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<Paper uid="C94-2204">
  <Title>TYPED FEATU19E STRUCTURES AS DESCRIPTIONS</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2. A FEATURE STRUCTURE
SEMANTICS
</SectionTitle>
    <Paragraph position="0"> A signatm.e provides the symbols from which to construc.t typed feature structures, and an interpretation gives those symbols meaning.</Paragraph>
    <Paragraph position="1"> Definition 1. E is a siguature iff E is a sextuple (~, %, ~, G, ffl, ~), is a set, (%,-&lt;} is a partial order, { foreachrE72, } = crC72. |fair thcna=r ' ~2t is a so/,, is a partial tbnction from the Cartesian product of 72 and ~2\[ to %, and for each r C 37, each r' C % and each o&amp;quot; C ~, if~(r, or) is defined aml r ~ r' then ~(r', ct) is defined, and a(~, ~) _-&lt; a(&lt;, .).</Paragraph>
    <Paragraph position="2"> \]Ienceforth, I tacitly work with a signature {Q, 72, ~, O, ~(, ~}. 1 call members of Q states, members of 37 types, ~ subsumption, members of ~ species, members of 9.1 attributes, and ~: appropriateness.</Paragraph>
    <Paragraph position="3"> Do.fil).itlon 2. 1 is an interpretation iff l is a triple (U, S, A), U is a set, S is a total time|ion from U to A is a total function from ~{ to the set of partial functions from U to U, tbr each (t C ~\[ and each u C U, if a((:~)(~,) is deC, ned then ~(S(u), a) is defined, and ;~(s'(~,), ,,) ~ ,V(A(~)(*O), and for each cY G ~( and each u E U, if~(X(u), a) is d(,Jined Suppose that 1 is an interpretation (U, PS', A). I call each member of U an object in I.  \]','a.ch type. denotes a set. of o\])jecl;s in \[. The denotations of the species partition U, and S assigns et*ch object iu 1 the ul|ique species whose denottttion contains the object.: ol)jcct u is in the denotation of species cr it\[' cr = ,~'(u). Subsumption &lt;m&lt;:odes t~ rel~tionship bcl;wccn the denotations of species and I,ypcs: object ,t is in the denotation of I;ype r if\[ r ~ 5;(u). So, if r~ _-j r2 then the denol:~ttiou o\[&amp;quot; type rt contMns the denotation of l;ylw, 7&amp;quot;2. Each at|;rilmte del~otes a. partial ft,nction from l;hc objects iu 1 to tim ob.icct.s iu i, aim A assigns e~clt artl;ribute the l&gt;m:t;ia\[ funcl.iol~ il, denol;es. Appropriateness encodes ~t rcbttionship between l;he dcnotaLions of species and atl:ributes: ifa(cr, ,v) is deliued then the den()tt~tion of a.ttributc (v acts upoi~ each ol~jecl, il, the, denota.l;ion of species cr to yield at, olLiect in the dcnol, ation of type ~(o-, ,v), but ifa((r, ,,,) is undefined then the denotati(m of al.l.ribul.e ~v ~tc/.s upon no object in the deuotalion of species or. So, if~(r,{v) is defined then the. (h&gt; uota.tion of a ll, ribute rt a.cl.s Ul~(m each objccl, in the denotation of tyl)c v 1;o yichl an object in the del|otal;iol~ of type a(r, ,').</Paragraph>
    <Paragraph position="4"> I call a linitc sequence of attribul,es a path, and write q3 for I,he set, of paths.</Paragraph>
    <Paragraph position="5">  Definition 3. 1' is the path interl~retati(m fimctlon under 1 ill&amp;quot; I is an interpretation (U, PS', A), 1' is a tol, al timctim~ l)'om q3 to the s.t ,f l)a, rtia, l fimctions from U 1,o U, alld lbr each ((vl ..... (v,,) 6 ~, /'(m,...,'v,~) is the timcti&lt;mal coml,o,siti,m of d ( m ) ..... A ( (~,, ).</Paragraph>
    <Paragraph position="6"> 1 write t~ for the path iute,'prctal.ion flu,orion mMer l.</Paragraph>
    <Paragraph position="7"> De.finition 4. l,' is a \[baturc structm.c ill&amp;quot;  I,&amp;quot; is a quadrulde (Q, q, 5, 0), Q is a tinite subset o1'~\], q~Q, 8 is a. finite pa.rtia.I function from the, Ca, rtesian l,rgduct ot&amp;quot; Q mM c2\[ to Q, 0 is a totM l)mction from Q to %, and for each q/ ~ Q, &amp;n&amp;quot; some re (5 q3, re rlm.s to q' in I c, where (,'vt,...,;M) z't/zzs l,o q' ill 1&amp;quot; ill' q' 6 Q, and ~.&amp;quot; son., {qo,..., q,} C- q,</Paragraph>
    <Paragraph position="9"> q,, -. q/.</Paragraph>
    <Paragraph position="10"> }&amp;quot;,;tch \['(!;tl;llr(! Stl'tlC\[,llr(~ is a COllllCC~,C(l f~\]001&amp;quot;(! machine (see \[MooRI,; 1(;56\]) with finitely mauy st~tes, input alphabet 9..\[, and output Mplm.bet X.</Paragraph>
    <Paragraph position="11"> Definition 5. 1; is true of u under 1 iff F is a featnre structure (Q, q, 5, O), 1 is a.n interpretation (U, S, A), u is an object in 1, and for each re1 6 q3, ca.oh rc 2 C q3 and each q' ~ (O, if rot runs to q/ in t&amp;quot;, and rr.2 runs to q~ in l&amp;quot; tl,,,,, :,,(~,)(,,) i,~ ,mi,,.a, J~(~)(,,) i~ ,&gt;t/,,.4 0(q') ~ s(v,(,~,)(u)).</Paragraph>
    <Paragraph position="12"> Definition 6. I,' is a satisfiable feature struc- null ture ill' I&amp;quot; i,s a feature ,~tructure, and for some interpretation I m,l some object u in 1, l&amp;quot; is true ol'u under 1.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="1251" type="metho">
    <SectionTitle>
3. MORPHS
</SectionTitle>
    <Paragraph position="0"> The M)undance of inLerpregations se.mns to preclude an effectiw~ algoriidml to decide if a fea.ture structure is s~tisfiabh~. However, I inserl; morl)hs I)eLweell \['ea, l, tlre sgrllCtllrCs ,q3ld objects \[.o yMd au iutm'prctaLion free charac~ tcrisat,ion of ~t saLislia.ble fcat;ure structure.</Paragraph>
    <Paragraph position="1"> Definition 7. M is a semi-morph ill&amp;quot; M is a triple (A, l', A), A is a nonemlH.y sulmet orgy, 1' Ls an effuiva, lcnce rehttJon over A,</Paragraph>
    <Paragraph position="3"> A i,~' a total function from A to ~'5,</Paragraph>
    <Paragraph position="5"> Detii,ition 8. M is a. morph ill&amp;quot; M is a semi-morph (A, 1', A), a.nd /br each (v 6 ~2\[ aim ca.oh n 6 q3, then rccv ~ A.</Paragraph>
    <Paragraph position="6"> \]:,a,ch nlorph is the Moshicr M~straction (see \[MosIIIER 1988\]) of a connected mtd totMly well-typed (see \[CARPt,;NTt,:I~ 1992\]) Moore machine with possibly intlnitely many slates, inpul a.ll)lla.bel; Q{, and oul:put Mphabet C/'~.  Definition 9. M abstracts u under l iff M is a morph (A, P, A), \[ is an interpretation (U, PS', A), u is an object in I,</Paragraph>
    <Paragraph position="8"> ifl'e,(re)(u) is defined, and = s(P,(re)O0).</Paragraph>
    <Paragraph position="9"> Proposition 10. l'br each interpretation I and each object u in I, some unique, morph ahstracts u under l. I thus write of the abstraction of u under \[. Definition 11. u is a standard ohject i\[r u is a quadruple (A, P, A, E), (&amp;, 1', A) is a morph, and E is an equivalence c/ass under 1'.</Paragraph>
    <Paragraph position="10"> \[ write U for the set of standard objects, write ~ for the total function fi'om U to ~, where for each a E O and each (A,I',A,E) C U, S(&amp;, F, A, E) = cr iff for some rr G E, Afro) = or, and write A for the total function fi'om ~t to the set of partial functions fi'om U to U, where for each &lt;v E 9.1, each (&amp;, F, A, F,) E U and  each (&amp;', F', A', E') G U, X(c~)(A, r, A, E) is defined, and /(,~)(a, r, A, E) = (a', r', A', E') iff (A, I',A) = (a',F',A*), and for some re G E, rea. E F,'.</Paragraph>
    <Paragraph position="11"> Lemma 12. (U, S, A) is an interpretation. I write 7 for (U, ,5', A}.</Paragraph>
    <Paragraph position="12"> Lemma 13. For each (A,I',A,E) E (), ea.ch (A', r', A', E') E 9 a,.~ each re C q~, ~'/~(re)(A, r, A, r.) is (le~.,e~l, a.,,t ~5~(re)(A, r, A, ~) = (a', r', A', ~') ia&amp;quot; (a, r, A) = (~', r', A'), a,,(~ for some re' G 1'3, re% G E'.</Paragraph>
    <Paragraph position="13"> ProoL By induction on the length of re. ', Lemma 14. For each ( A, F , A, E} EU,  if E is the equivalence class of the. empty path under 1' then the abstraction of (A, F, A, E) under is (A, F, A).</Paragraph>
    <Paragraph position="14"> Proposition 15. I'br each morph M, for some interl&gt;retation \[ and some object u in I, M is the abstraction ofu under I.</Paragraph>
    <Paragraph position="15"> Definition 16. 1;' approximates M iff F is a l}ature structure (Q,q,6,0), M is a morph (A, I', A), and for each re1 E e43, each re'2 C q3 and each q' EQ, il'rel runs to q~ in I&amp;quot;, and re2 runs to q' in F then (~rt, rr2) E r, and o(q') ~ a(~).</Paragraph>
    <Paragraph position="16"> A feature structure approximates a morph iff the Moshier abstraction of the feature structure abstractly subsumes (see \[CARPEN'PI,;lt 1992\]) the morph.</Paragraph>
    <Paragraph position="17"> Proposition 17. For each interpretation I, each ohject u in I and each feature structure F~ F is true of a under 1 iff 1;' approximates the abstraction of u under I.</Paragraph>
    <Paragraph position="18"> Theorem 18. For each feature structure I,', l i' is satisfiable iff 1,' approximates some morph.</Paragraph>
    <Paragraph position="19"> Proof. From prol&gt;ositions 15 and 17. B</Paragraph>
  </Section>
  <Section position="6" start_page="1251" end_page="1252" type="metho">
    <SectionTitle>
4. RESOLVED FEATURE
STRUCTURES
</SectionTitle>
    <Paragraph position="0"> Though theorem 18 gives an interpretation free eharacterisation of a satisfiable feature structure, the characterisation still seems to admit of no effective algorithm to decide if a feature structure is satisfiable, tlowever, I use theorem 18 and resolve.d feature structures to yield a less general interpretation free characterisation of a satisfiable feature structure that admits of such an algorithm.</Paragraph>
    <Paragraph position="1"> Definition 19. R is a resolved feature structure itr R is a feature structure (Q, q, a, p}, p is a total function from Q to 6, and for each ~ E 91 and each q' G Q, if ~(q I, ct) is defined then ~(p(q'), ~r) is defined, and (~(p(q'), oz) ~_ p(a(q', c~)).</Paragraph>
    <Paragraph position="2"> Bach resolved feature structure is a well-typed (see \[CARI'ENTF, R 1992\]) feature structure with output alphabet (%.</Paragraph>
    <Paragraph position="3"> Definition 20. IC/. is a resolvant off iff R is a resolved lbature structure (Q, q, 6, p), F is a feature structure (Q,q,~,O), and rot each q' e Q, o(q') ~_ p(q').</Paragraph>
    <Paragraph position="4"> Proposition 21. ~br each interpretation I, each object u in I an(/ each feature structure I a , 1;' is true of u under 1 ill&amp;quot;some resolwmt of J;' is true of u under I.  Definition 22. (~, %, -&lt;, 0,~2\[, ~) is rational iff for each er G 0 and each (v G ~2\[, ira(o-, ~) is defined then ~r some a' ~ O, ~(cr, a') :&lt;_ or'.</Paragraph>
    <Paragraph position="5"> Proposition 23. 1\[&amp;quot; (~, %, ~, O, ~21, ~) is rational then for each resolved tbature structure R, R is satisfiabh'..</Paragraph>
    <Paragraph position="6"> Proof. Suppose that N. = (Q, q, 6, p) mid fl is a bijection from ordinal ( to G. Let</Paragraph>
    <Paragraph position="8"> An U rrcr ~r ff An, and  (z:X, F, A) is a morph thud; 1~ approximates. By theorem 18, R is satisliable. ,&amp;quot; Theorem 24. If (.Q, %, ~, ~, '2\[, ~) is rati~mal  then tbr each feature structure l&amp;quot;, f&amp;quot; is satisfiable ifl&amp;quot; I,' has a. resoh'am. Proof. l?rom proposil, ious 21 and 23. *</Paragraph>
  </Section>
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