D A T A T Y P E S I N C O M P U T A T I O N A L
P H O N O L O G Y
Ewan Klein
University of E d i n b u r g h , C e n t r e for C o g n i t i v e Science 2 Buccleuch Place, l~;dinl)urgh E l l 8 91,W, Scotland
Entail: k l o in@od, a c . uk
A B S T R A C T
T h i s p a p e r exanfines c e r t a i n a s p e c t s of phono- logical s t r u c t u r e from t h e v i e w p o i n t of a h s t r a c t d a t a types, O u r i m n l e d i a t e goal is to find a for- m a t for l)honological r e p r e s e n t a t i o n which will be r e a s o n a b l y f,'fithful to t h e concerns of t h e o r e t i : cal p h o n o l o g y while I)eing rigorous enough to a(I- rail a c o m p u t a t i o n a l i n t e r l ) r e t a t i o n . The longer t e r m goal is to i n c o r p o r a t e such r e p r e s e n t a t i o n s
i n t o all a p p r o p r i a t e general framework for llatn- ral l a n g u a g e processing, i
1
I n t r o d u c t i o n
One of t h e d o m i n a n t p a r a d i g n l s ill cnrrell| coln- putat.ional linguistics is l)rovided by unification- b a s e d g r a m m a r f o r m a l i s m s . Such f o r m a l i s m s (of. IShieber 1986; K a s p e r t~ R o u n d s 1986)) describe hierarchic~d f e a t u r e s t l ' t l e t l l r e s , w h i c h iH i n a l l y
ways would a p p e a r to be an ideal s e l l i n g [br f o r m a l phonological analyses. 1,'eature bundles have l o n g been used l)y phonologists, and more recent w o r k on so-called feature geonletry (e.~. ( C l e m e n t s 1985; Sagey 19,~6)) has i n t r o d u c e d hi- e r a r c h y i n t o such r e p r e s e n l a t i o n s . Nevertheless. t h e r e are reasons to step back from s t a n d a r d f e a t u r e - b a s e d apl~roaches, and i n s t e a d to adopl the a l g e b r a i c p e r s p e c t i v e of a b s t r a c l d a t a types (AD'P) which h a s been widely a d o p t e d iu com- l)uter science. One general m o t i v a t i o n , which we shall n o t e.xplore here. is thai Ihe a c l i v i l y of g r a n t l n a r w r i t i n g , viewed as a process of pro- g r a m m e specification, should be a m e n a b l e Io sl~p- wise refinement in which the set of {sol neces- sarily i s o m o r p h i c ) n,odels a d m i t t e d by a loose IThe work reported in this paper has [)¢:~,1, ~;tl ried ollt its part of the research i)rf)glitli/lll(!S o{ l]l(' ].{llnl&l[ (~oFiin/llllic&[iOll |lesea.rch (}(:Illl'C. sl/ppOl'led })3
the OK Economic and Social Rescalch (:ouncil aml the project Computational l)houoh)gy: .,I ('onst~aint-fh~s¢d Approach, funded by the IlV. ~qcience and Engineering I(t. search Council, under grant (;R/(;-22081. 1 am glalt'ful to Steven Bird. Kimba Newton and 'l'm/v Simou [m di> cussions relating to this work.
AcrEs DE COLING-92, NAtc~S, 23-28 Ao~rr 1992
specilication is g r a d u a l l y narrowed down to a u,fiqtm 'algebra (cf. ( S a n n e l l a & Tarleeki 1987) for an overview, and (Newton in prep.) for the a p l d i c a t i o n to g r a m m a r writing). A second mo- t i v a t i o n , discussed in d e t a i l by (Beierle & P l e t a t
1988; Beierle K~ P l e t a t 1989; Beierle et al. 1988),
is to use e q u a t i o n a l ADTS to p r o v i d e a m a t h e m a t - ical foundation for h~ature s t r u c t u r e s . A t h i r d m o t i v a t i o n , d o m i n a n t in this pal)er , is to use the AI)T appl'oach lo p r o v i d e a richer a r r a y of ex- plicit d a t a types t h a n are readily a d m i t t e d by "p'tlre' feature s t r u c t u r e a p p r o a c h e s . Briefly, in t h e i r raw form, [eature t e r m s (i.e., fnrnlalislns for d e s c r i b i n g h~alure stru(:tures) do not a l w a y s pro- vide a p e r s p i c u o u s f o r m a t for r e p r e s e n t i n g strllc-
t II re.
On the ADT a p p r o a c h , complex d a t a types are built up from a t o m i c types by m e a n s of c o n - s t r u c t o r f u n c t i o n s . For e x a m p l e . . . . (where we use the underscore '_' to m a r k the position of the fimction's a r g u m e n t s ) creates e l e m e n t s of t y p e L i s t . A d a l a t y p e m a y also have s e l e c - t o r f u n c t i o n s for t a k i n g d a t a e l e m e n t s a p a r t . T h u s , selectors for l h e t y p e L ± s t are the func tions f i r s t and l a s t . S t a n d a r d feature-bossed e n c o d i n g of lisls uses only selectors for the d a t a type; i.e. the feature labels FIRST and LAST ill
( 1 ) FIRST : o" 1 17 LAST : (FIRST : o" 2 17 LAST : nil)
tlowever, the list c o n s t r u c t o r is left implicit, T h a t
is, the feature term e n c o d i n g tells you how lists are pulled a p a r t , but does not say how they are built up. W h e n we confine our a t l e n t i o n j u s t to lists, lhis is not much to worry a b o u t , ltowever, tile s i t u a t i o n becomes less s a t i s f a c t o r y when we atIelnpI' to encode a larger variety of d a t a struc- tures into one and the s a m e f e a t u r e term; say, for e x a m p l e , s t a n d a r d lis(s, associatiw~ lists (i.e.
strings), c o n s t i t u e n t s t r u c t u r e hierarchy, and au t o s e g m e n t a l association. In o r d e r to distinguish axtequately between e l e m e n t s of such d a t a types, we really need to know the logical p r o p e r t i e s of their respective c o n s t r u c t o r s , and this is a w l
ward w h e n t h e c o n s t r u c t o r s are n o t m a d e ex- plicit. For c o m p u t a t i o n a l phonoloKv, it is not an u n l i k e l y scenario to b e confronted w i t h such a va- riety of d a t a s t r u c t u r e s , since one m a y well wish to s t u d y t h e c o m p l e x i n t e r a c t i o n between, say, n o n - l i n e a r teml)oral relations and prosodic hier- archy. As a vehicle for c o m p u t a t i o n a l i m p l e m e n - t a t i o n , the u n i f o r m i t y of s t a n d a r d a t t r i b u t e / v a l u e n o t a t i o n is e x t r e m e l y usefld. As a vehicle for the- ory d e v e l o p m e n t , it can be e x t r a o r d i n a r i l y uu- perspicuous.
T h e a p p r o a c h which we p r e s e n t here t r e a t s phono- logical c o n c e p t s as a b s t r a c t d a t a types. A par- t i c u l a r l y convenient d e v e l o p m e n t e n v i r o n l n e n t is provided by the l a n g u a g e O B J (Goguen & Win- kler 1988), which is based on order sorted equa, tionaJ logic, and all the e x a m p l e s given below (except where e x p l M t l y i u d i c a t e d to the con t r a r y ) run in the version of O B J 3 released by sltI in 1988. T h e d e n o t a l i o n a l s e m a n t i c s of a.n OB.] m o d u l e is an algehra, while its o p e r a t i o n a l se- m a n t i c s is based on order sorted rewritiug. I 1 1.1 and 1.2 give a more d e t a i l e d i n t r o d u c t i o n into the formal framework, while § 2 a n d 3 i l h l s t r a t e the a p p r o a c h w i t h some phonological examples.
1 . 1 A b s t r a c t D a t a T y p e s
A d a t a t y p e consists of one or more d o m a i n s of d a t a i t e m s , of which certaiu e l e m e n t s are des- i g n a t e d as basic, t o g e t h e r w i t h a set of opera tious on t h e d o m a i n s which suffice to g e n e r a t e al] d a t a i t e m s in the d o m a i n s fl'om the I)asic items. A d a t a t y p e is a b s t r a c t if it is i n d e p e n d e n l of
a n y p a r t i c u l a r r e t ) r e s e n t a t i o n a l scheme. A fun- d a m e n t a l claim of the ADJ g r o u p (cf. (Goguen. T h a t c h e r ,~ W a g n e r 1976)) and llluch subsequent work (cf. ( E h r i g & MMn" 1985)) is t h a t a b s t r a c l d a t a t y p e s are (to be modelled as) algebras: and moreover, t h a t the models of a b s t r a c t data types are ilfitial alget)ras. ~
T h e s i g n a t u r e o f a m a u y - s o r t e d algebra is a l)air = <S,O } consistiug of a set S of sorts and a se~ O of c o n s t a n t and o p e r a t i o n symbols. A s p e c i - f i c a t i o n is a p a i r ( r E > consisting of a signal are t o g e t h e r w i t h a set g of e q u a t i o n s over t e r m s c o n s t r u c t e d from s y m b o l s in O and variables of the sorts in S. A m o d e l for a speciIica.tion is ~An initial algebra is characlerized uniquely up to |so morphism as the semantics of a specification: there is a unique homomorphisnl from the initial algebra inlo t'vely algebra of the specification.
an algebra over the s i g n a t u r e which satisfies all the equations £. I n i t i a l algebras play a special role as the s e m a n t i c s of an algebra. An i n i t i a l a l g e b r a is m i n i m a l , in the sense expressed by the principles "no j u n k ' and 'no confusion'. 'No j u n k ' m e a n s t h a t the a l g e b r a only c o n t a i n s d a t a which are denoted by variable-fl'ee t e r m s b u i l t up from ol)eration symbols in the s i g n a t u r e . 'No confu- sion' means t h a t two such t e r m s t a n d t ~ d e n o t e t h e s a m e object in the a l g e b r a only if t h e equa- tion t = F is derivable from the e q u a t i o n s of the specification.
Specifications are w r i t t e n in a convent|ohM for- m a t consisting of a d e c l a r a t i o n of s o r t s , opera- tion symbols (op), and e q u a t i o n s (oq). P r e c e d i n g the e q u a t i o n s we list all t h e variables ( v a r ) which figure in them. As an i l l u s t r a t i o n , we give below an OBJ sl)ecification of the d a t a t y p e LIST1.
( 2 ) obj LIST1 i s s o r t s Ell L i s t op n i l : - > L i s t .
o p . ~ : E l i L i s t - > L i s t . o p h e a d : L i s t - > E l i . o p t a i l : L i s t - > L i s t . v a r X : E l i .
v a t L : L i s t . e q ( X . n i l ) = X . e q h e a d ( X , L) = X . e q t a i l ( X . L) = L . e n d o
T h e sort list betweeu the : and the - > in an o p e r a t i o n d e c l a r a t i o n is called the a r i t y of the o p e r a t i o n , while the sort after t h e -> is its v a l u e s o r t . Together. tiw al'ity and value sort consti- l u t e the r a n k of an o p e r a t i o n . T h e d e c l a r a t i o n op n i l : -> E l t m e a n s t h a t n i l is a c o n s t a n t of sorl Ell,
The s p e c i t i c a t i o n ( 2 ) fails to g u a r a n t e e t h a t there are any objects of El/:. W h i l e we could of course add soule c o n s t a n t s of this sort, we would like to have a more general solution. In a p a r t i c u l a r application, we m i g h t w a n t to define phonologi- cal words as a L i s t of syllables (plus o t h e r con- s t r a i n t s , of course), and phonological p h r a s e s as a L i s t of words, rl'hat is, we need to p a r a m e - t e r l z e the t y p e LIST1 with r e s p e c t to t h e class of elements which c o n s t i t u t e the lists.
Before t u r n i n g to p a r a m e t e r i z a t i o n , we will first see how a m a n y - s o r t e d specification l a n g u a g e is generalized to an order sorted l a n g u a g e by intro- d u c i n g a subsort relation.
Sul)l)ose, for exanlple, t h a t we a d o p t the claim
t h a t all syllables h a v e ( ' l o n s e t s :(. Moreover. we wish to d i v i d e syllables i n t o the s u b c l a s s e s lmavy
a n d light. Obvimusly we wan! h e a v y a n d light
syllables t o i n h e r i t t h e l)roperties of the clas> of all s y l l a b l e s , e.g., t h e y h a w ' ( ' 1 o n s e t s . We use l t o a v y < S y l l t o s t a l e t h a t H e a v y is a s u b s o r l of tile s o r t S y l l . We inlerl)l'et this to m e a n t h a i lhe class of h e a v y syllables is a s u b s e ! of t h e class (if all s y l l a b l e s . Now, let o n s e t _ : S y l l - > N o r a
lie all o p e r a t i o n w h i c h selects tlle tits! m o r a of a syllable, anti let us i m p o s e t h e Iollowing con- s t r a i n t ( w h e r e Cv is a sul)sor! of N o r a ) :
( 3 ) v a r S : S y l l . v a r CV : Cv .
e q o n s e t S = C V ,
T h e n tile f r a m e w o r k of or(ler s o r t e d a l g e b r a ell- sures t h a t o n s e t is also delined for obje('l > of s{)i't
H e a v y .
llx~turlling to lists, the s p e c i I i c a t i o n ill (,I) (sli~hll.v simplified f r o m t h a t used h> ( ( ; o g u e n ,k: W i n k h q
| 9 8 8 ) ) i n t r o d u c e s E l i alld N e L i s t ( n o t l OlUl)t 3
lists) as s u b s o r t s of L i s t . a n d t h e r e b y !rein'ores on L I S T I in a n u m b e r of resi)ects, h, a d d i t i o n . tile s p e c i f i c a t i o n is p a r a m e t e r ! z e a l . T h a i is. il c h a r a c t e r i z e s a list of Xs, w h e r e the p a r a l n e l e r X
c a n b e i n s t a n t i a t e d tm a n y m o d u l e which satislies tile c o n d i t i o n TRIV; the l a l l e r is w h a t ((;oy;uell & W i n k l e r 1988} call a "requirenlenl t h e o r y ' , a n d in lhis case s i m p l y iml)oses on a n y inpul m o d u h , t h a t it h a v e a sot! w h i c h c a n be mal)p('(I to Ihe s o r t E l i .
(4) o b j L I S T [ X :: T R I V ] i s s o r t s L i s t N e L i s t ,
s u b s o r t s E l t < N e L i s t < L i s t , o p n i l : -> L i s t .
o p . _ : L i s t L i s t -> L i s t .
o p . : N e L i s t L i s t - > N e L i s t .
o p h e a d : N e L i s t - > E f t ,
o p t a i l : N e L i s t - > L l s t .
v a t X ; E l t .
v a t L : L i s t .
e q (X , n i l ) : X .
e q h e a d ( X . L ) = X , e q t a i l ( X . L) = L . e n d o
N o t i c e t h a t the list c o n s t r l l c t o r _._ llOW i)el'forllls t h e a d d i t i o n a l fluter!on o l
append,
a l l o w i n g Iwo lists tm lie c o n c a t e n a t e d , h, a d d i t i o n . ! h e se l e c t o r s llave beell m a d e ' s a f e ' , ill lhe Sellse t h a i t h e y o n l y a p p l y to o b j e c t s (i.e.. n o n e m p l y lisls)f o r w h i c h t h e y giwr sensible results: for w h a l . ill L I S T I , w o u l d h a v e been t h e m e a n i n g of h e a d ( n i l )?
allere, the term mNSET ief(!lS to lh(' inilal mma o[ a syllM)le in llyman's (198,t) velsion of tit(' iil(nai( th(!ol 3
2
M e t r i c a l T r e e s
:\s a f u r t h e r i l l u s t r a t i o n , we give below a speci- lit'at ion of the d a t a lyp(! B I N T R E E . T h i s m o d u l e has t w o p a r a m e t e r s , b o l h of w h o s e r e q u i r e l n e n t theories a r e TRIV. 4
(5)
o b 3 B I N T R E E [ N O N T E R M T E R M :: T R I V ] i ss o r t s T r e e N e t r e e .
s u b s o r t s E l i . T E R M N e t r e e < T r e e .
o p _ [ _ . _ ] : E I t . N O N T E R M T r e e T r e e - > N e t r e e .
o p _ [ _ ] : E I t . N O N T E R N E f t . T E R M - > T r e e .
o p l a b e l _ : T r e e - > E I t . N f l N T E R M . o p l e f t _ : N e t r e e - > T r e e ,
o p r i g h t _ : N e t r e e - > T r e e .
v a r s E l E 2 : T r e e ,
v a r s h : E I t . N O N T E R M .
e q l a b e l (h [ E l , E 2 ] ) = A ,
e q l a b e l (h [ El ]) = A .
e q l e f t ( h [ E1 , E2 ] ) : E1 .
eq r i g h t (h [ E1 , E2 ] ) = E2 . (~itdo
\Ve can l~mx~ inslanl, iale 1he f o r m a l p a r a n m t e r s of th0 m o d u l e in (5) with i n p m module.s whiEh sup- ply a l ) l n ' o p r i a l e sels of ilOlll, e r l n i n a ] a n d t e r m i n a l s y m b o l s , l,el us use u i ) p e r c a s e q u o t e d identifiers ( e M n e n l s of the OB.I i n o d u l e QID) for n o n t e r m i - nals. a n d lower case for t e r m i n a l s . T h e specitlca- lion in (5) allows us to t r e a t t e r m i n M s as trees, st; Ihal a b i n a r y tree. r o o t e d ill a n o d e 'A, c a n
have l e r m i n a l s as its d a u g h t e r s , l t o w e v e r , we ills() allow t e r m i n a l s to be d i r e c t l y d o m i n a t e d by a n ( m - b r a n c h i n g m o l h e r node. [Ioth possibilities o c c u r in the e x a m p l e s below. (6) i l l u s t r a t e s the i n s t a n t i a t i o n of tornlal p a r a m e t e r s by a n a c t u a l m o d u l e , n a m e l y QID. using t h e m a k e c o n s t r u c t .
16} m a k e B I N T R E E - Q I D i s B I N T R E E [ Q I D , Q I D ] endm
T h e nexl e x a l n p h , s h o w s Nellie r e d u c t i o n s in this m o d u l e , obt, aiued by t r e a t i n g t h e e q u a t i o n s as r e w r i t e rules a p p l y i n g fi'om left t o r i g h t .
~'l'hc n~tatir,a Elt .NONTERN. EIt. TEPd4 utilizes a qual- !lit:at!on M t he sort Eli by the input module's paranleter labch this is simply to allow disamlfigulttion.
(7)
l e f t ( ' h [ ' a , ' b ] ) .l e f t ( ' A [ ' B [ ' a ] , ' C [ ' b ] ] ) . " ~ ' B [ ' a ]
l e f t ( ' A [ ' B [ ' a , ' b ] , ' c ] ) .
~ * ' B [ ' a , 'b]
r i g h t ( l e f t ( ~ A [ ( ' B [ ' a , ' b ] ) , ' c ] ) ) .
l a b e l ( ' A [ ' a , ' b ] ) .
.x~ JA
l a b e l ( r i g h t ( ' A [ ' a , * B [ ' b , ' c ] ] ) ) .
~ 4 J B
S u p p o s e we n o w wish t o m o d i f y t h e definition of b i n a r y t r e e s t o o b t a l u m e t r i c a l trees, T h e s e are b i n a r y trees w h o s e b r a n c h e s a r e ortlered a c c o r d - i n g to w h e t h e r t h e y a r e labelled 's" ( s t r o n g ) or ' w ' ( w e a k ) .
• v
In a d d i t i o n , all trees h a v e a tlistinguishetl leaf n o d e c a l l e d t h e ' d e s i g n a t e d t e r m i n a l e l e m e n t ' ( d t e ) , w h i c h is c o n n e c t e d to t h e r o e ! of the tree I)y a p a t h of ' s ' n o d e s .
Let us define ' s ' a n d "w' to t>e o u r n o n t e r m i n a l s :
(8) obj M E T i s s o r t s L a b e l
o p s s w : -> L a b e l .
e n d o
In o r d e r to buihl tilt, d a t a i y p e of m e t r i c a l lr¢,e~ on t o p of b i n a r y trees, we c a n i m p o r t Ill(, mod- uh, BINTREE, s u i t a b l y i n s t a n t i a l e l l , using O B . l ' s e x t e n d i n g c o n s t r u c l . Notice t h a i we use MET to i n ~ t a n t i a t e t h e p a r a m e t e r w h i c h fixes BINTFLEE's ~et Of n o n t e r m i n a l s y m b o l s . ~
191 obj H E T T R E E is e x t e n d i n g
B I N T R E E [ M E T , Q I D ] * ( s o r t I d t o L e a f )
o p d i e : T r e e - > L e a f .
v a t L : L e a f .
v a t s T1 T 2 : T r e e .
'~'['he * construcl tells , s thai the i)ri,cipal ~.Ol~ of OlD. llalnely Id, is mappe({ (1)), a sig,tai,.e .;o*pl, isnl) to l llc sort Leaf in METTREE. ceq signals the presen(c o [ a (-otl- difionaI cquation. == is a buill-in I)olymou)hic cqualil> operation in OBJ.
Acres DECOLING-92. NAm ,'~% 23-28 Aor~r 1992
v a r s X : L a b e l . eq d t e ( X [ L ] ) = L .
c e q d i e ( X [ T1 , T2 ]) = d i e T1
if l a b e l TI == s . c e q d i e ( X [ T1 , T2 ]) = d i e T 2
if l a b e l T2 == s .
e n d o
T h e e q u a t i o n s s t a t e t h a t t h e d t e ( d e s i g n a t e d ter- m i n a l e l e m e n t ) o f a tree is t h e d t e of its s t r o n g s u b t r e e . A n o t h e r w a y o f s t a t i n g this is t h a t the i n f o r m a t i o n a b o u t d t e e l e m e n t of a s u b t r e e T is p e r c o l a t e d u p to its p a r e n t n o d e , .just in case T is tile "s' b r a n c h of t h a t n o d e .
T h e s p e c i f i c a t i o n METTREE c a n be criticised on a n u m b e r of g r o u n d s , i t h a s to use c o n d i t i o n a l e q u a t i o n s in a c u m b e r s o m e w a y t o t e s t w h i c h d a u g h t e r of a 1)inary tree is l a b e l l e d 's', More- over. it fails t o c a p t u r e t h e r e s t r i c t i o n t h a t no b i n a r y tree c a n h a v e d a u g h t e r s w h i c h are b o t h w e a k . or b o t h s t r o n g . T h a t is, it fails t o c a p t u r e t h e essential p r o p e r t y of m e t r i c a l trees, n a m e l y t h a t m e t r i c a l s t r e n g t h is a relational n o t i o n . W h a t we r e q u i r e is a m e t h o d for e n c o d i n g t h e fob l o w i n g i n f o r m a t i o n at a notle: " m y left (or r i g h t ) d a u g h t e r is s t r o n g " . O n e e c o n o m i c a J m e t h o d of d o i n g this is to label (all a n d o n l y ) b r a n c h i n g n o d e s in a b i n a r y tree w i t h o n e of t h e following t w o lahels: ' s w ' ( m y left d a u g h t e r is s t r o n g ) , ' w s ' ( m y r i g h t d a u g h t e r is s t r o n g ) . T h u s , we r e p l a c e M E T with t h e following:
obJ M E T 2 is
s o r t s L a b e l
ope sw ~s : -> L a b e l .
e n d s
W e can n o w simplify b o t h B I N T R E E a n d M E 'l'l{ l:;t']:
obj B I N T R E E 2 [ N O N T E R M T E R M :: T R I V ] is s o r t s T r e e N e t r e e ,
s u b s o r t s E I t . T E R M N e t r e e < T r e e .
o p _ [ _ , j : E I t . N O N T E R M T r e e T r e e -> N e t r e e .
o p l a b e l _ : T r e e -> E I t . N O N T E R M .
o p l e f t : N e t r e e -> T r e e .
o p r i g h t _ : N e t r e e -> T r e e . r a r e El E2 : T r e e .
v a r s A : EIt. N O N T E R M .
e q l a b e l (A [ El , E2 ] ) = A .
e q l e f t ( A [ E 1 , E2 ]) = El . e q r i g h t (A [ El , E2 ]) = E2 .
e u d o
obj M E T T R E E 2 is e x t e n d i n g
B I N T R E E 2 [ M E T 2 , Q I D ] * ( s o r t Id to L e a f ) .
o p h i e _ : T r e e -> L e a f .
v a r L : L e a f .
v a r s T I T2 : T r e e .
e q d t e L = L .
e q d t e T = i f l a b e l T = = s w
t h e n d i e ( l e f t T ) e l s e d t e ( r i g h t T ) f i . e n d s
3 F e a t u r e G e o m e t r y
T h e p ~ r t i c u l ~ r f e a t u r e g e o m e t r y we shM1 specify here is b a s e d o n t h e a r t i c u l ~ t o r y s t r u c t u r e de- fined in ( B r o w m a n & G o k l s t e i n 1989)Y T h e five a c t i v e a r t i c u l a t o r s a r e g r o u p e d i n t o a h i e r a r c h i cal s t r u c t u r e i n v o l v i n g a t o n g u e n o d e a n d an o r a l n o d e , an s h o w n in t h e f o l l o w i n g d i a g r a m .
r o o t
glot, tal relic oral
tongue labial
coronal dorsal
T h i s s t r u c t u l ' e is specilied via t e r m ('onstl'UC|Ol'~ ( _ _ } a , , a { . . . . } w h i c h ~i . . . s t a n d a r < i ),(,~iti ... t e n c o d i n g o f f e a t u r e s . F, a c h f e a l t t r e vahlc is ex- p r e s s e d as a llaturaJ ltUlnl)or [)o|w('ell 0 a n d 4. r e p r e s e n t i n g t h e c o n s t r i c t i o n d e g r e e of the ('or r e s l ) o n d i n g a r t i c u l a t o r . For examl)le, the tertu
{ 4 , 0 } : T o n g u e is an item of s o n T o n g u e con s i s t i e g of t h e value ,I for t h e Ioalure ('o)toN..\[.
a n d 0 for t h e DORSAl,; this in t u r u express(,> a s i t u a t i o n w h e r e t h e r e i s m a x i m a l c o a s t i ' i c t l o n o f
t h e t o n g u e tip, a n d m i n i m a l c o n s t r i c t i o n of t h e t o n g u e I)ody. O f c o u r s e , this e n c o d i n g is rat her c r u d e , a n d l)ossil)ly sacrifices c l a r i l y for cot,ci sion. H o w e v e r , it sultices as a w o r k i a g ex;,leple. We will r e t u r i / to constri(qioll degt'ees })el()w. T h e four s o r t s G e s t u r e , R o o t . O r a l a n d T o n g u e
ill (lO) atld the first t h r e e ol)erator~ cal)turo )h(' +;For spa(:(! reasons w( + hilVC Oltlittcd ;ItlX d i ~ ( I I ~ H ) I ) O[ ]~rowlnan (k! Gohistein's COllStricliolt ]o(;ttion (('[) .in(] (:Oltslfit:tioll She+l)(! ((;S) pltl'illlittlClt,. ~*'~ (' alSO hax,. omit
ted (lie supralaryllgea] node, since its i~hono]ogi(al lob t- somewhat dubious.
desired tree s t r u c t u r e , u s i n g a n a p p r o a c h w h i c h s h o u l d be f a m i l i a r b y uow. F o r e x a m p l e , t h e t h i r d c o n s t r u c t o r t a k e s t h e c o n s t r i c t i o n d e g r e e s
of G l o t t a l a n d V o l i c g e s t u r e s , a n d c o m b i n e s t h e m with a c o m p l e x i t e m o f s o r t O r a l to build all item of sort R o o t . T h e specifie~ttion i m p o r t s t h e u m ( h d e NAT of n a t u r a l n u m b e r s to p r o v i d e v a l u e s for c o n s t r i c t i o n degrees.
( 1 0 ) obj FEATS i s e x t e n d i n g N A T ,
s o r e s G e s t u r e R o o t O r a l T o n g u e .
s u b s o r t s R a t R o o t O r a l T o n g u e < G e s t u r e . o p { _ , } : N a t N a t - > T o n g u e .
o p { _ , _ } : T o n g u e N a t - > O r a l .
o p { . . . } : N a t N a t O r a l - > R o o t ,
o p _ ! c o r o n a l : T o n g u e --> N a t .
o p ! d o r s a l : T o n g u e - > N a t .
o f ! l a b i a l : O r a l - > N a t .
o p ! t o n g u e : O r a l - > T o n g u e .
o p _ ! g l o t t a l : R o o t - > N a t . o p ! r e l i c : R o o t - > N a t . o p _ ! o r a l : R o o t - > O r a l . v a t s C C I C 2 : N a t .
v a t s 0 O r a l .
v a t s T T o n g u e .
e q { C l , C 2 } ! c o r o n a l = C I .
e q { C I , C 2 } ! d o r s a l = C 2 ,
e q { T C } ! t o n g u e = T .
e q { T C } ! l a b i a l = C •
e q { C 1 , C 2 ) 0 } ! g l o t t a l = C 1 .
e q { C 1 , C?, , O } ! v e l i c = C 2 .
e q { C I , C 2 , 0 } ! o r a l = 0 .
elldo
AVe a d o p t the uol;atiollal coilventiOll of p r e p e n d int~ a '!" to the s a m e of seh:ctors w h i c h c o l r e s p o n d d i r e c t l y to f e a t u r e s . For e x a m p l e , t h e
! c o r o n a l seleC{or is a funct;ion d e f i n e d on conl- pIox ilenls of SOl"( Tollguo which rettzrlls air i t e m ()f sort Nat, r e l n ' e s e e t i u g t h e c o n s t r i c t i o n d e g r e e • ca]u(' for ('oronality.
Sonn' i l l u s t r a t i v e r e d u c t ions in t h e FEATS m o d u l e are given l)elow.
( I l l { 3 , 4 , { { 4 , 1 } , 1 1 } ! o r a l .
~-, { { 4 , 1 } , 1 1
{ 3 , 4 , { t 4 , 1 } , 1 } } ~ora~ !to~gue . .... {4,1}
{3,4,{{4,1},1}} ! o r a l !tongue ! c o r o n a l .
III 1 lie ..\ I)'1 ~pplo~-lch to leal Ill'e st rtlcI tires, feel'i- l e a n t 3 is r e p r e s e n t e d by e q t t a t i n g the v a l u e s of ~(']('('tol'~. I'IIIIS. Sill)pose Ih;ll two 5 e g n l e n t s S1, $2 '~hale ;t voicing sl)e(:ilication. We c a n w r i t e th],~ >t~ f~>llows:
( 1 2 ) S 1 ! g l o t t a l = S 2 ! g l o t t a l
T h i s s t r u c t u r e s h a r i n g is c o n s i s t e n t with one of t h e m a i n m o t i v a t i n g factors b e h i n d a u t o s e g m e n - tal phonology, namely, the u n d e s i r a b i l i t y of rules such as [~ voice] - - [~ nasal I.
Now we can i l l u s t r a t e the flmction of selectors in phonological rules. C o n s i d e r the case of 1';11- glish r e g u l a r p l u r a l f o r m a t i o n ( - s ) , where the voicing of t h e suffix s e g l n e n t agrees with t h a t of t h e i m m e d i a t e l y p r e c e d i n g s e g m e n t , unless it is a coronal fricative (in which case there m u s l be an i n t e r v e n i n g vowel). S u p p o s e we i n t r o d u c e the variables S1 $ 2 : Root, where S1 is the stem- final s e g m e n t a n d S 2 is the suffix. T h e rllle nllls[ also be able to access tile coronal node of $1. M a k i n g use of the selectors, this is simply $2 ! o r a l ! t o n g u e ! c o r o n a l (a nota.tion reminis- cent of p a t h s in feature logic. ( K a s p e r k l l o n n d s 1986)). T h e rule m u s t test w h e t h e r this coronal node c o n t a i n s a fi'icative specification. This ne- cessitates an e x t e n s i o n to our specification, which is described below.
B r o w m a n & Goldstein (19S9. 234ff) define "con s t r i c t i o n degree p e r c o l a t i o n ' , based on w h a l they call ~tube g e o m e t r y ' . The vocal t r a c | can b(, viewed as an interconnected set of tllbes, and t h e a r t i c u l a t o r s correspond to valves which have a m m l b e r of s e t t i n g s r a n g i n g from fiflly open to fiflly closed. As already m e n t i o n e d , these ~el- t i n g s are called constriction degrees ( ! cds). where fully closed is the m a x i m a l constriction and fully open is t h e m i n i m a l constriction.
T h e net constriction degree of the oral cavity m a y be expressed as t h e m a x i m u m of the con- s t r i c t i o n degrees of t h e lips, t o n g u e tip and Iongue body. T h e net c o n s t r i c t i o n degree of the oral and nasal cavities t o g e t h e r is s i m p l y the m i n i m m n of the two c o m p o n e n t c o n s t r i c t i o n degrees. To re cast this in the present f r a m e w o r k is straight[or- ward. However, we |teed l o first define t h e op- e r a t i o n s m a x a n d rain over pairs of n a l t l r a ] 1111111-
bers:
( 1 3 ) o b j M I N M A X
is p r o t e c t i n g N A T .
o p s m i n m a x : N a t N a t - > N a t . r a r e M N : N a t .
e x t e n d i n g F E A T S + M I N M A X .
o p _ ! c d : G e s t u r e - > N a t .
o p s c l o c r i t n a r r o w
m i d w i d e o b s o p e n : G e s t u r e - > B o o l .
v a t G : G e s t u r e .
v a t N ~ 1 N 2 : N a t .
v a r s 0 : O r a l .
v a r s T : T o n g u e .
e q N !cd = N ,
e q { N 1 , N 2 } !cd = m a x ( N 1 , N 2 ) .
e q { T , N } !cd = m a x ( T ! c d , N ) .
e q { N 1 , N 2 , 0 } !cd ~ m a x ( N l , m i n ( N 2 , 0 !cd)) .
e q clo(G) = S !cd == 4 . e q crit(G) = G !cd == 3 . e q n a r r o w ( G ) = G !cd = = 2 .
e q m i d ( G ) = G !cd = = I . e q w i d e ( G ) = G !cd == O .
e q o b s ( G ) = G !cd > 2 . e q o p e n ( G ) = G ! c d < 3 . e n d o
T h e specification C D allows classification into five basic constriction degrees (c'lo, c r i t , n a r r o w , mid, and w i d e ) by m e a n s of c o r r e s p o n d i n g one- place predicates, i.e. b o o l e a n - v a l u e d o p e r a t i o n s over gestures. For e x a m p l e , the fifth e q u a t i o n above s t a t e s t h a t G has the c o n s t r i c t i o n degree c t o (i.e. elm(G) is t r u e ) i f and only i f 6 !cd == 4.
T h e w o r k i n g of these p r e d i c a t e s is i l l u s t r a t e d be-
l o w :
(15) { 3 , 0 , { { 4 , 1 } , 1 } } !oral ! t o n g u e !cd . ~ 4
{ 3 , 0 , { { 4 , 1 } , 1 } } !oral !cd ~ 4
{ a , o , { 1 4 , ~ } , 1 } } ~cd .
~ 3
m i d ( { 3 , 0 , { { 4 , 1 } , 1 } } !oral l ~ i a l ) . t r u e
w i d e ( { 3 , O , { { 4 , 1 } , l } } ! o r a l ! l a b i a l ) . f a l s e
o p e n ( { 3 , 0 , { { 4 , 1 } , l } } ] o r a l ! l a b i a l ) . t r u e
c l 0 ( { 3 , 0 , { { 4 , 1 } , 1 } } !oral ! t o n g u e ) .
t r u e
R e f e r e n c e s
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