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<?xml version="1.0" standalone="yes"?> <Paper uid="C96-2152"> <Title>Disambiguation by Prioritized Circumscription</Title> <Section position="4" start_page="901" end_page="904" type="metho"> <SectionTitle> 2 Prioritized Circumscription </SectionTitle> <Paragraph position="0"> In this section, we briefly review prioritized circmnscritltion. For simI)licity sake, we modify the definition of prioritize(l circumscril)tiol, by (Mc-Carthy, 1986; Lifschitz, 1985). The difference is that we let all t)rcdical;es vary and lnaximize preference rules whereas Lifschitz mininfize abnormal predicates for prefercnc(', rlth~s.</Paragraph> <Paragraph position="1"> Let cI)(x) and ~P(x) be formulas with th(! same nulnller of fre(~ w,.riables x. We say (,hat C/) and ~P are similar. C/1) > ~I/ stands for Vx(~II(x) D (P(x)).</Paragraph> <Paragraph position="2"> We extend this notation to tuples of formubm ep, ~IJ wllQFC (I) 22: {\[)\[ .... {~,f,, alld ~1 / = @1 ..... ~\[,'., all(l and ~ are similar (each (I)j and @5 ~tre similar): (D > ~1) stands for A<' _ .~::~(I'j > ~Pj. We also write i1)> ~PA~P >(l) asCD =9 and 4)>~PA~(qJ_> (It) as ~P > ~P.</Paragraph> <Paragraph position="3"> Let a tuph', of formulas 4) be broken into disjoillt parts q51 , (1)2 .... , (1)a:. Let ~Iri be similar to 4;< We (te~ll('. q* ~ ~ d,Q&quot; t, (A i .1 (\[~j = @.i ~i =: A'i::Iv j--1 D ~ /)it. We also wril;e (1) ~ vii A ~(~It _ ~I)) as * ~ iP.</Paragraph> <Paragraph position="4"> Definition 1 Let A(P) be a formula, and (D(P) be a tuple of formulas wh, ich, is brokcn into 4il(P),4,2(P) ..... (1)~:(P) where P is a, tv, ple of predicates used in these formulas.</Paragraph> <Paragraph position="5"> The ,~ynl, act, ic d@',nition of prioritized circumscription i.s as follow.s:</Paragraph> <Paragraph position="7"> 1. p i.s a, t'uplc of predicate variabh',s ca, oh, of wh, ich, ha,s the same arity as th, e corresponding predicate constant in P, 2, A(p)((D(p)) is a formula obtained b!l replacing evcry occurrence, in A(4L respectively) of a predicate con.st(m,t in P by the, correspondinfl predicate varia, ble in p.</Paragraph> <Paragraph position="8"> At:cording 1,o l:he result of (\]Afschil,z, 1{)85). wc give a, model theorel,ic tlclinil,ion of l:hc ~d)ovc \[of nmla, (1) ~t,~ folh~ws.</Paragraph> <Paragraph position="9"> Definition 2 Wc dcfi~,~ ant, o'rdc':&quot; > ovc'r Ioqica, I i'.,l, crprcl, ation,,~ a,,s folio.,,<&quot; M~>M .wh,C/,rc I. M' a,',,d M have lb.c .~a,'m,c do'm,o,i,~,. 2. cm&quot;U/ co'n,.,#,an,/, a,'n,d, fu,'n, cl, io'n, .s!/'m, bol k,a,.s /k,e ,sa, w,c i',,tcrprclatio'n, in, M/ a,'H,d M. ,7. q,(p) >- *l'(q) i,s t'r.,, in, M' (o'r. cq.iva, lc'.,tlfl. i,, M) :,,,. M'\[p\] ,,.,~ p ,,,,a M\[P\] ,,.~ q .,/,,.,.,: M'\[P\] (M\[P\]) &quot;,:.s o. t,,.pl,: of tb,, <,:t,m,sio',,,.s .fo'v M' (M. '.,xst.:cl, i vcly ) of prcdica, t,#s i'.. P in l,he a, bovt! order, ~t grealcr ilttcrprel,a,t,ion is more prtd'cral)h;. 'l'he a,lmvc, order h~tuidvcly me,ms I,ha, l: logical int, cq)rtfl;~d,iotts which m~txima,lly sa,lis\[y a, subsel, ot7 ~li ~ ;brt~ prNTcrM,lt!, a,nd if l,htu'(! arc iul,crprel,a,l,ious which mLl,isf~g lit(', mmm ft)rllllll~ts ill q)l. l,hcll iltt,crl)rt'.l,;tt;iOllS which ma,ximaAly smisfy ~t subsel; of ~1)2 i~rc prcfvra,blc, aml... a,nd if l here arc intc.rprtfl;a,I;ions which sal,isf'y dtc sa,me t'orn)ul~ts in (1) z' ~, I:h(m it, l,('.rl)rel~tl:ions which m;~xitmdly sM,isfy ~t subs('.l, of il) \]'~ ~ue l>r(4'(~ral)h'.. Lcl; A I)c ~r \['Ol'ltlllla. Wc s,~y (:hal: a, logicM inl;erpr(,.l;Mion M is the &quot;m,o,st prcfc~'a, blc 'm, odcl M, I,k,c cla,~.~ of mml, cl,s of A w.r.t.. > if t, h0.re is no nIo(tcl M' of A in l, hc class such l:h,td, M' > M a,ml uo(; M > M'.</Paragraph> <Paragraph position="10"> According t,o t, hc result of (Lifschil,z. 198,5), wc ha,re l,hc followhlg ('OITt',S\[)OIII\[(:IIt;C I)(:l,ween syII.-La,cl.ic deli ldl,itm mM scnl~mlic dctinil,ion.</Paragraph> <Paragraph position="11"> Theorem 1 A Iogica,1 in.l, crp'.ct.,l,io~, M i,s a, &quot;m.odcl o,f (1) iJf M i,s I, hc &quot;mo,sl, l)'r@:rablc modal w,r.t. > h~, tlw cla,,~,s of'm, odcl,s of A.</Paragraph> <Paragraph position="12"> in ~my sil,md;ion. And, wc rcga,rtl ~I} a,s ~ hi)h: o\[ |)r(q~H't,,llCC, l'llh~s. Nol,c I;\]t;t,l, t)l'(!\[~H'Cllt:t~ l'lt\[Cs ~-l,i'c, plll, into hicraxchy ~mcording l,o ,St,F(!llgl;h of \]-II'('.t'~',I'-Cllt:l.! I'Illcs. :\['hcIl, l;\[IC IIIOSI, pl&quot;cfcr~dJc models corrcsl)ond wil;h l;hc most l)refcra,1)lc rcmliugs since C;-Lt;h tltodc\] s~4islies Sl,l'onp~(!r 1)rt4't',rtm(:e ruh~s a,s much a,s pt)ssil)lt' m.l l,herct'ore, t ht! syl~l,a,t:l,ic definiliou bt!ctmms a Slmt:ilit:M, ion o\[ l.he l~refcra,lflc rea,ding,~ by Fhc, mtm I.</Paragraph> <Paragraph position="13"> In l,hc subst!(lucnl, std),sct:l:ions, w(' lirsdy fix mt C, Xl)erimtml~d h)gica,l represenla, l,ion of S(!lli;elt(:es. I)a.t;l(grt)und kuowlcdgt: m.t prt:ferences. Then. we l,rea\], l,hc cx~mq~lc, in Set:don it by the logic;d repl'(~s(~tll,~LI,iO t|.</Paragraph> <Section position="1" start_page="902" end_page="903" type="sub_section"> <SectionTitle> a.1 Logical Representation of Sentences and Background Knowledge </SectionTitle> <Paragraph position="0"> Wc uso ;m axla~pbfl, ion of KowMski's evenl, caJculus (Kowalski and Sergot, 1986}. Howcwu'. l;hc ith'.;r of dis;unbiguat,ion in i;ltis 1)~rper does not tiepond ou ,t p;trl,icut~tr represcnt, al;ion. We ~tssutne {,\[lg-'L\[, CH.ch SClli,CIICC exl)ressc, s }gll eVCltI,. For CX~Llllpit:, a, SClJt,encc' &quot;John g~tw~ the l:ch'.scol)C t;t) t, ho, lnmt'&quot; is rcprc.scnLcd a,s the \[o\[lowing fornmlm ,u:t( I,\], ( ;ivc ) A actor(E, J,,h,'.,) A okjcct( t';, 51'clcscopc ) A &quot;rccipie,~.t( E. M .,~,) A (:OII/\[)I('.X sCIII;CIIC(! is Slll)\[)OSe( |t,O \[)C. dCCOlllposed inl;o ~ seL of siluple senI,e:u(:es which is t, rmmla,i,cd into t,hv. n, bovc rcprc.sentat,ion. Ambiguities a,rc expressed by disjunct;ions. For exmnplc., t, hc scntel.:C &quot;gohn s;tw a man wil;h sr l;t~'lcscope&quot; is cxpressed a.s follows.</Paragraph> <Paragraph position="2"> The laM: CoIIjllIICt; c.xpre.sst~s mnbiguity in l,he l)hra,sc ' &quot;wilah ~t I clcst:opc.&quot; (used ~ts ~L devit:e or cartied by 1,11(': ma,n).</Paragraph> <Paragraph position="3"> In tahiti,toll I:o l, he SClH;tttl,i(: rel)l'CSCld,a.l,ion, we also use synl,acl,it:M informal>ion fl't)n~ a, tmrser so l;ha\[; gr~tlttln~d,i(:aJ 1)r(~ft~r(,,lt(;(: rlll(',s (:~tH H(', exl)rcsscd. For ex~mq)le, we show some of tim gr,m> mal,it:;d inforlmtl:ion of l, hc s(HII;tHIcc &quot;',John gatve dm I clcscopt~ l;t) l,he ma, n'&quot; ;ts follows. (We assunm t,hM, stml;ct|cc munl)cr is 1).</Paragraph> <Paragraph position="4"> imtting stronger preferences into a stronger hierarchy of l)references.</Paragraph> <Paragraph position="5"> For example, consider the following two grammarital preferences.</Paragraph> <Paragraph position="6"> 1. If &quot;He&quot; appears in a sellteltce as the subject and the subject in the previous sentence is male, then it is 1)referal)le that &quot;He&quot; refers to the previous subject.</Paragraph> <Paragraph position="7"> 2. If &quot;He&quot; appears in a sentence as the subject ~tnd someone in the previous sentence is male, then it is preferable that &quot;He&quot; refers to the one in the t)revious sentence.</Paragraph> <Paragraph position="8"> Suppose that the former is stronger than the latter. This priority of the t)references means that the formula: (isa(a, Male) A subj(i, a)</Paragraph> <Paragraph position="10"> shouht be satisfied as much as possible for every a and i, and if it is maximally satisfied then the following forinnla:</Paragraph> <Paragraph position="12"> shouhl be satisfied as much as possible for every a and i.</Paragraph> <Paragraph position="13"> We can represent semantic preferences as well. For exalnple, a preference &quot;If al sees a2, then a2 and al are not equal&quot; means that the following expression shouhl be satisfied as nmch ~s possible</Paragraph> <Paragraph position="15"> Note that the Mmw; is a preference rule because there is a possibility of reflexive use of &quot;see&quot;.</Paragraph> </Section> <Section position="2" start_page="903" end_page="904" type="sub_section"> <SectionTitle> 3.3 Example </SectionTitle> <Paragraph position="0"> Now, we are ready to treat disamhiguation of the sentences used in Section 1 by prioritized circumscription. null We consider the following l)ackground knowledge which is always true. We denote the conjunctions of the following ;~ioms as A0(P) where p d~.f (eq, is, time, act, actor, object, recipient, device, sub j, in_the_sentence).</Paragraph> <Paragraph position="1"> 1. If al is equal to a2 then a2 is eqnM to az. ValVa2(eq(al, a2) D eq(a2, al)) 2. If al and o,2 are equal and a2 and aa are equal, then al and a3 are equal.</Paragraph> <Paragraph position="3"> 5. If al has o at time i, and al is not equal to a2, then a2 does not have o at time i.</Paragraph> <Paragraph position="4"> This is same as (2).</Paragraph> <Paragraph position="5"> We consider the following preferences.</Paragraph> <Paragraph position="6"> 1. If ax sees a2, then ax and a2 are not equM. * (P, e, =(5) 2. If a is lnale and a is the snbject of i-th sentence and &quot;He&quot; is in the next sentence, then a is equal to :'He&quot;.</Paragraph> <Paragraph position="7"> * 2(P, e, a,i) =(3) 3. If a is rome and a is in i-tll sentence and &quot;He&quot; is in the next sentence, then a is equM to &quot;He&quot;.</Paragraph> <Paragraph position="8"> C/Pa (P, a, i) =(4) 4. If someone gives o to a at time i, then a has o at time i + 1. This expresses inertia of ownership.</Paragraph> <Paragraph position="9"> = (act(e, Give) A object(e, o) Arecipient(e, a) A time(e, i)) D ?e I (act(el, Have) A actor(q, a) Aobjeet(el, o) A time(el,i + 1)) 5. If&quot; a buys o at tinle i, then a has o at time i + 2. This preference of another inertia of ownership is weaker than the former preference 1)ecause time interval is longer than the fornler t)reference.</Paragraph> <Paragraph position="11"> We assmne that ~ is a formula which should; be satisfied in the first place, O~ in the second place, (pa ~ in the third place, q54 in the fourth place and * ) in the fifth place.</Paragraph> <Paragraph position="12"> Example 1 We con.sider the following sentences.</Paragraph> <Paragraph position="13"> John just saw a man with a telescope.</Paragraph> <Paragraph position="14"> He bought the telescope yesterday.</Paragraph> <Paragraph position="15"> A logical representation of the above sentences ix as folh)ws and we denote it as AI(P).</Paragraph> <Paragraph position="16"> ti?ne (El, 2) A act(E1, See) A actor ( El, John)</Paragraph> <Paragraph position="18"> Ai'n,_thc_sc'n, tcncc( 2, He) NoLe thaL we represent &quot;just&quot; as Lime 2 and &quot;yesterday'&quot; a.s time 0.</Paragraph> <Paragraph position="19"> In t, he synLa(:tic deiinil;ion of the lliOSl; prefer-M,le reading (I), we let A(P) be &(P)A At(P) and /~: I)e 5.</Paragraph> <Paragraph position="20"> We show an intuitive ext)l;ulation of inferen(:e of geLLing tl,e most t)referM)le reading as \[i)llows. F'rom the preference 2, &quot;lie&quot; preferably refers t,() ,lohn. NoLe LhaL although t, he t)reference 3 seems Lo l)e alq)li(:able, iL is noL acLually used since the stronger prefcre, nce 2 overrides Lhe preferen(:c 3.</Paragraph> <Paragraph position="21"> Thell, from Lhe preference 5, John had l;he telescope el: Lime 2. Frolll Lhe t)reference l, .lohn is not equM to the, mau. Then, the man (:aunol: have l:he Leles(:ope, at Lime 2 front l;he l)a(:kground knowledge 5 and l;herefore, t;he t:eh;seope was used as a device fi-om the disjuncLiol~ iu A1 (P). We ca.n a(:l,ua.lly prove tha, t &:vice(l':l,telescop(:) is l,rue in t, he most 1)referM)le remlings.</Paragraph> <Paragraph position="22"> Example 2 Suppose we add the following sen/once t,o the p'rcvious scnl, e'n, ces.</Paragraph> <Paragraph position="23"> But, he gave the telescope to the man this morning.</Paragraph> <Paragraph position="24"> A logi(:al representation relate(t I;o this Sclll;ence is as follows. We denot;e the fornml;t as A~(P).</Paragraph> <Paragraph position="25"> t,i', 1,,C/&quot; (I'23, \]) A (l, dl;,( E3, (-~i',,e) A acl:or(\],':~, l\] e ) Aobjcct( \]'\]3, Telescope) A recipient( Ea, Man) Note thai; we represent %his morning&quot; a.s time 1. In l;his case, we h;t a(P) be A0(P) A A~(P) A A2(P) in the synta(:t:ie definil;ion. The, u, reading of &quot;'widl ~t t;elescol)e&quot; is (:hanged. From l:he pret: erence 4, Lhe 1,H-Ln shouhl have had Lhe l;eles(:ot)e a(; I;ilne 2. if the, (;eles(:ol)e were used as a device el; dnle 2, John wouhl Mso have Lhe Leles(:ot)e aL dm same time a(:(:ording to background knowledge 4 and it (:ontradiets background knowledge 5. Then, the weaker t)referen(-c 5 is rel;r~cl,ed Lo ~woid contradiction and the stronger preference 4 is survived. Therefore, in l;he mosL t)refera,1)le rea,ding, Lhe ll\[;l, li h~ul Lh(,' telescot)e at l;inle 2.</Paragraph> </Section> </Section> <Section position="5" start_page="904" end_page="904" type="metho"> <SectionTitle> 4 HCLP language </SectionTitle> <Paragraph position="0"> Now. we discuss an imt)lementation of prioritized (:iv(:ulns(:ritfl:ion by IICI~P. FirsLly, we briefly review ~t hi(,r;Lrchi(:M consLrainL logic l)rogr~ttn ming(HCM )) language. We follow t.he definition of (l~orning el; M., 1989).</Paragraph> <Paragraph position="1"> An HCI,I ) program consisLs of rules of (;lie form: h: -bl ..... b.., where h is a predicat, e and each of hi ..... b,,, is a predicate or a constraint or a 1M)eled (:(restraint. A lal)eh'.d (:onstrMnl; is of the form: label C whe,'e C is ~t constraint in specitie (lomaill and label is ~ label whi(:h expresses st, rengl,h of the (:onsl;rainL (/.</Paragraph> <Paragraph position="2"> The oper;d, ionM smmmties for HCLP is similar Lo CLP exeet)t manipulating a (:Ollstraint hierarchy. In \[\[CLP, we a(:cmnulate labeled consLrMnts to form a constraint hierarchy by each 1M)el while exe(:uLing CLP until CLP solves all goMs mM gives a, reduced required constrMnts. 'Phen, we solve constraint hierarchy wiLh required const, rMnLs. To solve (:onstrainL hierar(:hy, we firstly lind a m;~ximal subseL of constraints for the strongest level which is (:onsistent with the require(l constrMnl;s. Then, we try to find a inaximM subset of consLraints in the se(:ond strongest, level with respe(:l: to t, he union of the. required consLrMnt, s and Lhe lnaximal (:onsisl;ent subset for l.h(; sLrongest, level .... and so on until a maximM consisl;ent sub-set of COltsLraints in the k-th strongest, level is added. The.n, an assignment which satisfies t;1,e final seL of consl;r:tinl;s is eMled a sol,.tio'n,.</Paragraph> <Paragraph position="3"> O a.,t be assignm,;nts C0'(and t)e a se.t of constraints in the strongest, level of tl,e hierarchy sat, istied t)y 0(amt o), and C~(and ~2 CC/) l)e ~L set; of (:onstrMnts in the secon(l strongest level of t.he hierarchy satisfied by 0(and a) .... , an(l C~'(,md C a:).~ be a set of (:onstrMnt.s in f.he t,>f.il strongest level of Lhe. hierarchy satisfied by 0(an(t (7).</Paragraph> <Paragraph position="4"> 0 is locally-predicate-better (Borning el, el., 1989) Lluul (~ w.r.L, t, he (:onstrainl. hierarchy if there exists i(t < i < k) such that for (,'very j(:l < j < .i c:; : ,,na c:; c We can prove thaL if 0 is a solution, t, hen there is no assignment ~r which satisfies the required (:onstrMnLs and is locally-predicate-better than 0.</Paragraph> <Paragraph position="5"> Note l;hal; t;ll(: definition of loeally-t)redicatel)etter (:onlpm'~to,&quot; is similar to the definition of the orde, r over logical interpretation in the t)ri oritized cir(:umscription. The difference is that locally-1)redicate-better (:omparator (:onsiders assignment:s for variabh,,s in constraint;s in IICLP whereas t, he order over h)gical interpret.aLton COilsiders ~msignmenl;s of truth-value for formulas in 1)rioritized circumscril)tion.</Paragraph> </Section> <Section position="6" start_page="904" end_page="905" type="metho"> <SectionTitle> 5 Implementation by HCLP </SectionTitle> <Paragraph position="0"> language In order to use, HCLI ) l,~nguage h)r iml)lemen~ation of prioritizcd (:ireunmcripdon, we need Lo change t'ornnflas in 1)rioritized circumscription into (:onsl:raints in tlCLP. It is done as follows. We introduce a domain closure axiom so Lhat we only consider relevant constants used in the given senten(:es. Then, we inst;mtiztte universM-quandfied variM)le, s in background knowledge mM free variables in preferen(:es wit, h the relevmlt (:onsL;mts and iul~rodu(:e Skolean fimctions for existentialqua.ntified variables.</Paragraph> <Paragraph position="1"> For ex~mll)h'., we lu~ve the following fl)rmula l>y inst:anl,i~tting t)r(',f('rence 4 in Section 3.3 with Ea for c and t;h(', m;m for a ~md th(: t(;lescol)( ~. for o ,rod 1 for i and introducing a. Skolem functioll f: (act(E 3 , Give) A objcct(Ea, Tcle.scopc) Arccil)ic'~,l,( E a, Ma,7~,) A timc( Ea, 1)) D (a.,t(f(Ea, Man, Tclc,scop( , 1), Ilavc ) Aactcrr(f(Ea, Man, Tclcsco't)c, 1), Mw~r) aot, jcct( f ( \[~'a, Man, Telescope, 1), Tclc scW)c ) AI, imc( f ( Ea, Man, Telescope, 1), 2)) By this trm~slal;ion, every forlnula t)e(:om(',s ground a,nd we r(.g,'trd a, ditf(,r(mt ground atom as gt (liffcrent; propositional synlbol. '\]'11(',71, every fornlula, in t)rioritiz('d circumscription can 1)e rcg~r(h,'d ~ts a 1)ooh'mt (:onsi:raint in HCLP. We tra, nsla.te ~dl formulas in the syntactic detinitiol~ of l:h(.' background knowledge and t:he s(',nten(:(',s in \]~',X\[Llllples 1 &71(l 2 into boolean (:OllSl;l'a, ittt, s ill oTlr IICM ) la, ngu;~ge (Sat;oh, 1990). Then, fi'om 1,\]m two s('7~ten(:('s in I~\]xa, ml)le 1, our IICLP l~mgu~tgc givc's th(' following result as ~c part of a solution:</Paragraph> <Paragraph position="3"> a,,/,( 177. Scc) true de'vice( ET , ~l'clc scope) = true which m('~ms l;ha, t the t, eh',scope is used as a (h'~vi(:e. And, our ItCLP language gives l;|t(? following result for tim S{',IlI;(*,IIC(',S in Exa, nll)le 2: tim, c(ET, 2) = true a,(q~o~&quot; (El, .lob',,) true</Paragraph> <Paragraph position="5"> obje r't( \]5'{, TclcscW)e ) = true which lllCa,llS thai: the mint ha,s t, he t(~lescot)(~ (a, ll(1 il; is not used a,s ~ device).</Paragraph> </Section> class="xml-element"></Paper>