PARADIGMATIC MORPHOLOGY 
Jonathan Calder 
University of Edinburgh 
Centre for Cognitive Science 
2 Buccleuch Place 
Edinburgh 
Scotland 
EH8 9LW 
ABSTRACT 
We present a notation for the declarative 
statement of morphological relationships and lexieal 
rules, based on the traditional notion of Word and 
Paradigm (cf Hockett 1954). The phenomenon of 
blocking arises from a generalized version of 
Kiparsky's (1973) Elsewhere Condition, stated in 
terms of ordering by subsumption over paradigms. 
Orthographic constraints on morphemic alternation 
are described by means of string equations (Siekmann 
1975). We indicate some criticisms to be made of our 
approach from both linguistic and computational 
perspectives and relate our approach to others such as 
Finite-State Morphology (Koskenniemi 1983), 
DATR (Gazdar and Evans 1989) and object-oriented 
morphophonemics (de Smedt 1984, Daelemans 
1988). Finally, we discuss the questions of whether a 
system involving string equations allows a reduction 
to finite-state techniques. 
I Introduction 
A common assumption in linguistics is that the 
phonological, morphological and orthographic 
statements are most appropriately phrased in a 
fundamentally procedural way, (see for example 
Hoeksma and Janda 1988). Morphological analysis 
under the rubric of finite-state morphology 
(Koskenniemi 1983) has arguably tended to support 
the view that morphological alternation is best 
described by stating procedures for the destructive 
alteration of orthographic units. At the very least, it 
appears to have led to the view that morphological 
descriptions should be restricted to those with an 
immediate interpretation in terms of the operations of 
finite-state transducers. 
In this paper, we present a notation for the 
declarative statement of morphological relationships 
and lexieal rules, based on the traditional notion of 
Word and Paradigm (WP, Hockett 1954, see also 
Anderson 1982). The phenomenon of blocking arises 
from a generalized version of Kiparsky's (1973) 
Elsewhere Condition, stated in terms of ordering by 
subsumption over paradigms. Orthographic 
constraints on morphemic alternation are described by 
means of string equations (Siekmann 1975). 
We will first give a brief introduction to string 
equations and the other formal devices used in our 
model, namely lexical entries and rules, grammatical 
properties and paradigms. We give example 
paradigms and show how our interpretation of 
paradigms leads to the phenomenon of blocking. We 
will then indicate some criticisms to be made of our 
approach from both linguistic and computational 
perspectives. We discuss relations between our 
proposals and the approaches of Finite-State 
Morphology (FSM, Koskenniemi 1983), DATR 
(Gazdar and Evans 1989) and object-oriented 
morphophonemies (de Smedt 1985, Daelemans 
1988). One important question in the light of current 
work in morphology is whether a system involving 
string equations allows a reduction to finite-state 
techniques. We review some possible answers to this 
question. 
2 Components of the Model 
2.1 String Equations and String 
Unification 
This introduction is based on Siekmann (1975). 
A string a is a sequence of elements drawn from a 
finite alphabet C combined by the associative 
operator +, representing the concatenation of strings. 
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A string specification or string form is a sequence 
possibly containing variables drawn from the set of 
variables V, disjoint from C. Omission of the 
operator + increases legibility as shown in the right 
hand column of (1) which gives examples of strings 
(a,b) and string specifications (c-e) and where lower 
case alphabetics represent elements of C and upper 
case alphabetics elements of V. 
(1) a. w+a+l+k+s walks 
b. s+O+r sOr 
c. A A 
d. W+s Ws 
e. k+V+t+V+b kVtVb 
String specifications are partial descriptions of 
strings. As with the standard use of unification in 
computational linguistics (Shieber et al 1983, Pereim 
1987), we may take two partial descriptions to 
describe the same object. We use string equations to 
represent this situation. The examples in (2) show 
string equations and the assignments of values to 
variables which satisfy the equations. 
(2) a. walks = Ws 
a'. W/walk 
b. sOrAk = XYZYW 
b'. A/O, Y/O, X/s, Z/r, W/k 
c. kVtWb = CiDaE 
c'. V/i, W/a, C/k, D/t, E/b 
The operation which determines the assignments 
of values to variables (equivalently, determines the 
most general unifying substitution for a given 
problem) is string unification. While no complete, 
terminating algorithm for the general case of string 
unification exists, the class of problems described by 
Siekmann (op. cit., section 4.3.3.2) as P0.5, that is 
where repeated variables are only permitted on one 
side of an equation, are decidable and have only 
finitary substitutions (see also Siekmann 1984). 
Whether or not an equation falls within P0.5 is easily 
determined. The examples in (2) and in the rest of this 
paper all fall within this class. We will refer to the 
result of applying a unifying substitution to either 
side of a string equation as the unification of the two 
string specifications in question. 
The relation of subsumption defines a partial 
ordering on string specifications. A string 
specification S subsumes another S' (S _ S') if all 
ground instances (i.e. instances that contain no 
variables) of S' are also instances of S. Equivaleafly, 
S E S' if the unification of S and S' is S' (S' LJ s 
S = S'). If S and S' are inconsistent, S U s S' is 
undefined. 
It is worth noting that the use of string 
unification is widespread in the field of automatic 
theorem proving, as an extension of standard 
resolution techniques and typically as an instance of 
what Plotkin (1972) terms "building in equational 
theories", that is the extension of standard unification 
algorithms by axioms such as associativity and the 
development of a normal form for the resulting theory 
(Plotldn op.cit, p74). 
2.2 Lexical entries 
A lexical entry S:P associates a string S (i.e. S 
contains no variables) and a set of grammatical 
properties P. We will here treat grammatical 
properties as atomic. (Their ultimate interpretation is 
intended to be similar to that of templates in PATR-II 
(Shieber et al 1983)). A lexical specification ~:¢ 
subsumes another ~':¢' iff ~ _ ¢Y' and ¢ _< ¢' 
where < represents the partial ordering over sets 
defined by the relation of set inclusion. (In other 
words, c':~b ' contains at least as much orthographic 
and grammatical information as ~:~b). The lexicon 
consists of a finite set of lexical entries. 
2.3 Lexical rules 
A lexical rule is a triple <Name, IS:IP, 
OS:OP>, representing a mapping between a set of 
"input" properties IP and a set of "output" properties 
OP. The interpretation of a rule with respect to 
grammatical properties is as follows: 
(3) Given a lexical item S:P, and a lexical rule, 
<LR, IS:IP, OS:OP>, as before, L R 
relates P to another set of properties P' (its 
"output") in the following way: 
P' = (P~IP) u OP 
where OP ~ P' and IP ~ P. 
\ is set complement. Equivalently P = (P'\OP) 
- 59 - 
v IP 1. The use of set complement allows a general 
ceteris paribus statement 2. That is, properties not 
mentioned in the rule are unchanged. The relationship 
between the string specifications IS and O S is 
mediated by a paradigm. 
2.4 Paradigms 
A paradigm is a quadruple, <Name, ¢~:~, 
\[LR1 ... LRn\], \[Sl ... Sn\]>,n --- 1,which 
relates string forms ¢~ and Si via the lexieal rule 
LRi under conditions 0 where the set of string 
variables in ¢~ and Si are not disjoint. Si is a derived 
string form. Any variables in Si also occur within ¢~ 
(this restriction will be reformulated shortly). Name 
is the (unique) name of the paradigm. ¢~:07 is the 
lexical specification of the paradigm. (Alternatively, 
it is the underspecified word whose behaviour the 
paradigm describes). The interpretation of a paradigm 
is given in (4). 
(4) If a paradigm <Name, ¢~:~, LR, S> is 
applicable, lexical items S:P and Si:P' are 
related by lexical rule <LRi, IS:IP, 
OS:OP>, with P, P', as in (3). 
For a paradigm n <Name, ¢~:~, LR, S> to 
be applicable to a lexical item S:P, two conditions 
must hold: 
(5) a. a_S and¢~ <_P. 
b. There is no paradigm ~' <Name ), ¢~':~', 
LR', S'> such that¢~ ~_¢~°, ~ _<~',¢~' 
S and ~) < P. 
(5a) requires that the lexical specification of the 
paradigm subsume the lexical item. (5b) requires that 
there be no paradigm whose lexical specification is 
more specific than that of ~ which is also applicable 
to the lexical item. The effect of (5b) is to enforce a 
generalized Elsewhere Condition (Kiparsky 1973), 
under which a morphological operation is only 
allowed if there is no more specific statement which 
also holds. This also captures the notion of nested 
'lIn fact, for this relationship to hold, we have to 
add the following restrictions over the properties 
mentioned in the rule and lexical item: P c~ OP < 
IP and symmetrically P' c~ IP < OP. We thank 
Marc Moens for this observation. 
2Related proposals are made by Flickinger (1987, 
ch. 5). 
generalizations or stratification (Flickinger et al 1985, 
Gazdar and Evans 1988). Given a lexicon containing a 
finite number of lexical entries, paradigms and lexical 
rules, the set of lexical items is defined as the closure 
of the lexicon under the application of lexical rules 
mediated by the paradigms. Nothing in the basic 
formalism we propose constrains this closure to be 
finite or requires that the set of lexical items contain 
the set of lexical entries. Note that the restrictions we 
have imposed on lexical entries and variables in 
derived strings imply that, in the lexical items that 
result from the application of paradigms, there will be 
no string variables. 
3 Abbreviatory conventions 
We introduce three abbreviatory conventions, the 
first two trivial, the third less so. First, as lexical 
rules always make reference to input and output string 
forms and these forms can be determined by reference 
to the paradigm that relates them, we do not need to 
state string forms in lexical rules. This leads to the 
concrete syntax 0P, OP as before): 
(6) lexical_mle(Name, IP -~ OP) 
The second convention allows us to state more 
complex constraints on string forms. In a paradigm 
<~, q:~, ... >, we allow arbitrary equations over 
string forms to be included in ~, including negative 
and disjunctive constraints 3, and a syntax for 
allowing the expression of character classes 
(effectively these arc just a special case of disjunctive 
constraint). This allows statements of the following 
kind, where + represents string concatenation, 
represents an inequality between strings and PROLOG 
conventions for marking variables are followed. 
(7) Stem+Affix = Word, 
Stem = Prefix+C+V, 
V ¢ vowels, C E consonants, V ~ y, Affix = d 
(7) might be taken to describe the behaviour of 
vowel-final verbs under affixation of the past tense 
morpheme. Note that statements which do not 
constrain the value of ~ do not take part in the 
calculation of subsumption relations over paradigms. 
3We make the restriction that any such 
constraints do not contain variables. 
-60- 
We also have to revise our restrictions on the 
occurrences of variables in derived string forms given 
in 2.4. Any variable in a derived string form Si 
must either occur in the string specification or of the 
paradigm or must be equated, directly or indirectly, to 
some form consisting of variables drawn solely from 
or and ground forms. 
The third convention is considerably more 
complex and effects a rapprochement between our 
scheme and those of default logics for lexical 
description (Gazdar and Evans 1988, Flickinger 1987) 
and object-oriented morphophonemics (Daelemans 
1988). Given a paradigm <re, or:C, LR, S>, if 
there is only one directly subsuming paradigm <~ ', 
or':¢ ', LR', S'>, and for some i, LRi ffi LR'i and 
or u or' --~ Si = S'i (i.e. we would get the same 
result if we used either rc or rt'), we are allowed to 
omit the references to LRi and Si in re. In other 
words, we allow the inheritance of a string form and 
associated lexical rule from the more general 
paradigm. In the case of n directly subsuming 
paradigms, the same convention applies if or u orl 
... u On --~ Si = S'i,1 .... S'i,n. That is, 
multiple inheritance must be consistent. In the case 
where this condition doesn't hold, we allow the 
equivalence of inheritance by delegation (Daelemans 
op. cit.), so enforcing orthogonality of inheritance 
(Touretzky 1986); corresponding to some LRi, we 
allow Si to be of the form paradigm(n"), where 
g" is the name of a paradigm, such that n" _ n, 
with the interpretation that the paradigm n" relates o 
and Si via LRi. 
4 Example paradigms and lexical rules 
We are now in a position to give some example 
paradigms. These llave the concrete syntax: 
(8) table(Name, String:Properties, LR, S) 
where Properties may also include string 
equations and LR and S are fists of names of lexical 
rules and of string forms respectively, subject to the 
conventions described above. 
In Table 1 below, (9) is the most general 
paradigm for English verbs. (10), (11), and (12) are 
instances of that paradigm under (Sb) above. (12) is 
also an instance of (11). 
(9) table(verb, Verb: \[verb, base, Past = Verb+ed\], 
\[base 3sg non3sg past..participle past passive 
\[Verb Verb+s Verb Past Past Past 
(10) table(verb_age, age:\[verb, base\], 
\[progressive\] 
\[ageing\] ) 
(11) 
(12) 
progressive\] 
Verb+ing\] ) 
table(verb_strong, S+in+C:\[verb, base,strong, Past=S+un+C, C= {g~k}\], 
\[past_.participle past passive\] 
\[Past S+an+C Past\] ) 
table(verb_bring, bring: \[verb, base,strong, Past = brought\], 
\[past..participle past passive\] 
\[Past Past Past\] ) 
Table 1: Example paradigms 
- 61 - 
The abbreviatory mechanisms allow us roughly 
the same amount of compaction for these descriptions 
as found in Daelemans' (1988) approach. Lexical rules 
that might be associated with such paradigms are: 
(13) lexical_rule(3sg, 
\[verb, base\] ---> \[verb, finite, 3sg\]) 
lexical rule(non3sg, 
\[verb, base\] ---> \[verb, finite, non3sg\]) 
5 Criticisms of the framework 
There is one major criticism of our approach 
from a linguistic point of view, namely that in cases 
such as "ageing", there is no reason why the form of 
the participle ending "ing" should be the same in all 
verbal paradigms. Likewise we cannot make the 
generalizations that passive and past participle forms 
are identical in every verbal paradigm and that the 
orthographic behaviour of the verbal singular affix 
and nominal plural affix is identical. Defining 
subsumption on the basis of lexical specifications of 
paradigms alone leads to a very simple statement of 
the conditions of inheritance of derived string forms, 
but disallows the possibility of inheritance of partial 
derived string forms. The restriction of inheritance by 
delegation or stipulation to subsuming paradigms, 
while natural, is not motivated by more general 
considerations. 
This problem becomes much more obvious and 
acute in analysing non-concatenative morphology, as 
in Semitic (McCarthy 1981). It is not the 
intercalation of the consonantal roots and vocalic 
melodies which leads to difficulties, as this is easily 
described in our framework. Rather, the problem lies 
in having to choose which of the root and melody 
should be expressed as the word with which a 
paradigm is associated. On the one hand, traditional 
grammar would suggest that the consonantal root has 
some claim to this status. However, there are clearly 
relationships between the vocalic melodies which 
indicate syntactic regularities on the basis of Ablaut 
(McCarthy 1981, p403), and these regularities cannot 
be captured if we choose the consonantal root as the 
paradigmatic word and disallow inheritance of partial 
derived string forms. In any case, such regularities 
should presumably be stated independently of any 
roots with which they happen to combine. 
The above criticism provides an interesting 
illumination of Matthews' (1974, p163) claim that 
different styles of morphological analysis are required 
by different language types, word and paradigm 
morphology being particularly suited to inflecting 
rather than isolating and agglutinating languages. 
Current work is investigating how we may alter some 
of the assumptions in the definitions in section 2, to 
allow for some degree of parametricity in the 
languages that such systems may describe. 
A second criticism is both computational and 
linguistic. String unification is a very powerful 
operation and, while it is arguable that strings in our 
sense and orthographic constructs are indeed the same 
kind of object, one may justifiably have reservations 
about introducing string equations into linguistic 
description. The resolution of this point awaits 
further work on the formal properties of finitary 
theories. We return to this point in our discussion of 
phonological theory below. The computational 
aspects of this problem will have less force if we can 
show that there is a reduction from descriptions 
involving string unification to some less powerful 
mechanism such as finite state transduction. This 
point is also discussed further below. 
6 Comparison with other frameworks 
That our approach is more general than standard 
FSM is clear from the fact that string variables can 
represent an unbounded amount of orthographic 
material. In this way, we can, for example, model 
unbounded reduplication. The problematic cases of 
Finnish ambifixation described by Karttunen (1983 
p181, citing Koskenniemi) are handled 
straightforwardly, although this raises immediate 
questions about the formalism's interpretation in 
terms of finite-state transduction, discussed shortly. 
G0rz and Paulus's (1988) algorithm for the location 
of stem vowels in German which undergo Ablaut can 
reconstructed declaratively. Our approach also solves 
the problem noted by Bear (1988) of the 
overgenerality of rules in an FSM system. He 
introduces the mechanism of a negative rule feature to 
prevent the application of an epenthesis rule which 
would otherwise insert an "e" before the suffix "s" in 
- 62 - 
the case of words such as "banjo" and "tango". The 
need for negative rule features and their 
quasiprocedural implications are avoided in the system 
we propose. The following paradigm correctly states 
the facts and will apply only in the case of those 
items that violate the subgeneralization concerning 
nominal plurals in "o". 
(14) table(piano, 
S:\[noun, singular, 
S = {piano, piccolo .... }\], 
\[singular, plural\] 
\[S, S+s\]) 
Free variation, such as that between "o" and "oe" 
forms in words like "cargo", is treated analogously by 
allowing the derived string specification to be 
S+{e, O}+s in this case (where Et represents the 
empty string), although this obviously fails to 
capture the fact that the alternants are precisely those 
found in the most closely related paradigms. Finally, 
having the level of properties over which to state 
generalizations means that our lexical representations 
of strings are not cluttered with non-realizable 
diacritic characters whose function is simply to 
trigger morphological processes (of Bear 1988). 
The main advance we would claim for our 
system is that we have provided a calculus for 
orthographic forms, bringing the treatment of 
orthography within the same kind of logical 
framework now accorded to our treatment of semantic 
and syntactic objects. The fully declarative 
interpretation of our system and the similarity of 
statements within it to work by Daelemans (1988) 
offer a way of giving a formal treatment of object- 
oriented morphophonemics. 
Finally, recent work by Gazdar and Evans (1988) 
may offer techniques for extending the formalism to 
more complex grammatical descriptions, in line with 
many current views on the nature of syntactic 
categories as recursively structured objects. Current 
work is examining this possibility. 
7 Reduction to finite.state transducers 
While the declarative nature of our system means 
that it is not tied to a particular computational 
interpretation, the fact that the closure of the lexicon 
may be infinite argues in favour of an interpretation 
that does not presuppose computation of that closure, 
and considerations of efficiency in favour of one in 
terms of finite-state transduction (Koskenniemi 
1983) 4 . It is immediately clear that heavy restrictions 
have to be placed on systems involving string 
equations in order to have any chance of a reduction to 
finite-state transduction at all. One difficulty will lie 
in representing paradigm subsumption induced by 
non-orthographic properties. The assumption that 
there is only a finite set of such properties would 
make this problem easier. (We assume the 
subsumption relations engendered by orthographic 
properties are readily handled by the standard notion of 
priority of transition in FSTs). A much graver 
problem lies in eliminating non-finite-state 
constructions. Example (15) below exhibits (at least) 
context-frecness: 
(15) table(weak_verb, 
S+en:\[verb, weak, root\], 
\[ .... past_participle .... \] 
\[ .... ge+S+et .... \]) 
Here any string may be substituted for S. If we 
allow pastparticiple to represent the null lexical rule, 
we may generate the string set (ge)nS(et) n which is 
clearly homomorphie to the context-free language 
anb n. Similar demonstrations can be given of 
context-sensitive 5 and even more powerful languages. 
In order for the reduction to go through in this case, 
we should have to demonstrate the finitariness of 
substitutions for S. This is clearly impossible in the 
general case. The kinds of restriction to be imposed 
would include at least the prohibition of copying over 
unbounded domains and of affixation at both margins. 
Under the optimistic assumption that appropriate 
restrictions can be found, we might proceed by 
computing tree-structured lexicons on the basis of 
lexical entries and of affixes introduced under 
paradigms. Continuation classes, and possibly further 
4This section and the following have benefited 
greatly from discussions with Mike Reape. 
5Such as the string specification WW, where W 
ranges over strings from some alphabet and which 
clearly shows context-sensitivity (Aho and Ullman 
1972, p198). 
- 63 - 
partitions and duplications of the lexicons, could be 
computed on the basis of the properties mentioned in 
paradigms and lexical rules. Information from the 
distribution of derived forms with respect to string 
specifications in paradigms could then be used to 
construct the FSMs that mediate surface and lexieal 
tapes. Problematic cases might only be handled by 
indicating an ordering over paradigms where context- 
freeness is implicated, effectively indicating that those 
paradigms may only apply to non-derived forms and 
allowing incompleteness in the computation of the 
closure of the lexicon. 
The above paragraphs are mostly speculation. As 
Gazdar (1985) notes, it is not certain that 
morphological phenomena in natural language are 
best characterized by f'mite-state devices. Depending 
on one's view of the data cited by Culy (1985) and the 
ambifixation cases mentioned above, the formal 
power of our framework might be interpreted as a 
virtue rather than a vice and future work should also 
look at introducing (at least) context-free devices into 
our computational interpretation of morphology. 
Unsurprisingly, this is an area for further research. 
8 Implementation 
All aspects of the system described above have 
been implemented, primarily by Mike Reape. The 
implementation of lexical rules differs somewhat 
from the presentation given here, in that we allow the 
association of a PATR-II style lexical rules with the 
lexical rules we describe above and, as such, lexical 
rules may perform arbitrary mappings over feature 
structures. The work considerably extended published 
string unification algorithms to handle identity and 
arbitrary constraints over string forms. The system is 
a subcomponent of a polytheoretic lexicon system 
(Calder and te Lindert 1987, cf. also Zaenen 1988) and 
is currently being used to generate English lexical 
entries for use with Unification Categorial Grammar 
(Zeevat et al 1986, Calder et al 1986). The system 
generates lexical entries off-line; the lexicons used to 
date have provably finite closures under the 
application of lexical rules. Current work is focused 
on the computational interpretation of paradigms and 
on descriptive work in languages other than English. 
ACKNOWLEDGEMENTS 
This work is supported by ESPRIT project 
P393: ACORD "The Construction and Interrogation 
of Knowledge Bases using Natural Language Text and 
Graphics". Thanks are due in particular to Mike 
Reape and also to Ewan Klein, Claire Gardent, Enrico 
te Linden, Mare Moens, Antonio Sanfilippo and 
Henk Zeevat for discussions and criticism of previous 
proposals. All errors are of course my own. 
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