Backwards Phonology 
John Bear 
Artificial Intelligence Center 
SRI International 
Abstract 
This paper constitutes an investigation into the gener- 
ative capabilities of two-level phonology with respect 
to unilevel generative phonological rules. Proponents 
of two-level phonology have claimed, but not demon- 
strated, that two-level rules and grammars of two- 
level rules are reversible and that grammars ofnnilevel 
rules are not. This paper makes "reversibility" ex- 
plicit and demonstrates by means of examples from 
Tunica and Klamath that two-level phonology does 
have certain desirable cababilities that are not found 
in grammars of unilevel rules. 
1 Introduction 
Since Koskenniemi proposed using two-level phonol- 
ogy in computational morphological analysis in 1983, 
it has enjoyed considerable popularity \[Koskenniemi, 
1983\]. It seems to be both expressiyely powerfid 
and computationaily tractable. Two-level phonologi- 
cal granntmars have been written for a dozen or more 
languages, and written in a form that is interpretable 
by a program. One question that arises fairly fre- 
quently however, at least in the context of discussion 
about two-level morphology, is roughly, "Why don't 
you use normal generative phonological rules?" i.e., 
rules of the type that are taught in elementary linguis- 
tics classes. A slightly more positive way to ask the 
question is, "In what way or ways does Koskenniemi's 
notion of two-level pholmlogical rule represent a the- 
oretical advance?" This paper addresses that ques- 
tion by extending the notion of unilevel rule system 
to cope with tim same types of phenomena that two- 
level rule systems were designed to handle, and then 
contrasting the two different systems. 
At the annual meeting of the Linguistic Society of 
America (LSA) in 1981, Ron Kaplan and Martin Kay 
presented a paper describing results about equiva- 
lences between what they call a cascade of finite-state 
transducers and a set of normal, ordered phonologi- 
cal rules \[Kaplan and Kay, 1981\]. At the I, SA's 1987 
annual meeting, Lauri Karttunen gave a paper at- 
tempting to show that, when viewed a certain way, 
Koskenniemi's two-level rules possess a certain ele- 
gance that cannot be ascribed to ordered sets of rules, 
namely their independence from order per se \[Karto 
tunen, 1986\]. 
In spite of Karttunen's paper and Koskenniemi's, 
and perhaps to some extent because of Kaplan and 
Kay's paper, it is still not obvious to people who are 
interested in this field what, if anything, two-level 
phonology offers that cannot already be found in tile 
linguistic literature under the heading of generative 
phonology. Koskenniemi has made some claims about 
grammars of two-level rules being reversible whereas 
sets of ordered rules are not. However these claims 
are not backed up by solid argumentation, and the 
Kaplan and Kay paper seems to argue otherwise. 
From a linguistic point of view, there may be good 
reason to think that people use two different sets of 
rules or procedures for generation and recognition. 
From a computational point of view, however, it is 
interesting to ask, "What needs to be done in order 
to use the same grammar for generation and recogni- 
tion; does a single reversible grammar lead to more 
or less work in terms of writing the grammar and in 
terms of run-time speed; and finally, does a reversible 
grammar lead to a more or less elegant presentation 
of the phenomena?" Another reason for asking about 
reversibility is to make a comparison of these two rule 
formalisms possible. The main novelty in Kosken- 
niemi's system is the reversibility of the system, so we 
may well question what would be necessary to view 
unilevel rules as reversible. 
In short, there are very. good reasons for being inter- 
ested in properties of reversibility, and these proper- 
ties will serve as the basis tot this paper's comparison 
between the two different types of phonological rule 
formalisms mentioned above. The discussion here will 
focus more on concrete examples of generative capac- 
ity, and much less on issues of what is involved in 
building an acceptable linguistic theory. \[For more on 
global concerns of linguistic theory, see, for example, 
Ellasson, 1985\]. The questions addressed here will be, 
"What assumptions need to be made to use a gram- 
mar of unilevel generative rules to do recognition?" 
1 13 
and "Ilow does tim resulting combination of grammar 
plus rules-of-interpretation compare with a two-level 
style grammar?" 
2 Reversibility of Unilevel 
Rule Systems 
The question of grammar reversibility involves two 
interrelated but separate issues. The first is whether 
the notational or descriptive devices of a grammar 
are in general amenable to being reversed, and what 
is involved in the reversal. The second is whether 
individual accounts of the phenomena" of a particu- 
lar language are reversible, and, again, if so, what is 
involved in the reversal. 
Tim remarks in this paper are mainly concerned 
with the general paradigm of generative phonology, 
in particular, segmental phonology as is described in 
elementary texts - e.g., Kenstowicz arid Kisseberth 
(1979), IIalle and Ciements (1983), Schane (1973), 
Mohanan (1986) - rather than any particular linguis- 
tic theory. The main techniques discussed are rewrite 
rules, orderings of rules, features, and variables for 
feature values (e.g., the alpha and beta of assimila- 
tion rules). The problems of suprasegmental phonol- 
ogy will be left for another paper. 
3 Backwards Rules 
I shall start by making explicit what it means to apply 
a phonological rule in the backwards direction. The 
basic idea is extremely straightforward and will be, I 
think, uncontroversial. 
a-+ b / ot_fl (1) 
A rule like the one in (I) transforms the string/o~afl/ 
into the string /abfl/. Here c~ and fl are strings of 
characters over some alphabet, e.g., the phonemes of 
a language. I take it that such a rule can also be in- 
terpreted as mapping the string/o~bfl/into the string 
/o~afll, when it is applied backwards. 
To take a more linguistically realistic rule, let us 
consider the simple rule in (2). 
. / _ g (2) 
From a recognition point of view, this means that 
if we have the sequence trig\] in a surface form of a 
word, then the underlying sequence could be /n g/. 
In slightly more general terms, we look for the seg- 
ment on the right side of the arrow to see whether it 
appears in the context given in the rule. If so, we can 
transform that segment into the segment on the left 
side of the arrow. 
4 Obligatory Versus Optional 
The rule in (2) says nothing about whether it is op- 
tional or obligatory in the backwards direction. Op- 
tionality in the backwards direction is entirely inde- 
pendent of optionality in the forward direction. In 
English the rule in (2) seems to be obligatory in the 
reverse direction, i.e., every surface \[t3\] seems to come 
from an underlying/n/. In the forward direction, it 
does not always apply. This is demonstrated by the 
pair: co\[~l\]gress vs. co\[n\]gressional, l 
In a language that had phonemic/Ij/and/n/, the 
rule might be obligatory in the forward direction and 
optional in the backward direction. 9" That is, if \[rj\] 
on the surface can come from either/n/or/(I/, then 
the rule would necessarily be optional in the reverse 
direction. 
The point here then is that one needs to specify in 
the grammar not just whether a rule is obligatory or 
optional in the forward direction, but also whether it 
is obligatory or optional in the backwards direction. 
5 Reversibility and Rule Or- 
dering 
The previous example describes the case of a single 
rule and points out that attention must be paid to 
whether a rule is optional or obligatory in the back- 
wards direction as well as in the forward direction. 
The following case of rule ordering shows that there 
is more to the issue of reversibility than the distinc- 
tion between "optional" and "obligatory." 
There is a beautiful example in the Problem Book 
in Phonology by lIalle and Clements (1983) of the ele- 
gance of rule ordering. In this section I will show that 
the device of ordered rules is not generally reversible 
using their example from Klamath. 
Tile data from Kiamath together with five rules 
are taken from llalle and Clements (1983), who in 
turn give their source as being Klamath Grammar by 
Barker (1964): 
IMohanan (1986) p. 151. 
2That obligatory rules need not be obligatory when applied 
in the backwards direction has been pointed out by Ron l(aplan 
(in a course at tile LSA Summer Institute at Stanford, 1987) 
14 2 
nl ---+ II 
/honli:na/--+ holli:na `flies along the bank' 
hi. ~ ih 
/hon!y / --+ holhi '\]ties into' 
nl' --} I? 
\[honl'a : l'a\[ ---, hoi?a : i'a 'flies into the fire' 
i t --+ lh 
/pa : I~a/--+ pa : iha 'dries on' 
!1' --+ I? 
/yalyali'i/--~ yalyal?i 'clear' 
Halle and Clements also say that Barker assumes 
that all phonological rules are unordered and that all 
rules apply simultaneously to underlying representa- 
tions to derive surface representations. 3 They then 
give the following exercise: "Show how Barker's set of 
rules can be simplified by abandoning these \[Barker's\] 
assumptions and assuming that phonological rules ap- 
ply in order, each rule applying to the output of the 
preceding rule in the list of ordered rules. Write the 
rules sufficient to describe the above data, and state 
the order in which they apply. ''4 
,The rules that one is supposed to arrive at are 
roughly these: (} 
n --. i / _ } (3) 
t.-, h / l_ (4) 
? / l_ (5) 
The ordering to impose is that Rule (3) applies be- 
fore Rules (4) and (5), and that Rules (4) and (5) 
are unordered with respect to each other. The reader 
can verify that the rules give the correct results when 
applied in the forward (generative) direction. In the 
backwards (recognition) direction, the derivations for 
the five forms are as given below. The rule numbers 
are superscripted with a minus one to indicate that 
these rules are inverses of the rules listed above. 
holli:na -+ honli:na 
Rule 3 -I 
holhi -+ boll) -+ hon{i 5 
Rule 4-1 Rule 5-1 
3ltalle and Clements (1983) p. 113 
4 Ibid. 
holfa:l'a ---+ holi'a:l'a ~ honl'a:l'a 
Rule 5 -1 Rule 3 -1 
pa:lha -4 pa:il.a ---+ *pa:nla 
Rule d -t Rule 3-t 
yalgal?i ---* yalyali'i --4 *yalyani'i 
Rule 5 -1 Rule 3 -1 
What we~see here is that in order to recognize the 
form holli:na correctly, Rule (3) must be obligatory 
in the reverse direction. However, in order to get the 
correct results for the forms pa:lha and yalyalfi, Rule 
(3) may not apply at all; i.e., it is not correct to say 
that the results can be obtained by correctly stipulat- 
ing whether a rule is optional or obligatory. Rule (3) 
works well in the forward direction, but gives incorrect 
results when applied in the backwards direction. In 
short, the elegant set of ordered rules makes incorrect 
predictions about recognition. In contrast, Barker's 
original unordered set of rules correctly describes the 
data regardless of direction of application (i.e., gener- 
ation vs. recognition). 
This is a result about ordering of rules. I have not 
shown that a set of ordered rules is never reversible, 
only that such a set is not necessarily reversible. 
6 Variables and Deletion 
The previous example used extremely plain rules: no 
features, no alphas or betas, and no deletion. The 
next example I shall present involves some of these 
commonly used devices. I shall try to make clear when 
they can be used in a reversible way (though they need 
not be), and when they just do not seem amenable 
to reversal. Before discussing reversal further, I will 
present the data and the set of rules for describing 
the data ill the generative framework. The data and 
analysis were taken from Kenstowicz and Kisseberth 
(1979). 6 Their data come from the language Tunica. 
The rules and data deal with two phenomena: vowel 
assimilation and syncope. The rules, given below, are 
ordered, with (6) occurring before (7). \[Note on tran- 
scription: the question mark represents glottal stop.\] 
SThls is correct modulo the change of i back into y which 
Halle and Clements assure us is not, part of the issue at hand. 
For purposes of discussing reversibility it merely provides more 
support for the argument that unilcvel rules are not easily 
reversed. 
6p. 292. They cite their source as IIaas (1940). 
3 15 
--* a back .¢ +low /~ round fl round -- 
(6) 
+ / -" / syllabic 1 0 ? - stress i -- " 
Rule (7) says (or was meant to say) that unstressed 
vowels are deleted before glottal stops. Rule (6) was 
intended to mean that /a/ assimilates to \[el or \[hi 
when it is separated by a glottal stop from a preceding 
/i/ or /u/ respectively. 
In addition to the two rules just given, Kenstowicz 
and Kisseberth mention but do not formulate a rule of 
Right Destressing that follows both rules. The rules 
are in accord with the following data, also taken from 
Kenstowicz and Kisseberth. The following forms show 
assimilation. 
To verb He verbs She verbs She is v-ing Gloss 
pd pdfuhki p6C.aki p6hk~, aki look 
p~ pf?uhki p#eki pfhkfaki emerge 
yd yd ?uhki yd ?aki ydhk ?aki do 
~d 6d?uhki ~d?aki ~dhk?aki take 
These forms show syncope and assimilation. 
To verb He verbs She verbs She is v-ing Gloss 
hdra hdr?uhki hdr?aki hdrahk?dki sing 
h(pu h(p?uhki h~paki h\[pnhkfdki dance 
ndgi ndgfuhki ndg?eki ndgihkfdki lead 
As a sample derivation, Kenstowicz and Kisseberth 
give the following: 
/ndgifdki/ 
1 
ndgiC, gki 
1 
ndg?(ki 
1 
For the purpose of going through a backwards deriva- 
tion, 1 will make explicit a few assumptions. First, 1 
assume that the Vowel Assimilation rule is really as 
in (8) below. 
Vowel Assimilation (Modified) 
Vowel Assimilation 
Syncope 
Right Destressing 
\[+svU 
+low \] 
+syli 
+low 
a back 
round /\[ 1 
ot back f 
round 
(a) 
It is a matter of style that the features \[ + syll, + low\] 
were left out of the feature bundle to the right of tile 
arrow in Kenstowicz and Kisseberth's formulation of 
tile rule. Although it is considered good style to do 
so, the omission of such information makes it unclear 
how the rule should be applied for recognition. Hence 
I have included this information in Rule (8). v 
Another assumption I will make is that the unfor- 
mulated rule of Right Destressing lends nothing to my 
argument herd. I assume that the rule when applied 
in the reverse direction puts stress on the appropriate 
syllable and nowhere else. s 
Finally, I will spell out what I consider to be a 
reasonable interpretation of how to use the rules for 
recognition. When interpreted backwards, Rule (8) 
says that a low vowel that is separated by a glottal 
stop from another vowel with which it agrees in back- 
heSS and rounding might have come from some other 
low vowel. The syncope rule in (7), when interpreted 
backwards, says to insert an unstressed vowel before 
glottal stops. As was pointed out. before, there is no 
way to deduce whether these rules are obligatory or 
optional in the reverse direction. Indeed, it is not at 
all obvious what "obligatory" even means in terms of 
the assimilation rule t~ken backwards. 
Given these assumptions, we can now produce a 
reverse derivation for \[na's?ekq. 
\[n~?eki\] ~ nfi~?~ki 
/n~i?Eki 
/n~i?~ki~i\] 
/ Xnfisi?Ski 
/ /n~?~ki 
//n~E?gki ~n~?~ki 
// n~?5ki ~/n~a?gki 
~-- nfi~a?~ki 
\'~ nfi~u?~ki 
\x n ~o?~ki 
~n~?~ki 
First Reverse Destressing is applied to give ndg?gki. 
Then Reverse Syncope applies to insert various hy- 
pothesized vowels in forms in the column to the right. 
Finally, the rightmost column shows the results of 
7Presumably Kenstowlcz and Kisseberth want to treat \[¢\] 
as being \[+ low\] to keep the rule simple and still contrast \[el 
with \[i\]. If they treat \[e\] as \[- low\] and \[a\] as \[+ low\], the 
assimilation rule becomes messier. This assumption about \[el 
becomes important later. 
sit seems clear that segmental accounts will fall short when 
dealing with suprasegmental issues like stress. The goal of 
this paper is to contrast two different ways of doing segmental 
phonology. Both would presumably benefit from autosegmental 
extensions. 
16 4 
applying the reverse of the Assimilation rule to the 
preceding forms. A box is drawn around the correct 
underlying form. 
What we end up with are 14 or 15 possible forms 
- clearly too many. One problem is that the assim- 
ilation rule in (6) and (8) was formulated with only 
generation in mind. If we change it slightly, adding 
the features \[+back, -round\] to the bundle to the left 
of the arrow as in (9), 
+syll +syll 
--* c~back ? +back c~back ~round -- 
-round flr ound (9) 
we have a better rnle. Now it says that \[e\] and \[~\], 
when they result from assimilation, come specifically 
from/a/. This makes the results better. The previous 
version of the rule just mentions low vowels, of which 
there are three that we know about: s,a, ~.s When 
we specify that of these three we always want /a/, 
we have a more accurate grammar. Now instead of 
recognizing 14 or 15 possible underlying forms for the 
word ndg?eki, the grammar only recognizes ten. 
There iis a very simple but subtle point at issue 
here, havihlg to do with writing reversible rules. The 
grammar writers knew when they were formulating 
the assimilation rule that \[e\] and \[3\] were never go- 
ing to come up as input to the rule because these two 
vowels do not exist in the underlying representations. 
They also knew that there were no other rules ap- 
plying before the assimilation rule which would intro- 
duce \[¢\] or \[~\]. Hence they did not need to distinguish 
between tim various possibilities for low vowels. In 
short, the grammar writers made use of fairly subtle 
information to write a rule which was as pared down 
as possible. Leaving out the features in (9), as Ken- 
stowicz and Kisseberth do, looks elegant, but turns 
the two-way rule into a one-way rule that works only 
for generation. This is a case where leaving out some 
features obscures the content of the rule and prevents 
one from correctly applying the rule for recognition. 
In short, this is a case where the rule could have been 
written in a way that was reversible, or at least more 
reversible, but in the name of "brevity" or "elegance" 
it was not. 
The vowels \[e\] and \[~\] also provide complications for 
the revcrqal of the vowel deletion rule. We have no 
reason to believe from the data given that the deleted 
vowel is ever \[~\] or N. IIowever there is not a good 
way of saying, using standard rule writing techniques, 
that any vowel that is introduced in the recognition 
9As mentioned in an earlier footnote, Kenstowicz and Kisse- 
berth seem t,o treat \[e,\] as \[+ low\]. 
must be one of the underlying ones. In ordered sets of 
rules, there is not lypically a distinction made between 
the segments that can occur as input to a rule and 
segments that can only occur as output. One of the 
unhappy consequences is that \[e\] and \[~\] have the same 
status with respect to the rules of Tunica as the other, 
underlying, vowels in the language. 
An even more serious problem revealed by this Tu- 
nica example is the inability of the standard genera- 
tire rule-writing mechanism to specify the interrela- 
tionship between rules. The rules apply based only on 
strings of characters they get as input, not oll what 
rules came before. In the case at hand, however, we 
would like to be able to relate the two rules to one 
another. What we would really like to be able to 
say is that when in the course of recognition it be- 
comes necessary to reintroduce the deleted vowel, if 
there iu an \[e\] on the surface the reintroduced vowel 
must be \[i\], and if there is an \[~\] the reintroduced 
vowel must be \[u\] or \[o\]. This is a problem with alpha 
(assimilation) rvdes. There is no way to say that if 
there is an Is\] or \[~1 on the surface, then the reverse 
of the syncope rule must apply, when doing recogni- 
tion, and, furthermore, that it must apply in such a 
way that the assimilation rule can then apply (again 
in reverse) and, lastly, that the reverse of the assim- 
ilation rule must then apply. In simpler terms, there 
is no way to say that if there is an \[~\] (respectively 
\[~\]) on the surface, then it must be preceded by an 
underlying/i/(respectively/u/or/o/). 
When dealing with cases of deletion, and mergers 
in general, it is not generally possible to write a set of 
rules that maps surface forms unambiguously to a sin- 
gle underlying form. In the ease of the ~hmica vowel 
deletion, there are occurrences of surface forms in 
which the phonological rules cannot tell which vowel 
to reintroduce when doing recognition. There are, 
however, cases where it is clear which vowel should be 
reintroduced, e.g., the case above, and in these cases, 
both the grammar formalism and the individual anal- 
ysis should be able to express this information. The 
mechanism of using alphas and betas, for instance in 
assimilation rules, does not appear to have this ex- 
pressive capacity. 
The problem could be ameliorated by writing less 
elegant rules. For instance, the syncope rule in (7) 
could be written as in (1O). 
+syllabic \] 
+underlying --* 0/_ q. (10) 
-stress 
This would ensure that the nommderlying vowels \[~\] 
and \[.~\] would not be introduced when applying the 
rules in the reverse direction. It still would not be as 
5 17 
restrictive as one could be using two-level rules. 
One could argue that all one needs to do is use the 
lexicon to weed out the forms that are wrong. Yet 
one would not consider suggesting the same thing if a 
grammar generated too many surface forms, although 
one could imagine using a surface lexicon as a filter. 
The technique of using the lexicon to weed out the 
forms that are wrong is a perfectly good efficiency 
measure, but has no bearing on the question of how 
well a formalism maps underlying forms to surface 
forms and vice versa. 
In the rest of this paper I will present and dis- 
cuss two-level accounts of phonological phenomena 
described earlier, and show the merits of such an ap- 
proach. 
7 Two-level Rules 
In the two-level accounts that have been proposed 
\[Koskenniemi 1983, Karttunen and Wittenburg 1983, 
Bear 1986, etc.\], there are two alphabets of segments, 
underlying and surface. There are constraint-rules 
about which underlying segments may b'e realized as 
which surface segments, and vice versa, based on con- 
text. The rules' contexts are strings of pairs of seg- 
ments, each underlying segment paired with a sur- 
face segment. Deletions and insertions are handled 
by pairing a segment with a null segment. What is 
crucial about the rules is that each element of a con- 
text is actually a pair of segments, an underlying and 
a surface segment. The ability to refer to both sur- 
face and underlying contexts in a rule allows the rule 
writer to describe phenomena that are handled with 
ordered rules in the unilevel approach. 
The other powerful device in two-level phonology is 
an explicit listing of the two alphabets and the feasible 
mappings between them. These mappings are simply 
pairs of segments, one surface segment paired with 
one underlying segment. This list of feasible pairs 
typically contains many pairs of identical segments 
such as (a,a) or (b,b), representing that there are seg- 
nmnts that are the same underlyingly as on the sur- 
face. The list also contains pairs representing change. 
For the Tunica example, (a,¢) and (ao) would be in 
the list, but (a,u) and (i,u) for example would not be. 
The feasible pairs can be thought of as machinery for 
generating strings of pairs of segments that the rules 
either accept or reject. An accepted string of segment 
pairs constitutes a mapping from an underlying form 
to a surface form and from surface to underlying form. 
8 Rule Ordering 
In a paper presented at the 1986 annual meeting of 
the Linguistic Society of America, Lauri Karttunen 
proposed this solution for the Klamath data above: 1° 
I i':= } 
,-,il_ l..= (11) i°:_ 
h / =:t_ (12) 
I'--, ? /=:l_ (13) 
The contexts of tile rules should be read as follows. 
Each pair separated by a colon is a lexical segment 
followed by a surface segment. The equals sign is 
a place holder used when the rule writer does not 
want to make any commitment about what some seg- 
ment must be. So, for instance, 1':= is an underlying 
/1'/paired with some surface segment, and the rule 
doesn't care which. Similarly, =:1 is a way of stil~u- 
lating that there is a surface \[I\] in the context, and 
we don't care, for the purposes of this rule, which 
underlying segment it corresponds to. The right ar- 
row, ---~, is being used in the way described in Bear 
\[1986, 1988 a,b\]. For example, Rule (11) should be 
construed as allowing the pair of segments n:! (un- 
derlying n corresponding to surface l) to occur in the 
rule's environment, while disallowing the pair n:n. Al- 
though the right arrow rule is reminiscent of' the arrow 
in unilevei rules, this interpretation is nondirectional. 
There are two other kinds of constraints to allow one 
to deal effectively with the asymmetries involved in 
pairing underlying forms with surface forms. In Bear 
\[1986, 1988\] the two other kinds of constraints are 
(1) to allow a pair of segments to occur in a certain 
context without disallowing the default pair (e.g. n:n 
in the previous example is a default pair), and (2) to 
disallow a pair in some context without allowing some 
other pair. For example, the rule types in (14) and 
(15) are allowed. 
a:b allowed here: a _ fl (14) 
a:b disallowed here: a _ fl (15) 
In Koskenniemi \[1983, 1984\] tile constraints are 
slightly different, but have roughly the same func- 
tionality. I!1 Koskenniemi's system, one may stipu- 
late that if a lexical segment occurs in some context, 
then it must correspond to some particular surface 
segment. One may also stipulate that a certain lexi- 
cal/surface segment pair may only occur in a certain 
environment. 
1°I'm using an amalgamation of notations from Koskenniemi, 
Karttunen and Wittenburg, and Bear. 
18 6 
Karttunen \[1986\] pointed out that the three rules in 
(ll), (12), and (13) work correctly to give the right re- 
suits when generating surface forms from underlying 
forms, and made the point that they do so without re- 
course to the device of rule ordering. Another point he 
could have made about these rules which I will make 
here is that they are just as effective in producing 
the right underlying forms from surface forms. There 
is not the problem of multiple intermediate levels of 
representation, where one is faced with the choice of 
whether to continue applying \[reversed\] rules or to 
stop and call the form a result. 
9 Combining Assimilation 
With Deletion 
One solution for the Tunica data is given below) 1 
Vowel 
-stress \] 
?_ 
a --, ~1 i:= .~ _ 
--~ V/7 where Vowel e { 
(16) 
(17) 
(18) 
Kules (16) and (17) say that/a/assimilates to the 
underlying vowel preceding it, with a glottal stop in- 
tervening. One other crucial element of the two-level 
way of doing things is that in addition to rules, a 
grammar contains a list of feasible segment pairs. For 
this Tunica case, there presumably would not be a 
feasible pair/e/:\[e\], nor would there be /~/:\[~\] since 
\[el and \[3\] do not seem to occur as underlying vowels. 
Itence the surface Is\] in our example word \[ndg?ekz\] 
would be forced unambiguously to correspond to an 
underlying/a/. This is exactly what we want. 
Rule (18) specifies that unstressed vowels are 
deleted when they occur before a glottal stop. The 
rule makes clear that only the four vowels i, a, o, and 
u are deleted, and also that when doing recognition, 
only those vowels are allowed to be inserted. 
These rules make it clear that the underlying form 
for \[ndg?ekt\] must be/ndgi?dki/modulo details of the 
rule of Right Destressing. 
10 Analysis by Synthesis 
There is one system for doing computational morphol- 
ogy, specifically for recognizing Turkish, which uses 
11 It is a common abbreviatory convention that any pair of 
idendical segments, e.g., a:a, can be written simply as a single 
segment, e.g., a. So, in these rules the glottal stop character 
represents the pair: ?:?. 
unilevel rules \[Hankamer, 1986\]. The system first in- 
vokes an ad hoc procedure to find the first heavy syl- 
lable of a Turkish word. This substring and perhaps 
a few carefully constructed variants of it are consid- ' 
ered as possible stems for the word. Next, based on 
the morphotactic information about the stem found 
in the lexicon, assuming one of the possible stems is 
in the lexicon, several possible suffixes are proposed 
as possible. A set of phonological rules is applied 
to the hypothesized underlying forms consisting of 
stem+suffix. Whichever of them results in a string 
that matches the input surface form is considered to 
be right. The process is repeated until the entire 
string is analyzed. 
Since "l~lrkish is exclusively suffixing and has strong 
phonotactic constraints on what can be a stem, it is 
possible to write an ad hoc routine to pick the stem 
out. It remains to be seen how this method of anal- 
ysis can be made general enough to be applied suc- 
cessfully to other languages. While Hankamer's paper 
is interesting in its own right, it would be a mistake 
to construe it ms demonstrating anything very general 
about reversibility of unilevel rule systems. 
11 Conclusion 
The question has been asked, "What is so good about 
Koskenniemi's two-level phonology?" The answer is 
that it allows one to write reversible, nonprocedural 
descriptions of phonological phenomena with much 
more accuracy than does the conventional unilevel 
formalism. The point I have stressed here is the re- 
versibility. From a computational point of view, this 
represents a step forward. There are no published 
accounts of reversible grammars written in a unilevel 
formalism so far as I know and there are many written 
in two-level rules. Koskenniemi's proposal was made 
with computation ill mind as opposed to linguistic 
theory. It may, in the long run, have an impact on 
linguistic theory. It definitely has had a large impact 
on computational morphology. 
Acknowledgements 
The bulk of this work was done while I was a visit- 
ing scientist at the IBM LILOG project in Stuttgart, 
Federal Republic of Germany, in the summer of 1988. 
This work was also made possible by a gift from the 
System Development Foundation as part of a coordi- 
nated research effort with tile Center for the Study 
of Language and Information, Stanford University. I 
would like to thank the people at IBM, Stuttgart, SRI, 
and CSLI for supporting this work. I would also like 
7 19 
to thank the following people for many helpful discus- 
sions and comments: Meg Withgott, Martin Emele, 
Mary Dalrymple, Petra Steffens, Bob Mugele, and 
IIans Uszkoreit. 
I would not have been able to produce this paper 
had it not been for Emma Pease who has done con- 
siderable work defining phonetic fonts and graphics 
macros for "l~X which she made available. I would 
also like to thank Mary Dalrymple for helping me with 
IbTEX. 

References 

\[1\] Barker, M.A.R. (1964) Klamath Grammar, Uni- 
versity of California Press, Berkeley and Los Ange- 
les, Calilbrnia. 

\[2\] Bear, John (1985) "Interpreting Two-Level Rules 
Directly," presented at a Stanford workshop on 
finite-state morphology. 

\[3\] Bear, John (1986) "A Morphological Recognizer 
with Syntactic and Phonological Rules," COLING 
86, pp. 272-276. 

\[4\] Bear, John (1988) "Two-Level Rules and Negative 
Rule Features," COLING 88, pp. 28-31. 

\[5\] Eliasson, Stig (1985) "Turkish k-Deletion: Sim- 
plicity vs. Retrieval," in Folia Linguistiea )(IX, 3-4, 
pp. 289-311, Mouton Publishers, The IIague. 

\[6\] Gazdar, Gerald (1985) "Finite State Morphology: 
A Review of Koskenniemi (1983)," Technical Re- 
port No. CSLI-85-32 of the Center for the Study 
of Language and Information, Stanford University, 
Stanford, California. 

\[7\] Ilaas, Mary (1940) Tunica. Handbook of Ameri- 
can Indian Languages, Vol. 4. Smithsonian Institu- 
tion, Bureau of American Ethnography, Washing- 
ton, D.C. 

\[8\] lialle, Morris, and G.N. Clements (1983) Problem 
Book in Phonology: A Workbook for Introductory 
Courses in Linguistics and in Modern Phonology, 
The MIT Press, Cambridge, Massachusetts, and 
London, England. 

\[9\] IIankamer, Jorge (1986) "Finite State Morphol- 
ogy and Left-to-Right Phonology," in Proceedings 
of the West Coast Conference on Formal Linguis- 
tics, published by Stanford Linguistics Association, 
Stanford, California. 

\[10\] Kaplan, Ronaid, and Martin Kay (1981) Paper 
presented at the annual meeting of the Linguistic 
Society of America. 

\[11\] Karttunen, Lauri (1983) "Kimmo: A General 
Morphological Processor," in Texas Linguist.ic Fo- 
rum #22, Dairymple et al., eds., Linguistics De- 
partment, University of Texas, Austin, Texas. 

\[12\] Karttunen, Lauri (1986) "Compilation of Two- 
Level Phonological Rules," presented at the Annual 
Meeting of the Linguistic Society of America in San 
Francisco, California. 

\[13\] Karttunen, Lauri, Kimmo Koskenniemi and 
Ronald Kaplan (1987) "TWOL: A Compiler for 
Two-Level Phonological Rules," distributed at the 
1987 Summer Linguistic Institute at Stanford Uni- 
versity, Stanford, California. 

\[14\] Karttunen, Lauri and Kent Wittenburg (1983) 
"A Two-Level Morphological Analysis Of English," 
in Texas Linguistic Forum #22, Dalrymple et al., 
eds., Linguistics Department, University of Texas, 
Austin, Texas. 

\[15\] Kay, Martin (1983) "When Meta-rules are not 
Meta-rules," in K. Sparck-Jones, and Y. Wilks, eds. 
Automatic Natural Language Processing, John Wi- 
ley and Sons, New York, New York. 

\[16\] Kay, Martin (1987) "Nonconcatenative Finite- 
State Morphology," paper presented at a workshop 
on Arabic Morphology, Stanford University, Stan- 
ford, California. 

\[17\] Kennstowicz, Michael, and Charles Kisseherth 
(1979) Generative Phonology, Academic Press, Inc., 
IIarcourt, Brace, Jovanovich, Publishers, Orlando, 
San Diego, New York, Austin, Boston, London, 
Sydney, Tokyo, Toronto. 

\[18\] Koskenniemi, Kimmo (1983) Two-Level Mor- 
phology: A General Computational Model for 
Word-form Recognition and Production. Publica- 
tion No. 11 of the University of IIelsinki Depart- 
ment of General Linguistics, tIelsinki, Finland. 

\[19\] Koskenniemi, Kimmo (1983) "Two-Level Model 
for Morphological Analysis," IJCAI 83, pp. 683-685. 

\[20\] Koskenniemi, Kimmo (1984) "A General Com- 
putational Model for Word-form Recognition and 
Production," COLING 84, pp. 178-181. 

\[21\] Mohanan, K.P. (1987) A Theory of Lexical 
Phonology, D. Reidel Publishing Company, Dor- 
drecht, Itolland. 

\[22\] Schane, Sanford (1973) Generative Phonology, 
Prentice Hall, Englewood Cliffs, New Jersey. 
