A Logic You Can Count On
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Logic You Can Count On Silvano Dal Zilio — Denis Lugiez — Charles Meyssonnier. N° 5022 November 2003. ISSN 0249-6399. ISRN INRIA/RR--5022--FR+ENG. THÈME 1. apport de recherche.
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(35) * 1 5 . , 98 ' =c$4U &m ;B = , AC)=)" ' ,-f98J = 8$,-A 4Q2* ¬((title, year )[>]) | >) * #6'5 0 a $# e>24U 1@ =6,-.CAC. 4 # @ =() 5 , a 4 12# 8$,?) # (; 5 :0,- 4 ) ! = ' '= , # ' =c04U #R5 &1 * ' ,R>V 1E &1 8 5 $#$ a 4 1# AJ( 5 $# title , year B. ;. article[title[>] | author [Cardelli[ ]] | year [1998[ ]] | ¬((title, year )[>] | >)] .. éÛ.×éÅè.
(36)
(37) M. ¡ qn]aGlMHUVWYK;M½jhqlM+df[]VGJdYXl?G?]D M+o dYXJmmlM JqlM+W*yzWfhm[h]`_noLcVhj]VM>cVhdYq}d?cC{lh{lhjd?YcpWfklGlhoLWYYM>q?]acVhM>U W?U+]VGlM}d?cVKXlW5F `O¦ article[title[>] | author [>] | ¬((title[>] ∨ (year [>] | year [>])) | >)] . ©JdYcaKXlW'`O¦ UVk#MDo hjJM>U]VG#Wz];Wiy¦WYjhm¢{lh{lhjd?YcpWfklGJho>Wf9M>q?]acV_§KXJU`];o dYq1]pWfhq \ $64U. d?qlM#MLm Wf{!MLM>m title 4 # dYqJMJM>mWf{!MLM>m author ¥WfqJm AC # dYqlMn#MLmWY{#M>jMDm year !k#d1UVUVhj{Jj_ WYjd?ql?UVhm[M hj]VG§Ud?K;M;df]aGlMLcNXJqJUk!M>oLh#M>mJ ^JM>mlU>FHGlMDUMNo d?qJU]VcpWfhq?]pU^oLdYXlmi{!MNM Z[klcaM>UaUMDmKNdYcaM mlhjcaM>o ]V_XJUVhjqlN]aGlMUVGlM>W¦y?M>mo dYKNk!d?UVh]ahjd?q0qJdf]aWf]VhdYqdf<UMDo]ahjd?qvljO?[{_\UVW¦_hql;]VGJWf]^WNyzWfhmMLq1]acV_ KXJU]H{!Mnhqd?qlM}df.]VGlMUVM ]p0U F (nt , na , ny , no ) ([[title[>]]], [[author [>]]], [[year [>]]], [[(title, author , year )⊥ [>]]]) ,. JG MLcaM n , n¡ , n , n WYcVM}hq1]VML?MLc5yzWfcahWf{lM>UHUVXJopG]VGJWf] (n = 1) WfqJm (n > 1) WYqJm (n 6 1) WfqJm qlMNdf9]aGlMNK¤WfhqoLdYq1]Vcahj{JX[]VhdYqJUdf]VGlhU}caMLk!dYcV]hUQ]Vd(UGld ]VGJWf]nWYq_(1XJWYq1]VhjJMLcV,caMLM (n. F ",>d?cVK0)XlWo>Wfq§{!MM Z[klcaM>UaUMDmhjq§]aGlM\UaWfKNM W¦_?]aGJWz]hU>.W?U}]VGlM\klcVd[m[X#o]df5hq1]VML?MLcpUnyYMDo]adYcpU ,m[M #qJWf{lMQhjqb9caM>UV{lXlcaYMLcHWfcahj]VGlKNM ]aho>Ua ½{_UMD1XlMLqJoLM>U+dfMLMLKNMLq1]+dYcaKXlWYUQÅ]aGJWz]5hULd?cVKXlW?U½dY ]aGlMnhjq#m α[A] P²q]VGlMNqlMLZ]nUVM>o ]VhdYq M;cVMDoLWf.]VGlMN{JW?opYcadYXlqJmidYqwb½cVMDU{lXJcV?MLc}Wfcah]aGlKNM ]Vho;WfqJmUM>KNhjhjqJM>Wfc UVM ]pUH]VG#Wz] MnqlM>M>m(hqdYcpm[ MLc5]ad\m[MLJqlMn]aGlM ,GlMDW¦yYMDU "d?Yho¤ , ". +dYcaK¤Wfj_?#Wfq#m\]aGlMLqi]VdklcadzyYM}hj]aU MD1XlhjyzWYjM>qJo M hj]VG]VGlM1XJWYq1]VhjJMLcV,caMLMcpWfYKNM>q?]5dY. F "9 t. a. y. o. t. a. y. o. % %p + . b½cVMDU{lXJcV?MLc^WYcVhj]VGlKNML]VhohU*]aGlMJcpU]÷dYcpm[MLc5]aGlMLd?cV_df9MD1XJWfh]`_dzyYM>c5]VGJMYcadYXJk +) df9qJWf]VXlcpWf q XlK{#M>caU hj]VGW?mlm[hj]VhdYq K v"T N nb½cVMDU{lXJcV?MLc*d?cVKXlWYU; WfUVd0oLWYjM>mo d?qJU`]acaWYhjq1]pUa 5(N, WYcVM;m[M>Uao cah{#MDmihq ]aGlMdYjd hqln]pWf{lMY GlM>cVM M, N, . . . cpWfql?M*dzyYM>c½k!d?UVh]ahjy?Mhq1]VM>YMLcyzWfcahWY{ljMDUWYqJm m, n, . . . caWYqlYM dzy?MLcHk!d?UVh]ahjy?MQhq1]VM>YMLc5oLdYqJU]aWYq?]pUL #¬ ° >Lz > ¬ > P²q?]aML?MLcHM Z[klcaM>UaUhdYq Exp ::= k!d?UVh]ahjy?MQhq1]VM>YMLc5oLdYqJU]aWYq?] n k!d?UVh]ahjy?MQhq1]VM>YMLcHyzWfcahWf{lM N WYmJm[h]ahjd?q Exp + Exp 9 b a c > M UV{lXlcaYMLc5WYcVhj]VGlKNML]Vho^d?cVKXlWYU φ, ψ, . . . ::= V ] D M U`]5d?c5M>1XJWYjhj]`_ (Exp = Exp ) l q > M ?Wz]ahjd?q ¬φ [ m h U e`Xlq#o]VhdYq φ∨ψ. M [ Z h U`]aMLq1]VhWf1XJWYq1]Vhj#oLWf]VhdYq ∃N.φ 1. 1. 2. 2. b9caM>UV{lXlcaYM>c<o dYq#U`]acaWYhjq1]aU.K¤W¦_{!M5XJUVM>m]Vdm[MLJqlM*W}UVXl{JU]aWfq1]ahWYlo WYUaUdY%,m[M>oLhmJWf{lMD .klcadYk!MLcV]VhM>U dzy?MLck!d?UVhj]VhyYM9hq1]VML?MLcpULDhj?M9dYcM ZlWfKNklM `]VGlMyzWfXlM½dY M hU.U`]acVho]aj_QYcaM>Wf]VMLc]VGJWYqn]VGlMyzWfXlM½dY N
(38) p ×× Ýo=p.
(39) B 74)4U * B <EE B
(40) . #&# = ,. ODu. . X#Uhql}]VGJMHdYcaKXlW _?dYc hU9WYqNdmJmqXlK{#M>c
(41) !P²q ]aGlhUCcaMLk!dYcV]> M½XJUVM½∃X.(M b9caM>UV{lXlca=YM>Nc¥+dYcaXKXJ+W?1)U¥]VdM Z[klMcaM>UaUCWYcVhj]VGlKNML]VhoLWf1o d?qJU`∃X.(M ]acaWYhjq1]pUdz=yYM>XcK+XJX]ahj+kl1)ho hj]VhM>U dYKXl]ahUVM ]pU5df%MLMLKNM>q?]pUL FHGlcadYXJYGld?X[]9]VGJM*]VMLZ] M^XJUVM*]aGlM^y?M>o]adYcqldf]pWz]VhdYqC ?d?c9]aXlklM>U+df¥hq?]aML?MLcpULWfqJm dYc]VGlM UVh dLMNqXlK{!MLc5dYM>jM>K;M>q1]aUp ½dY S M}XJUM]VGJMnqldf]pWz]VhdYq#nU φ(N) d?cHW;b9caM>UV{lXlcaYM>c½d?cVK|S| XlW Gld1UM caMLM^y¦WYcVhWf{JjMDU½WfcaM^WYj#hjq N = (N , . . . , N ) WYqJm |= φ(n , . . . , n ) GlM>q φ{N ← n } . . . {N ← hU^UVWf]VhU`JMDm¥ n } FHGlMQm[MLqJdf]aWf]VhdYq [[φ(N)]] dfCWb9cVMDU{JXlcV?MLc<d?cVKXlW φ(N) hU½]VGlM}UML]dYhjq1]VM>YM>c½yYMDo]adYcpU n UVXJopG ]aGJWz] |= φ(n) +b9cVMDU{JXlcV?MLc5Wfcahj]VGlKNM ]aho}hU*WYqhq1]VM>cVMDU`]ahjqlNMLZ[WYKNkljMnhqo d?K;kJX[]aWf]VhdYqJWYCo d?K;kJjMLZhj]`_ ]aGlMLd?cV_{!M>oLWYXJUVMhj]QhU^d?qlMdY9]VGJMM klcVd?{lMLK ]VGJWf]QkJcVdzyzWf{Jj_qlMLMDmKNdYcaM]aGJWfqik!dY_qldYKNhWf.cVXJq ]ahjKNM KjOVO2N3F.M>yYM>cV_¤WfYd?cVhj]VGJK GlhopG0mlM>o hm[MDU9]aGlM^]acVXl]VGdfCWb9cVMDU{JXlcV?MLc+o dYq#U`]acaWYhjq1] φ ]VG#Wz]HhU]VMDU`] d?cUd?K;M¤oLdYqJU]aWYq?] c GlMLcaM n hU}]VGlM MLqJGJfM ]V]VGJGMLdfc [[φ]].FH=GJML∅caM½GJhUWYWYUnUVWd^caW*Xlq1]Vqlhd KNM¤q}df]a+cVhWfkl]_}jMDM WYZ[Uk#] d?2ˆ(2ˆ(cn)) l q L M 1 q a ] h Y W ? l X J k # k M>c÷{#d?XlqJmnhq]aGlM dYcpU]oLW?UM K Ev D2N ¦]aGJWz] hUdYc<WfqXlφq{#d?XlqJm[MDmWY]aMLcaqJWz]ahjd?qdY#1XJWfq1]VhjJM>caU F¥]VGJMHoLdYKNklM Z[hj]`_ndf#opGlMDophjql^ ]VGlM5UaWz]ahU#Wf{Jhjh]`_ndY W^dYcaKXJW hU%hjq]VhKNMHWf].KNd1U`] <FHGlM+klcadY{lMLKhU%x*b9÷oLdYKNklM ]VM+d?c]aGlM5M Z[hU`]aMLq1]VhWf cpWf?KNMLq1]+dYφ.b½cVMDU{lXJcV?MLc5Wfcah]aGlKN2ˆ(2ˆ(2ˆ(pn))) M ]ahoY 1. p. 1. p. 1. 1. p. p. . " %
(42) ) . M>oLhmJWf{lhjhj]`_dY#b9caM>UV{lXlcaYM>c%WYcVhj]VGJK;ML]Vho+KNW¦_{#M5klcadzyYMDmXJUVhqlnWQoLdYqlqJM>o]ahjd?q hj]VG`#0 A )4U) , #0 # dYqJWf]VXlcpWf<qXlK{#M>caU> ¨ 4U) , #0 dY N L(b, P ) ChUQ]VGlMUVM ]ndfHyYMDo]adYcpU^ ?MLqlM>caWf]VM>m{_ihjqlMDWfc oLdYK{lhjq#Wz]VhdYq(dY]aGlMk!MLcahjd[mlU P = p , . . . , p h]aG p ∈ N dYcWf i ∈ 1..p hj]VG(]aGlM{JW?UM. b∈N. n. n. F. 1. L(b, P ) =. . k. b+. X. n. i. λi p i λ1 , . . . , λ k ∈ N .. ¨ #0 A )4U) , #0 hUW\Jqlhj]VM;XlqlhdYqdf½hqlM>WYcQUVM ]pUL ,M>K;hhjqlMDWfc}UML]aU}WfcaMMLZlWYo]aj_]aGlMKNd[m[M>U dY*b9caM>UV{lXlcaYMLcWfcah]aGlKNM ]Vho\dYcaKXlWYU>C]aGJWz]NhUL%]aGlMUVM ];df*hq1]VML?MLcyYM>o ]Vd?caUUVWf]VhU`_hqlwWd?cVKXlW hU;WiUVMLKNhjhqlM>WYcUVM ];df 9WfqJm¢oLdYqyYM>caUVML_Y ,M>KNhjhjqJM>Wfc;UML]aUWfcaM\o d?UVM>m§Xlq#m[MLc φ(N , . . . , N ) W?mlm[hj]VhdYqA h]aG L + M = x + y Nx ∈ L, y ∈ M 9XlqJmlMLcN{JWYUVho\UVM ]NdYk!MLcpWz]VhdYq#U0 XlqlhdYq hq1]VM>caUVM>o ]VhdYqJUVM ]Qo dYKNklMLKNM>q?]DC>La \WfqJm(XJqJm[MLc ) ,R6 hj]VG L = L + · · · + L n ]VhKNM>U> S WYqJm P²q]VGlM0oLWYUVMNdf5hj]VMLcpWz]ahjd?q]aGlMUVMLKNhjhqlM>WYcUML] K¤W¦_i{!M\W(XlqJhjd?qdY MLZ[k#d?LqlMLq1=]VhWf _K¤Wfq_LhjqlMDWfc5UVM ]aUhq0]aGlMnqXlK{!MLc5dY.hqlM>WYcHUVM ]pU5hq L L 98 N * = J#0 # L + M * ° ° °
(43) ° , =. #0 A )4U) , #$ # L, M $ 8 , . . , 4 7 A ) 4 ) , 8 B 1 L # #0 #0 # N L k∈N b9caM>UV{lXlcaYM>c\o d?qJU]VcpWfhq?]pUWfq#m UM>K;hhjqlMDWfc\UVM ]pU\WYcV M $6)> 4U. M>1XlhyzWfMLq1]>H]aGJWz]0hU>5YhyYM>q W b½cVMDU{lXJcV?MLcNo dYq#U`]acaWYhjq1] φ +hj]hUNk!d?UaUh{lM]Vd¢o d?K;kJX[]VM]VGJM{#WYUVM>UNWfqJmÿk!MLcahjd[mlUNdYQW§UM>KNhjhjqJM>Wfc UVM ]¤caMLklcaM>UVMLq1]ahjql HWfq#mÿoLdYqyYM>caUVML_Y'Fd{JXlhjm WUVMLKNhjhqlMDWfc¤UML]o dYcacaM>UVk#d?qJm[hqli]adWYhyYM>q b½cVMDU{lXJcV?MLcHo dYq#U`]aca[[φ]] WYhjq1]>hj]*hUHM>qldYXJYG0]VdNk#M>cd?cVK ?X#Wfq1]VhjJMLcHM>jhKNhjqJWf]VhdYqdYq0]VGlMoLdYqJU]VcpWfhq1] Kï\v TVN W;k!d?UaUh{l_M Z[k!MLqJUVhyYM}klcadoLM>m[XJcVM? 1. i∈1..k. p. p. n. ∗. n∈N. ∗. n. p. k. ∗. p. éÛ.×éÅè.
(44)
(45) . ) -% . . O?O. %)
(46) ). +Xlhm[hjqJQXJk#d?q]aGlM*o d?qlqlMDo]VhdYqN{!M ] MLM>q;UVMLKNhjhqlM>WYc9UML]aU<WYqJm;b9cVMDU{JXlcV?MLc<Wfcahj]VGlKNM ]ahoY M*oLWfq;hÅ] ]aGlMUVXlK WfqJm(hj]VM>caWf]VhdYqd?k#M>caWf]VdYcpUHdY qiUVMLKNhjhqlMDWfcUVM ]pU5]ad¤]VGlMjM>yYM>Cdf%]aGlMdY?hoYl]VG#Wz] hU> MoLWYq mlMLcahjy?M^o d?qlqlM>o ]Vd?caU WfqJm UX#opG¤]VGJWf] lWfqJm <P²q]aGlMdYjd hqlJ 0:-+ [[ψ]] 06 ∧ [[φ)A ' ]]4U = &[[φ]] 6 → JWYqJm )> , # 4 hjy?M>U>= &[[φ]] =M}K¤ mWf a YM} &XJ6 UVMn df]VGl+MnX#UXJWYC∗m[M>cVhyYMDm0oLdYql[[φqlMD+o]aψ]] ∀ ¨ UaUXlKNM N, N WYqJm N WfcaMQm[hU e`dYhq?]^UVM>1XlM>qJo MDU+df%y¦WYcVhWf{JjMDU F ∗. 1. ∗. 2. (φ + ψ)(N) = ∃N1 , N2 . (N = N1 + N2 ) ∧ φ(N1 ) ∧ ψ(N2 ) .. P÷] hj<WfUVd0klcadzyYMXJUMLXl%]Vd(m[MLJqlMYW?UQhjq F "9E]VGJM;W?m¦e`XlqJo ]}dYUVXlK(¥]VGJWf]QhU}W0oLdYqlqJM>o]adYc VU XJopG(]aGJWz] ξ ` φ B ψ hj½WfqJmd?ql_hj ξ + φ ` ψ GlMLcaM]aGlMMLq1]pWfhjKNMLq1]QcVM>Wf]VhdYq φ ` ψ !KNM>Wfq#BU ]aGJWz] [[φ]] ⊆ [[ψ]] FHGJMd?jd hql0m[MLJqlhj]VhdYq§hU}cVM>K;hqlhUVoLMLq1]}df½]aGlM¤o d?cVcaM>UVk#d?qJm[M>qJo M{#ML] M>MLq B WYqJm\]VGlMnhjqJM>Wfc5hKNklho>Wz]VhdYq(o d?qlqlMDo]Vd?c5df.jhqlM>WYc5jd?Yhof GlhopG W?U5WfcVMDWYm[_¤KNMLq1]ahjd?qlM>m0hjqK MN÷ ¨ UaUXlKNM. N, N1. WYqJm. N2. WfcaMQm[hU e`dYhq?]^UVM>1XlM>qJo MDU+df%y¦WYcVhWf{JjMDU F. (φ B ψ)(N) = ∀N1 . φ(N1 ) → ∃N2 .(N2 = N1 + N) ∧ ψ(N2 ) .. FHGlMd?jd hjqJklcadYk!MLcV]`_¢U`]pWz]aM>U]VGlMUVdYXlq#m[qlM>UaU;df]VGlMDUM(MLq#o d[m[hql?UNWfq#m'caMLWz]VMDU B h]aG W UVXl{[]acaW?o]ahjd?q0d? k#M>caWf]VhdYq\dzy?MLc+]VGJMnk#d MLcpUML]+dY N 1 ψ , , -# c0,m ,8$,-A74 # *f = [8$, > ,-. #0 S 8 ° ° ° ## A φ p. ) , > $(, # * % > . S ⊆ [[φ + ψ]] ⇔ S ⊆ [[φ]] + [[ψ]] , S ⊆ [[φ B ψ]] ⇔ S + [[φ]] ⊆ [[ψ]] .. °°
(47) l© d?c½]aGlM^#caU]MD?XJhjyzWfMLq#o MY M^klcadzyYM*]VGJWf] [[φ]] + [[ψ]] = [[φ + ψ]] ¨ UVUVXlKNM n hU+WyYMDo]adYc MG#W¦yYM n ∈ [[φ]] + [[ψ]] hj^Wfq#m§dYql_hjH]VGlM>cVM0M Z[hU`]] dyYMDo]adYcpU n Wfq#m n UXJopG§]aGJWz] N Wfq#m Wfq#m l]VG#Wz]5hU>lhj%WYqJm0dYql_\hj n ∈ [[φ + ψ]] n=n +n ©JdYc%]VGJM*UMDo dYq#mnMD?∈XJhjyz[[φ]] WfMLq#o MYYhn]9hU%∈MLql[[ψ]]d?XlYG]VdnkJcVdzy?M9]VG#Wz] n hq [[φBψ]] hjEWYqJm;dYql_hj n+[[φ]] ⊆ . + _ [ m L M J l q j h V ] h Y d q Y d M J G ¦ W Y y M j h q j h 9 f W J q m ? d l q \ _ j h < Y d c Y W j !]aGlMLcaMhU n hq [[ψ]] UVXJopG0]VGJWf] B ]aGJWz]5hULn [[φ B ψ]] #W?UqJMLM>mlM>m¥ n ∈ [[φ]] [[ψ]] n+n =n n + n ∈ [[ψ]] P÷]hUWfUVd0k!d?UaUh{lM;]Vdm[MLcahyYM¤W0dYcaKXlWd?cnh]aMLcpWz]ahjd?q φ UVXJopGw]aGJWz] [[φ ]] = [[φ]] ]VG#Wz]hUL h9WYqJm(dYql_\hj hU5]aGlMUVXlK dY%W¤Jqlhj]VMqXlK{#M>c^df%yYMDo]adYcpU5UVWf]VhU`_hql M]pWf?M}]VGlM |= φ (n) M>KNk[]`_UVXlK ]Vd0U`]pWfqJmd?nc*]VGJMqXlCyYMDo]adYcD 0 = (0, . . . , 0) *FHGlMoLdYqJU]VcaXJo ]VhdYqhU^φ1Xlhj]VMoLdYKNklM Z¥ WYqJm0cVMD?XJhjcaM>U+hq0]aGlM}JcpU`]5klWYoLM}]Vd¤o d?KNklX[]VM}]aGlMn{JW?UMDU+WYqJmk#M>cVhd[mlUHdf]VGlMUVMLKNhjhqlMDWfc*UVM ] [[φ]] . hq. p. 1. 1. 2. 1. 2. 2. 1. 1. 2. ∗. ∗. ×× Ýo=p. 2. 1. ∗. ∗.
(48) B 74)4U * B <EE B
(49) . #&# = ,. O¦v. . ¨ UaUXlKNM φ hU*WdYcaKXJWNUVXJopG\]aGJWz]. F. hj]VG. Pi = pi,1 , . . . , pi,li P P ∃µi , λi,j . N = i6l µibi + j6li λi,j pi,j V W ∧ i6l j6li λi,j 6= 0 → µi 6= 0. φ∗ (N) = . [[φ]] =. S. L(bi , Pi ). i∈1..l. P²q1]VXJh]ahjy?ML_Y#]VGlMnd?cVKXlW oLdYqJU]VcpWfhq1]aU+]aGlMy?M>o]adYc ]Vd{!Mn]aGlMUVXlK df<MLMLKNMLq1]pU5]aWYYM>q cadYK S L(b , P ) GlMLcaφMw(n) WYq MLMLKNMLq1]0hq L(b , P ) Ånd?c0WYj i ∈ 1..l ¤hU\{1_ m[MLJqlhj]VhdYq W UVdYX[]ahjd?q0dY φ ° ° ° ## A # , $# c$,m ,8$,-A 4Q ) 8&, & >2,- Ec04 $# *f 5 8` 1 =4U. 8e 5 , [ # # φ a =) e#0 ; (n , . . . , n ) 8 p 4 A # ) N # = |=φn(n)= P 8$, 4)4 B ∗. i. i∈1..l. 1. ni. i. i. i. ∗. 1. |= φ(ni ). p. k. i ∈ 1..k. °°
(50) "CM ] φ {#M^W}b½cVMDU{lXJcV?MLc.dYcaKXlW}m[MLqJdf]VMDmN{_n]VGlMUM>K;hhjqlMDWfc9UVM ] S L(b , P ) hj]VG dYc*WYj P = ¨ UVUVpXlKNM, .|=. . ,φp (n) *FHGlMLcaMiWf∈caMn1..l hjq1]aMLYM>caU µ dYc i ∈ 1..l 5WYqJm λ Åd?c i ∈ 1..l WfqJm j ∈ 1..l WfqJm W P P UVXJopGw]VGJWf] d?cnWf i ∈ 1..l → µ 6= 0 "CM ]V]Vhql I =n =i ∈ 1..l µµb 6=+0 MGJλW¦y?Mp µ = λ = 0 d?λcWf 6=i ∈0 1..l !WYqJm]VGXJU \ I j ∈ 1..l ©ld?c P P P EjML] EWYqJm¥¥hj n= b + M ] n = . .µ. b= n+ = b뽩lpd?c^Wf i ∈ Ii#∈Wfq#Im\d?c*Wfn j =∈ 1..µ MnGJW¦λyYM np ∈ L(b , Pµ) #>Wfq#2m ]aGXJU |= φ(n ) GJhjM n = P µ b + P λ p = P P n ©JdYc}]VGJMo dYqy?MLcpUM?WYUaUXlKNM;]VGJMLcaMNWYcVM n , . . . , n UVXJopG]VGJWf] n = P n WYqJm |= φ(n ) d?c5Wf h ∈ 1..k <©ld?c+MDWYopG h ∈ 1..k n KX#U`]H Wf¥hq?]ad;Wf]5jMDWYU]Hd?qlMQdfC]aGlM L(b , P ) dYc i ∈ 1..l UVd]aGlMLcaMQhU+WUVM>1XlM>qJo MdfChq1]VM>YMLcpU µ UXJopG\]VGJWf] M}oLWfq0m[M>oLdYKNk#d1UM^]VGlM}UVXlK P n hq1]Vd n = P P n GlMLcaM n ∈ L(b , P ) dYcnWf i ∈ 1..l h ∈ 1..µ FHGXJU>E]VGlM>cVM WYcVMQhq?]aML?MLcpU λ d?c i ∈ 1..l h ∈ 1..µ Wfq#m j ∈ 1..l lUVXJopG0]VGJWf] n = b + P λ p _hMLm[hql n = P P b + P λ p ]aGJWz]5hU n = P µ b + P λ p hj]VG λ d?=c*PWf λ M}o d?qJo XJm[MQUhKNkl_¤{1_NqldY]Vho hql]VGJWf]d?c+WYj i ∈ 1..l 1hj µ = 0 ]aGlMLq λ =0 j ∈ 1..l P²q\]VGlMQWfK{JhjM>q?]+jd?Yhofo d?K;kJX[]Vhql]aGlM}m[MLqldY]aWf]VhdYqdY A B B cVMD1XlhjcaM>UWXlqJhjy?MLcpUVWY#1XJWfq1]ah#o>Wz ]ahjd?qdzy?MLc [[A]] Wfq#mhj]9hU.]aGlMLcaM d?cVM5W}o d1U`]aj_dYk!MLcpWz]VhdYqCFHGlhU<o d?K;kJjMLZhj]`_hUVUVXlM5m[dM>U<qJdf]9Wfklk!M>WYc9WYU oLjMDWfcaj_0hjq(]aGlMmlM Jqlhj]VhdYqwdf9UXJ{[]VcpWYo ]VhdYq XQhyYMLq(]aGJWz]}WfqXlq#o dYq1]acVd?jM>m(Wfj]VM>cVq#Wz]VhdYq(dY½UXlK Wfq#m qJML?Wf]VhdYqNK¤W¦_M>W?m;]Vdb9caM>UV{lXlcaYM>c%dYcaKXlWYU h]aGWfq¤Xlq{!dYXlqJmlM>m¤Wfj]VM>cVq#Wz]VhdYqNdY¥1XJWYq1]VhjJMLcpULY]VGlM X#UMdf mld1MDU^qJdf]nWfklk!M>WYcKNdYcaMklcadY{JjM>KNWf]Vhon]aGJWfqw]VGlMNo d?K{lhqJWz]ahjd?qidf+o dYKNk!d?UVh]ahjd?q£ WfqJm qJML?Wf]VhdYBqngd?cVM;UXJcVklcahUhqlY_Y GlM>qoLdYqJUVhmlMLcahjql0dYk!MLcpWz]adYcpU*m[MLcahyYM>micVd?K F "9Eh]WYklk#MDWf+cpU*]aGJWz] ]aGlMo d?KNkljMLZ[h]`_hU5m[d?K;hqJWf]VM>m{_hj]VM>caWf]VhdYq i∈1..k. . i. i∈1..l. i. i,1. i,li ∗. i. i i. i6l. j6li. i,j. i∈I. i i. i,2. j6li. i. i,j. j6li. i,µi. i,j. i,j. i. i. i,j. i. i,j. i. i,1. i,j. i. i,j. i. j6li. i,j. i. i∈I. i i. i. j6li. i,j. 1. i,j. i∈I. i. i,j. i,j. h. h∈1..k. h. i. h. i. i i∈1..l. i6l h6µi i,h,j i6l. i,j. i,j. h6µi. i,h. h6µi i,h,j. i. i. j6li. i. i. i. i,h,j. h. h∈1..k. i,h. i. i. i,j. j6µi. k. i. i,j. i,h. i6l. i. i i. j6li. j6li i. i,h,j i,j. i,j. i,j. i. HF GJM ,GlM>W¦y?M>U "CdYYho} , ". .hU½W}qlM NK dmJWfldY?hodYc½hjq[d?cVK¤Wf]VhdYq]VcaMLM>U.]aGJWz]+m[hjcaM>o ]V_M>qJo d?K;k#WYUaUMDU . . 5. b½cVMDU{lXJcV?MLc5o d?qJU`]acaWYhjq1]pUL[Wfq#m\hU5{JW?UMDm\d?q\]aGlM UVGlMDW¦yYM>m0o d?K;k!d?UVhj]VhdYq
(51) qJdf]aWf]VhdYqdf*,M>o ]VhdYqv[OY éÛ.×éÅè.
(52)
(53) . OD|. " ¬ . ¬ ° " > +9MLKNMLq1]5dYcaKXJW E ::= MLMLKNMLq1] hj]VG(WY{#M>hq α α[A] W Y d Xlq1]ahjql;d?cVKXlW A ::= V ] caXlM > U GJM>W¦yYMDm\oLdYKNk#d1Uhj]VhdYq§ hj]VG |N| = |E| ∃N φ E MWY?WfhqNXJUVM5]VGlM^jML]]VM>caU ]adcpWfql?M5dzyYMLc%d?cVKXlWYU<dY , "91{lX[]9]aGlhU+UVGldYXlmNqldY]oLWYXJUM WYq_WfK{lhYXlhj]`_ hj]VG F ",d?cVKA,XlW?B,UL . . .M^cVMLMLc<]ad}]VGlM^UVM>1XlM>qJo M E df¥MLMLKNM>q?]½dYcaKXlWYU<WYklk#MDWfcahjqJ hq0WUGJM>W¦yYMDm¤o d?K;k!d?UVhj]VhdYq WYU½]VGl7M # 'E' ,-> $(, df P²qldYcaKNWYj_YWfq0MLMLKNMLq1]dYcaKXlW hqWnUVXlklk!dYcV]9yYM>o ]Vd?c<KNWf]aopGlAMDU<YcadYXJkJU%df¥MLMLKNM>q?]pU<df!]VGJM5AdYcaK a[d] hj]VG a ∈ α WfqJm d |= A <α[A] FHGlM oLdYKNk!d?UVh]ahjd?q ∃Nφ E hU<WQ1XJWYq1]Vhj#oLWf]VhdYq;dzyYMLc.]aGlMHq1XJK{!MLc9dY#MLMLKNM>q?]pU%hjqNMDWYopGdY#]aGlM>UVMHYcadYXlk#UL oLdYqJU]VcpWfhqlMDm({1_]aGlMb9caM>UV{lXlcaYM>c*d?cVKXlW φ nFHGlMdYcaKXlW > m[dM>UqldY]Qo d?qJU]VcpWfhqh]pUKNd[m[M>Uhjq WYq_ W¦_? " ¬ ¬ ° " ¬ £ ,Y . [[α[A]]] [[(E1 , . . . , Ep )]]. = =. a[d] a ∈ α , d ∈ [[A]] ([[E1 ]], . . . , [[Ep ]]). [[>]] [[∃N φ E]]. = =. IT S. n∈[[φ]] n. [[E]]. FHGlM5UVMLK¤Wfq1]ahoHm[MLJqlhj]VhdYq;df!UVGlMDW¦yYM>mo d?K;k!d?UVhj]VhdYq;hU%klcadY{#Wf{l_}M>WYUVhMLc.]Vd}XlqJmlMLcpU`]pWfqJmhjq]VM>cVK¤U Yd n]VGlMWYUaUd[oLhWf]VM>m UaWz]ahU WYo ]VhdYq caMLWz]VhdYqC ¨ UaUXJK;M E hU]aGlMUVXlklk!dYcV] (E , . . . , E ) WfqJm n = ¨ qihq[dYcaK¤Wz]ahjd?q(]VcaMLMhU^hq]aGlMUVM ] hj½WfqJmd?ql_0hj9h]QK¤W¦_0{!M;mlM>o d?KNk#d1UMDm (n , . . . , n ) hq1]Vd;]VGJMnklcVd[m[X#o]*df n MLMLKNMLq1]pU5UVWf]VhU`_hql E ¥>nL>J[[E]] WfqJm n M>jM>K;M>q1]aU5UaWz]VhU_1hql E 1. 1. p. p. 1. 1. p. p. dYc*WYj FHGlM>q K¤Wz]popGlM>U¤]VGlMwoLdYKNk!d?UVh]ahjd?q φ E hWfq#mÿd?ql_£hjQ]aGlMLcaMiMLZhU]aUW¢UVM>1XlM>qJo MdY KXlj]Vhkljho hj]VdhM>U n UX#opG\]aGJWz] |= φ(n) WYqJm ∃N d ∈ n [[E]] d ∈ n [[E]] ⇔. . . . d ≡ i∈1..p (ei1 | . . . | eini ) eij |= Ei i ∈ 1..p, j ∈ 1..ni Q. . HF GJMiU_q1]aWfZ£df , "
(54) m[dM>Uqldf]caM>U]Vcaho ]¤]VGlMiUVXlklk!dYcV]yYMDo]adYcpU]aGJWz]0KNW¦_£{#MiX#UMDmÿhq WdYcaKXlWl x^dYqlML]VGlM>jMDUVU>UVdYKNMUVXlklk!dYcV]aU;GJW¦yYM{!M ]]aMLckJcVd?k#M>c]ahjMDU]aGJWfq dY]VGlM>caU> ©JdYcNM ZlWYK;kJjM?W§UXJklk#d?c] y?M>o ]VdYc}K¤W¦_w{!M ,Rm)/ ]VG#Wz]hUL S n [[E]] = IT d?cnM>1XlhyzWfMLq1]V_ S [[E ]] = E ¨ qldf]aGlMLcnhq1]VM>cVMDU`]ahjqlklcadYk!MLcV]`_dfHUXlkJk#d?c]nhUQ]Vd(GJW¦yYM;MLMLKNM>q?]}d?cVKXlW?U h]aG§mlhU e`dYhq1]nmlMLqldY]aWz ]ahjd?qJU> P²q0]aGlMnWz]]aMLc^o>WYUVMY MnUVW¦_¤]aGJWz]5]aGlMnMLMLKNM>q?]5d?cVKXlWYU5WfcaM 4U) ,-4 .e ) 1\ 3'= 1E = %P²q]VGJhU o>WYUVMY¦]aGlM5m[M>oLdYKNk#d1Uhj]VhdYqdf#W]acVM>MhU%Wf W¦_[UXlqlh?XJMYz]aGJWz]<hULfhj n 6= m ]aGlMLq n[[E]]∩m[[E]] = ∅ cpW hjqlWkJWfcpWfjM> hj]VG§hjqJM>WfcnWYj?ML{lcpWl M;m[MLJqlMW0qldf]ahjd?qwdY Ydd[m
(55) ;UVXlklk!dYcV]Qy?M>o]adYcpULE]aGJWz] MnoLWYj c& #0 $# GlhopG(WYcVMQYM>qlMLcpWz]ahjqJ;UVM>1XlM>qJo MDUHdf%jhqlM>WYcV_¤hqJm[MLk!MLq#m[MLq1]^M>jM>K;M>q1]+d?cVKXlWYU> n∈Np. ×× Ýo=p. i∈1..p. i.
(56) B 74)4U * B <EE B
(57) . #&# = ,. O. .
(58) l > 03, (E , . . . , E ) # ec& # # 8f 1 =4U.C 8 i 6= j )A ' 4U $# [[E ]] ∩ $ 8 , 4)4 * 1 S Be c& # # # ' R, '5 ,C 8e 1 4U. 8 > ,-. [[E @) # =c 8$,-A74Q[[E@ 8 ]] E= E# 4 # ` c& # E # B # '\' ]] =,- ∅> $(, 'Ei,'= j,-∈)"1..p. ¬ °. 1. j. p. i. i. i∈1..p. FHGlM5UVhjKNklM>U]<M ZlWfKNklM+df#WQ{JW?UhUChU.]VGJMHUVhqlYM ]adYqUVM>1XlM>qJo M ¨ qldY]VGlM>c9UhKNkljM+M ZlWYK;kJjM hU*WJqJh]aMUXl{#UML]*df hU+]VGJMUMD?XJMLqJoLM (a [>], . . . , a [>], Σ [>]) GlMLcaM Σ = a , . . .AnyE FHGlMnqJM Z]*klcadYk!d?UVh]ahjd?qU]aWf]VM>U+]aGlM}XlqJmJWfKNMLq1]aWYklcadYk!MLcV]`_¤df%,{JaWYUVM>U> GlhopG0hUH]aGJWz]^WYq1_\hjΛq[d?c K¤Wf]VhdYq0]VcaMLMnWYm[KNhj]aU*W;XlqJh1XlMm[MDo d?K;k!d?UVhj]VhdYq0dYjd hqlNW?hjy?MLq{JWYUVhUL # c # # *? 5 [8$,`=.
(59) ) 8$,-AJ6 6, & * d* ° ° ° ## A 5 , # J= ) , # > E $(=,&* (En ∈, .N. . ,* E# =) d ∈ n [[E]] B °°
(60) "CM ] {#M*]VGlMQo dYKNk!d?UVh]ahjd?q hj]VG dYc½WYj ∈ 1..l ,hqJo M E hU9W{JWYUVhU> WYqMLMLKNM>q?] e dhq¤]VGJMQm[MDo d?K;k!d?UVhj]VhdYqedY |d. {!. .ML|dYeqJ?U< ]adMLeZ[W?∈o]ajE_NdYqJMUVM ] i [[E "M ] m[M>qldf]aM>U½]VGlM qXlK{#M>c½dYMLMLKNMLq1]aU½hjq d ]aGJWz]+{#M>jd?ql?U9]Vd [[E ]] +_No d?qJU`]acVX#o]VhdYqC M*GJW¦y?]]M d ∈ nn [[E]] Wfq#mN]VGlhU hU+]VGJMndYql_ W¦_¤]Vd¤m[MDo dYKNk!d?UVM d dYjd hql E 1. ⊥. p. 1. 1. p. p. p. 1. l. i. i. k. k. k. . %-. +M hjq1]Vcad[m[XJoLM9]VGJMqldY]VhdYqdY , ma A dYJWUVXlklk!dYcV]y?M>o]adYc.WfqJmmlM>Uao cahj{!M+W^KNM ]VGJdmdYc.{lXlhmlhjql WNoLdYKNKNdYq{JWYUVhU+cadYK GlM ]aMLcadYYM>qlMLd?XJUHUXJklk#d?c]pUL P²q[d?cVK¤WYj_YJ]VGlMUVXlklk!dYcV] F hU^W¤caM JqJMLKNMLq1]*df E hj<cadYK WYq_0m[MDo dYKNk!d?UVh]ahjd?qdf9W;]VcaMLMdzyYMLc h } ] hU*k!d?UaUh{lM]Vd\M Z]VcpWYo ]QW¤KNdYcaMklcaM>oLhUVMm[MDo d?K;k!d?UVhj]VhdYqdzyYM>c F ©JdYc^MLZlWfKNklMY#]VGlM;UXlkJk#d?c] E hU5WNcaM JqJMLKNMLq1]5df (α[>]) (α[A], α[¬A]) ¬ °
(61) \¬ .¬ ¬ , ma A 8&,RA 5 e# 'E' ,- E = (E , . . . , E ) )=([ 5 . # '\' ,-. B. SF = (F1 , . . . , Fq ) [[Ei ]] = (i,j)∈R [[Fj ]]. # , 4Qm . R. 8. # = 8$,`4)4 i ∈ 1..p % e > 1. 1..p × 1..q. p. MiXJUVM]VGlMqldY]aWf]VhdYq GlM>q ]VGlM>cVM(M Z[hU`]pUWcVMLJqlM>K;M>q1] cadYK hq1]Vd WfqJm VU hK;kJj_ cah]aM F E GlMLq\F]VGlMcVM>WfE]VhdY q R hU+dY{yhdYXJU9cVd?K ]aGlM}o dYq1]aM Z]><RFHGlMKNWYEhjq0klcadYk!MLFcV]`_dY caM #qlMLKNMLq1]pUhU]VGJWf] GlMLq F caM JqJM>U E lWYq1_ , ",d?cVKXlW hj]VG(UXJklk#d?c] E oLWYq\{!M}cVM cVhj]]aMLq\hq1]Vd WYq0MD1XlhjyzWYjM>q?]Hd?cVKXlW hj]VGUVXlklk!dYcV] F 1 , # '\' ,- > 0(, # ° ° ° ## A , . . .* , F ) ) # = F E * = `8$E,@==.(E ,, -.# .c0., ,mE ), & # F6,R=)=(Fφ(N) N = (N , . . . , N ) * % R. > != , M = (M , . . . , M ) 1. 1. p. 1. q. R. 1. ∃ Ni. q. . i∈1..p. 1. # 5 & # 6,R)=. p. [[∃N φ E]] = [[∃M ϕR F]] , ϕR. . P V Mj = (i,j)∈R Xji Vj∈1..q P , Xj (i,j)∈R . ∧ i∈1..p Ni = (i,j)∈R Xji . ∧φ i. éÛ.×éÅè.
(62)
(63) . O¦t. °°
(64) MQklcadzyYM 9FHGlMkJcVddf¥d?c+]VGlMQdf]aGlMLcHm[hcVMDo]VhdYq0hUHUVhjKNhWYc> φ E]] ⊆ [[∃M ϕ F]] ¨ UVUVXlKNM d ∈ [[∃N φ[[∃N H F l G > M q h N U V ] J G w M. o ? d ; K k!d?UVhj]VhdYq df n M>jM>KNMLq1]aU0UVWf]VhU`_hql E ,UVW¦_ E]] d L L a f W J q m > M j > M N K L M 1 q a ] 5 U a U z W a ] h U _ j h l q a U ¦ W _ h]aG e ,...,e +_£m[MLJqlhj]VhdYq df nF E MG#W¦yYM [[EE]] = Se , .[[F. . ,]]e FHGJMLcaM dY|=caMφ(n) MLyYM>cV_¢M>jM>KNMLq1] e hq ]aGlhUm[MDo d?K;k!d?UVhj]VhdYqwUVGldYXJmiWYUVd¤ WYj%hq?]ad\d?qlMdY<]VGlMNo d?K;k!dYqJMLq1]aUdf [[F]] E]VG#Wz]hU>#]aGlMLcaMMLZ[hU]aU WYqÿhqJm[MLZ r hq 1..q UVXJopGÿ]VGJWf] i R r WfqJm e ∈ [[F ]] dYcWf i ∈ 1..p, j ∈ 1..n AFHGJhU ?hjy?M>U+W;qlM m[M>oLdYKNk#d1Uhj]VhdYq0df d hjq1]Vd]VGJMUXlkJk#d?c] F UVXJopG\]aGJWz]H]aGlMnoLWfcpm[hqJWfhj]`_ m df (i, j) ∈ hU<]aGlM*qXlK{#M>c9df¥MLMLKNM>q?]pU9K¤Wz]popGlhql .FHGJM*cVM>Wf]VhdYq d ∈ [[∃M ϕ F]] 1..p × 1..n r = k F d?jd U+{_\klcadzyhjql]VG#Wz] ϕ (m , . . . , m ) Gld?mJUL "CM ] x {!M*]VGJM^oLWYcam[hqJWYjhj]`_dYE]VGlMQUML] j ∈ 1..n r = k Y]aGJWz]+hU>Y]aGlM^qXlK{#M>c½dYEM>jM>KNMLq1]aU [ d?c j ∈ 1..n lWYUaUVhj?qlM>m\]Vd F MnGJW¦yYM n = P dYc*WYj i ∈ 1..p GlhopG(hjKNklhM>U x e P P P # f W J q m ? d * c f W lWYUHcaM>1XlhcVMDm¥ x ) x ,..., x |= φ( m = k ∈ 1..q R. 1 1. 1 n1. p. p 1. p. R. i. i,j. p np. iRj. i j. i,j. 1. 1. i j. j. ri,j. i. k. i. i,j. R. i k. i j. 1. q. i,j. i. i. k. 1 j. 1Rj. R. k. p j. pRj. i k. i. iRk,k∈1..q i iRk,i∈1..p k. k. ## A E 1 F , ' ,R '= , c #0 -# B & c0)4 1 ' ,R '= , c # # * % * , a $# c& E 1 F B E×F. ° ° °. . 1\ ; &1. °°
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