TYPED FEATU19E STRUCTURES AS DESCRIPTIONS 
Paul .John }ring* 
Seminar fiir Sprachwissenschaft, Ebcrha,rd-t(arls-UniversitSot l 
ABSTRACT 
A description is an entity that can be inter- 
preted as true or false of an object, and us- 
ing feature structures as descriptions accrues 
several computational benefits. In this paper, 
1 create an explicit interpretation of a typed 
feature structure used as a description, define 
the notion of a satisfiable fe.ature structure, 
and create a simple and effective algorithm to 
decide if a fe.ature structure is satisfiable. 
1. INTRODUCTION 
Describing objects is one of several purposes 
for which linguists use fe.at, ul:e structures. A 
description is an er, tity that can be interpreted 
as true or false of an object. For example, the 
conventional interpretation of the description 
'it is black' is true of a soot particle, but false 
of ~ snowtlake, q'herefore, any use of a feature 
structure to describe an object delnands that 
the feature structure can be interpreted as true 
or false of the object. In this paper, I tailor 
the semantics of \[K~NG 1 989\] to suit the typed 
feature structures of \[CAII.I'F, NTFA{ 1 992\], and 
so create an explicit interpretation of a typed 
feature structure used as a descript, ion. I then 
use this interpretation to define the notion of 
a satisfiable feature structure. 
Though no featm'e structure algebra provides 
descriptions as expressive as those provided 
by a feature logic, using feature structures to 
describe objects profits from a large stock of 
available computational techniques to repre- 
sent, test and process feature structures. In 
this paper, I demonstrate the computational 
benefits of marrying a tractable syntax and 
an explicit semantics by creating a simple and 
effective algorithm to decide the satisfiability 
*The research presented in this paper was sl)on- 
sored by '\]'eilprojekt B4 "Constraints on (h'ammar for 
Efficient Generation" cff the Sonderforschungsbereich 
340 of the Deutsche ForschungsgemeinschafL I also 
wish to thank Bob Carpenter, Dale (lerdemmm, q'hile) 
GStz and Jennifer King for their invalualAe hel l) with 
this paper. 
tWilhehnst.r. 113, 72t17,1 ~l~{ilfingen, (\]el'Ill\[|fly. 
Einaih klng®sfs.nphil.unl- t uebingen.de. 
of a feature structure. Gerdemann and Ggtz's 
'Doll type resolution system implements both 
the sen,antics and an efficient refinement of 
the satisfiability algorithm I present here (see 
\[C,5TZ 1993\], \[GEItDF, MANN AND I(ING 1994\] 
and \[G~m)EMA~N (VC)\]). 
2. A FEATURE STRUCTURE 
SEMANTICS 
A signatm.e provides the symbols from which 
to construc.t typed feature structures, and an 
interpretation gives those symbols meaning. 
Definition 1. E is a siguature iff 
E is a sextuple (~, %, ~, G, ffl, ~), 
is a set, 
(%,-<} 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\[ to %, and 
for each r C 37, each r' C % and each o" C ~, 
if~(r, or) is defined aml r ~ r' 
then ~(r', ct) is defined, and 
a(~, ~) _-< a(<, .). 
\]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. 
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 ~\[ and each u C U, 
if a((:~)(~,) is deC, ned 
then ~(S(u), a) is defined, and 
;~(s'(~,), ,,) ~ ,V(A(~)(*O), and 
for each cY G ~( and each u E U, 
if~(X(u), a) is d(,Jined 
Suppose that 1 is an interpretation (U, £', A). 
I call each member of U an object in I. 
"1250 
\]','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 <m<: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\[" type rt 
contMns the denotation of l;ylw, 7"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>m:t;ia\[ funcl.iol~ il, 
denol;es. Appropriateness encodes ~t rcbttion- 
ship 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> 
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, ,'). 
I call a linitc sequence of attribul,es a path, 
and write q3 for I,he set, of paths. 
Definition 3. 1' is the path interl~retati(m 
fimctlon under 1 ill" 
I is an interpretation (U, £', 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<mal 
coml,o,siti,m of d ( m ) ..... A ( (~,, ). 
1 write t~ for the path iute,'prctal.ion flu,orion 
mMer l. 
De.finition 4. l,' is a \[baturc structm.c ill" 
I," is a quadrulde (Q, q, 5, 0), 
Q is a tinite subset o1'~\], 
q~Q, 
8 is a. finite pa.rtia.I function from the, 
Ca, rtesian l,rgduct ot" Q mM c2\[ to Q, 
0 is a totM l)mction from Q to %, and 
for each q/ ~ Q, 
&n" some re (5 q3, re rlm.s to q' in I c, 
where (,'vt,...,;M) z't/zzs l,o q' ill 1" ill' 
q' 6 Q, and 
~." son., {qo,..., q,} C- q, 
q = qo, 
for each i <., 
8(qi,o'i41) iS de, lined, and 
3(qi, (Vi4-1) := qi+l, ~ltl(l 
q,, -. q/. 
}",;tch \['(!;tl;llr(! Stl'tlC\[,llr(~ is a COllllCC~,C(l f~\]001"(! 
machine (see \[MooRI,; 1(;56\]) with finitely 
mauy st~tes, input alphabet 9..\[, and output 
Mplm.bet X. 
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", and 
rr.2 runs to q~ in l" 
tl,,,,, :,,(~,)(,,) i,~ ,mi,,.a, 
J~(~)(,,) i~ ,>t/,,.4 
0(q') ~ s(v,(,~,)(u)). 
Definition 6. I,' is a satisfiable feature struc- 
ture ill' 
I" i,s a feature ,~tructure, and 
for some interpretation I m,l some object u 
in 1, l" is true ol'u under 1. 
3. MORPHS 
The M)undance of inLerpregations se.mns to 
preclude an effectiw~ algoriidml to decide if a 
fea.ture structure is s~tisfiabh~. However, I in- 
serl; 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. 
Definition 7. M is a semi-morph ill" 
M is a triple (A, l', A), 
A is a nonemlH.y sulmet orgy, 
1' Ls an effuiva, lcnce rehttJon over A, 
\[br each ~v 6 2(, each ~l ~- ~ and each 
~r.e ~ q3, 
il'Tq(v ~ A and (Trt,Tr~) (5 1' 
theql (Trim rr,2,~) ~ I', 
A i,~' a total function from A to ~'5, 
for each rq ~ q3 and each 7r.e C q3, 
i\['(7fl, 71"2) ~_ l' ~,h(?l, A(TII)= A(TF2) , ~tlld 
tbr each (~ C ~21 and e:rch 7r 6 9f3, 
if fro: 6 A 
O,.,, ~ 6 A, ~(A(,0, "9 i,' ,teti,.,d, ~u.l 
a(A(~),-) ~ a(~(0. 
Detii,ition 8. M is a. morph ill" 
M is a semi-morph (A, 1', A), a.nd 
/br each (v 6 ~2\[ aim ca.oh n 6 q3, 
then rccv ~ A. 
\]:,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 ¢'~. 
1251 
Definition 9. M abstracts u under l iff 
M is a morph (A, P, A), 
\[ is an interpretation (U, £', A), 
u is an object in I, 
for each rq G 9,3 and each re2 C ~, 
(re1, ~r2) E I' 
itr P/rr,)(, 0 is ddi,,ed, 
P,(~)(,,) is Jea,,e~, ~,,d 
for each cr E 0 and each re C q~, 
(re, ~) c a 
ifl'e,(re)(u) is defined, and 
= s(P,(re)O0). 
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), 
(&, 1', A) is a morph, and 
E is an equivalence c/ass under 1'. 
\[ 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(&, 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 <v E 9.1, each (&, F, A, F,) E U and 
each (&', 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,'. 
Lemma 12. (U, S, A) is an interpretation. 
I write 7 for (U, ,5', A}. 
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" (a, r, A) = (~', r', A'), a,,(~ 
for some re' G 1'3, re% G E'. 
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). 
Proposition 15. I'br each morph M, 
for some interl>retation \[ and some object u 
in I, 
M is the abstraction ofu under I. 
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", and 
re2 runs to q' in F 
then (~rt, rr2) E r, and 
o(q') ~ a(~). 
A feature structure approximates a morph iff 
the Moshier abstraction of the feature struc- 
ture abstractly subsumes (see \[CARPEN'PI,;lt 
1992\]) the morph. 
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. 
Theorem 18. For each feature structure I,', 
l i' is satisfiable iff 1,' approximates some 
morph. 
Proof. From prol>ositions 15 and 17. B 
4. RESOLVED FEATURE 
STRUCTURES 
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 charac- 
terisation of a satisfiable feature structure that 
admits of such an algorithm. 
Definition 19. R is a resolved feature struc- 
ture 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~)). 
Bach resolved feature structure is a well-typed 
(see \[CARI'ENTF, R 1992\]) feature structure 
with output alphabet (%. 
Definition 20. I¢. 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'). 
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"some resolwmt of J;' is true of u under I. 
1252 
Definition 22. (~, %, -<, 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') :<_ or'. 
Proposition 23. 1\[" (~, %, ~, O, ~21, ~) is ra- 
tional then for each resolved tbature structure 
R, R is satisfiabh'.. 
Proof. Suppose that N. = (Q, q, 6, p) mid fl is 
a bijection from ordinal ( to G. Let 
A0 = {71" f('r S°l\]le q' ~ il~l ' } 71" l'llnS to q! 
P0 = (rq,~r2) ~r~ runs 1.o q' in 1~, and , 
~r2 r~s toq!in Ie 
and 
An= (~r, cr) ~r runs l.o q! in lg,~md . 
cr -p(q ) 
For each n ~ IN, let 
An+l = 
An U rrcr ~r ff An, and 
~(A,,(rr), ,e) is defined 
l'n+l : 
l'~,z U (Trl~V , "/r2Cg ) 7Cl('g ~ An+l' gtll(t 7r.,ev ~ An4.1, and ' 
(;1,7r2) C \['n, 
An+l =z .'< 'a, \] 
~r~v ~ A,~+, \ A., and t 
A,~U (Trcr,fl(~)) c is the least ordinal f' 
ill ( such |,hat \[ 
a(A,,(~),,*) ~ ;~(~) J 
For each n ~ IN, (A,,, I',,,A,) is a semi-morph. 
lint 
,x = U{A. I,, ~ ,N), 
r= U{r,~ I- e,N}, ~.,d 
A-- U{A, I" < ~N}. 
(z:X, F, A) is a morph thud; 1~ approximates. By 
theorem 18, R is satisliable. ," 
Theorem 24. If (.Q, %, ~, ~, '2\[, ~) is rati~mal 
then tbr each feature structure l", 
f" is satisfiable ifl" I,' has a. resoh'am. 
Proof. l?rom proposil, ious 21 and 23. • 
5. A SATISFIABILITY 
ALGORITHM 
In this section, I use theorem 24 to show how 
- given a rationM signature I.lmt meets reason 
M~le comput~tiona.l conditions 1.o construct 
an effective Mgorithm to decide if a f<tture 
sl.ructure is s~tisfial)h~. 
Dctinition 25. (£), %, <, G, ~, ~) is com- 
putable iff 
Q, '~ and ~2\[ are counta.lde, 
0 is finite, 
l'or .some (;Hb(:tiw,' fimction SUB, 
for each T1 ~ ~£ arid each r.2 ~ T., 
if rl ~ v2 
then SUB(rl, ru) =: 'true' 
otherwise SUB(rl, r2) = 'tklse', mid 
lbr some ellbctive function APP, 
for each r G • and each c~ G 92, 
if ~(r, cY) is defined 
then APP(T, ¢v) --=- ~(r, ~v) 
otherwise APP(r, (v) ='undefined', 
Proposition 26. lf(~,%, ~, 0,~1,~) is con> 
puta.lde then for some effective fimction RES, 
\[br each feature structure I a, 
RES(\]") -- a list of the resolvants o1' \[". 
Proof. Since (k~, '12, y, '0, ~, ~) is computabh'. 
for some elfeetiw; function GEN, 
for e~ch linite Q c O, 
GER(Q) = a list of the total functions 
from Q to (_'~, 
for some effectiw~ fimction TESTI, 
for each finite set Q, each tlnite partiM 
function ~ from the Caxtesmn product, of Q 
and ~\[ 1.o Q, ~md each total flmction 0 from 
O to %, 
if for (,,.'h (q, ~) in the domMn of 5, 
N(O(q), ev) is de.fined, arm 
a(0(q), ,,,) ~ off(q, .)) 
then TESTI(~, O) ~ 'true' 
otherwise TEST1(6, 0) = 'false', 
and for seine et\['ectiw~ flmction TEST2, 
lbr e;~ch tinite set Q, each total function 01 
from Q to % ~md e~Lcll total fimction 0~ 
from Q to %, 
if fo,' e\[tch (l ~ (~}, 01(q) ~-~ 02((1) 
thell TEST2(01,02) 7-_- 'tr/10' 
otherwise TESTu(0t, 0~) = 'false'. 
Construct RES ,as follows: 
for each fe;tl.ure s|;rtl(:\[,llre ((v), q, {5, 0), 
while Ei, , az (p, Pl,..., fli) is not clnl)ty 
do set, )21,, = (pt,...,pi) 
if TESTI ((5, p) _--. %rue', 
TEST2(0, p) = 'true', and 
>~,,,,~ = <,i,,..,,}) 
th,,.,, ~,t ~o,,~ = (,,,I,...,4> 
if Eo,,t -- (\[q .... , p.) 
t,h~. o,,q.. ((O, q, ~, .,>,,.,, (q, q, ~,.,.)). 
RES is an effect.ive algorithm, and 
for e4u;h foalAIl:e s\[;i?llCtltr(~ 1", 
RgS(/") -- ~ list of the resolwml, s of l c. 
ii 
1253 
12,54 
Theorem 27. /f (k~,~, ~, ~,~,~) is rational 
and computable then for some eflbctive func- 
tion SAT, 
for each feature structure F, 
if F is satisfiable 
then SAT(F) = 'true 
otherwise SAT(F) = 'false'. 
Proof, From theorem 24 and proposition 26. 
Gerdemann and G6tz's Troll system (see 
\[G6Tz 1993\], \[GFRDEIVIANN AND KING 1994\] 
and \[GERDEMANN (FC)\]) employs an efficient 
refinement of RES to test the satisfiability of 
feature structures. In fact, Troll represents 
each feature structure as a disjunction of the 
resolvants of the feature structure. Loosely 
speaking, the resolvants of a feature structure 
have the same underlying finite state automa- 
ton as the feature structure, and differ only 
in their output fllnction. Troll exploits this 
property to represent each feature structure 
as a finite state automaton and a set of output 
flmctions. The '1¥oll unifier is closed on these 
representations. Thus, though RES is compu- 
tationally expensive, Troll uses RES only dur- 
ing compilation, never during run time. 
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