$/35)&0$3(..&$1&)0&$15,)35),&$%+9#&$%7+#52,J5< Y#$0(9+4)&6 #()1$#&$#()1$+#52,&11(#1$'/1$M$3&

4)+$&15$*,/+$()$-/);$%/#1$)#&$15,)35),&$'/,5+3)0+>,& G #()1$#()1$+#52,&11(#1$M$3&$4)+$&15$*,/+$%/#1$5()5&1 0&1$15,)35),&1$@/);$5/)5(0(9+&1C$&5$/);$,&0/5+(#1$4)7+0$:$/$&#5,&$0&1$*2,+521$%/#1$%+X2,&#5&1$15,)35),&1$@/);

,&0/5+(#1$%&$3(#124)&#3&$0(9+4)&C<

W(),$/,,+*&,$M$)#&$#(5+(#$%&$*2,+52$0(9+4)&6 #()1$%&*(#1$%(#3$92#2,/0+1&,$1),$5()5&1$0&1$15,)35),&1 G

<1=.%&%7. GD@h/0+%+52C: P01AL⁺,-( * $#-.,( * +( * $# * $0.12,( * +(3 * %45+16#)3: V0,3*+130-3*2,",-(*J04E,$/φ +/L⁺(3)*7#$1+(*0,*2,"(X(*(3)*,-(*7541)5*$0.12,(*;+(*$#*$0.12,(*+(3*%45+16#)3<*31*()*3(,$(E(-)*31φ(3) * 74#1/

+#-3*)0,)(3*$(3*3)4,6),4(3*%0,4L⁺8

|qφ :⇐⇒ %0,4*)0,AA : A |qφ

=&$3&55&$#(5+(#$%&$*/0+%+526 #()1$(85&#(#1$)#&$#(5+(#$%&$3(#124)&#3&$0(9+4)& G

<1=.%&%7. GF@T(#124)&#3&$0(9+4)&C: P01AL⁺,-( * $#-.,( * +( * $# * $0.12,( * +(3 * %45+16#)3: V0,3*#%%($0-3 ,-( * J04E,$/φ,-( * 60-352,(-6( * $0.12,( * +",- * (-3(E'$( * +( * J04E,$(3 * 31 * () * 3(,$(E(-) * 31φ(3)*74#1*+#-3*)0,)(3

$(3*3)4,6),4(3*0Y*)0,)(3*6(3*J04E,$(3*30-)*74#1(3 8

Σ|qφ :⇐⇒ %0,4*)0,AA:31A |qψ%0,4*)0,)(3*$(3*J04E,$(3ψ∈Σ@ #$043A |qφ

T&55&$%2"#+5+(#$%&$*/0+%+52$#()1$'&,.&5$%&$3(.'5&,$3(..&$*/0+%&1$()$0(9+4)&.&#5$*,/+&1$%&1$'D,/1&1 4)+$#&$1(#5$'/1$%&1$5/)5(0(9+&1$%&$0/$0(9+4)&$',('(1+5+(##&00&< W,&#(#1$'/,$&;&.'0&$0/$'D,/1&$()*&,5&

1)+*/#5& G

@NC (x!y)→(Rxb→Ryb)

Q#5&,',252&$%/#1$)#&$3&,5/+#&$15,)35),&6 @NC %+56 '/,$&;&.'0&6 4)&$1+$0/$*/0&),$%&$0/$*/,+/80&$HxI$&15$0/

.J.&$4)&$0/$*/0&),$%&$0/$*/,+/80&$HyI6 /0(,1$1+x/+.&$j/,+&6 /0(,1y /+.&$j/,+&< L/$*2,+52$%& @NC #&

%2'&#%$'/1$%&$0/$*/0&),$%&1$*/,+/80&$HxI$&5$%(#3$%&$07/11+9#/5+(#$%&$*/0&),1$&#$4)&15+(#$\$4)&00&$4)&

1(+5$07/11+9#/5+(#$'/,5+3)0+>,&6 1+$&00&$/11+9#&$0&$.J.&$+#%+*+%)$M$3&1$%&);$*/,+/80&16 /0(,1$07/.(),$%&

j/,+&$#&$%+15+#9)&,/$'/1$0&),1$,2-2,&#51< E()1$*(:(#1$29/0&.&#5$4)&$0/$*2,+52$%& @NC #&$%2'&#%$'/1$#(#

'0)1$%&$07+#5&,',25/5+(#$'/,5+3)0+>,&$%&$HRI6 #+$%&$3&00&$%&$HbI$\$1+$#()1$07+#5&,',25(#1$3(..&$H 1+$0&1

*/0&),1$%&$HxI$&5$%&$HyI$1(#5$0&1$.J.&16 /0(,1$1+$)#$%&1$%&);$%2'&#%$%&$=+&)6 /0(,1$07/)5,&$&#$%2'&#%

29/0&.&#5 I6 &00&$&15$29/0&.&#5$*,/+&< L/$'D,/1& @NC &15$%(#3$*,/+&$%/#1$5()5&1$0&1$15,)35),&16 &5$1()1 5()5&1$0&1$/11+9#/5+(#16 %(#3$0(9+4)&.&#5$*,/+&$()$*/0+%&<

(11)

*+:> ;(0)04N0&%&4&%7.0 I+G

i)7&15F3&$4)+$3D/#9&$1+$#()1$',2";(#1 @NC %&$%&);$4)/#5+"3/5&),1$)#+*&,1&01$4)+$0+&#5$0&1$*/,+/80&1$HxI

&5$HyI$4)+$(#5$%&1$(33),,&#3&1$0+8,&1$%/#1 @NC o

@ZC ∀x∀y((x!y)→(Rxb→Ryb))

E()1$/*(#1$%2KM$*)$4)&$0/$*/0+%+52$%7)#&$'D,/1&$#&$%2'&#%$K/./+1$%7)#&$/11+9#/5+(#$%&$*/0&),1< E()1 /''&0(#1= 6$Z),4( * ,-17(43(X/ >%7)#&$-(,.)0&φ0/$-(,.)0&$(85&#)&$&#$/%K(+9#/#5$Mφ%&1$4)/#5+"3/F 5&),1$)#+*&,1&01$3(,,&1'(#%/#5$M$3D/4)&$*/,+/80&$%(#5φ3(#5+&#5$)#&$(33),,&#3&$0+8,&< L/$*/0+%+52$#&

%+15+#9)&$'/1$&#5,&$)#&$-(,.)0&$&5$1/$30U5),&$)#+*&,1&00& G

K"17'O2( GH: T-( * J04E,$( * (3) * 7#$1+( * 31 * () * 3(,$(E(-) * 31 * 3# * 6$Z),4( * ,-17(43(X( * (3) * 7#$1+(:

Wuvwxv ](+5A)#&$15,)35),&$/,8+5,/+,&<φ&15$*/0+%&$1+$&5$1&)0&.&#5$1+φ&15$*,/+&$%/#1A1()1$5()5&1 0&1$/11+9#/5+(#1$%&$*/0&),1$'(11+80&1< y5/#5$%(##2,H6 3&55&$3(#%+5+(#$&15$#23&11/+,&$&5$1)k1/#5&$'(),

0/$*2,+526 %/#1A6 %&$0/$30U5),&$)#+*&,1&00&$%&φ< !

E()1$*(:(#1$%(#3$4)& @NC 1&,/$*/0+%&$1+$&5$1&)0&.&#5$1+$1/$30U5),&$)#+*&,1&00& @ZC &15$29/0&.&#5$*/0+%&<

T&$5D2(,>.&$#()1$/'',&#%$4)&$#()1$'()*(#1$#()1$0+.+5&,$M$0/$3(#1+%2,/5+(#$%&$'D,/1&1$'(),$%23+%&,

%&1$4)&15+(#1$%&$*/0+%+52<

E()1$/*(#1$3(..&$&;&.'0&$%7)#&$24)+*/0&#3&$0(9+4)&$@3(#124)&#3&$0(9+4)&$,23+',(4)&C$0&$1)+*/#5 G ](+5φ)#&$-(,.)0&$3(#5&#/#5n(33),,&#3&1$0+8,&1$%&$*/,+/80&1$%+X2,&#5&1$Hx1I6 V HxnI G

A |qφ ⇐⇒ A |q"∀x1∀x2. . .∀xn(φ)#

@[C

Q0$&15$M$,&./,4)&,$4)&$3&55&$24)+*/0&#3&$&15$*,/+&$1&)0&.&#5$/)$#+*&/)$%&$0/$*/0+%+52 G 1+φ&15$*/0+%&6 37&15FMF%+,&$*,/+&$%/#1$5()5&1$0&1$15,)35),&1$&5$1()1$5()5&1$0&1$/11+9#/5+(#1$%&$*/0&),16 /0(,1$1/$30U5),&

)#+*&,1&00&$07&15$/)11+6 &5$*+3&$*&,1/< !)$#+*&/)$%7)#&$15,)35),&$&5$%7)#&$/11+9#/5+(#$'/,5+3)0+>,&6 )#&

-(,.)0&$&5$1/$30U5),&$)#+*&,1&00&$'&)*&#5$5,>1$8+&#$%+X2,&,< ](+5A6 '/,$&;&.'0&6 )#&$15,)35),&$%(#5 07)#+*&,1$%&$%+13(),1$1(#5$5()1$0&1$/#+./);$&5$HP xI$&15$+#5&,',252$'/,$07&#1&.80&$%&$5()1$0&1$'+#9()+#1<

]()1$)#&$/11+9#/5+(#$'/,5+3)0+>,&$\$)#&$/11+9#/5+(#$4)+$/11+9#&$)#$'+#9()+#$M$HxI6 HP xI$1&,/$*,/+&<

j/+1$3&3+$#&$*&)5$'/1$%+,&$4)&$5()1$0&1$/#+./);$1(#5$%&1$'+#9()+#16 #+$4)&6 1()1$3&55&$/11+9#/5+(#6 H∀x(P x)I$1&,/$29/0&.&#5$*,/+&<

*+:> ;(0)04N0&%&4&%7.0

L/$#(5+(#$%7/11+9#/5+(#$%&$*/0&),1$*/,+2&$M$0/$'0/3&$HxiI$@3-< '<ZBZC$#()1$'&,.&5$%&$%+,&$4)/#%$)#&

/11+9#/5+(#$%&$*/0&),1$1/5+1-/+5$)#&$'D,/1&$)#+*&,1&00&.&#5$()$&;+15&#5+&00&.&#5$4)/#5+"2&< Q.+5/#5 0/$',(32%),&$%&$1)815+5)5+(#$'(),$0/$0(9+4)&$',('(1+5+(##&00&$@3-< '<NZBC6 #()1$'()*(#1$%2"#+,$%&1

#(5+(#1$/#/0(9)&1$&#$1:#5/;& G 0/$1)815+5)5+(#$%7)#$5&,.&$'/,$)#$/)5,&$5&,.&$%/#1$)#$5&,.&$()$)#&

-(,.)0&<

<1=.%&%7. GC@])815+5)5+(#1$%/#1$%&1$5&,.&1C: P1*=s>*()*=t>*30-)*+(3*)(4E(3*()*=xi> * ,-( * 7#41#'$(@-0,3 +5C-1330-3*,-*-0,7(#,*)(4E(@2,(*-0,3*#%%($0-3*= $# *3,'3)1),)10-*+(*=xi>*%#4*=t>*+#-3*=s> >*0,*= =s(xi/t)> >@

+( * E#-1M4( * 456,4317(@ 60EE( * 3,1A 8

9: P1*=s> * (3) * $# * EUE( * 7#41#'$( * 2,( * =xi>@ #$043*=s(xi/t)>*(3)*=t>:

I: P1*=s> * (3) * ,-( * 7#41#'$( * #,)4( * 2,( * =xi>@ #$043*=s(xi/t)>*(3)*=s>:

K: P1*=s> * (3) * ,-( * 60-3)#-)(@ #$043*=s(xi/t)>*(3)*=s>:

R: P1*=s> * (3) * ,- * )(4E( * %0,4 * ,-( * 7#$(,4 * +( * J0-6)10- * =fj(t1, . . . , tµ(j))>*%0,4*+(3*)(4E(3*=t1>@ Q@ =tµ(j)>@

#$043*=s(xi/t)>*(3)*=fj(t1(xi/t), . . . , tµ(j)(xi/t))>:

(12)

I+D *+: ,-.&#/()(&)012#.&%34()5()6#)678%34()5(0)$'15%9#&0

Y#$8,&-6 5()5&$(33),,&#3&$1'4/%&$HxiI$%/#1$HsI$&15$,&.'0/32&$'/,$HtI< E()1$'()*(#1$./+#5&#/#5

%2"#+,$3&$4)7&15$0/$1)815+5)5+(#$%7)#$5&,.&$'/,$)#$/)5,&$%/#1$)#&$-(,.)0& G

<1=.%&%7. D+@])815+5)5+(#1$%/#1$%&1$-(,.)0&1C: P1φ(3) * ,-( * J04E,$(@ =t> * ,- * )(4E( * () * =xi>*,-/

N#41#'$(@ -0,3 * +5C-1330-3 * ,-( * -0,7(X( * J04E,$(@ 2,(*-0,3*#%%($0-3*= $( * 453,$)#) * +( * $# * 3,'3)1),)10- * +( * =xi>

%0,4*=t>*+#-3φ>*0,*="φ(xi/t)#>@ +( * E#-1M4( * 456,4317( * 60EE( * 3,1A 8

9: P1φ(3) * =t1!t2>*%0,4*+(,L*)(4E(3*=t1>*()*=t2>@ #$043"φ(xi/t)#(3) * =t1(xi/t)!t2(xi/t)>:

I: P1φ(3)*=Ri(t1, . . . tλ(i))>*%0,4*,-*31.-(*+(*4($#)10-*=Ri>*()*+(3*)(4E(3*=t1>@Q=tλ(i)>@ #$043"φ(xi/t)# (3) * =Ri(t1(xi/t), . . . tλ(i)(xi/t))>:

K: P1φ(3A"¬ψ#%0,4 * ,-( * J04E,$/ψ@ #$043"φ(xi/t)#(3A"¬ψ(xi/t)#:

R: P1φ(3A"ψ∧χ#@"ψ∨χ#@"ψ → χ#0,"ψ ↔ χ#%0,4*+(3*J04E,$(3ψ(Aχ@ #$043"φ(xi/t)#(3A

"ψ(xi/t)∧χ(xi/t)#@"ψ(xi/t)∨χ(xi/t)#@"ψ(xi/t)→χ(xi/t)#0,"ψ(xi/t)↔χ(xi/t)#4(3%(6?

)17(E(-):

[: P1φ(3A"∀xj(ψ)#0,"∃xj(ψ)#%0,4 * ,-( * J04E,$/ψ() * ,-( * 7#41#'$( * =xj>@ #$043

"φ(xi/t)#:=% "∀xjψ(xi/t)# i(=j

φ i=j "φ(xi/t)#:=% "∃xjψ(xi/t)# i(=j

φ i=j

Y#$8,&-6 #()1$,&.'0/?(#1$5()5&$(33),,&#3&$0+8,&$%&$0/$*/,+/80&$HxiI$'/,$0&$5&,.&$HtI< ]+$HxiI$#7/$/)3)#&

(33),,&#3&$0+8,&$%/#1φ6"φ(xi/t)#&15$0/$.J.&$-(,.)0&$4)&φ< ]+$#()1$'()*(#1$J5,&$/11),21$9,z3&$/) 3(#5&;5&$4)&φ3(#5+&#5$/)$.(+#1$)#&$(33),,&#3&$0+8,&$%&$HxI6 #()1$23,+*(#1"φ(a)#'(),"φ(x/a)#G

"φ(a)#&15$07#%%$16#)10F%)$',2%+3/5"φ(x)#Ma<

Y#$1)815+5)/#5$%&1$5&,.&1$'(),$%&1$*/,+/80&1$M$0&),1$(33),,&#3&1$0+8,&16 #()1$%&*(#1$-/+,&$/55&#5+(#$M

#&$'/1$0+&,$)#&$*/,+/80&$4)+$#&$0725/+5$'/1$+#+5+/0&.&#5< ](+5φ0/$-(,.)0&$H∃x(x(!y)I< ]+$)#&$15,)35),&

/$)#$%(./+#&$3(#5&#/#5$'0)1$4)7)#$1&)0$+#%+*+%)6 +0$:$/),/$5()K(),1$)#&$/11+9#/5+(#$%&$*/0&),1$M$HyI 4)+$1/5+1-/+5$3&55&$'D,/1&$()*&,5&< ]+$#()1$1)815+5)(#1$)#&$/)5,&$*/,+/80&$HzI$'(),$HyI$%/#1φ6 #()1 (85&#(#1$H∃x(x(!z)I6 -(,.)0&$4)+$&15$29/0&.&#5$1/5+1-/+1/80&$%/#1$5()5&1$0&1$15,)35),&1$M$'0)1$%7)#

+#%+*+%)$\$0&$,21)05/5$%&$0/$1)815+5)5+(#"φ(y/z)#1&,/$*,/+$1+$&5$1&)0&.&#5$1+φ07&15$29/0&.&#5<

Y#$1)815+5)/#5$HyI$'/,$HxI$#()1$(85&#(#1$)#$/)5,&$,21)05/5 G 0/$-(,.)0&$H∃x(x(!x)I$&15$)#&$'D,/1&

3(.'0>5&$&5$4)+$#7&15$*,/+&$%/#1$/)3)#&$15,)35),&< L&$%+/9#(15+3$%&$3&$3D/#9&.&#5$&15$4)&$07(33),,&#3&

%&$0/$*/,+/80&$HyI6 0+8,&$%/#1φ6 /$252$0+2&$%/#1"φ(y/x)#\$)#$1+.'0&$3D/#9&.&#5${5&,.+#(0(9+4)&7$/$,2F

%)+5$0&$#(.8,&$5(5/0$%7(33),,&#3&1$0+8,&1$%&$*/,+/80&1< E()1$%&*(#1$&;30),&$3&$3/1$'(),$0&1$1)815+5)5+(#1 /%.+11+80&1<

E()1$%+1(#1$/0(,1$4)&$0/$*/,+/80&$HxI$%/#1φ#725/+5$'/1$1'4(*%0,4*=y>&5$/%('5(#1$0/$%2"#+5+(#$1)+*/#5& G

<1=.%&%7. D*@])815+5)5+(#1$/%.+11+80&1C: P01Aφ,-( * J04E,$(@=t>*,-*)(4E(*()*=xi> * ,-( * 7#41#'$(:V0,3 +130-3*2,(*=t>*(3)*$1'4(*%0,4*=xi>*+#-3φ31*$",-(*+(3*%0331'1$1)53*3,17#-)(3*(3)*$(*6#3 8

9: φ(3) * ,-( * J04E,$( * #)0E12,/ B

I: φ(3A"¬ψ#() * =t>*(3)*$1'4(*%0,4*=xi>*+#-3ψB

K: φ(3A"ψ∧χ#@"ψ∨χ#@"ψ→χ#0,"ψ↔χ#%0,4*+(3*J04E,$(3ψ(Aχ() * =t>*(3)*$1'4(*%0,4*=xi>

+#-3ψ() * +#-3χB

R: φ(3A"∀xj(ψ)#0,"∃xj(ψ)#() * =xi>*-"#*%#3*+"066,44(-6(*$1'4(*+#-3ψ:

[: φ(3A"∀xj(ψ)#0,"∃xj(ψ)#@ =xj>*-"#*%#3*+"066,44(-6(*+#-3*=t>*()*=t>*(3)*$1'4(*%0,4*=xi>*+#-3ψ:

T&55&$%2"#+5+(#$#()1$/'',&#%$4)7)#$5&,.&$HtI$&15$0+8,&$'(),$)#&$*/,+/80&$HxiI$%/#1$)#&$-(,.)0&φ17+0

#7:$/$'/1$%&$*/,+/80&$HxjI$%/#1$HtI$&5$/)3)#$4)/#5+"3/5&),$H∀xjI$%/#1φ5&0$4)7)#&$(33),,&#3&$0+8,&$%&

HxiI$&15$%/#1$0/$'(,52&$%&$H∀xjI< |&',&#(#1$0&$3/1$%&$%&);$*/,+/80&16 (t$0/$1)815+5)5+(#$%&$HyI$'/,$HxI

,-.&#/()(&)012#.&%34()5()6#)678%34( 5(0)$’15%9#&0

!"#$%&'()*+

,-.&#/()(&)012#.&%34()5()6#)678%34(

5(0)$'15%9#&0

*+:* ;()6#.8#8( L

*+:I ;#)012#.&%34()5()6#)678%34()5(0)$'15%9#&0

*+:? ;#)M#6%5%&1

*+:> ;(0)04N0&%&4&%7.0

+: ;()6#.8#8( L