Optimal Morphology 
Jochen Trommer 
Institut fuer Linguistik/Allgemeine Sprachwissenschaft 
Universitaet Potsdam Postfach 601553 D-14415 Potsdam 
Abstract 
/ 
Optimal morphology (OM) is a finite state for- 
malism that unifies concepts from Optimality 
Theory (OT, Prince ~: Smolensky, 1993) and 
Declarative Phonology (DP, Scobbie, Coleman 
Bird, 1996) to describe morphophonological 
alternations in inflectional morphology. Can- 
didate sets are formalized by inviolable lexi- 
cal constraints which map abstract morpheme 
signatures to allomorphs. Phonology is imple- 
mented as violable rankable constraints select- 
ing optimal candidates from these. Both types 
of constraints are realized by finite state trans- 
ducers. Using phonological data from Albanian 
it is shown that given a finite state lexicaliza- 
tion of candidate outputs for word forms OM 
allows more natural analyses than unviolable fi- 
nite state constraints do. Two possible evalu- 
ation strategies for OM grammars are consid- 
ered: the global evaluation procedure from E1- 
lisou (1994) and a simple strategy of local con- 
straint evaluation. While the OM-specific lex- 
icalization of candidate sets allows straightfor- 
ward generation and a simple method of mor- 
phological parsing even under global evaluation, 
local constraint evaluation is shown to be prefer- 
able empirically and to be formally more re- 
strictive. The first point is illustrated by an 
account of directionality effects in some classi- 
cal Mende data. A procedure is given that gen- 
erates a finite state transducer simulating the 
effects of local constraint evaluation. Thus local 
as opposed to global evaluation (Frank & Satta, 
1998) seems to guarantee the finite-stateness of 
the input-output-mapping. 
1 C/O-Alternations in Alba- 
nian Verbs 
In many cases C/O-alternations in the Albanian 
verb paradigm are triggered by the tendency to 
avoid sequences of vowels (VV, hiatus) or con- 
sonants (CC), e.g. 
(1) 
(a) (b) (c) (d) 
p~,je *pie hape *hapje 
CVCV CVV CVCV CVCCV 
'drink it!' 'open it!' 
(b) and (d) seem to be out because they 
contain illicit sequences of VV and CC respec- 
tively 1. For illustration of a declarative account 
I implement the morphological candidate set for 
each form and the phonological constraint as 
regular expressions The corrrect forms are then 
obtained by intersecting the two expressions: 
(2) 
Morphology:2 (pil hap) (j)e 
Phonology: 
(CV)*, for C = (p\]hlj) and V = (ilale) 
1 Alternatively these alternations can be viewed as a 
strategy for arriving at "perfect" syllable structures, i.e. 
syllables with onsets and without codas. This probably 
more adequate view can be formalized using the same 
methods, but for expositional ease I will ignore syllable 
structure and assume the simpler analysis above. 
~A more adequate morphological characterization fol- 
lows in (2') 26 
Intersection(Morphology, Phonology): 
(pije\]hape) 
Intriguing as this simple model is; it runs into 
difficulties with occuring CC and VV sequences 
in Albanian verbs: 
(3) 
(a) (b) 
bie hapje 
CVV CVCCV 
'he falls' 'you opened' 
Since hapje is ungrammatical as V+OBJ but 
grammatical as V+IMF:2SG there seems no 
purely phonological way to predict the ob- 
served contrasts.By lexicalizing the alternations 
though, say by stating that OBJ is -je after vow- 
els and -e after consonants we lose the insight 
that the process is motivated by the perfectly 
natural and crosslinguistically observable avoid- 
ance of of CC and VV (cf. Prince & Smolensky, 
199:3). The solution I propose to this dilemma 
is the following: 
(4) 
a. Replace the CV-constraint by a soft 
constraint marking CC and VV as con- 
straint violations. 
b. Annotate the candidate set with 
independently needed morphoseman- 
tic interpretation and choose as the 
correct form for certain morphological 
features (e.g.PI+OBJ) the phonolog- 
ically optimal form annotated with it. 
3 
More concretely (2) is replaced by (2'): 
3The idea to use optimality to decide between allo- 
morphs with respect to an morphological characteriza- 
tion can also be found in Kager (1995) See Russel (1995) 
for some discussion of morpheme signatures. 
(2') 
Morphology: 
(PI IHAP~ (OBJIOBJIIMF~ pi I hap \] \ e I je I je / 
Phonology: 
• 
1 (oCC'))I (:V *)? 
The phonological constraint following Elli- 
son (1994) has the form of a regular relation 
mapping phonological strings into sequences of 
constraint violation marks ('1') and 'O's which 
stand for no violation. The effect of the con- 
straint can best be seen in tableau form familiar 
from OT: 
(5) 
PI+OBJ *CC/VV 
pije 
pie ,t 
HAP+OBJ 
hape 
hapje *v 
PI+IMF 
pije 
HAP+IMF 
hapje * 
Differently from OT optimal forms are not 
computed with respect to an underlying phono- 
logical form, but directly referring to a morpho- 
logical characterization of the generated word 
form. 
2 Formalism 
An OM grammar is a quintuple {MA, 
PA,M,P,(0,1}} where MA and PA are sets of 
symbols, the morphological and the phonologi- 
cal alphabets. M is a finite set of regular rela- 
tions, each mapping MA into PA, while P is a 
finite sequence of regular relations mapping MP 
into {0, 1}. 27 
3 Generation Morphology 
Specific word forms are characterized as strings 
of abstract morphems, e.g. PI+OBJ 4. Specific 
candidate sets are obtained by the crossover 
product of word forms 5 with the candidate rela- 
tion. For ease of exposition I give the algorithm 
for finite state automata and transducers which 
are formally equivalent to regular expressions 
and binary regular relations respectively. For a 
transducer T and an automaton A the crossover 
product AxT is generated in the following way. 
IT and Ih are the initial, FT and FA the final 
states: 
(6) Crossover product(A,T) 6 
1 make (IA,IT) initial in AxT 
2 make (FA,FT) final in AxT 
3 for each arc from z to t in T labeled M2/P 
4 for each arc from x to y in A labeled M1 
5 if h~I1 = h'I2 
6 then add to AxT an arc 
7 from (x,z) to (y,t) labeled P. 
Obviously, the resulting automaton contains 
all and only the phonological output candidates 
for the given morphological input. (7) shows the 
application of the algorithm to PI+OBJ and the 
candidate set from (2'): 
(7) 
Input 
Initial State: 0 
Final State: 2 
Transitions: (0, 1, PI), (1, 2, OBJ) 
4 For sake of readability concatenation of morpholog- 
ical symbols is represented by '+'. 
5Strings are a trivial case of regular expressions. 
GCrossover product(A, T) is equivalent to the image 
of A under T (Kaplan & Kay, 1994:340-42), defined as 
the range of the composition Id(A) o R, where Id(A) 
is the identity relation that carries every member of A 
into itself. See Frank & Satta (1998:5-6) for the same 
concepts under different terminology. 
Initial State: A 
Final State: C 
Transitions: 
(i, B, PI/pi), (i, B, HAP/hap), 
(B, C, OBJ/je),(S, C, OBJ/e), 
(B, C, IMF/je) 
Resulting Candidates 
Initial State: (0, A) 
Final State: (2, C) 
Transitions: 
((0, A), (1, B) pi), ((1, B), (2, C), e) 
((1, B), (2, C), je) 
Since the candidate set and the constraint in 
(2') are regular Ellisons (1994) algorithms for 
getting an automaton containing only optimal 
candidates, as long as candidate set and evalu- 
ation transducer are regular, can be used. For 
details I refer the interested reader to Ellisons 
paper. 
4 Parsing 
The candidate transducer constitutes a kind of 
backbone for parsing phonological strings into 
strings of abstract morphemes. For example the 
candidate transducer in (2') will allow correct 
parsing of hape into HAP+OBJ. A complica- 
tion arises when the transducer maps phonolog- 
ical forms to more than one morphological form, 
possibly including incorrect ones. E.g. the ex- 
ample transducer will map hapje onto (correct) 
HAP+IMF and (incorrect) HAP+OBJ. Then 
given the generation procedure above for every 
obtained morphological string it can be checked 
if it generates the input phonological form. A 
special case of this are not existing word forms. 
For example we will get PI+OBJ as the only 
possible parse for pie. But since the optimal 
output for PI+OBJ isn't pie but pije there is 
no actual parse. 28 
5 Comparison with other ap- 
proaches 
6 Regular phonological pro- 
cesses 
Optimal Morphology clearly is a hybrid. Can- 
didate sets are given by lexicalized monotonic 
constraints as in DP. Phonological constraints 
on the other side are violable and ranked as in 
OT z. 
Looking at constraint based formalisms as 
tentatives to restrict the rather rich theoretical 
inventory of generative SPE theory (Chomsky 
& Halle, 1968) it becomes obvious that OM in- 
deed fills a conceptual gap: 
(s) 
SPE OT DP OM 
arbitrary rule/ yes yes no yes 
constraint order 
language specific yes no yes yes 
rules/constraints 
underlying yes yes no no 
representations 
Neglecting here the difference between rules 
and constraints, OT has chosen to eliminate 
language specific constraints s while maintain- 
ing underlying representations. It is not a priori 
clear that this is favorable to eliminating arbi- 
trary rule (constraint) ordering or underlying 
representations, and indeed this last choice is 
what happens in OM, while DP has eliminated 
two of SPEs theoretical instruments. In this re- 
spect it is the most restrictive framework, but 
we have seen empirical and conceptual problems 
with its assumptions in the preceeding section. 
Consider the following: Lexicalized candidate 
sets in both DP and OM are given by language- 
specific constraints, thus this point makes no 
difference. Now, when the toy example above 
proves to be representative, OM (like OT) al- 
lows to maintain universal constraints where DP 
cannot. 
7See section 6 for an example of ranking. 
SNote that the OT-claim to use only language- 
specific constraints is crucially weakened by the family 
of alignment-constraints that can be instantiated in a 
language-specific way. 
It might seem that OM can handle regular 
phonological processes only by stipulation of 
allomorphs with a high degree of redundancy. 
Thus for the case of German final devoicing 
we would have to assume two allomorphs for 
each morpheme showing the alternation, e.g. 
{Tod, Tot} 9 as in Tot, 'death(sig)' and Tod- 
e,'death-plu'. The choice between these allo- 
morphs could be accomplished by the two con- 
straints !DEVOICE that marks voiced obstru- 
ents in coda position and !VOICE that marks 
unvoiced obstruents generally. The ranking 1° 
!DEVOICE ~> !VOICE will make 'Tot'- the op- 
timal candidate in Tot and 'Tod-' in Tode. Tot, 
'dead' with an "underlying" t will remain the 
same in every context, assuming that this is 
the only allomorph. Though this works tech- 
nically it seems highly undesirable to have two 
allomorphs for Tod one of which can be pre- 
dicted.But the regular expression (Tod \[ Tot) 
can equally well be written as To(d\[t) or even as 
To\[-continuant -sonorant +coronal\] since regu- 
lar languages can be enriched straightforwardly 
by bundles of finite-valued features (Kaplan 
Kayi 1994:349-351). Thus allomorphy in this 
and comparable cases reduces to underspecifi- 
cation. 11 
7 Locality of Constraint Eval- 
uation 
Ellisons (1994) algorithms offer a way of glob- 
ally finding optimal candidates out of regu- 
lar candidate sets. This however is not the 
9 German orthography is used except for the phonemic 
writing of voiceless 't') 
1°Ranking can be implemented along the lines of Elli- 
son (1994). 
nlt might be argued that this move doesn't explain 
the nonexistence of voiced Coda-obstruents in German, 
since nonaiternating voiced obstruents could be accom- 
plished by fully specifying them as voiced in morphology. 
But there are languages like Turkish (Inkelas:1994:3), 
where certain morphemes resist otherwise regular final 
devoicing and this descriptive possibility thus seems to 
29be well motivated. 
only possible method to generate word forms 
given a particular OM-grammar. In the exam- 
ple above the optimal word form can also be 
found in a much more local way: by traversing 
at once the candidate automaton and the eval- 
uation transducer and choosing at each state 
the '0'-transition, when possible. The reader 
can check this by constructing finite state au- 
tomata out of the regular relations in (2'). 
Though this is a case of extreme locality, proba- 
bly not representative for phonological phenom- 
ena in general, it seems promising to study, how 
far constraint evaluation can work locally. An 
argument that constraint evaluation not only 
can but SHOULD happen locally can be con- 
structed out of the following well known data 
from Mendel2: 
(9) 
H: k5 'war' pdld 'house' 
L: @d, 'debt' b~l~ 'trousers 
HL: mb~ 'owl' ngihl 'dog' 
LH: mbd 'rice' fhndd 'cotton' 
LHL: rob5 'companion' nyhhti woman' 
are the following logical possibilities of tonal re- 
alization: 
(lO) 14 
mbu: mbfi 
felama: f~ldrnh, fdldrnf, 
fdlhmtl, fdhlmh, fdldmd 
nyaha: nydhtl, nyhh5, nydh5 
The problem of selecting out of these the cor- 
rect for is solved by Goldsmith (1976) through 
the following mapping procedure: 
(11) 
Tone mapping 
a. Associate the first tone with the 
first syllable, the second tone with the 
second syllable and so on. 
b. Tones or syllables, not associated 
as a result of (a) are subject to the 
wellformedness condition. 
Well-Formedness Condition 
H: hdwdmd 'waistline' 
L: kphkSl~ 'tripod chair' 
HL: fdldmh 'junction' 
LH: ndhvSld 'sling' 
LHL: n~ki'l~ 'groundnut' 
Ill Mende nouns have one of five specific tone 
patterns (indicated in the left column). Note 
that atoms of tone patterns like L in HL can 
be realized as sequences of high tone syllables 
as in fe:lhmtl or as part of contour tones 13 as in 
mb£ Hence for tabu, nyaha and felama, there 
12The ideas presented here owe much to the concept of 
incremental optimization developed in Walther (1997). 
Locality in Constraint evaluation is also invoked in Tesar 
(1995}, but there it isn't conceived as empirically differ- 
ent from global evaluation. For an interesting hybrid 
of directional rule application/constraint evaluation see 
Kager (1993) 
13Contour tones are the falling tone (mbfi) the ris- 
ing tone(tubal) and the falling rising tone (mbd). Acute 
stands for H, grave for L tone. As Leben, I analyze con- 
tours as sequences of Hs and Ls. 
a. Every tone is associated with some 
syllable. 
b. Every syllable is associated with 
some tone. 
c. Association lines may not cross. 
This mapping procedure is (apart from its use 
in a wealth of other phenomena) stipulation. 
I will sketch how the same effects can be de- 
rived from plausible constraints on phonological 
wellformedness and locality of constraint evalu- 
ation. 
Let's suppose that the candidate sets in (10) 
are given by lexical constraints realized as trans- 
ducers 1S. It's natural to assume that contour 
tones are a marked case in human language, vi- 
olating the constraint *Contour and that there's 
a ban on adjacent syllables with the same tone, 
14In Goldsmiths system these possibilities would cor- 
respond to the possible noncrossing exhaustive associa- 
tions of the stems with their respective tone pattern. For 
details see Trommer (1998). 
lSFor an implementation see Wrommer (1998). 3O 
an instance of the Obligatory Contour Principle 
(!OCP(Tone)). 
(12) 
*Contour 16 
,where C stands for contour tones and S for 
simple tones. 
!OCP(Tone) 
• (0--) 
, where L = Low Tone, H = High Tone 
mbfi violates *Contour, but being the only 
candidate for tabu its optimal. For felama 
filSmh and fdldmd will be excluded since they 
violate both constraints and the other candi- 
dates only one. But 1~ fdldmh, fdl~rnfi and 
fdh~mh for that reason are all on a par. Nydhd 
is out because it violates two times *Contour, 
but there's no way to decide between nydhh and 
nyfh5 that violate it only once. 
However once locality of constraint evaluation 
is taken seriously only the correct candidates re- 
main: Suppose we are following the candidate 
automaton for felama, for 'e' 'e" will be cho- 
sen since the word has to start with H and a 
contour tone would violate *Contour. Next, 'h' 
will be selected since 'd' violates !OCP(Tone) 
and 'd' again *Contour. For the same reason 
'd' would be best for the next 'a'. But now ac- 
cording to the unviolable lexical constraint that 
requires the tone pattern for this noun, only 
'd' is possible, even if it violates !OCP(Tone). 
a~Obviously H and L in (12) have to be spelled out seg- 
mentally, i.a. in their shape as contour tones. In Trom- 
mer (1998) I present a procedure to convert constraints 
on tones in constraints on corresponding segments. A 
more autosegmental treatment following the lines of Eis- 
ner (1997) is also considered there and shown to be com- 
patible with the results presented here. Consonants are 
ommitted for ease of exposition. 
17For simplicity I ignore possible results out of rank- 
ing the constraints and treat them like one complex con- 
straint 
More or less the same story can be told about 
nyaha. Hence the left to right-assymmetry in 
the mapping convention emerges out of locality 
or, put in a more placative way, the tendency to 
choose marked options as late as possible. Note 
that the use of violable constraints is necessary 
in this account to get contour tones and tone 
plateaus at all, but only when necessary.For ex- 
ample an unviolable constraint prohibiting con- 
tours would exclude correct forms as rnb~i and 
nyhhd. 
8 Locality and Directionality 
In Mende tone patterns marked (tonal) struc- 
ture appears as far to the right as possible. Al- 
though this seems to be the unmarked case in 
tone languages the mirror image too seems to 
exist (e.g. Kannakuru, Odden, 1995:460). To 
describe such cases it's sufficient to use candi- 
date automata that are interpreted from right 
to left. 
Directionality effects like the ones found in 
tone mapping are also typical in many other 
phonological domains, e.g. foot construction 
(Hayes, 1985; Kager, 1995), root and pattern 
morphology (McCarthy, 1979), and syllabifica- 
tion (Ito 1988). Even if a locality-based reanal- 
ysis isn't possible in all of these cases tradition- 
ally handled by derivational mapping and pars- 
ing devices, this is a promising area for further 
research on locality and markedness. 
9 Locality and Generative 
Power 
Frank & Satta (1998) show that input-output 
mapping in OT isn't regular in the general case 
even if GEN and the constraints are imple- 
mented as finite state transducers. This result 
carries over immediately to OM under global 
constraint evaluation. However if local evalua- 
tion is used there seems to be an algorithm to 
construct an equivalent finite state transducer. 
Since the complete algorithm requires a more 
elaborate representation of morpheme structure 
3than I can give here I illustrate only the basic 
idea. is Let's first consider a slightly modified 
version of (2') and the corresponding transduc- 
erslg: 
(2") 
Morphology: 
(13) 23-product(A,O) 2° 
1 make (IA,Io) initial in Ax2aO 
2 make (FA,F¢) final in Ax230 
3 for each arc from z to t in A labeled M/P1 
4 for each arc from x to y in O labeled P2/N 
5 if P1 = P2 
6 then add to Ax230 an arc 
7 from (x,z) to (y,t)labeled M/P1/N. 
Applied to the transducers in (2") we get: 
As Transducer: (14) 23-product (Morphology, Phonology) 
Initial State: 0 
Final State: 6 
Transitions: 
(0,1,hl), (1,3,a1), (3,4,pl), (0,2@2), 
(2,4,i2),(4,5, j3),(5,6, e3), (4,6,e3) 
Phonology: 
Initial State: A 
Final States: C,V 
Transitions: 
(a, C, C/0), (V, C, C/0),(C, C, C/0) 
(A, \7, V/0), (C, V, V/0),(V, V, V/0) 
For convenience abstract morphemes are rep- 
resented by numbers that are repeated for each 
segment realizing the respective morpheme. 
The following algorithm yields from a candi- 
date transducer and an evaluation transducer a 
transducer with three tapes integrating caMP 
date generation and their mapping to marked- 
ness values. The first tape of this new trans- 
ducer (Kx23C) corresponds to the candidate set, 
the second one to the morphological signatures 
and the third one to the evaluation under the 
constraint: 
XSproblems for the described method arise with recur- 
sivity since '33' mapped to 'aa' is ambiguous between 
two aomorpemes and one aa-morpheme. Further states 
that can be reached and left by transitions of the same 
morphological index would need a seperate treatment in 
the procedure described on p. 9. 
19the regular expression for the phonological part is 
exactly as in (2'). 
Initial State: Ao 
Final State: V6 
Transitions: 
(Ao,Cl,hl/0), (Ao,C2,p2/0), (C1,V3,al/0), 
(C2,V4,i2/0), (V3, C4, pl/O), (C4, C5,j3/1), 
(C4, V6, e3/0), (V4,C5,j3/O), (V4,V6, e3/1), 
(C5,V6, e3/0) 
Again we can find the optimal output for a 
certain morphological input in a local way. E.g. 
traversing (14) and the automaton correspond- 
ing to '1+3+' (HAP+OBJ) at once we will ar- 
rive without choice at , C4 ('111'). There we 
choose locally to go to V6 over e30 since this is 
less marked than the transition over j3/1 to C5. 
We get 1113/hape. 
Alternatively we can also dispense with the 
evaluation tape altogether: We delete each tran- 
sition from every state S if there is another tran- 
sition from S with the same morphemic index 
and a smaller number on the third tape. For 
(14) this means that we remove the transition 
from V4 to V6 over e3/1 since there is a "bet- 
ter" transition namely over j3/0 to C5. Simi- 
larly the transition from C4 to Cs over j3/1 is 
eliminated in favour of the transition over e3/0 
to V6. Since for each state and every index there 
remains only one transition the third tape be- 
comes superfluous and is removed. The result is 
an input/output transducer mapping '1113' (i.e. 
HAP+OBJ) to hape and '2233' (i.e. PI+OBJ) 
to pije: 
~°A is mnemonic for candidate, O for constraint trans- 
ducer. IA, Io and FA, Fo are the initial and final states 
3,~f A and O respectively 
(15) 
Initial State: Ao 
Final State: V6 
Transitions: 
(Ao,Cl,hl), (ho,C2,P2), (Cl,V3,al), (C2,V4,i2), 
(V3, C4, p,), (C4, V6, e3) 
(V4,C5,j3), (C5,V6,e3) 
10 Final Remarks and Conclu- 
sions 
It's quite probable that locality as developed in 
the preceeding three sections is too simple to 
account for phonological data that require eval- 
uation of segments at a certain distance from 
each other. But the point was to show that lo- 
cality of constraint evaluation in the introduced 
framework even in this form is empirically sup- 
ported and preferable on theoretical grounds. 
A promising extension would be to evaluate lo- 
cally in lnultiple phonological domains using au- 
toseglnental representations along the lines of 
Eisner (1997), but the technical realization of 
this still has to be worked out. As the Albanian 
data the tonal patterns in Mende reveal the ad- 
vantages of using OM's violable constraints in 
the context of lexicalized candidate sets. On 
the other hand this lexicalization allows a sim- 
ple generation procedure. Parsing phonologi- 
cal output forms onto morphological signatures 
in OM is relatively straightforward while the 
question is not even adressed seriously in finite 
state formalizations of OT. (Ellison, 1994; Eis- 
ner, 1997). Both parsing and generation are 
even simpler if OM is interpreted as a finite state 
transducer under local constraint evaluation. It 
remains to be seen, if the specific mixture of 
OM borrowing from DP AND OT will give rise 
to further linguistically worthwhile analyses and 
to efficient computation of morphophonology. 

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