Directional Constraint Evaluation in Optimality Theory
Jason Eisner
Departmentof Computer Science /University of Rochester
Rochester, NY 14607-0226 (U.S.A.) / [email protected]
Weightednite-stateconstraintsthatcancount
un-boundedlymanyviolationsmakeOptimalityTheory
morepowerfulthannite-statetransduction(Frank
andSatta,1998). Thisresultisempiricallyand
com-putationally awkward. We propose replacing these
unbounded constraints, as well as non-nite-state
GeneralizedAlignmentconstraints,withanewclass
of nite-state directional constraints. We give
lin-guisticapplications,resultsongenerativepower,and
algorithmstocompilegrammarsintotransducers.
1 Introduction
Optimality Theory is a grammar framework
thatdirectlyexpressesconstraintson
phonolog-icalforms. Roughly,thegrammarprefersforms
thatviolateeachconstraint aslittleaspossible.
Most constraints used in practice describe
disfavored local congurations inthe
phonolog-ical form (Eisner, 1997a). It is therefore
possi-bleforagivenformtooendasingleconstraint
at several locations in the form. (For example,
a constraint against syllable codas will be
of-fendedbyevery syllablethathas acoda.)
When comparingforms, then,howdo we
ag-gregate a form's multiplelocal oenses into an
overallviolation level?
A constraint couldanswerthisquestioninat
leastthree ways,thethird being ourproposal:
Unbounded evaluation (Prince and
Smolensky,1993). A form's violation level
is given by the number of oenses. Forms
withfeweroensesare preferred.
Bounded evaluation (Frank and Satta,
1998; Karttunen, 1998). A form's
viola-tion level is min(k;numberof oenses ) for
some k. This islike unboundedevaluation
exceptthat theconstraint does not
distin-guishamongformswith k oenses. 1
Directional evaluation. A form's
vio-lation level considers the location of
of-fenses,nottheirtotal number. Under
Iamgratefultothe3anonymousrefereesforfeedback.
1
Notethatk=1gives\binary"constraintsthatcan
bedescribedsimplyaslanguages. Anyk-bounded
con-to-rightevaluation, theconstraint prefers
formswhoseoensesareaslateaspossible.
To compare two forms, it aligns them
(ac-cordingto theircommonunderlying
repre-sentation),andscans theminparallelfrom
left to right, stopping at the rst location
whereoneformhasanoenseandtheother
does not (\sudden death"); it prefers the
latter. Right-to-leftevaluationissimilar.
x2ofthispapergiveslinguisticand
computa-tionalmotivationfortheproposal. x3formalizes
theideaandshowsthatcomposingatransducer
withadirectionalconstraintyieldsa transducer.
Thusdirectionalconstraints,likeboundedones,
keep OT within the class of regular relations.
(Butwealsoshowthemtobemore expressive.)
2 Motivation
2.1 Intuitions
RecallthatOT'sconstraintrankingmechanism
isan answerto thequestion: How can a
gram-mar evaluate a form by aggregating its
viola-tions of several constraints? Above we asked
the same question at a ner scale: How can a
constraintevaluateaformbyaggregatingits
of-fensesat several locations? Figure 1 illustrates
thatouranswerisjustconstraintrankingredux.
Directional evaluation strictly ranks the
im-portance of the locations within a form, e.g.,
from left to right. This exemplies OT's \do
only when necessary" strategy: the constraint
prefers to postpone oenses until they become
strictlynecessary towardthe right of theform,
even at thecost ofhavingmore of them.
One might thinkfrom Figure 1that each
di-rectional constraint could be decomposed into
several binary or other bounded constraints,
yielding a grammar using only bounded
con-straints. However, no single such grammar is
general enough to handle all inputs: the
num-berofconstraintsneededforthedecomposition
ban.to.di.bo
ban.ton.di.bo
ban.to.dim.bon
ban.ton.dim.bon
1
*! *
☞
*****!
***!*
1 1 2 3 4
*! *
* *!
☞
* * ** *! * *
1 4 3 2 1
*! *
☞
* **! * *
*! * * *
Figure1: Directionalevaluationassubconstraintranking. Allcandidateshave4syllables;wesimplify here
byregardingtheseasthelocations. C
1
issomehigh-rankedconstraintthateliminatesban.to.di.bo;NoCoda
isoendedbysyllablecodas. (a)TraditionalunboundedevaluationofNoCoda. (b)Left-to-rightevaluation
ofNoCoda,shownasifitweresplitinto4constraintsevaluatingthesyllablesseparately. (c)Right-to-left.
2.2 Iterative and oating phenomena
The main empirical motivation for
direction-allyevaluatedconstraintsistheexistenceof
\it-erative" phenomena such as metrical footing.
(Derivationaltheoriesdescribedthesewith
pro-cedures that scanned a form from one end to
theotherandmodiedit;see(Johnson,1972).)
For most other phenomena, directional
con-straints are indistinguishable from traditional
unboundedconstraints. Usually,thecandidates
with the fewest oenses are stillthe ones that
survive. (Since their competitors oend at
ex-actly the same locations, and more.) This is
precisely because mostphonology is local:
sat-isfyingaconstraintatonelocationdoesnot
usu-allyblock satisfyingit at another.
Distinguishing cases, like the articial Fig.
1|where theconstraint canonlytrade oenses
at one location for oenses at another|arise
only under special conditions involving
non-localphenomena. Justasdirectionalevaluation
would predict, such a forced tradeo is always
resolved (toourknowledge)byplacingoenses
aslate, orasearly,ashigherconstraintsallow:
Prosodic groupings force each segment or
syllable to choose which constituent (if
any) to associate with. So-called
left-to-rightdirectionalsyllabication(Mesterand
Padgett, 1994) will syllabify /CVCCCV/
greedily as CVC.CV.CV rather than
CV.CVC.CV, postponing epenthetic
ma-terial until as late as possible.
Simi-larly, left-to-right binary footing (Hayes,
1995)prefers()()over()()or
()(), postponingunfootedsyllables.
Floating lexical material must surface
somewhere in the form. Floating features
(e.g., tone) tend to dock at the leftmost
orrightmostavailablesite, postponing the
appearance of these marked features.
In-as possible (McCarthy and Prince, 1995),
postponingtheappearance ofanaÆxedge
orother aÆx material withinthe stem. 2
Floating non-lexical material mustalso
ap-pear somewhere. If a high-ranked
con-straint, Culminativity, requires that a
primary stress mark appearon each word,
then a directional constraint against
pri-marystresswillnotonlypreventadditional
marksbutalsopushthesinglemarktothe
rst or last available syllable|the
tradi-tional \EndRule"(Prince,1983).
Harmony must decide how far to spread
features, and OCP eects such as
Grass-man's Law must decide which copies of
a feature to eliminate. Again, directional
evaluation seemsto capture thefacts.
2.3 Why not Generalized Alignment?
In OT, following a remark by Robert
Kirch-ner, ithasbeentraditionaltoanalyzesuch
phe-nomena using highly non-local Generalized
Alignment (GA) constraints (McCarthy and
Prince, 1993). For example, left-to-right
foot-ing is favored by Align-Left
(Foot, Stem),
which requires every foot to be left-aligned
with a morphological stem. Not only does
each misalignedfoot oend theconstraint, but
the seriousness of its oense is given by the
2
\Availablesite"and\possible"amountofinxation
are dened here by higher-ranked constraints. These
might restrict the allowed tone-bearing units and the
allowedCV shape afterinxation, butdonotfully
de-terminewheretheoatingmaterialwillsurface.
Arefereeaskswhycodasdonotalsooat(topostpone
NoCodaoenses). Answer: Flotationrequiresunusual,
non-localmechanisms. Genoraconstraint mustensure
thatananchoredtonesequenceresemblestheunderlying
oatingtonesequence,whichmayberepresentedonan
auxiliary inputtapeor (ifbounded) asaninputprex.
Butordinaryfaithfulnessconstraintscheckonlywhether
These numbers are summed over all oending
feet to obtain the violation level. For
exam-ple, [()()()]
Stem
has 1+3+6=10
vio-lations,and [()()]
Stem
isequallybad
at 4+6=10 violations. Shifting feetleftward or
eliminating themreduces theviolation level.
(Stemberger, 1996) argued that GA
con-straints were too powerful. (Ellison, 1995)
showed that no single nite-state unbounded
constraint could dene the same violation
lev-els asa GA constraint. (Eisner, 1997a) showed
more strongly that since GA can be made to
center a oating tone on a phrase, 3
no
hierar-chy of nite-state unbounded constraintscould
even dene the same optimal candidates as a
GA constraint. Thus GA cannot be simulated
inEllison's(1994)nite-stateframework(x3.2).
For this reason, as well as the awkwardness
and non-locality of evaluating GA oenses, we
propose to replace GA with directional
con-straints. Directionalconstraintsappeartomore
directly capturetheobserved phenomena.
Wedonotethatanother,trickierpossibilityis
to eliminateGAinfavorofordinaryunbounded
constraints that are indierent to the location
of oenses. (Ellison,1994)noted that GA
con-straints that evaluated the placement of only
one element (e.g., primary stress) could be
re-placedbysimplerNoInterveningconstraints.
(Eisner,1997b)givesaGA-freetreatmentofthe
metrical stress typologyof(Hayes, 1995).
2.4 Generative power
It has recently been proposed that for
compu-tational reasons, OTshouldeliminate notonly
GA but all unbounded constraints (Frank and
Satta, 1998; Karttunen, 1998). As with GA,
we oer theless extremeapproach of replacing
them withdirectionalconstraints instead.
Recall that a phonological grammar, as
usu-ally conceived, is a description of permissible
(UR, SR) pairs. 4
It has long been believed
thatnaturallyoccurringphonologicalgrammars
are regular relations (Johnson, 1972; Kaplan
and Kay, 1994). This means that they can
be implemented as nite-state transducers
(FSTs) that accept exactly the grammatical
pairs. FSTs are immensely useful in
perform-3
Thisisindeedtoopowerful: centeringisunattested.
4
tainingall possibleSRs fora UR),
comprehen-sion(conversely),characterizingthesetofforms
onwhichtwogrammars(perhapsfromdierent
descriptiveframeworks)woulddier,etc.
More-over, FSTscan be appliedinparallelto regular
sets of forms. For example, one can obtain a
weighted set of possible SRs (a phoneme
lat-tice) from a speech recognizer, pass it through
the inverse transducer, intersect the resulting
weighted setof URswiththe lexicon,and then
extract the best survivingURs.
(Ellison, 1994; Eisner, 1997a) frame OT
within thistradition, bymodeling Genand the
constraints as weighted nite-state machines
(see x3.2). But although those papers showed
howtogeneratethesetofSRsforasinglegiven
UR,theydidnotcompiletheOTgrammarinto
an FST, orobtaintheother benetsthereof.
In fact, (Frank and Satta, 1998) showed
that such compilationis impossiblein the
gen-eral case of unbounded constraints. To see
why, consider the grammar Max, Dep,
Har-mony[height]Ident-IO[height]. This
gram-mar insists on height harmony among surface
vowels, but dislikeschanges from the UR. The
result is the unattested phenomenon of
\ma-jority assimilation" (Bakovic, 1999; Lombardi,
1999): a UR with more high vowels than low
willsurfacewithallvowelshigh,andvice-versa.
SoOTmaycompare unbounded countsinaway
thatan FSTcannotand phonology doesnot.
This suggests that OTwith unbounded
con-straints is too powerful. Hence (Frank and
Satta, 1998; Karttunen, 1998) propose using
only bounded constraints. They show this
re-ducesOT'spowerto nite-state transduction.
The worry is that bounded constraints may
notbe expressiveenough. A2-boundedversion
of NoCoda would not distinguish among the
nalthreeformsinFigure1: itisagnosticwhen
theinputforcesmultiplecodasinallcandidates.
To be sure,a k-bounded approximation may
workwellforlargek. 5
Butitsautomaton(x3.2)
willtypicallyhavektimesasmanystatesasthe
unbounded original, since it unrollsloops: the
5
Using theapproximate grammarfor generation,an
output is guaranteed correct unless it achieves k
vio-lations for some k-bounded constraint. One can then
raise k,recompile the grammar,and try again. Butk
tersecting manysuch largeconstraints can
pro-duceverylargeFSTs|whilestillfailingto
cap-ture simple generalizations, e.g., that all codas
aredispreferred.
In x3, we will show that directional
con-straints are more powerful than bounded
constraints, as they can express such
generalizations|yet they keep us within
theworldof regularrelations and FSTs.
2.5 Related Work
Walther (1999), working with intersective
con-straints, denes a similar notion of Bounded
LocalOptimization(BLO).Trommer(1998;
1999)appliesavariantofWalther'sideatoOT.
The motivation inbothcases islinguistic.
Wesketchhowourideadiersvia3examples:
UR uuuuu uu uuu uuuuu
candidate X vvvbb vv vbb vvvbb
candidate Y vvbaa vvvvbaa vzbaa
Considerb,aleft-to-rightconstraintthatis
of-fended byeach instance ofb. On ourproposal,
candidate X wins in each column, because Y
alwaysoendsb rst,at position3 intheUR.
But under BLO, this oense is not fatal. Y
can survive b byinserting epentheticmaterial
(column 2: Y wins by postponing b relative to
theSR),orbychangingvtoz(column3: Yties
X, sincevv 6=vz and BLOmerely requires the
cheapestchoicegiventhesurfaceoutputsofar).
In the same way, NoCoda under BLO would
trigger many changes unrelatedto codas. Our
denitionavoidstheseapparentinconveniences.
WaltherandTrommerdonotconsiderthe
ex-pressive power of BLO (cf. x3.3) or whether
grammarscanbecompiledintoUR-to-SRFSTs
(ourmainresult; seediscussioninx3.4).
3 Formal Results
3.1 Denition of OT
An OT grammaris apair (Gen; ~
C) where
thecandidategeneratorGenisarelation
thatmapseachinputtoanonemptysetof
candidateoutputs;
thehierarchy ~
C =(C
1 ;C
2
;::: ) is a nite
tupleofconstraintfunctionsthatevaluate
outputs.
We write ~
C(Æ) forthetuple(C (Æ);C (Æ);:::).
asitsSRsalltheoutputsÆsuchthat ~
C(Æ)is
lex-icographicallyminimalinf ~
C(Æ) :Æ2Gen()g.
The values taken by C
i
are called its
viola-tion levels. Conventionally these are natural
numbers,butanyorderedset willdo.
Ourdirectionalconstraintsrequirethe
fol-lowinginnovations. Each input is a stringas
usual,butthe outputsare notstrings. Rather,
each candidate Æ 2Gen() is a tupleof jj+1
strings. We write Æ for the concatenation of
these strings (the \real" SR). So Æ species an
alignment of Æ with . The directional
con-straint C
i
maps the tuple Æ to a tuple of
nat-ural numbers (\oense levels") also of length
jj+1. Itsviolation levelsfC
i
(Æ):Æ2Gen()g
are comparedlexicographically.
3.2 Finite-state assumptions
Wenowconneourattentionto nite-stateOT
grammars,following(Ellison,1994;Tesar,1995;
Eisner, 1997a; Frank and Satta, 1998;
Kart-tunen, 1998). Gen
is a regular
relation, 6
and may be implemented as an
un-weightedFST.Eachconstraintisimplemented 7
as a possiblynondeterministic, weighted
nite-state automaton(WFSA)thataccepts
and
whose arcs areweightedwithnatural numbers.
An FST, T, is a nite-state automaton in
whicheacharcislabeledwithastringpair:.
Without loss of generality, we require jj 1.
This lets us dene an aligned transduction
that maps strings to tuples: If = a
1 a
n ,
we dene T() as the set of (n + 1)-tuples
Æ =(Æ
0 ;Æ
1 ;:::Æ
n
) suchthatT hasapath
trans-ducing :Æ along which Æ
0 Æ
i 1
is the
com-pleteoutput beforea
i
is readfrom theinput.
We nowdescribehowtoevaluateC(Æ)where
C is aWFSA. Consider thepath inC that
ac-cepts Æ. 8
In (un)bounded evaluation, C(Æ) is
the total weight of this path. In left-to-right
evaluation,C(Æ)isthen+1tuplegivingthe
re-spectivetotal weightsofthesubpathsthat
con-sumeÆ
0 ;:::Æ
n
. In right-to-left evaluation, C(Æ)
isthereverse of theprevioustuple. 9
6
Ellisonrequiredonlythat Gen()beregular(8).
7
Space prevents giving the equivalent
characteriza-tionasalocallyweightedlanguage(Walther,1999).
8
Iftherearemultipleacceptingpaths
(nondetermin-ism),taketheonethatgivestheleastvalueofC(Æ).
9
This is equivalent to C R
(Æ R
n ;:::;Æ
R
0
) where R
ThankstoGen,nite-stateOTcantrivially
im-plementanyregular input-outputrelationwith
noconstraintsatall! Andx3.4belowshowsthat
whether we allow directional or bounded
con-straintsdoesnotaect thisgenerative power.
But inanother sense, directional constraints
arestrictlymoreexpressivethanboundedones.
If Gen is xed, then any hierarchy of bounded
constraintscan be simulatedbysomehierarchy
ofdirectionalconstraints 10
|butnotvice-versa.
Indeed, we show even more strongly that
di-rectional constraints cannot always be
simu-lated even byunbounded constraints. 11
Dene
basinx2.5. Thisrankstheset(ajb) n
in
lexico-graphic order, so it makes 2 n
distinctions. Let
Genbethe regularrelation
(a:ajb:b)
equivalentto(Gen;C
1 ;:::;C
s
)foranybounded
or unbounded constraints C
1
So candidatesÆ of lengthnhave at most
(kn) s
dierentviolationproles ~
C(Æ). Choosen
suchthat2 n
must contain two distinct strings, Æ =
x
Let i be minimal such that x
i 6= y
i
, and
with-outlossofgeneralityassumex
i
and Æ is lexicographically minimal in Gen().
So the grammar (Gen ;b) maps to Æ only,
whereas(Gen ; ~
C) cannot distinguish between Æ
and Æ 0
,soitmaps toneither orboth.
3.4 Grammar compilation: OT = FST
ItistrivialtotranslateanarbitraryFST
gram-mar into OT: let Genbe theFST, and ~
C =().
The rest of this section shows, conversely, how
to compile a nite-state OT grammar(Gen ; ~
C)
into an FST, provided that the grammar uses
onlyboundedand/ordirectionalconstraints.
10
How? By using states to count, a bounded
con-straint'sWFSAcanbetransformedsothatalltheweight
ofeachpathfallsonitsnalarc. Thisdenesthesame
optimalcandidates,evenwheninterpreteddirectionally.
11
Norvice-versa,sinceonlyunboundedconstraintscan
implementnon-regularrelations(x2.4,x3.4).
12
Applyx3.4.4toeliminateany'sfromtheconstraint
WDFAs(regardedasoutputlesstransducers),thentake
Let T
0
= Gen. For i > 0, we will construct
an FST T
i
that implements the partial
gram-mar (Gen;C
(x) contains the forms
y 2T
i 1
(x)for whichC
i
(y) isminimal.
IfC
i
isk-bounded,weusetheconstructionof
(Frank and Satta, 1998; Karttunen,1998).
IfC
i
isaleft-to-rightconstraint,we compose
T
i 1
withtheWFSAthatrepresentsC
i
,
obtain-ingaweightednite-state transducer(WFST),
^
T
i
. Thistransducermayberegardedas
assign-ing a C
i
-violation level (an (jj+1)-tuple) to
each :Æ itaccepts. We mustnowprune away
the suboptimal candidates: using the DBP
al-gorithm below, we construct a new unweighted
FSTT
i
thattransduces :Æ itheweighted ^
is right-to-left, wedo justthesame,
ex-cept DBP isusedto construct T R
3.4.2 Directional Best Paths: The idea
All that remains is to give the construction of
T
, which we call Directional Best
Paths (DBP). Recall standard best-paths or
shortest-paths algorithms that pare a WFSA
downto itspaths ofminimumtotalweight
(Di-jkstra, 1959; Ellison, 1994). Our greedier
ver-sion does not sum along paths but always
im-mediately takesthe lightest\available"arc.
Crucially,availablearcsaredenedrelativeto
theinputstring,because wemustretainoneor
more optimaloutputcandidatesforeachinput.
So availability requires \lookahead": we must
take a heavier arc (b:x below) just when the
restoftheinput(e.g., abd)cannototherwisebe
acceptedon anypath.
1
a:a
2
(xcabbreviates(;x;c))
Onthisexample,DBPwouldsimplymakestate
6non-nal(forcingabctotakethelightarc
un-availableto abd),butoften itmust add states!
This relativization is what lets us compile a
hierarchyofdirectionalconstraints,onceandfor
all,intoansingleFSTthatcanndtheoptimal
in-tion forunboundedconstraints, and previously
proposedconstructionsfordirectional-style
con-straints(see x2.5) onlyndtheoptimaloutput
fora singleinput,orat best anitelexicon.
3.4.3 Dir. Best Paths: A special case
x3.2restrictedourFSTssuchthatforevery arc
label:,jj1. Inthissectionweconstruct
under the stronger assumptionthat
jj=1, i.e., ^
T
i
is-free ontheinputside.
IfQisthestateset of ^
T
i
,thenletthestateset
of T
i
be f[q;R ;S] : R S Q;q 2 S R g.
This has size jQj 3 jQj 1
. However, most of
these states are typically unreachable from the
startstate. Lazy\on-the-y"construction
tech-niques(Mohri, 1997)can beused to avoid
allo-cating states or arcs until they are discovered
duringexplorationfrom thestartstate.
For 2
;q 2 Q, dene V(;q) as the
minimum cost (a jj-tuple of weights) of any
-reading pathfrom ^
T
i
's startstate q
0 to q.
ThestartstateofT
i is[q
0 ;;;fq
0
g]. Theintent
is that T
i
have a path from its start state to
[q;R ;S]that transduces :Æ
reads , it \follows" ^
T
i
's cheapest
-readingpathstoq,whilecalculatingR ,towhich
yetcheaper(butperhapsdead-end)pathsexist.
Let[q;R ;S]beanalstate(inT
). So an accepting
pathin ^
T
i
survivesinto T
i
ithere is no
lower-cost acceptingpath in ^
T
i
forthesame input.
The arcs from [q;R ;S] correspond to arcs
from q. For each arc from q to q 0
labeled a:
and with weight W, add an unweighted a :
arc from [q;R ;S] to [q
], provided that
thelatter state exists(i.e.,unless q 0
2R 0
,
indi-catingthatthere is acheaperpathto q 0
). Here
R 0
is theset of states that are either reachable
fromRbya(single)a-readingarc,orreachable
from S by an a-reading arc of weight <W. S 0
isthe unionof R 0
and allstates reachable from
S byan a-readingarcof weight W.
3.4.4 Dir. Best Paths: The general case
To apply theabove construction, we must rst
transform
to insert unbounded amounts of surface
mate-rial (to be pruned back by the constraints). 14
To eliminate 'swhile preserving these
seman-tics, we are forced to introduce FST arc labels
of the form a: where is actually a regular
set of strings, represented as an FSA or
regu-lar expression. Following -elimination, we can
applythe construction of x3.4.3 to get T
i , and
nally convert T
i
back to a normal transducer
byexpanding each a: into a subgraph.
When we eliminate an arc labeled : , we
must push and the arc's weight back onto
a previous non- arc (but no further; contrast
(Mohri,1997)). The resultingmachinewill
im-plementthesamealignedtransductionas ^
T
i but
moretransparently: inthenotationof x3.2,the
arcreadinga
i
willtransduce itdirectly to Æ
i .
15
Concretely, suppose ^
T
i
can get from state q
to q 00
via a pathof total weight W that begins
with a :
to substitute an arc from q to q 00
be innitely many such q{q 00
paths, of varying
weight, so we actuallywrite a: ,where
de-scribesjustthoseq{q 00
pathswithminimumW.
The exact procedure is as follows. Let G be
thepossiblydisconnectedsubgraphof ^
T
i
formed
by -reading arcs. Run an all-pairs
shortest-paths algorithm 16
on G. This nds, for each
state pair (q 0
;q 00
) connected by an -reading
path, the subgraph G
q 0
;q 00
of G formed by the
minimum-weight -readingpaths from q 0
to q 00
,
as well as the common weight W
q 0
;q 00
of these
paths. So for each arc in ^
(). (G() denotes the regular
languagetowhichGtransduces.) Havingdone
this,we can deleteall -readingarcs.
The modied -free ^
T
i
is equivalent to
14
Asisconventional. Besidesepentheticmaterial,Gen
oftenintroducescopious prosodicstructure.
15
Thatarcislabeleda
i
: whereÆ
i
2 . Butwhatis
a0? A specialsymbolE 2thatweintroduceso that
Æ
0
canbepushedbackontoit: Before -elimination,we
modify ^
Ti by giving it a new start state, connected to
theoldstartstatewithanarcE:. After-elimination,
weapplyDBPandreplaceEwithintheresultT
i .
16
(Cormen et al., 1990) cites several, including fast
of the suboptimal subpaths. Here is a
graph fragment before and after -elimination:
1
a:a
2
3
ε
:b
4
ε
:d
ε
:b
ε
:c
1
2
a:a
3
a:ab+
4
a:ad
Note: Right-to-leftevaluationappliesDBPto
^
T R
i
,soconsistencywithourpreviousdenitions
meansitmustpush'sforward, notbackward.
4 Conclusions
This paper has proposed a new notion in OT:
\directionalevaluation,"whereunderlying
loca-tionsarestrictlyranked bytheirimportance.
Traditional nite-state OT constraints have
enough power to compare arbitrarily high
counts; Generalized Alignment is even worse.
Directionalconstraintsseemtocapturethepros
of these constraints: they appropriately
mili-tate againstevery instanceof a disfavored
con-guration in a candidate form, no matter how
many,and they naturallycaptureiterative and
edgemost eects. Yet they do nothave the
ex-cess power: wehave shownthatan grammarof
directional and/or bounded constraints can be
compiledinto a nite-state transducer. That is
bothempiricallyandcomputationallydesirable.
Themostobviousfutureworkcomesfrom
lin-guistics. Can directional constraints do all the
work of unbounded and GA constraints? How
do they change thestyleof analysis? (E.g.,
di-rectionalversionsofmarkednessconstraintspin
down the locations of marked objects, leaving
lower-rankedconstraintsnosay.) Finally,
direc-tional constraints can be variously formulated
(is *Cluster oended at the start or end of
each cluster? or of its enclosing syllable?). So
what conventionsorrestrictionsshouldapply?
References
EricBakovic. 1999. Assimilationto the unmarked.
Ms.,RutgersOptimalityArchiveROA-340.
T. H. Cormen, C. E. Leiserson, and R. L. Rivest.
1990. IntroductiontoAlgorithms. MITPress.
EdsgerW.Dijkstra. 1959. A noteontwoproblems
in connexion with graphs. Numerische
Mathe-matik, 1:269{271.
Jason Eisner. 1997a. EÆcientgeneration in
primi-tiveOptimalityTheory. InProc. of the 35th
An-nualACLand8thEACL,Madrid,July,313{320.
ening, editor, Proc. of SCIL VIII,MIT Working
Papersin Linguistics31,Cambridge,MA.
T. Mark Ellison. 1994. Phonological derivation in
optimalitytheory. InProc. ofCOLING.
T.MarkEllison. 1995. OT,nite-state
representa-tions and procedurality. In Proc. of the
Confer-ence onFormal Grammar,Barcelona.
Robert Frank and Giorgio Satta. 1998.
Optimal-ityTheoryandthegenerativecomplexityof
con-straintviolability. Comp. Ling.,24(2):307{315.
BruceHayes. 1995. Metrical StressTheory:
Princi-ples andCaseStudies. U.ofChicagoPress.
C. Douglas Johnson. 1972. Formal Aspects of
Phonological Description. Mouton.
RonaldM.KaplanandMartin Kay. 1994. Regular
models of phonological rule systems.
Computa-tionalLinguistics,20(3):331{378.
Lauri Karttunen. 1998. The proper treatment of
optimality in computationalphonology. InProc.
ofFSMNLP'98,1{12,BilkentU.,Ankara,Turkey.
LindaLombardi. 1999. Positional faithfulness and
voicing assimilationin OptimalityTheory.
Natu-ral LanguageandLinguistic Theory,17:267{302.
JohnMcCarthyandAlanPrince. 1993. Generalized
alignment. In Geert Booij and Jaap van Marle,
editors,Yearbookof Morphology, 79{153.Kluwer.
JohnMcCarthyandAlanPrince. 1995. Faithfulness
andreduplicativeidentity. InJillBeckmanetal.,
editor,Papers inOptimalityTheory,259{384.U.
ofMassachusetts,Amherst: GLSA.
ArminMesterandJayePadgett. 1994. Directional
syllabicationinGeneralizedAlignment.
Phonol-ogyatSantaCruz3,October.
Mehryar Mohri. 1997. Finite-state transducers in
language&speechprocessing. Comp.Ling.23(2).
Alan Prince and Paul Smolensky. 1993.
Optimal-ity Theory: Constraintinteraction in generative
grammar. Ms.,RutgersU. andU.ofColorado.
AlanPrince. 1983. Relating tothegrid. Linguistic
Inquiry,14:19{100.
J. P. Stemberger. 1996. The scope of the theory:
Wheredoesbeyondlie? InL.McNair,K.Singer,
L.M.Dobrin,andM.M.Aucoin,eds.Papersfrom
the Parasession on Theory andData in
Linguis-tics,CLS23,139{164.ChicagoLinguisticSociety.
BruceTesar. 1995. Computational Optimality
The-ory. Ph.D.thesis,U.ofColorado,Boulder.
Jochen Trommer. 1998. Optimal morphology. In
T. Mark Ellison, editor, Proc. of the 4th ACL
SIGPHONWorkshop,Quebec,July.
JochenTrommer. 1999. Mendetonepatterns
revis-ited: Tonemappingaslocalconstraintevaluation.
InLinguisticsin PotsdamWorking Papers.
MarkusWalther. 1999. One-levelprosodic
