Computational structure of generative phonology 
and its relation to language comprehension. 
Eric Sven Ristad* 
MIT Artificial Intelligence Lab 
545 Technology Square 
Cambridge, MA 02139 
Abstract 
We analyse the computational complexity of 
phonological models as they have developed over 
the past twenty years. The major results ate 
that generation and recognition are undecidable 
for segmental models, and that recognition is NP- 
hard for that portion of segmental phonology sub- 
sumed by modern autosegmental models. Formal 
restrictions are evaluated. 
1 Introduction 
Generative linguistic theory and human language 
comprehension may both be thought of as com- 
putations. The goal of language comprehension 
is to construct structural descriptions of linguistic 
sensations, while the goal of generative theory is 
to enumerate all and only the possible (grammat- 
ical) structural descriptions. These computations 
are only indirectly related. For one, the input to 
the two computations is not the same. As we shall 
see below, the most we might say is that generative 
theory provides an extensional chatacterlsation of 
language comprehension, which is a function from 
surface forms to complete representations, includ- 
ing underlying forms. The goal of this article is 
to reveal exactly what generative linguistic theory 
says about language comprehension in the domain 
of phonology. 
The article is organized as follows. In the next 
section, we provide a brief overview of the com- 
putational structure of generative phonology. In 
section 3, we introduce the segmental model of 
phonology, discuss its computational complexity, 
and prove that even restricted segmental mod- 
els are extremely powerful (undecidable). Subse- 
quently, we consider various proposed and plausi- 
ble restrictions on the model, and conclude that 
even the maximally restricted segmental model is 
likely to be intractable. The fourth section in-. 
troduces the modern autosegmental (nonlinear) 
model and discusses its computational complexity. 
"The author is supported by a IBM graduate 
fellowship and eternally indebted to Morris Halle 
and Michael Kenstowicz for teaching him phonol- 
ogy. Thanks to Noam Chomsky, Sandiway Fong, and 
Michael Kashket for their comments and assistance. 
235 
We prove that the natural problem of construct- 
ing an autosegmental representation of an under- 
specified surface form is NP-hard. The article 
concludes by arguing that the complexity proofs 
are unnatural despite being true of the phonolog- 
ical models, because the formalism of generative 
phonology is itself unnatural. 
The central contributions of this article ate: 
(i) to explicate the relation between generative 
theory and language processing, and argue that 
generative theories are not models of language 
users primarily because they do not consider the 
inputs naturally available to language users; and 
(ii) to analyze the computational complexity of 
generative phonological theory, as it has developed 
over the past twenty years, including segmental 
and autosegmental models. 
2 Computational structure 
of generative phonology 
The structure of a computation may be described 
at many levels of abstraction, principally includ- 
ing: (i) the goal of the computation; (ii) its in- 
put/output specification (the problem statement), 
(iii) the algorithm and representation for achiev- 
ing that specification, and (iv) the primitive opera- 
tions in which terms the algorithm is implemented 
(the machine architecture). 
Using this framework, the computational struc- 
ture of generative phonology may be described as 
follows: 
• The computational goal of generative phonol- 
ogy (as distinct from it's research goals) is to 
enumerate the phonological dictionaries of all 
and only the possible human languages. 
• The problem statement is to enumerate the 
observed phonological dictionary of s particu- 
lax language from some underlying dictionary 
of morphemes (roots and affixes) and phono- 
logical processes that apply to combinations 
of underlying morphemes. 
• The algorithm by which this is accomplished 
is a derivational process g ('the grammar') 
from underlying forms z to surface forms 
y = g(z). Underlying forms are constructed 
by combining (typically, with concatenation 
or substitution) the forms stored in the under- 
lying dictionary of morphemes. Linguistic re- 
lations are represented both in the structural 
descriptions and the derivational process. 
The structural descriptions of phonology are 
representations of perceivable distinctions be- 
tween linguistic sounds, such as stress lev- 
els, syllable structure, tone, and articula- 
tory gestures. The underlying and surface 
forms are both drawn from the same class 
of structural descriptions, which consist of 
both segmental strings and autosegmental re- 
lations. A segmental string is a string of 
segments with some representation of con- 
stituent structur. In the SPE theory of Chom- 
sky and Halle (1968) concrete boundary sym- 
bols are used; in Lexical Phonology, abstract 
brackets are used. Each segment is a set of 
phonological features, which are abstract as 
compared with phonetic representations, al- 
though both are given in terms of phonetic 
features. Suprasegmental relations are rela- 
tions among segments, rather than properties 
of individual segments. For example, a syl- 
lable is a hierarchical relation between a se- 
quence of segments (the nucleus of the syl- 
lable) and the less sonorous segments that 
immediately preceed and follow it (the onset 
and coda, respectively). Syllables must sat- 
isfy certain universal constraints, such as the 
sonority sequencing constraint, as well as lan- 
guage particular ones. 
a The derivntional process is implemented by 
an ordered sequence of unrestricted rewriting 
rules that are applied to the current deriva- 
tion string to obtain surface forms. 
According to generative phonology, comprehen- 
sion consists of finding a structural description for 
a given surface form. In effect, the logical prob- 
lem of language comprehension is reduced to the 
problem of searching for the underlying form that 
generates a given surface form. When the sur- 
face form does not transparently identify its cor- 
responding underlying form, when the space of 
possible underlying forms is large, or when the 
grammar g is computationally complex, the logical 
problem of language comprehension can quickly 
become very difficult. 
In fact, the language comprehension problem 
is intractable for all segmental theories. For ex- 
ample, in the formal system of The Sound Pat. 
tern of English (SPE) the comprehension prob- 
lem is undecidable. Even if we replace the seg- 
mental representation of cyclic boundaries with 
the abstract constituents of Lexical Phonology, 
and prohibit derivational rules from readjusting 
constituent boundaries, comprehension remains 
PSPACE-complete. Let us now turn to the tech- 
nical details. 
3 Segmental Phonology 
The essential components of the segmental model 
may be briefly described as follows. The set of 
features includes both phonological features and 
diacritics and the distinguished feature segment 
that marks boundaries. (An example diacritic is 
ablaut, a feature that marks stems that must 
undergo a change vowel quality, such as tense- 
conditioned ablaut in the English sing, sang, sung 
alternation.) As noted in SPE, "technically speak- 
ing, the number of diacritic features should be at 
least as large as the number of rules in the phonol- 
ogy. Hence, unless there is a bound on the length 
of a phonology, the set \[of features\] should be un- 
limited." (fn.1, p.390) Features may be specified 
q- or - or by an integral value 1, 2,..., N where N 
is the maximal deg/ee of differentiation permitted 
for any linguistic feature. Note that N may vary 
from language to language, because languages ad- 
mit different degrees of differentiation in such fea- 
tures as vowel height, stress, and tone. A set of 
feature specifications is called a unit or sometimes 
a segment. A string of units is called a matriz or 
a segmental string. 
A elementary rule is of the form ZXAYW 
ZXBYW where A and B may be ~b or any unit, 
A ~ B; X and Y may be matrices (strings of 
units), and Z and W may be thought of a brack- 
ets labelled with syntactic categories such as 'S' 
or 'N' and so forth. A comple= rule is a finite 
schema for generating a (potentially infinite) set 
of elementary rules. 1 The rules are organised into 
1Following 3ohnson (1972), we may define schenm 
as follows. The empty string and each unit is s schema; 
schema may be combined by the operations of union, 
intersection, negation, kleene star, and exponentiation 
over the set of units. Johnson also introduces variables 
and Boolean conditions into the schema. This "schema 
language" is a extremely powerful characterisation of 
the class of regular languages over the alphabet of 
units; it is not used by practicing phonologists. Be- 
cause a given complex rule can represent an infinite set 
of elementary rules, Johnson shows how the iterated, 
exhaustive application of one complex rule to a given 
segmental string can "effect virtually any computable 
mapping," (p.10) ie., can simulate any TNI computa- 
tion. Next, he proposes a more restricted "simultane- 
ous" mode of application for a complex rule, which is 
only capable of performing a finite-state mapping in 
any application. This article considers the indepen- 
dent question of what computations can be performed 
by a set of elementary rules, and hence provides loose 
lower bounds for Johnson's model. We note in pass- 
ing, however, that the problem of simply determining 
whether a given rule is subsumed by one of Johnson's 
schema is itself intractable, requiring at least exponen- 
236 
lineat sequence R,,R2, ...Rn, and they ate ap- 
plied in order to an underlying matrix to obtain a 
surface matrix. 
Ignoring a great many issues that are important 
for linguistic reasons but izrelevant for our pur- 
poses, we may think of the derivational process as 
follows. The input to the derivation, or "underly- 
ing form," is a bracketed string of morphemes, the 
output of the syntax. The output of the derivation 
is the "surface form," a string of phonetic units. 
The derivation consists of a series of cycles. On 
each cycle, the ordered sequence of rules is ap- 
plied to every maximal string of units containing 
no internal brackets, where each P~+, applies (or 
doesn't apply) to the result of applying the imme- 
diately preceding rule Ri, and so forth. Each rule 
applies simultaneously to all units in the current 
derivations\] string. For example, if we apply the 
rule A --* B to the string AA, the result is the 
string BB. At the end of the cycle, the last rule 
P~ erases the innermost brackets, and then the 
next cycle begins with the rule R1. The deriva- 
tion terminates when all the brackets ate erased. 
Some phonological processes, such as the as- 
similation of voicing across morpheme boundaries, 
are very common across the world's languages. 
Other processes, such as the atbitraty insertion 
of consonants or the substitution of one unit for 
another entirely distinct unit, ate extremely rate 
or entirely unattested. For this reason, all ade- 
quate phonological theories must include an ex- 
plicit measure of the naturalness of a phonologi- 
cal process. A phonological theory must also de- 
fine a criterion to decide what constitutes two in- 
dependent phonological processes and what con- 
stitutes a legitimate phonological generalization. 
Two central hypotheses of segmental phonology 
are (i) that the most natural grammaxs contain 
the fewest symbols and (ii) a set of rules rep- 
resent independent phonological processes when 
they cannot be combined into a single rule schema 
according to the intricate notational system first 
described in SPE. (Chapter 9 of Kenstowicz and 
Kisseberth (1979) contains a less technical sum- 
maty of the SPE system and a discussion of sub- 
sequent modifications and emendations to it.) 
3.1 Complexity of segmental 
recognition and generation. 
Let us say a dictionary D is a finite set of the 
underlying phonological forms (matrices) of mor- 
phemes. These morphemes may be combined by 
concatenation and simple substitution (a syntactic 
category is replaced by a morpheme of that cate- 
gory) to form a possibly infinite set of underlying 
forms. Then we may characterize the two central 
computations of phonology as follows. 
tial space. 
The phonological generation problem (PGP) is: 
Given a completely specified phonological matrix 
z and a segmental grammar g, compute the sur- 
face form y : g(z) of z. 
The phonological recognition problem (PRP) is: 
Given a (partially specified) surface form y, a dic- 
tionary D of underlying forms, and a segmental 
grammar g, decide if the surface form y = g(=) 
can be derived from some underlying form z ac- 
cording to the grammar g, where z constructed 
from the forms in D. 
Lenuna 3.1 The segmental model can directly 
simulate the computation of any deterministic~ 
Turing machine M on any input w, using only 
elementary rules. 
Proof. We sketch the simulation. The underlying 
form z will represent the TM input w, while the 
surface form y will represent the halted state of M 
on w. The immediate description of the machine 
(tape contents, head position, state symbol) is rep- 
resented in the string of units. Each unit repre- 
sents the contents of a tape square. The unit rep- 
resenting the currently scanned tape square will 
also be specified for two additional features, to 
represent the state symbol of the machine and the 
direction in which the head will move. Therefore, 
three features ate needed, with a number of spec- 
ifications determined by the finite control of the 
machine M. Each transition of M is simulated by 
a phonological rule. A few rules ate also needed to 
move the head position around, and to erase the 
entire derivation string when the simulated m~ 
chine halts. 
There are only two key observations, which do 
not appear to have been noticed before. The first 
is that contraty to populat misstatement, phono- 
logical rules ate not context-sensitive. Rather, 
they ate unrestricted rewriting rules because they 
can perform deletions as well as insertions. (This 
is essential to the reduction, because it allows 
the derivation string to become atbitatily long.) 
The second observation is that segmental rules 
can f~eely manipulate (insert and delete) bound- 
ary symbols, and thus it is possible to prolong the 
derivation indefinitely: we need only employ a rule 
R,~_, at the end of the cycle that adds an extra 
boundary symbol to each end of the derivation 
string, unless the simulated machine has halted. 
The remaining details are omitted, but may be 
found in Ristad (1990). \[\] 
The immediate consequences are: 
Theorem I PGP is undecidable. 
Proof. By reduction to the undecidable prob- 
lem w 6 L(M)? of deciding whether a given TM 
M accepts an input w. The input to the gen- 
eration problem consists of an underlying form 
z that represents w and a segmental grammar 
237 
g that simulates the computations of M accord- 
ing to \]emma 3.1. The output is a surface form 
y : g(z) that represents the halted configuration 
of the TM, with all but the accepting unit erased. \[\] 
Theorem 2 PRP is undecidable. 
Proof. By reduction to the undecidable prob- 
lem L(M) =?~b of deciding whether a given TM 
M accepts any inputs. The input to the recog- 
nition problem consists of a surface form y that 
represents the halted accepting state of the TM, 
a trivial dictionary capable of generating E*, and 
a segmental grammar g that simulates the com- 
putations of the TM according to lemma 3.1. The 
output is an underlying form z that represents the 
input that M accepts. The only trick is to con- 
struct a (trivial) dictionary capable of generating 
all possible underlying forms E*. \[\] 
An important corollary to lemma 3.1 is that we 
can encode a universal Turing machine in a seg- 
mental grammax. If we use the four-symbol seven- 
state "smallest UTM" of Minsky (1969), then the 
resulting segmental model contains no more than 
three features, eight specifications, and 36 very 
simple rules (exact details in Ristad, 1990). As 
mentioned above, a central component of the seg- 
mental theory is an evaluation metric that favors 
simpler (ie., shorter) grammars. This segmental 
grammar of universal computation appears to con- 
tain significantly fewer symbols than a segmental 
grammar for any natural language. Therefore, this 
corollary presents severe conceptual and empirical 
problems for the segmental theory. 
Let us now turn to consider the range of plau- 
sible restrictions on the segmental model. At 
first glance, it may seem that the single most 
important computational restriction is to prevent 
rules from inserting boundaries. Rules that ma- 
nipulate boundaries axe called readjustment rules. 
They axe needed for two reasons. The first is to 
reduce the number of cycles in a given deriva- 
tion by deleting boundaries and flattening syntac- 
tic structure, for example to prevent the phonol- 
ogy from assigning too many degrees of stress 
to a highly-embedded sentence. The second is 
to reaxrange the boundaries given by the syn- 
tax when the intonational phrasing of an utter- 
ance does not correspond to its syntactic phras- 
ing (so-called "bracketing paradoxes"). In this 
case, boundaries are merely moved around, while 
preserving the total number of boundaries in the 
string. The only way to accomplish this kind of 
bracket readjustment in the segmental model is 
with rules that delete brackets and rules that in- 
sert brackets. Therefore, if we wish to exclude 
rules that insert boundaries, we must provide an 
alternate mechanism for boundary readjustment. 
For the sake of axgument--and because it is not 
too hard to construct such a boundary readjust- 
ment mechanism--let us henceforth adopt this re- 
striction. Now how powerful is the segmental 
model? 
Although the generation problem is now cer- 
taiuly decidable, the recognition problem remains 
undecidable, because the dictionary and syntax 
are both potentially infinite sources of bound- 
aries: the underlying form z needed to generate 
any given surface form according to the grammar 
g could be axbitradly long and contain an axbi- 
traxy number of boundaries. Therefore, the com- 
plexity of the recognition problem is unaffected 
by the proposed restriction on boundary readjust- 
ments. The obvious restriction then is to addi- 
tionally limit the depth of embeddings by some 
fixed constant. (Chomsky and Halle flirt with 
this restriction for the linguistic reasons mentioned 
above, but view it as a performance limitation, 
and hence choose not to adopt it in their theory 
of linguistic competence.) 
Lernma 3.2 Each derivational cycle can directly 
simulate any polynomial time alternating Turing 
machine (ATM) M computation. 
Proof. By reduction from a polynomial-depth 
ATM computation. The input to the reduction is 
an ATM M on input w. The output is a segmen- 
tad grammar g and underlying form z s.t. the sur- 
face form y = g(z) represents a halted accepting 
computation iff M accepts ~v in polynomial time. 
The major change from lemma 3.1 is to encode 
the entire instantaneous description of the ATM 
state (ie., tape contents, machine state, head po- 
sition) in the features of a single unit. To do this 
requires a polynomial number of features, one for 
each possible tape squaxe, plus one feature for the 
machine state and another for the head position. 
Now each derivation string represents a level of 
the ATM computation tree. The transitions of the 
ATM computation axe encoded in a block B as fol- 
lows. An AND-transition is simulated by a triple 
of rules, one to insert a copy of the current state, 
and two to implement the two transitions. An OR- 
transition is simulated by a pair of disjunctively- 
ordered rules, one for each of the possible succes- 
sor states. The complete rule sequence consists 
of a polynomial number of copies of the block B. 
The last rules in the cycle delete halting states, 
so that the surface form is the empty string (or 
reasonably-sized string of 'accepting' units) when 
the ATM computation halts and accepts. If, on 
the other hand, the surface form contains any non- 
halting or nonaccepting units, then the ATM does 
not accept its input w in polynomial time. The 
reduction may clearly be performed in time poly- 
nomial in the size of the ATM and its input. \[\] 
Because we have restricted the number of em- 
beddings in an underlying form to be no more than 
238 
a fixed language-universal constant, no derivation 
can consist of more than a constant number of 
cycles. Therefore, lemma 3.2 establishes the fol- 
lowing theorems: 
Theorem 3 PGP with bounded embeddings is 
PSPA CE.hard. 
Proof. The proof is an immediate consequence of 
lemma 3.2 and a corollary to the Chandra-Kosen- 
Stockmeyer theorem (1981) that equates polyno- 
mial time ATM computations and PSPACE DTM 
computations. \[\] 
Theozem 4 PRP with bounded embeddings is 
PSPA CE-hard. 
Proof. The proof follows from lemma 3.2 and 
the Chandra-Kosen-Stockmeyer result. The dic- 
tionary consists of the lone unit that encodes the 
ATM starting configuration (ie., input w, start 
state, head on leftmost square). The surface string 
is either the empty string or a unit that represents 
the halted accepting ATM configuration. \[\] 
There is some evidence that this is the most 
we can do, at least for the PGP. The requirement 
that the reduction be polynomial time limits us 
to specifying a polynomial number of features and 
a polynomial number of rules. Since each feature 
corresponds to a tape square, ie., the ATM space 
resource, we are limited to PSPACE ATM compu- 
tations. Since each phonological rule corresponds 
to a next-move relation, ie., one time step of the 
ATM, we are thereby limited to specifying PTIME 
ATM computations. 
For the PRP, the dictionary (or syntax- 
interface) provides the additional ability to 
nondeterministically guess an arbitrarily long, 
boundary-free underlying form z with which to 
generate a given surface form g(z). This ability 
remains unused in the preceeding proof, and it is 
not too hard to see how it might lead to undecid- 
ability. 
We conclude this section by summarizing the 
range of linguistically plausible formal restrictions 
on the derivational process: 
Feature system. As Chomsky and Halle noted, 
the SPE formal system is most naturally seen 
as having a variable (unbounded) set of fea- 
tures and specifications. This is because lan- 
guages differ in the diacritics they employ, as 
well as differing in the degrees of vowel height, 
tone, and stress they allow. Therefore, the set 
of features must be allowed to vary from lan- 
guage to language, and in principle is limited 
only by the number of rules in the phonol- 
ogy; the set of specifications must likewise be 
allowed to vary from language to language. 
It is possible, however, to postulate the ex- 
istence of a large, fixed, language-universal 
set of phonological features and a fixed upper 
limit to the number N of perceivable distinc- 
tions any one feature is capable of supporting. 
If we take these upper limits seriously, then 
the class of reductions described in lemma 3.2 
would no longer be allowed. (It will be pos- 
sible to simulate any ~ computation in a 
single cycle, however.) 
Rule for m__At. Rules that delete, change, ex- 
change, or insert segments--as well as rules 
that manipulate boundaries--are crucial to 
phonological theorizing, and therefore cannot 
be crudely constrained. More subtle and in- 
direct restrictions are needed. One approach 
is to formulate language-universal constraints 
on phonological representations, and to allow 
a segment to be altered only when it violates 
some constraint. 
McCarthy (1981:405) proposes a morpheme 
rule constraint (MRC) that requires all mor- 
phological rules to be of the form A ---, B/X 
where A is a unit or ~b, and B and X are 
(possibly null) strings of units. (X is the im- 
mediate context of A, to the right or left.) 
It should be obvious that the MRC does 
not constrain the computational complexity 
of segmental phonology. 
4 Autosegmental Phonology 
In the past decade, generative phonology has 
seen a revolution in the linguistic treatment of 
suprasegmental phenomena such as tone, har- 
mony, infixation, and stress assignment. Although 
these autosegmental models have yet to be for- 
malised, they may be briefly described as follows. 
Rather than one-dimensional strings of segments, 
representations may be thought of as "a three- 
dimensional object that for concreteness one might 
picture as a spiral-bound notebook," whose spine 
is the segmental string and whose pages contain 
simple constituent structures that are indendent 
of the spine (Halle 1985). One page represents the 
sequence of tones associated with a given articu- 
lation. By decoupling the representation of tonal 
sequences from the articulation sequence, it is pos- 
sible for segmental sequences of different lengths 
to nonetheless be associated to the same tone se- 
quence. For example, the tonal sequence Low- 
High-High, which is used by English speakers to 
express surprise when answering a question, might 
be associated to a word containing any number 
of syllables, from two (Brazi 0 to twelve (floccin- 
auccinihilipilification) and beyond. Other pages 
(called "planes") represent morphemes, syllable 
structure, vowels and consonants, and the tree of 
articulatory (ie., phonetic) features. 
239 
4.1 Complexity of autosegmental 
recognition. 
In this section, we prove that the PRP for au- 
tosegmental models is NP-hard, a significant re- 
duction in complexity from the undecidable and 
PSPACE-hard computations of segmental theo- 
ries. (Note however that autosegmental repre- 
sentations have augmented--but not replaced-- 
portions of the segmental model, and therefore, 
unless something can be done to simplify segmen- 
tal derivations, modern phonology inherits the in- 
tractability of purely segmental approaches.) 
Let us begin by thinking of the NP-complete 
3-Satisfiability problem (3SAT) as a set of inter- 
acting constraints. In particular, every satisfiable 
Boolean formula in 3-CNF is a string of clauses 
C1, C2,..., Cp in the variables zl, z=,..., z, that 
satisfies the following three constraints: (i) nega- 
tion: a variable =j and its negation ~ have op- 
posite truth values; (ii) clausal satisfaction: every 
clause C~ = (a~VbiVc/) contains a true literal (a lit- 
eral is a variable or its negation); (iii) consistency 
of truth assignments: every unnegated literal of 
a given variable is assigned the same truth value, 
either 1 or 0. 
Lemma 4.1 Autosegmental representations can 
enforce the 3SAT constraints. 
ProoL The idea of the proof is to encode negation 
and the truth values of variables in features; to 
enforce clausal satisfication with a local autoseg- 
mental process, such as syllable structure; and 
to ensure consistency of truth assignments with 
a nonlocal autosegmental process, such as a non- 
concatenative morphology or long-distance assim- 
ilation (harmony). To implement these ideas we 
must examine morphology, harmony, and syllable 
structure. 
Morphology. In the more familiar languages 
of the world, such as Romance languages, mor- 
phemes are concatenated to form words. In other 
languages, such as Semitic languages, a morpheme 
may appear more that once inside another mor- 
pheme (this is called infixation). For example, the 
Arabic word katab, meaning 'he wrote', is formed 
from the active perfective morpheme a doubly in- 
fixed to the ktb morpheme. In the autosegmental 
model, each morpheme is assigned its own plane. 
We can use this system of representation to ensure 
consistency of truth assigments. Each Boolean 
variable z~ is represented by a separate morpheme 
p~, and every literal of =i in the string of formula 
literals is associated to the one underlying mor- 
pheme p~. 
Harmony. Assimilation is the common phono- 
logical process whereby some segment comes to 
share properties of an adjacent segment. In En- 
glish, consonant nasality assimilates to immedi- 
ately preceding vowels; assimilation also occurs 
240 
across morpheme boundaries, as the varied surface 
forms of the prefx in- demonstrate: in+logical -, 
illogical and in-l-probable --, improbable. In other 
languages, assimilation is unbounded and can af- 
fect nonadjacent segments: these assimilation pro- 
cesses are called harmony systems. In the Turkic 
languages all sutFtx vowels assimilate the backnesss 
feature of the last stem vowel; in Capanshua, vow- 
els and glides that precede a word-final deleted 
nasal (an underlying nasal segment absent from 
the surface form) are all nasalized. In the autoseg- 
mental model, each harmonic feature is assigned 
its own plane. As with morpheme-infixation, we 
can represent each Boolean variable by a harmonic 
feature, and thereby ensure consistency of truth 
assignments. 
Syllable structure. Words are partitioned into 
syllables. Each syllable contains one or more vow- 
ds V (its nucleus) that may be preceded or fol- 
lowed by consonants C. For example, the Ara- 
bic word ka.tab consists of two syIlabhs, the two- 
segment syllable CV and the three-segment dosed 
syllable CVC. Every segment is assigned a sonor- 
ity value, hrhich (intuitively) is proportional to the 
openness of the vocal cavity. For example, vowels 
are the most sonorous segments, while stops such 
as p or b are the least sonorous. Syllables obey a 
language-universal sonority sequencing constraint 
(SSC), which states that the nucleus is the sonor- 
ity peak of a syllable, and that the sonority of 
adjacent segments swiftly and monotonically de- 
creases. We can use the SSC to ensure that every 
clause C~ contains a true literal as follows. The 
centred idea is to make literal truth correspond to 
the stricture feature, so that a true literal (repre- 
sented as a vowel) is more sonorous than a false 
literal (represented as a consonant). Each clause 
C~ - (a~ V b~ V c~) is encoded as a segmental string 
C - z, - zb - zc, where C is a consonant of sonor- 
ity 1. Segment zG has sonority 10 when literal 
at is true, 2 otherwise; segment =s has sonority 9 
when literal bi is true, 5 otherwise; and segment zc 
has sonority 8 when literal q is true, 2 otherwise. 
Of the eight possible truth values of the three lit- 
erals and ~he corresponding syllabifications, 0nly 
the syllabification corresponding to three false lit- 
erals is excluded by the SSC. In that case, the 
corresponding string of four consonants C-C-C-C 
has the sonority sequence 1-2-5-2. No immediately 
preceeding or following segment of any sonority 
can result in a syllabification that obeys the SSC. 
Therefore, all Boolean clauses must contain a true 
literal. (Complete proof in Ristad, 1990) \[\] 
The direct consequence of this lemma 4.1 is: 
Theorem 5 PRP for the autosegraental model is 
NP-hard. 
Proof. By reduction to 3SAT. The idea is to 
construct a surface form that completely identi- 
ties the variables and their negation or lack of 
it, but does not specify the truth values of those 
variables. The dictionary will generate all possi- 
ble underlying forms (infixed morphemes or har- 
monic strings), one for each possible truth as- 
signment, and the autosegmental representation 
of lemma 4.1 will ensure that generated formulas 
are in fact satisfiable. \[\] 
5 Conclusion. 
In my opinion, the preceding proofs are unnatural, 
despite being true of the phonological models, be- 
cause the phonological models themselves are un- 
natural. Regarding segmental models, the unde- 
cidability results tell us that the empirical content 
of the SPE theory is primarily in the particular 
rules postulated for English, and not in the ex- 
tremely powerful and opaque formal system. We 
have also seen that symbol-minimization is a poor 
metric for naturalness, and that the complex no- 
rational system of SPE (not discussed here) is an 
inadequate characterization of the notion of "ap- 
propriate phonological generalisation. "2 
Because not every segmental grammar g gener- 
ates a natural set of sound patterns, why should 
we have any faith or interest in the formal system? 
The only justification for these formal systems 
then is that they are good programming languages 
for phonological processes, that clearly capture 
our intuitions about human phonology. But seg- 
mental theories are not such good programming 
languages. They are notationally-constrained and 
highly-articulated, which limits their expressive 
power; they obscurely represent phonological re- 
lations in rules and in the derivation process it- 
self, and hide the dependency relations and inter- 
actions among phonological processes in rule or- 
dering, disjunctive ordering, blocks, and cyclicity, s 
Yet, despite all these opaque notational con- 
straints, it is possible to write a segmental gram- 
mar for any decidable set. 
A third unnatural feature is that the goal of 
enumerating structural descriptions has an indi- 
rect and computationally costly connection to the 
goal of language comprehension, which is to con- 
struct a structural description of a given utter- 
ance. When information is missing from the sur- 
face form, the generative model obligates itself 
to enumerate all possible underlying forms that 
might generate the surface form. When the gen- 
erative process is lengthy, capable of deletions, or 
capable of enforcing complex interactions between 
nonlocal and local relations, then the logical prob- 
lem of language comprehension will be intractable. 
Natural phonological processes seem to avoid 
complexity and simplify interactions. It is hard 
to find an phonological constraint that is absolute 
and inviolable. There are always exceptions, ex- 
ceptions to the exceptions, and so forth. Deletion 
processes like apocope, syncopy, cluster simplica- 
tion and stray erasure, as well as insertions, seem 
to be motivated by the necessity of modifying a 
representation to satisfy a phonological constraint, 
not to exclude representations or to generate com- 
plex sets, as we have used them here. 
Finally, the goal of enumerating structural de- 
scriptions might not be appropriate for phonology 
and morphology, because the set of phonological 
words is only finite and phrase-level phonology is 
computationally simple. There is no need or ra- 
tional for employing such a powerful derivational 
system when all we are trying to do is capture 
the relatively little systematicity in a finite set of 
representations. 
6 References. 
2The explication of what constitutes a "natural 
rule" is significantly more elusive than the symbol- 
minimization metric suggests. Explicit symbol- 
counting is rarely performed by practicing phonolo- 
gists, and when it is, it results in unnatural rules. 
Moreover, the goal of constructing the smallest gram- 
mar for a given (infinite) set is not attainable in prin- 
ciple, because it requires us to solve the undecid- 
able TM equivalence problem. Nor does the symbol- 
counting metzlc constrain the generative or computa- 
tional power of the formalism. Worst of all, the UTM 
simulation suggested above shows that symbol count 
does not correspond to "naturalness." In fact, two 
of the simplest grammars generate ~ and ~', both of 
which are extremely unnatural. 
3A further difficulty for autosegmental models (not 
brought out by the proof) is that the interactions 
among planes is obscured by the current practice of 
imposing an absolute order on the construction of 
planes in the derivation process. For example, in En- 
glish phonology, syllable structure is constructed be- 
Chandra, A., D. Kozen, and L. Stockmeyer, 1981. 
Alternation. 3. A CM 28(1):114-133. 
Chomsky, Noam and Morris Halle. 1968. The 
Sound Pattern of English. New York: Harper 
Row. 
Halle, Morris. 1985. "Speculations about the rep- 
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Linguistics, Essays in Honor of Peter Lade- 
\]oged, V. Fromkin, ed. Academic Press. 
Johnson, C. Douglas. 1972. Formal Aspects of 
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Kenstowicz, Michael and Charles Kisseberth. 
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sis of the resulting stress assignment. A more natural 
approach would be to let stress and syllable structure 
computations intermingle in a nondirectional process. 
241 
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nonconcatenative morphology." Linguistic 
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242 
