DATA TYPES IN COMPUTATIONAL PHONOLOGY 
Ewan Klein 
University of Edinburgh, Centre for Cognitive Science 
2 Buccleuch Place, l~;dinl)urgh Ell8 91,W, Scotland 
Entail: klo in@od, ac. uk 
ABSTRACT 
This paper exanfines certain aspects of phono- 
logical structure from the viewpoint of ahstract 
data types, Our imnlediate goal is to find a for- 
mat for l)honological representation which will 
be reasonably f,'fithful to the concerns of theoreti: 
cal phonology while I)eing rigorous enough to a(I- 
rail a computational interl)retation. The longer 
term goal is to incorporate such representations 
into all appropriate general framework for llatn- 
ral language processing, i 
1 Introduction 
One of the dominant paradignls ill cnrrell| coln- 
putat.ional linguistics is l)rovided by unification- 
based grammar formalisms. Such formalisms (of. 
IShieber 1986; Kasper t~ Rounds 1986)) describe 
hierarchic~d feature stl'tletllres, which iH inally 
ways would appear to be an ideal selling \[br 
formal phonological analyses. 1,'eature bundles 
have long been used l)y phonologists, and more 
recent work on so-called feature geonletry (e.~. 
(Clements 1985; Sagey 19,~6)) has introduced hi- 
erarchy into such represenlations. Nevertheless. 
there are reasons to step back from standard 
feature-based apl~roaches, and instead to adopl 
the algebraic perspective of abstracl data types 
(AD'P) which has been widely adopted iu com- 
l)uter science. One general motivation, which 
we shall not e.xplore here. is thai Ihe aclivily 
of grantlnar writing, viewed as a process of pro- 
gramme specification, should be amenable Io sl~p- 
wise refinement in which the set of {sol neces- 
sarily isomorphic) n,odels admitted 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 gradually narrowed down to a 
u,fiqtm 'algebra (cf. (Sannella & Tarleeki 1987) 
for an overview, and (Newton in prep.) for the 
apldication to grammar writing). A second mo- 
tivation, discussed in detail by (Beierle & Pletat 
1988; Beierle K~ Pletat 1989; Beierle et al. 1988), 
is to use equational ADTS to provide a mathemat- 
ical foundation for h~ature structures. A third 
motivation, dominant in this pal)er , is to use the 
AI)T appl'oach lo provide a richer array of ex- 
plicit data types than are readily admitted by 
"p'tlre' feature structure approaches. Briefly, in 
their raw form, \[eature terms (i.e., fnrnlalislns for 
describing h~alure stru(:tures) do not always pro- 
vide a perspicuous format for representing strllc- 
t II re. 
On the ADT approach, complex data types are 
built up from atomic types by means of con- 
structor functions. For example .... (where 
we use the underscore '_' to mark the position 
of the fimction's arguments) creates elements of 
type List. A dala type may also have selec- 
tor functions for taking data elements apart. 
Thus, selectors for lhe type L±st are the func 
tions first and last. Standard feature-bossed 
encoding of lisls uses only selectors for the data 
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 constructor is left implicit, That 
is, the feature term encoding tells you how lists 
are pulled apart, but does not say how they are 
built up. When we confine our atlention just to 
lists, lhis is not much to worry about, ltowever, 
tile situation becomes less satisfactory when we 
atIelnpI' to encode a larger variety of data struc- 
tures into one and the same feature term; say, 
for example, standard lis(s, associatiw~ lists (i.e. 
strings), constituent structure hierarchy, and au 
tosegmental association. In order to distinguish 
axtequately between elements of such data types, 
we really need to know the logical properties of 
their respective constructors, and this is awl 
1 4 9 PRec. oF COLING-92. NANTES. AUG. 23-28. 1992 
ward when the constructors are not made ex- 
plicit. For computational phonoloKv, it is not an 
unlikely scenario to be confronted with such a va- 
riety of data structures, since one may well wish 
to study the complex interaction between, say, 
non-linear teml)oral relations and prosodic hier- 
archy. As a vehicle for computational implemen- 
tation, the uniformity of standard attribute/value 
notation is extremely usefld. As a vehicle for the- 
ory development, it can be extraordinarily uu- 
perspicuous. 
The approach which we present here treats phono- 
logical concepts as abstract data types. A par- 
ticularly convenient development environlnent is 
provided by the language OBJ (Goguen & Win- 
kler 1988), which is based on order sorted equa, 
tionaJ logic, and all the examples given below 
(except where explMtly iudicated to the con 
trary) run in the version of OBJ3 released by sltI 
in 1988. The denotalional semantics of a.n OB.\] 
module is an algehra, while its operational se- 
mantics is based on order sorted rewritiug. I 1 1.1 
and 1.2 give a more detailed introduction into 
the formal framework, while § 2 and 3 ilhlstrate 
the approach with some phonological examples. 
1.1 Abstract Data Types 
A data type consists of one or more domains of 
data items, of which certaiu elements are des- 
ignated as basic, together with a set of opera 
tious on the domains which suffice to generate al\] 
data items in the domains fl'om the I)asic items. 
A data type is abstract if it is independenl of 
any particular ret)resentational scheme. A fun- 
damental claim of the ADJ group (cf. (Goguen. 
Thatcher ,~ Wagner 1976)) and llluch subsequent 
work (cf. (Ehrig & MMn" 1985)) is that abstracl 
data types are (to be modelled as) algebras: and 
moreover, that the models of abstract data types 
are ilfitial alget)ras. ~ 
The signature ofa mauy-sorted algebra is a l)air 
= <S,O } consistiug of a set S of sorts and a se~ 
O of constant and operation symbols. A speci- 
fication is a pair (rE> consisting of a signal are 
together with a set g of equations over terms 
constructed from symbols in O and variables of 
the sorts in S. A model 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 signature which satisfies all 
the equations £. Initial algebras play a special 
role as the semantics of an algebra. An initial 
algebra is minimal, in the sense expressed by the 
principles "no junk' and 'no confusion'. 'No junk' 
means that the algebra only contains data which 
are denoted by variable-fl'ee terms built up from 
ol)eration symbols in the signature. 'No confu- 
sion' means that two such terms t and t ~ denote 
the same object in the algebra only if the equa- 
tion t = F is derivable from the equations of the 
specification. 
Specifications are written in a convent|ohM for- 
mat consisting of a declaration of sorts, opera- 
tion symbols (op), and equations (oq). Preceding 
the equations we list all the variables (var) which 
figure in them. As an illustration, we give below 
an OBJ sl)ecification of the data type LIST1. 
(2) obj LIST1 is sorts Ell List 
op nil : -> List . 
op .~ : Eli List -> List . 
op head : List -> Eli . 
op tail : List -> List . 
var X : Eli . 
vat L : List . 
eq (X . nil) = X . 
eq head(X , L) = X . 
eq tail(X . L) = L . 
endo 
The sort list betweeu the : and the -> in an 
operation declaration is called the arity of the 
operation, while the sort after the -> is its value 
sort. Together. tiw al'ity and value sort consti- 
lute the rank of an operation. The declaration 
op nil : -> Elt means that nil is aconstant 
of sorl Ell, 
The specitication(2) fails to guarantee that there 
are any objects of El/:. While we could of course 
add soule constants of this sort, we would like 
to have a more general solution. In a particular 
application, we might want to define phonologi- 
cal words as a List of syllables (plus other con- 
straints, of course), and phonological phrases as 
a List of words, rl'hat is, we need to parame- 
terlze the type LIST1 with respect to the class 
of elements which constitute the lists. 
Before turning to parameterization, we will first 
see how a many-sorted specification language is 
generalized to an order sorted language by intro- 
ducing a subsort relation. 
Sul)l)ose, for exanlple, that we adopt the claim 
Aeries DE COLING-92, NANTES, 23-28 ^offr 1992 1 5 0 PROC. OF COLING-92, NAN'rEs, AUo. 23-28, 1992 
that all syllables have ('lonsets :(. Moreover. we 
wish to divide syllables into the subclasses lmavy 
and light. Obvimusly we wan! heavy and light 
syllables to inherit the l)roperties of the clas> of 
all syllables, e.g., they haw' ('1 onsets. We use 
ltoavy < Syll to stale that Heavy is a subsorl of 
tile sort Syll. We inlerl)l'et this to mean thai lhe 
class of heavy syllables is a subse! of the class (if 
all syllables. Now, let onset_ : Syll -> Nora 
lie all operation which selects tlle tits! mora of 
a syllable, anti let us impose the Iollowing con- 
straint (where Cv is a sul)sor! of Nora): 
(3) var S : Syll .var CV : Cv . 
eq onset S = CV , 
Then tile framework of or(ler sorted algebra ell- 
sures that onset is also delined for obje('l > of s{)i't 
Heavy. 
llx~turlling to lists, the speciIication ill (,I) (sli~hll.v 
simplified from that used h> ((;oguen ,k: Winkhq 
|988)) introduces Eli alld NeList (notl OlUl)t 3 
lists) as subsorts of List. and thereby !rein'ores 
on LISTI in a number of resi)ects, h, addition. 
tile specification is parameter!zeal. Thai is. il 
characterizes a list of Xs, where the paralneler X 
can be instantiated tm any module which satislies 
tile condition TRIV; the laller is what ((;oy;uell 
& Winkler 1988} call a "requirenlenl theory', and 
in lhis case simply iml)oses on any inpul moduh, 
that it have a sot! which can be mal)p('(I to Ihe 
sort Eli. 
(4) obj LIST\[X :: TRIV\] is sorts List NeList , 
subsorts Elt< NeList < List , 
op nil : -> List . 
op ._ : List List -> List . 
op . : NeList List -> NeList . 
op head : NeList -> Eft , 
op tail : NeList -> Llst . 
vat X ; Elt . 
vat L : List . 
eq (X , nil) : X . 
eq head(X . L) = X , 
eq tail(X . L) = L . 
endo 
Notice that the list constrllctor _._ llOW i)el'forllls 
the additional fluter!on ol append, allowing Iwo 
lists tm lie concatenated, h, addition. !he se 
lectors llave beell made 'safe', ill lhe Sellse thai 
they only apply to objects (i.e.. nonemply lisls) 
for which they giwr sensible results: for whal. ill 
LISTI, would have been the meaning of head(nil )? 
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 Metrical Trees 
:\s a further illustration, we give below a speci- 
lit'at ion of the data lyp(! BINTREE. This module 
has two parameters, bolh of whose requirelnent 
theories are TRIV. 4 
(5) ob 3 BINTREE\[NONTERM TERM :: TRIV\] is 
sorts Tree Netree . 
subsorts Eli.TERM Netree < Tree . 
op _\[_._\] : EIt.NONTERM Tree Tree -> Netree . 
op _\[_\] : EIt.NONTERN Eft.TERM -> Tree . 
op label_ : Tree -> EIt. NflNTERM . 
op left_ : Netree -> Tree , 
op right_ : Netree -> Tree . 
vars El E2 : Tree , 
vars h :EIt. NONTERM . 
eq label (h \[ El , E2 \] ) = A , 
eq label (h \[ El \]) = A . 
eq left (h \[ E1 , E2 \]) : E1 . 
eq right (h \[ E1 , E2 \] ) = E2 . 
(~itdo 
\Ve can l~mx~ inslanl, iale 1he formal paranmters of 
th0 module in (5) with inpm module.s whiEh sup- 
ply al)ln'opriale sels of ilOlll, erlnina\] and terminal 
symbols, l,el us use ui)percase quoted identifiers 
(eMnenls of the OB.I inodule QID) for nontermi- 
nals. and lower case for terminals. The specitlca- 
lion in (5) allows us to treat terminMs as trees, 
st; Ihal a binary tree. rooted ill a node 'A, can 
have lerminals as its daughters, ltowever, we 
ills() allow terminals to be directly dominated by 
a n(m-branchingmolher node. \[Ioth possibilities 
occur in the examples below. (6) illustrates the 
instantiation of tornlal parameters by an actual 
module, namely QID. using the make construct. 
16} make BINTREE-QID is BINTREE\[QID,QID\] endm 
The nexl exalnph, shows Nellie reductions in this 
module, obt, aiued by treating the equations as 
rewrite rules applying fi'om left to right. 
~'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. 
ACRES DE COLING-92, NAMES, 23-28 AO~' 1992 1 5 1 Paoc. OF COLING-92, NANTES, AUG. 23-28, 1992 
(7) left ('h\['a,'b\]) . 
left ('A\['B\['a\],'C\['b\]\]) . 
"~ 'B\['a\] 
left ('A\['B\['a,'b\],'c\]) . 
~* 'B\['a, 'b\] 
right(left (~A\[('B\['a,'b\]),'c\])) . 
label ('A\['a,'b\]) . 
.x~ JA 
label(right ('A\['a,*B\['b,'c\]\])) . 
~4 JB 
Suppose we now wish to modify the definition of 
binary trees to obtalu metrical trees, These are 
binary trees whose branches are ortlered accord- 
ing to whether they are labelled 's" (strong) or 
'w' (weak). 
• v 
In addition, all trees have a tlistinguishetl leaf 
node called the 'designated terminal element '(dte), 
which is connected to the roe! of the tree I)y a 
path of 's' nodes. 
Let us define 's' and "w' to t>e our nonterminals: 
(8) obj MET is 
sorts Label 
ops s w : -> Label . 
endo 
In order to buihl tilt, data iype of metrical lr¢,e~ 
on top of binary trees, we can import Ill(, mod- 
uh, BINTREE, suitably instantialell, using OB.l's 
extendingconstrucl. Notice thai we use MET to 
in~tantiate the parameter which fixes BINTFLEE's 
~et Of nonterminal symbols. ~ 
191 obj HETTREE is extending 
BINTREE\[MET,QID\]*(sort Id to Leaf) 
op die : Tree -> Leaf . 
vat L : Leaf . 
vats T1 T2 : Tree . 
'~'\['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 
vars X : Label . 
eq dte( X \[ L \] ) = L . 
ceq die ( X \[ T1 , T2 \]) = die T1 
if label TI == s . 
ceq die ( X \[ T1 , T2 \]) = die T2 
if label T2 == s . 
endo 
The equations state that the dte (designated ter- 
minal element) of a tree is the dte of its strong 
subtree. Another way of stating this is that the 
information about dte element of a subtree T is 
percolated up to its parent node, .just in case T 
is tile "s' branch of that node. 
The specification METTREE can be criticised on 
a number of grounds, it has to use conditional 
equations in a cumbersome way to test which 
daughter of a 1)inary tree is labelled 's', More- 
over. it fails to capture the restriction that no 
binary tree can have daughters which are both 
weak. or both strong. That is, it fails to capture 
the essential property of metrical trees, namely 
that metrical strength is a relational notion. 
What we require is a method for encoding the fob 
lowing information at a notle: "my left (or right) 
daughter is strong". One economicaJ method of 
doing this is to label (all and only) branching 
nodes in a binary tree with one of the following 
two lahels: 'sw' (my left daughter is strong), 'ws' 
(my right daughter is strong). Thus, we replace 
MET with the following: 
obJ MET2 is 
sorts Label 
ope sw ~s : -> Label . 
ends 
We can now simplify both BINTREE and ME 
'l'l{ l:;t'\]: 
obj BINTREE2\[NONTERM TERM :: TRIV\] is 
sorts Tree Netree , 
subsorts EIt.TERM Netree < Tree . 
op _\[_,j : EIt.NONTERM Tree Tree -> Netree . 
op label_ : Tree -> EIt.NONTERM . 
op left : Netree -> Tree . 
op right_ : Netree -> Tree . 
rare El E2 : Tree . 
vars A : EIt. NONTERM . 
eq label (A \[ El , E2 \] ) = A . 
eq left (A \[ E1 , E2 \]) = El . 
eq right (A \[ El , E2 \]) = E2 . 
eudo 
obj METTREE2 is extending 
BINTREE2\[MET2,QID\]*(sort Id to Leaf) . 
op hie_ : Tree -> Leaf . 
1 5 2 Pgoc. OF COLING-92, NArCrES, AUO. 23-28, 1992 
var L : Leaf . 
vars TI T2 : Tree . 
eq dte L = L . 
eq dte T = if label T == sw 
then die(left T) 
else dte(right T) fi . 
ends 
3 Feature Geometry 
The p~rticul~r feature geometry we shM1 specify 
here is based on the articul~tory structure de- 
fined in (Browman & Goklstein 1989)Y The five 
active articulators are grouped into a hierarchi 
cal structure involving a tongue node and an oral 
node, an shown in the following diagram. 
root 
glot, tal relic oral 
tongue labial 
coronal dorsal 
This structul'e is specilied via term ('onstl'UC|Ol'~ 
(__} a,,a { .... } which ~i ...... standar<i ),(,~iti ...... t 
encoding of features. F, ach fealttre vahlc is ex- 
pressed as a llaturaJ ltUlnl)or \[)o|w('ell 0 and 4. 
representing the constriction degree of the ('or 
resl)onding articulator. For examl)le, the tertu 
{4,0} : Tongue is an item of son Tongue con 
sistieg of the value ,I for the Ioalure ('o)toN..\\[. 
and 0 for the DORSAl,; this in turu express(,> a 
situation where there is maximal coasti'ictlon of 
the tongue tip, and minimal constriction of the 
tongue I)ody. Of course, this encoding is rat her 
crude, and l)ossil)ly sacrifices clarily for cot,ci 
sion. However, it sultices as a workiag ex;,leple. 
We will returi/ to constri(qioll degt'ees })el()w. 
The four sorts Gesture, Root. Oral and Tongue 
ill (lO) atld the first three ol)erator~ cal)turo )h(' 
+;For spa(:(! reasons w( + hilVC Oltlittcd ;ItlX di~(II~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 structure, using an approach which 
should be familiar by uow. For example, the 
third constructor takes the constriction degrees 
of Glottal and Volic gestures, and combines 
them with a complex item of sort Oral to build 
all item of sort Root. The specifie~ttion imports 
the um(hde NAT of natural numbers to provide 
values for constriction degrees. 
(10) obj FEATS is 
extending NAT , 
sores Gesture Root Oral Tongue . 
subsorts Rat Root Oral Tongue < Gesture . 
op {_, } : Nat Nat -> Tongue . 
op {_,_} : Tongue Nat -> Oral . 
op { ..... } : Nat Nat Oral -> Root , 
op _ ! coronal : Tongue --> Nat . 
op !dorsal : Tongue -> Nat . 
of !labial : Oral -> Nat . 
op !tongue : Oral -> Tongue . 
op _!glottal : Root -> Nat . 
op !relic : Root -> Nat . 
op _ ! oral : Root -> Oral . 
vats C CI C2 : Nat . 
vats 0 Oral . 
vats T Tongue . 
eq { Cl , C2 } !coronal = CI . 
eq { CI , C2 } !dorsal = C2 , 
eq { T C } !tongue = T . 
eq { T C } !labial = C • 
eq { C1 , C2 ) 0 } !glottal = C1 . 
eq { C1 , C?, , O } !velic = C2 . 
eq { CI , C2 , 0 } !oral = 0 . 
elldo 
AVe adopt the uol;atiollal coilventiOll of prepend 
int~ a '!" to the same of seh:ctors which col 
respond directly to features. For example, the 
!coronal seleC{or is a funct;ion defined on conl- 
pIox ilenls of SOl"( Tollguo which rettzrlls air item 
()f sort Nat, reln'eseetiug the constriction degree 
• ca\]u(' for ('oronality. 
Sonn' illustrative reduct ions in the FEATS module 
are given l)elow. 
(Ill {3,4,{{4,1},11} !oral . 
~-, {{4,1},11 
{3,4,{t4,1},1}} ~ora~ !to~gue . 
.... {4,1} 
{3,4,{{4,1},1}} !oral !tongue !coronal . 
III 1 lie ..\ I)'1 ~pplo~-lch to leal Ill'e st rtlcI tires, feel'i- 
leant 3 is represented by eqttating the values of 
~('\]('('tol'~. I'IIIIS. Sill)pose Ih;ll two 5egnlents S1, 
$2 '~hale ;t voicing sl)e(:ilication. We can write 
th\],~ >t~ f~>llows: 
ACN:'S DE COLING+92, NANTES, 23-28 AO~r t992 1 5 3 PRec. of COLING°92, NAN'rEs, AUG. 2.3-28, 1992 
(12)S1 !glottal = S2 !glottal 
This structure sharing is consistent with one of 
the main motivating factors behind autosegmen- 
tal phonology, namely, the undesirability of rules 
such as \[~ voice\] -- \[~ nasal I. 
Now we can illustrate the flmction of selectors 
in phonological rules. Consider the case of 1';11- 
glish regular plural formation (-s), where the 
voicing of the suffix seglnent agrees with that of 
the immediately preceding segment, unless it is 
a coronal fricative (in which case there musl be 
an intervening vowel). Suppose we introduce the 
variables S1 $2 : Root, where S1 is the stem- 
final segment and S2 is the suffix. The rllle nllls\[ 
also be able to access tile coronal node of $1. 
Making use of the selectors, this is simply $2 
!oral !tongue !coronal (a nota.tion reminis- 
cent of paths in feature logic. (Kasper k llonnds 
1986)). The rule must test whether this coronal 
node contains a fi'icative specification. This ne- 
cessitates an extension to our specification, which 
is described below. 
Browman & Goldstein (19S9. 234ff) define "con 
striction degree percolation', based on whal they 
call ~tube geometry'. The vocal trac| can b(, 
viewed as an interconnected set of tllbes, and 
the articulators correspond to valves which have 
a mmlber of settings ranging from fiflly open to 
fiflly closed. As already mentioned, these ~el- 
tings are called constriction degrees ( ! cds). where 
fully closed is the maximal constriction and fully 
open is the minimal constriction. 
The net constriction degree of the oral cavity 
may be expressed as the maximum of the con- 
striction degrees of the lips, tongue tip and Iongue 
body. The net constriction degree of the oral and 
nasal cavities together is simply the minimmn of 
the two component constriction degrees. To re 
cast this in the present framework is straight\[or- 
ward. However, we |teed lo first define the op- 
erations max and rain over pairs of naltlra\] 1111111- 
bers: 
(13) obj MINMAX 
is protecting NAT . 
ops min max : Nat Nat -> Nat . 
rare M N : Nat . 
extending FEATS + MINMAX . 
op _!cd : Gesture -> Nat . 
ops clo crit narrow 
mid wide obs open : Gesture -> Bool . 
vat G : Gesture . 
vat N ~1 N2 : Nat . 
vars 0 : Oral . 
vars T : Tongue . 
eq N !cd = N , 
eq {N1,N2} !cd = max(N1,N2) . 
eq {T,N} !cd = max(T !cd,N) . 
eq {N1,N2,0} !cd ~ max(Nl,min(N2,0 !cd)) . 
eq clo(G) = S !cd == 4 . 
eq crit(G) = G !cd == 3 . 
eq narrow(G) = G !cd == 2 . 
eq mid(G) = G !cd == I . 
eq wide(G) = G !cd == O . 
eq obs(G) = G !cd > 2 . 
eq open(G) = G !cd < 3 . 
endo 
The specification CD allows classification into five 
basic constriction degrees (c'lo, crit, narrow, 
mid, and wide) by means of corresponding one- 
place predicates, i.e. boolean-valued operations 
over gestures. For example, the fifth equation 
above states that G has the constriction degree 
cto (i.e. elm(G) is true)if and only if6 !cd == 
4. 
The working of these predicates is illustrated be- 
l o w: 
(15) {3,0,{{4,1},1}} !oral !tongue !cd . 
~4 
{3,0,{{4,1},1}} !oral !cd 
~4 
{a,o,{14,~},1}} ~cd . 
~3 
mid({3,0,{{4,1},1}} !oral l~ial) . 
true 
wide({3,O,{{4,1},l}} !oral !labial) . 
false 
open({3,0,{{4,1},l}} \]oral !labial) . 
true 
cl0({3,0,{{4,1},1}} !oral !tongue) . 
true 
ACRES DE COLING-92. NANTES, 23-28 AOt~'r 1992 1 5 5 PROC. OF COLINGO2. NANTES, Aua. 23-28, 1992 
eq min(M,N) = if M <= N then M else N fi . 
eq max(M,N) = if M >= N then M else N fi . 
AcrEs DE COLING-92. NANTEs, 23-28 ho~" 1992 1 5 4 PROC. OF COLING-92, NAhn'ES, AUG. 23-28, 1992 

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