Preprtnt No. 6 Classification: TG 2.1 
Some formal properties of phonological redundancy rules. 
Stephan Braun* 
i. Introduction. 
Redundancy is a well-known phenomenon of phonemes 
or phonological matrices within the framework of the 
distinctive-feature theory of JAKOBSON and HALLE Ill Q 
Redundancy in this theory means that the specification 
(either + or -) of certain features of a phoneme is 
predictable given the specifications of certain other 
features of the same phoneme and/or of neighbouring 
phonemes of a phoneme sequence. These restrictions on 
feature specifications are usually expressed by 
"redundancy rules". E.g. in English all nasal phonemes 
voiced which is expressed by a rule ~+nasal\] --~ are 
~voice~, to be read as "each phoneme which is specified 
~nasaq must also be specified E+voice~ ". Among the 
redundancy rules usually two main types are distinguishech 
Those like the one just mentioned which express a 
restriction valid for each phoneme of a language, in- 
dependent of possible neighbouring phonemes, will be 
called "phoneme-structure rules" (P-rules) in this paper. 
Besides them, there are rules expressing restrictions on 
the admissible phoneme sequences of the language, e.g. 
English no ~+consonanta~ segment can follow a in 
morpheme-initial nasal; they will be called (as usual) 
"morpheme-structure rules" (M-rules). In the paper of 
STANLEY ~2\] the former are called segment structure 
rules and the latter sequence structure rules. 
The aim of the present paper is to investigate the 
properties of phonoiogical redundancy rules on a mathe- 
8 MUnchen 2 
Arcisstrasse 21 
Technische Hochschule 
West-Germany 
matical basis. Some of the problems arising in connec- 
tion with redundancy rules in phonology have been men- 
tioned already in the work of HALLE ~3\] where they are 
treated essentially on a linguistically intuitive basis. 
The paper of UNGEHEUER r#\] on the mathematical proper- 
ties of the distinctive feature system (using Boolean 
algebra by virtue of the fact that every feature can 
have exactly two specifications) mentions redundancy 
without going, however, into details. A very thorough 
and comprehensive treatment of the subject is given in 
the already mentioned paper of STANLEY where a formal 
way of arguing is used though no mathematical proofs are 
given. At any rate, STA~EY's results show that a forma- 
lized treatment of phonological redundancy is sensible. 
Most recently, redundancy rules have been discussed in 
the work of CHOMSKY and HALLE ~5\] 
The results of the present paper essentially con- 
firm - as far as the questions are the same - the re- 
sults of STANLEY being, however, somewhat more precise 
than his. The main result is that the complete set of 
P-rules for a set of fully specified phonemes can be 
derived from the prime implicants of a certain Boolean 
function and thus computed without recurrence to 
linguistic intuition, given only the set of phonemes. 
Algorithms for this task can be found in the mathemati- 
cal literature (e.g. MoCLUSKEY ~6\] ). ,This formulation 
then also allows in a simple way to test intuitively 
found P-rules for compatibility with a given set of 
phonemes. No hierarchy of the features need be assumed 
for this. Moreover, it is shown that phoneme sequences 
can be treated formally like single phonemes (with a 
higher number of features); thus all results for single 
phonemes hold for phoneme sequences as well, and M- 
rules are not essentially different from P-rules. 
Furthermoret some ideas are given how to compute from 
a set of P-rules another set of rules which generate 
just the non-redundantly specified matrices, i.e. the 
lexicon; these rules are called "lexicon rules" (L- 
rules). Finally, two questions connected with the intro- 
duction into phonological matrices of blanks for redun- 
dant specifications are discussed, viz. "When do diffe- 
rent matrices remain distinct - in the technical sense 
of \[2\] , p.#08 - after introduction of blanks?" and 
the position of blanks in matrices uniquely determined 
by the redundancy rules alone or has an order of appli- 
cation of the rules to be taken into account?". It is 
shown that both distinctness and uniqueness are guaran- 
teed if a hierarchy (a total ordering) is introduce~ 
among the features and if the feature on the right hand 
side of a rule is required to have higher rank with 
respect to this hierarchy - e.g. usually \[voicedJ is 
given higher rank than EvocalicJ - than any feature on 
the left hand side of the rule. Counterexamples show 
that neither distinctness nor uniqueness necessarily 
hold if this requirement is not met. 
Phoneme-structure rules are discussed in Sec. 2, 
morpheme-structure rules in Seco3, lexicon rules in 
Sec.4 and matrices with blanks in Sec.5. 
2. Phoneme-structure rules. 
As mentioned in the Introduction a phoneme-struc- 
ture rule (P-rule) is a statement predicting certain 
feature specifications of a single phoneme given other 
feature specifications of this phoneme. In order to 
formalize this concept some notational conventions will 
= B ,. ,B a set p be introduced. Let m ~ 1 "" pJ of fully 
specified phonemes and ~ = {fl .... 'fnl the set of n 
features and ~ = {+,-J the set of the two possible 
s~ecifications. Any phoneme B~ can then be written 
set of n ordered pairs: B = {~lfl,.O.~Bnfn~ with as a 
~iE~ for i=1,...,n. Every set of m~nordered pairs 
~ifi containing each feature only once will be called 
"phonemic set"; the phonemes of~ are thus special pho- 
nemic sets. This set-theoretic notation for phonemes is 
almost identical to the usual linguistic n~ation and 
will be mainly used throughout this paper; the only 
difference is that no ordering of the features is con- 
sidered so far. It turns out that ordering of the fea- 
tures need be introduced only much later; for the time 
being it would only unnecessarily complicate the proofs. 
Another notation for phonemes stems from the ob- 
servation that there are exactly two specifications for 
each feature. The features can, therefore, be conceived 
of as Boolean variables taking the values true and false 
and a phoneme B can be written as a conjunction of these 
variables. E.g. B = -~~+fl'-f2'+f3~ in set-theoretic 
notation is replaced by the conjunction B(fl,fy,fs) = 
flAf--2^f3 (~ is the complement of f taking the value 
tru_~e if f takes the value false and vice versa) which 
takes the value tru__~e if and only if fl takes the value 
tru___~e, f2 takes the value false and f5 takes the value 
true. Thus true corresponds to the specification +,false 
to the specification - and B is formed from B by writing 
instead of +fi and ~ instead of -fi" This correspon- fi 
A dence of B and B evidently is biunique. The whole set 
of phonemes is in this notation described by the Boolean 
function 
(I) g(fl' 'fn ) V 
A 
.°o = B(fl,...,fn) 
(Vdenotes disjunction - the logical or) which takes 
the value tru____~e if and only if at least one of the 
B(fl,...,f n) takes the value true, i.e. if the B cor- 
A responding to B is a phoneme of~ . For the following 
the complement function ~ of g given by 
V ^ (2) g(fl,...,fn) = C(fl,...,f n) 
will be of some importance, g describes the set of those 
phonemic sets with n features which are not phonemes of 
~. This set which will be denoted by~ is in practice 
much larger than the set~ since there are 2 n phonemic 
sets with n features while the number p of phonemes of a 
natural language is much smaller than 2 n for usual 
values of n (e.g. n=12). 
A prediction for a feature specification of a single 
phonem e (a P-rule) is, in the set-theoretic notation, a 
statement of the form 
(3) {~irl,.O.,~krk~ --~ ~r 
with ,r F, r@r i for i:l,..,k, 
0~k~n-l, which is to be read as "if the phonemic set 
a =~irl,...,~krk ~.J on the left hand side of (5) is a 
part (a subset) of some phoneme B of~ then the feature *) 
r is in B necessarily specified as~" . Note that the 
condition at'B corresponds to STA~EY's "submatrix inter- 
pretation of rule application" (cp, E2J,p.413). 
Now, in order for (3) to be called a prediction in 
a sensible sense of this word two obvious requirements 
must be fulfilled: 
(i) a must occur in at least one phoneme of 
(ii) ~ must be uniquely determined by a and r 
~or simplicity we add a further requirement 
(iii) a must be minimal, i.e. there is no phonemic set 
b~a such that b and r already suffice to uni- 
quely determine the specification of r in B. 
Since by (ii) a uniquely predicts~ as specifica- 
tion of r there is no phoneme P~ such that the phone- 
h = au~r~ (i.e. a plus the feature r specified mic set 
as~, written ~ "as set-theoretlc union) **) is a subset of 
P. Any phonemic set with n features containing h is, 
therefore, an element of ~. A phonemic set h with this 
property is called im~licant of~ . ~ore specifically, 
we define the notion of prime implicant of ~ : 
Definition l 
A phonemic set h = {~irl , .... ~mrm~ (l~m~n) is 
called prime implicant of ~ if and only if 
h ' (a) there is no BE~ such that ~B. 
~) The case k:O means "r is specified as ~in each 
phoneme of~". 
**)~= + for ~= - and ~ = - for ~ = +. 
(b) for every proper subset b~h there exists a 
B~ such that b C B. 
Condition (b) of Def.1 expresses a minimality require- 
ment on h which will turn out to be closely related to 
requirement (iii) above. 
The name "prime implicant" for h was chosen because 
in the Boolean notation of eqs.(1) and (2) the conjunc- 
tion h corresponding to h is a prime implicant (in the 
technical sense of the theory of Boolean functions) of 
the function ~, eq.(2): An implicant of a Boolean func- 
tion v of n variables is a conjunction q of m~n of these 
variables such that v is tru____~e whenever q is true; equi- 
valently, if t is any conjunction of the n variables 
which contains q then t = true implies v = true. q is a 
prime implicant of v if it is an implicant of v and if 
every proper part s of q is not an implicant of v; 
equivalently, if there is at least one conjunction w of 
the n variables containing s such that w = tru___~e implies 
v = false (or V = tru___~e). By condition (a) of Def.1 P~I~ 
for every phonemic set P with n features with h~P; in 
Boolean notation ~ is any n_place conjunction containing 
h and P~ ~ means P = tru__~e implies ~ = tru___~e. Thus h is an 
implicant of ~. Condition (b) of Def.1 in Boolean 
notation reads "if ~ then there is a B with bOB such 
A that B = true implies g = tru___~e (or ~ = fals__~e)". Thus 
is a prime implicant of ~. 
The remarks following conditions (i) through (iii) 
together with Def.1 suggest a connection between prime 
implicants of l~ and P-rules. This is expressed by 
Theorem 1 
1. From each prime implicant h = ~lrl,...,~mrm~ of~ 
2. 
Proof: 
I. 
2. 
m P-rules 
Pj = aj --~ ~jrj (j=l,...,m) 
with aj = h\~jrj~ (i°e. ~j is formed from h by 
omitting ~r~) can be derived which comply with 
con itione lil through (iii  
If P = a--~r (a =~lrl,...,~krk~ ,kS0) is 
a P-rule complying with (i) through (iii) then 
h = au{~ = ~lrl'''''~krk '~r~ 
is a prime implicant of~ . P is derived from h 
by 1., and h is uniquely determined by P. 
Pj evidently has the --f°rm of eq.(3). Since h is 
a prime implicant of ~ and ajch there is, by 
Def.l(b), a Bi~such that aj C B. Thus, Pj 
complies with (i)° The feature rj omitted in aj 
is in B necessarily specified as ~j since it 
must be specified somehow and cannot be specified 
as OCj because then h~B contrary to Def.l(a). 
Thus aj and rj uniquely determine ~j and (ii) is 
met. Suppo$6 there is a bcaj such that b and rj 
already uniquely determine ~j. Then there is, by 
Def.l(a), no B~containing c = bU~jrj~ . But 
this contradicts Def.l(b) since c is a proper 
part of h. Thus there is no such b and P complies 
with (iii), too. 
There is no BG~such that h~B. Pot, otherwise, 
r is specified as ~ instead of ~ in some phoneme 
of~ containing a which contradicts (ii). Thus h 
is, by Def.l(a), an implicant of ~ . Each proper 
subset of h is part of a B~: By (i) and (ii) 
there exists a BE~ such that c = a~r~ is a 
part of B. Each proper subset of c is, therefore, 
also a part of this B. Each proper subset of h 
which does not container is a subset of a, thus 
a proper subset of c, thus a part of B. Let 
d = b U ~r~ with boa be a proper subset of h con- 
taining ~r. Suppose there is no B~such that d 
is a part of B. Then r is never specified as~ in 
all those phonemes of~ which contain b (since 
bCa and (i) there are such phonemes) but always 
as~. Thus b~a and r suffice to uniquely deter- 
mine ~which contradicts (iii) for P. Therefore, 
also d is a part of some B~. Thus h is, by 
Def.l(b), a prime implicant of~ and, by 1., P is 
derived from h. 
Let h' = ~lSl,...,~s~ a prime implicant of~. 
Every P-rule derived from h' has the form P' = 
aj'-@~js .~ For P to be one of these P' a compari- 
son shows that necessarily a\] = a, y~ = ~ and 
sj = r. Then h' = a\]u~jsj~ = au{~r} = h; thus 
h is uniquely determined by P. 
According to Theorem 1 every P-rule for~ 
complying with requirements (i) through (iii) - it seems 
rather obvious that a P-rule should meet these require- 
ments - is derived from a corresponding prime implicant 
of~ o The task of finding all the P-rules for~ is, 
therefore, equivalent to the task of finding all the 
prime implicants for ~ or, equivalently, the prime 
implicants of the Boolean function ~. This is a well- 
known mathematical problem which can be more or less 
efficiently solved on a computer using e.g. the 
McCLUSKEY algorithm E6S . (The efficiency of this algo- 
rithm depends rather strongly on the number n of featu- 
res; n must not be too large). Moreover, this result 
means that, given only the set ~ of fully specified 
phonemes, the discovery of P-rules for this set need not 
depend on linguistic intuition; the complete set of P- 
rules can be computed via the prime implicants of 
which is, in turn, directly determined bye. 
By their connection to the prime implicants of~ 
the P-rules are divided into equivalence classes: two 
P-rules will be called equivalent if and only if they 
are derived from the same prime implicant of~ . By 
Theorem 1.2 the connection between P-rule and correspon- 
ding prime implicant is extremely simple; thus equiva- 
lence of P-rules is easily tested by comparing the prime 
implicants. Moreover, the compatibility of an intuitive- 
ly found P-rule with a given set of phonemes can also 
easily be tested: if a-@Mr is the P-rule then au(~r 3 
must be a prime implicant of~ ; in particular, no pho- 
neme of the set may contain au{~r) • 
Conditions (i) through (iii) for P-rules or, equi- 
valently, the requirement that P-rules are to be derived 
from prime implicants of~ are essentially identical to 
the "true generalization condition" of STANLEY ( K2S, 
p.421). In our set-theoretic notation this condition for 
a rule a-~r reads 
( ~ means logical implication). By the rules of Boolean 
algebra this is equivalent to 
--I(acBA~r~cB) for every B ~ 
(-I means negation, A means conjunction), i.e. there is 
no B such that h = a U(~r) CB which by Def.l(a) means 
I0 
that h is implicant of l~ . Note that the true genera- 
lization condition is thus not equivalent to h being a 
prime implicant of ~ ; it does, in other words, not 
meet the minimality condition (iii). Because this con- 
dition has turned out in the proof of Theorem 1 to be 
rather convenient it is proposed that (iii) is added to 
the true generalization condition. 
As an example consider the five labial consonants 
IPl,lbl,/m/,/f/,/v/ of English as given in HA~LE \[7\] , 
see tab.l. For simplicity only the four features 
Strid + + 
has - • - + - • - 
tab.l 
cont ~ _ ._ _ + + 
Voiced - + + - + 
\[strident\] , \[nasal\] , \[continuant\] and \[voiced\] are 
considered and the specifications \[-vocalic 3 , 
\[+consonantal 3 , ~grave 3 and \[+diffuse\] common to the 
five consonants are omitted. ~or this small example the 
prime implicants of ~ can be computed directly by means 
of Def.l: Assuming for convenience a fixed order of the 
features (e.g. that of tab.l) one has ordered sequences 
of the specifications + and - instead of the sets used 
so far. Then for each k in l~k~n=4 all - (~1.2 k possible 
specification sequences of length k are formed and 
matched with tab.1. If such a specification sequence 
does not occur in tab.1 it is an implicant of ~ , and 
it is a prime implicant of~ if it does not contain any 
shorter implicant already found. Thus one gets five prime 
implicants of ~ 
11 
4) \[+strid, +nasa, {-strid, +cont', {+strid,-cont~, 
{+nas, +cont I , {+nas ,-voic edJ 
and from them ten P-rules (two for each of the prime 
implieants) 
+strid--@ -nas 
-strid--+ -cont 
(5) +strid---~ +cont 
+nas ---@ -cont 
+nas ~ +voiced 
+nas ~ -strid 
+cont ~ +strid 
-cont ~ -strid 
+cont ~ -nas 
-voiced--@ -nas 
This is the complete set of P-rules for tab.l; any other 
redundancy rule is not a P-rule for this set. 
3. Morpheme-structure rules 
According to the Introduction morpheme-structure 
rules (M-rules) are predictions for feature specifica- 
tions of single phonemes within phoneme sequences. The 
only difference between P-rules and M-rules then is that 
M-rules may (but not must) contain features of more than 
one phoneme of the sequence (see the example in the 
Introduction). It will be shown that M-rules can like 
P-rules be derived from the prime implicants of a 
suitable Boolean function. This is done by formally re- 
ducing the case of phoneme sequences to the case of 
single phonemes. 
For sake of simplicity at first only sequences 
consisting of two phonemes of~ are considered, i.e. 
sequences BIB 2 with B l~and B 2~1~ . The n features of 
B 1 are denoted by fl,...,fn as before and the n features 
f' Of course, fi and f~ denote the of B 2 as f~,..., n " 
same phonological feature; they are distinguished merely 
formally to indicate their position in the phonemes of 
the sequence. For formal purposes, however, fi and f~ 
12 
may be considered to be different features, and thus we 
have two sets ~= (fl''''' I , , fn~ and ~ = {fl'''''fn~ of 
features. Uniting ~ and ~l to form the set F(*)= ~l 
we can conceive of the sequence BIB 2 as a "phoneme of 
2 nd degree" B ¢2)= B 1UB~ with the 2n features of ~cz) 
where B~ is formed from B 2 by replacing in it fi by fl. 
E.g. if ~ ={+fl,+f2~ and B 2 = ~-fl,+f2~ then 
B~) = {+fl'+f2'-f~ '+f2~ Let~)be the set of all pho- 
nemes of 2 nd degree (i.el the set of all admissible 
phoneme sequences of length two) then 1~ca)is a subset of 
~X~ s ( X denotes the set-theoretic product) with~ 
the original set of phonemes and~ I identical to~ 
except that fi is replaced everywhere by fl. If every 
sequence of two phonemes is admissible (this probably is 
an only theoretical limiting case) then~)=~Ml~ t. 
After this formal reduction of phoneme sequences to 
phonemes of higher degree it appears natural to assume 
that the M-rules will be nothing but the P-rules for the 
higher-order phoneme set, i.e. they are derivable from 
the prime implioants of~. This assumption is suppor- 
ted by the following: 
A natural requirement for M-rules is that they re- 
flect the restrictions on possible phoneme sequences of 
a language. In other words, if every sequence of phone- 
mes is admissible then the M-rules should coincide with 
the P-rules for the set~ . The following theorem shows 
that this is indeed the case: 
Theorem 2 
Let ~(~)=~X~ Then any implicant of ~) which 
contains features of both ~ and ~l is not a prime 
implicant of ~z). 
13 
Proof: 
Let T~*)= T UT'.with T = \[~irl ,...,~krk \] and 
T' =\[ ~lS{ .... ,~mSm ~ a phonemic set containing 
the features ri~ ~ (i=l,...,k) and s\] 6 ~' 
(j=l,...,m); ~i~ ,~jE~. T and T' contain 
features only from ~ and from ~' , respectively. 
If neither T nor T' is an implicant of ~c.)then 
there are elements BC2)and COn)of ~(2J such that T 
is a subset of Ben)and T' is a subset of C ~). B Cz) 
and COn)can be written a5 BCm= BUB', C~Z)= CUC' 
with B and C from ~ , B' and C' from'~ a . Since 
the features of T are all from ~ and the featu- 
res of T' are all from ~ T is a subset not only 
of B Cz) but even of B; likewise, T' is a subset of 
C'. Therefore, To2)= TUT' is a subset of the set 
D c~)= BUC'. Since B~ and C'~' we have 
D¢~)E~x~ l , i.e. Dc~)EI~) by assumption, and 
thus T L2) is not an implicant of~ ¢~). That is, if 
T cz) is an implicant of~ c~) then necessarily one 
of its proper subsets T and T' is an imp\]icant 
of ~(~)which shows that T c,) is not a prime 
implicant of~C~). 
Thus, if every sequence of phonemes is admissible 
then the prime implicants of ~¢*) contain features only 
from \[ or only from ~i , i.e. they are prime impli- 
cants of ~ ; because any prime impli£ant of ~ evidently 
is a prime implicant of ~¢~) the sets of prime implicants 
of ~ and ~cz) are identical which means that the M- 
rules for~x~ l coincide with the P'rules of ~ . Prime 
implicants of ~c*) other than those of ~ , in particular 
such with features from both phonemes of a two-phoneme 
14 
sequence, consequently occur only if ~c2) is a proper 
subset ofS~ I , i.e. if not every sequence of phonemes 
is admissible. Thus the concept of M-rules as P-rules of 
a set of phonemes of higher degree is sensible, and M- 
rules are to be derived from the prime implicants of ~ cz) 
in exactly the same manner as P-rules are derived from 
the prime implicants of ~ . 
Without proof we mention two special cases~ if oH 
the first or second position of the sequence the whole 
set ~ (or ~ l ,resp.) is possible then all the prime 
implicants of ~ occur among those of ~c~)~ if on the 
first or second position only a single phoneme is 
possible then M-rules with more than one feature contain 
features only from ~i or ~ ,respectively. 
As an example for M-rules assume that from the five 
phonemes of tab.~ the three sequences /pf/,/bv/ and /mb/ 
can be formed. Denote the features by strid 1,...,voiced~, 
strid 2,...,voiced 2 for the first and second phoneme 
of the sequence, resp. In this set of phoneme pairs the 
specifications of the features strid l, cont 1 and nas 2 
are all fixed as -, and the remaining M-rules are 
+nas i--~ {+voiced l,-strid 2,-cont 2,+voiced 2~ 
-nas 1---~ ~+strid 2,+cont 2~ 
+voiced 1-~ +voiced 2 
(6) -voiced l~+strid 2,+cont 2~-voiced 2\] 
+strid 2 --~ +cont 2 
-strid 2 --~ ~-cont 2,+voiced 2~ 
-cont 2 ~ +voiced 2 
where for each of the prime implicants only one rule has 
been given and for rules with the same left hand side the 
15 
right hand sides have been combined for abbreviation. 
From the preceding it is clear how to extend the 
definitions given to the case of sequences of more than 
two phonemes; in order to get the M-rules one has to 
find the prime implicants of ~¢~) with k~3 (k is the 
length of the sequence), i.e. the prime implicants of a 
BoJlean function of kn variables. The practical diffi ~ 
culty of this task for larger values of k and n should 
not be underestimated and here probably further research 
is necessary. In principle, however, all the M-rules of 
a language can be computed given only the set of all 
admissible phoneme sequences of this language (each 
phoneme being fully specified); furthermore, the M-rules 
in the literature, e.g. in HALLE E3~ , can be thus given 
tested for compatibility with each other and for con- 
formity with the occurring phoneme sequences. 
4. LexicOn rules. 
Having computed a set of P-rules (or M-rules) 
predicting the specifications of certain features the 
rules can be used to remove these "redundant" specifi- 
cations from the phonemes. It is common in linguistic 
practice to replace redundant specifications by blanks. 
In the set-theoretic notation used here complete removal 
of redundant elements ~r from the phonemes seems to be 
more adequate. In this section some ideas will be given 
t 
how to generate the remaining "non-redundant" subsets of 
phonemes, i.e. the lexicon, by a set of new rules called 
lexicon rules (L-rules). 
According to Sec.3 it suffices to consider the case 
of P-rules. Let r ~F be a feature and let 
i6 
(7) a 1--@ ~lr,°.., a k'~k r 
be k P-rules specifying r in different environments; no 
a contains feature r. Using Boolean notation each 
~ (j=l,...,k) corresponding to the set aj is a conjunc- 
tion of some of the Boolean variables fl,...,fn and each 
of the P-rules is a logical implication saying that the 
value of the variable r on the right hand side of the 
rule has a certain unique value if the left hand side 
has the value true .The value of r is, therefore, pre- 
dictable if at least one of the conjunctions aj has the 
value tru_._~e, i.e. if the Boolean expression 
(8) a~v~ 2 V ... V ~k 
has the value true. The value of r is, therefore, not 
predictable (i.e. the specification of r can be either 
+ or -) if the Boolean expression 
A (9) ~r = -'~1 A "-' ~2 A ...A -~a~ 
which is the negation of (8) has the value true. 
This makes possible the formulation of a rule 
(using again set-theoretic notation) 
(10) a --~ +r r 
called lexicon rule (L-rule) for ~ , to be read as 
"if a is a subset of some phoneme of ~ then both 
r 
a r U {+r~ and a r U ~-r} are subsets of phonemes of~ "~ 
These L-rules can then be used to generate the non- 
redundant phonemic sets of~ 9 starting with the empty 
set ~, by the following prescription: if a phonemic set 
b occurring in this process of generation contains a r 
then it is replaced by the two new phonemic sets 
b u b U  -rJ. 
I? 
In order to make this process straightforward some 
additional conventions are introduced: 
I. By the usual submatrix interpretation of rule 
application L-rule (10) is applicable to b if and only 
if a r is a subset of b. To test b for applicability of 
(l~) it is, therefore, useful to have b already non- 
redundantly specified in all the features occurring in 
a in order to avoid having (10) not applicable to b r 
only because the specification of b in one of these 
features has not yet been filled in. The simplest way of 
th~s ensuring applicability, whenever possible at all, 
is to introduce an ordering relation ( among the featu- 
res of ~ such that for f,g ~ either f~g or g4f 
holds. Ordering of the features is quite common in 
phonology though it is usually introduced at an earlier 
stage than here. In every prime implicant h of I~ there 
is, then, one feature f with the highest rank according 
to this ordering and we can require that from all the 
P-rules derivable from h only the single rule having f 
on the right hand side shall be chosen. Since by Theorem 
1.2 h is uniquely determined by each of its 2-rules no 
generality is lost by this special selection. In every 
P-rule aj --~ ~jr of (7) the left hand side then contain~ 
only features of a rank less than r, and thus also a r in 
(10) contains only features of a rank less than r. All 
the L-rules are then ordered in a nathral way: they are 
applied in the order of their right hand sides, and the 
non-redundant specifications are thus filled in "from 
top to bottom" starting with the lowest-order feature 
and ending with the highest-order feature. 
2. Since any b occurring in the process of generation 
t8 
contains only non-redundant specifications all specifi- 
cations in a r predictable via the P-rules from other 
specifications of a r must be removed from a r. If a r 
consists of a single specified feature occurring in a 
P-rule c--~a r then c --~±r is also an L-rule. 
3. If ~r for some r is always false then the specifi- 
cation of r is always predictable and no L-rule concern- 
A ing r exists; if, on the other hand, a r is always tru___~e 
or if - which is the case e.g. with the lowest-order 
feature - no P-rule concerning r exists then the 
specification of r is never predictable which is 
expressed by the L-rule ~--~r • 
With these additional conventions a set of L-rules 
is computed by (9) from the iP-rules such that for each 
L-rule there is at least one phonemic set to which it is 
applicable. 
As an example consider the P-rules for tab.1. For 
the order of features as in tab.1 they are given by the 
first column in eq.(5). ~or the lowest-order feature 
strid there is the L-rule ~@±strid since for this 
order no P-rule concerning strid exists. For has there 
is only the P-rule +strid --~-nas, thus -strid--~±nas 
is an L-rule. The feature cont is predictable from 
-strid or from +strid (or from +nas), i.e. it is always 
predictable and no L-rule concerning oont exists. The 
feature voiced occurs only in the P-rL~le +nas-~+voiced, 
thus --~as--~ +voiced is an L-rule, and since 
+strid --~-nas is a P-rule we get the additional L-rule 
+strid--~voiced. Since all a r consist of only one 
specified feature no further redundancies have to be 
19 
removed. Starting from ~, application of these rules 
gives tab.2 containing the lexicon forms of the five 
phonemes (with respect to the four features). It can be 
directly verified that tab.2 is filled up by the rules 
of eq.(5), first column, to give the complete phonemes 
of tab.1. 
For another order of the features one has a diffe- 
rent set of P-rules and, consequently, of L-rules. For 
the n! different orders of the n features there are n! 
different sets of L-rules each of which gives a diffe- 
rent set of lexicon segments (or lexicon matrices). Each 
of these sets is then filled up by the corresponding set 
of P-rules to give the complete set of fully specified 
phonemes (or matrices). 
strid + 
nas 
eont 
voiced + 
+ 
+ 
+ -- 
tab.2 
5. Matrices with blanks. 
5.1. Distinctness. 
As noted in Seo.4 P-rules can be used to remove 
redundant specifications from phonemes: if a --~ ~r is 
a P-rule and a is a subset of a phoneme B~ then the 
element ~r is removed from B. Instead of removing the 
element ~r from B, thus removing the feature r altog~he~ 
a common practice in linguistics is to leave the feature 
r in B but to change its specification into a blank (or 
zero). It is stressed very much in the literature 
(e.g. ~23 ,p.410) that this blank is not a specification 
20 
like + or -. This circumstance has been underlined by 
introduction of the notion of distinctness of phonemes 
(or phonemic matrices) - see \[2J ,p.408. Two phonemes B 
and C of ~ are called distinct if and only if there is 
at least one feature f E~ such that B is in f specified 
as + and C as - (or vice versa); conversely, B and C are 
said to be not distinct if and only if for every f~ 
either the specification of f in B is identical to that 
in C or one of both specifications is blank. There has 
been some discussion about this concept of distinctness 
e.g. \[2\] ,p.408 f.) and it has been argued that it (see 
is not completely sensible; for the present, however, 
we will accept it as existing and turn to the question 
"When do phonological matrices remain distinct after the 
introduction of blanks?". 
As before, we consider only the case of single 
phonemes. Fully specified phonemes are, of course, 
distinct but they do not necessarily remain so after the 
introduction of blanks. Taking tab.1 and its P-rules, 
e.g.(5), as an example the three rules +has 
~strid,-cont,+voiced~ - the right hand sides have been 
combined for abbreviation - applied to /m/ leave the 
phonemic set ~+nas~ or, using the symbol 0, the set 
~strid,+nas,Ocont,Ovoiced~ whereas the two rules 
+cont --~ ~+strid,-nas~ applied to /f/ leave ~+cont, 
-voiced} or ~0strid,Onas,+cont,-voiced~ which is not 
distinct from the result for /m/° 
It is possible, however, to have the phonemes of~ 
pairwise distinct after the introduction of blanks if 
(as already in Sec.4) an ordering of the features is 
introduced and if of all the P-rules derivable from a 
prime implicant of ~ only the single one with the 
2~ 
feature of highest rank on the right hand side is 
chosen: 
Theorem 3 
Let the features of ~ be totally ordered by an 
ordering relation ~ (i.e. for f,g ~ ~ either f~g or 
g~! and let rj ~r for all rj occurring within the left 
hand side a of a P_rule a --~r. Then the phonemes of 
are pairwise distinct after introduction of blanks. 
Proof: 
Let B and C be two (fully specified) phonemes of 
~, B~0. Then there is a certain number of 
features of ~ (at least one) such that B is 
specified contrary to C in exactly these 
"distinguishing" features and identical to C in 
the remaining features. Let f be that of the 
distinguishing features with the lowest rank. 
Then there is no P-rule a--~f which is appli- 
cable to both B and C: by assumption all the 
features in a are of lower rank than f, thus B 
and C coincide in all features of a. Since the 
rule is assumed to be applicable to both B and C, 
a is a subset of both B and C, and since B and C 
differ in f the set a U~f3 is a subset of eith~ 
B or C, whatever,. Thus this set is not an 
implieant of ~ and, therefore! a--~f cannot be 
a P-rule. This means that no blank can occur on 
f in B and C, i.e. B and C remain distinct even 
after the introduction of blanks. 
Without ordering of the features two phonemes can 
become not distinct as is shown by the examples above. 
Ordering of the features is, however, only sufficient 
22 
for pairwise distinctness, not necessary, i.e. a set of 
phonemes with blanks can remain pairwise distinct even 
without ordering of the features. For an example take 
the set 
~ +nas--~,strid, -strid--~-cont, +strid-~ +cont, 
(ll) L+c°nt--~-nas' +nas-~+voiced 
of P-rules for tab.1. (One P-rule has been chosen for 
each of the prime implicants of eq.(4)). This set is not 
comDatible with any ordering of the features since it 
would require nas(strid, strid~ cont and cont~ nas 
which is impossible. Applied to tab.1 in the order given 
in eq.(ll) we get tab.3 with pairwise distinct phonemes. 
strid - .- 0 + + 
nas - - + - .- 
cont 0 0 - 0 0 tab.3 
voiced - + 0 - + 
Unfortunately, there does not seem to be a simple 
and general necessary condition for pairwise distinct- 
ness of phonemes with blanks. 
5.2. Uniqueness. 
The result of tab.3 depends on the order in (ll) of 
the P-rules.The same P-rules, applied in the order 
~ -strid--~-cont, +nas--~-strid, +cont--~ -nas, 
(12) L+strid--~+c°nt' +nas--~+voiced 
give tab.4 which is different from tab.3. 
23 
strid 0 + + 
has -.- + 0 0 
(~ont 0 6 0 0 0 tab.4 
voiced - ÷ 0 - ÷ 
11~ ther words, the phonemes with blanks (or, for hl- 
rules, the matrices containing blanks) are not uniquely 
determined by the P-rules alone but also by the order in 
which the P-rules are applied to put in blanks. 
This situation can be described as follows: 
Let PI = a--~r with a =~irl,...,~krk~ be a P-rule 
which is applicable to a phoneme Bg~ , i.e. a is a 
subset of B. P1 can then be used to put a blank on r in 
B. This is, however, impossible if there is already a 
blank in B on one of the features of a because then a 
is no longer a subset of B. This blank on a feature r i 
of a (l~i@k) can be caused only by a P-rule P2 = 
b --~ir i which was applied before PI" Thus the posi- 
tion of blanks can - and indeed sometimes does, as the 
examples show - depend on the order of application of 
the P-rules. 
This order dependence somewhat complicates the 
situation and one can look for ways to avoid it. One way 
is to give up the submatrix criterion for rule appli- 
cation and to use the non-distinctness criterion instead. 
Then, the blank on r i in B would lear@ a not distinct 
from B, and P1 would remain applicable. A serious draw- 
back of this solution is, however, that a blank does not 
tell which of the specifications + and - has been re- 
moved by it. Thus, P1 would be applicable also to a 
phoneme B with a blank on r i which in its full form has 
24 
specified as~i instead of ~i" Thus the non-d~stinct- r i 
ness criterion alone is useless for rule applieation~ it 
must be amended by criteria ensuring the correct speci- 
fication of B on r i. ! 
There is, however, uniqueness even if we use the 
submstrix criterion for rule application if the features 
are, as before, totally ordered and only the special P- 
rules are chosen. For this case we have~ 
Theorem ¢ 
Let PI = a-~ ~r with a = ~irl ,...,~krk~ and 
P2 = b-'~iri be two P-rules applicable to the same 
phoneme B~B . Let rj<r and s<r i for each s occurring 
in b. Then there is a further P-rule P5 = z--~r 
applicable to B such that r i does not occur in ~zo 
Proof: 
The P-rules PI and \[?2 are derived from the prime 
implicants ql = a U~r\] and q2 = bulairi~ of~. 
Since P1 and P2 are both applicable to B both a 
and b are subsets of B; this means, in particular, 
that a and b are identically specified in featu- 
res common to both. Thus, a Ub is a phonemic set 
and a subset of B. Let c = aX\[~iri~ and let 
ho=C ubv~r\] ; since f<r for all features f 
occurring in c or in b!. h o is a phonemic set~, i\!,.~,,,~ 
h e is an implieant of 7~ : the feature ~'.j. r~ ,.~ .~,,,~ 
neither in c (by definition) nor in b (single ~ll 
features of b are of a rank less th;=~n r l\]) ~ t;hu~ 
r i does not occur in hoo If h o f~:_ C for some 
phoneme C of ~ -then e!tne.~ ~i~'i ~.~ or ~jr.L@C ~ 
t erefore, either o o{ irdoI    = or 
bU~iri~ = q2_~C which is both impossib\].e sLn~e 
Z5 
ql and q2 are implican~s of ~ . 
If h o is a prime implicant of ~ then we 
have the P-rule P3 = Zo-'~r with z o = cub 
~utting a blank on r, and P3 is applicable to B 
since (cub)c(a~b) ~B. If h o is not prime then 
there is a proper subset h I of h o such that h I is 
an implioant of ~ . Since each subset of z o is a 
subset of B necessarily ~r E h l, i.e. 
h I = ZlU~r I with ZlCS o. If h I is prime then 
P3 = Zl--~r is the requiredMP-rule; if not then 
there is an implicant h 2 of ~ with h 2Ch I and, 
similar as before, h 2 = z 2 u~r~ with z 2 ~ z 1. 
Thus we get a sequence of implicants h~ of 
with h i = ziU~ ~ and zi~zi_ 1 C °.. ~z o = bvc. 
If one of the h i i s prime then the sequence 
terminates with the i~-rule P3 = hi-*~r" Since 
the z i become smaller and smaller the sequence 
terminates in any case with h = z~and 
z = ~t~ ,~ ,t~, and h is prime since 
zcz o ~ B and ~r~ Cql and ql is prime, thus 
there is a C~ such that ~r~CC. 
Thus, even if the blank on r in B cannot be put 
there by P1 because it is "blocked" by P2 there is al- 
ways P3 which cannot be blocked by P2 and which puts the 
blank on r in B. Thus, the position of blanks in the 
phonemes of ~ is uniquely determined by the P-rules 
alone independent of the order in which they are applied. 
Z6 

References

~\] R.JAKOBSON and ~.HALLE: Fundamentals of 
Language. Den Haag, 1956. 

~23 R.STANLEY: Redundancy Rules in Phonology. 
Language 43, 393 (1967). 

\[3J M.HALLE: The Sound Pattern of Russian. 
's Gravenhage, 1959. 

E4\] G.UNGEHEUER: Studia Linguistica XIII, 60 (1959). 

\[5\] N.CHOMSKY and M.HALLE: The Sound Pattern of 
English. New York, 1968. 

\[6\] I.B.PYNE and E.J.McCLUSKEY,Jr.: The Reduction of 
Redundancy in Solving Prime Implicant Tables. 
IRE Trans. EC-II, 4 (473), Aug. 1962. 

\[71 M.HALLE: On the Bases of Phonology. In: J.A. 
FODOR and J.J.KATZ (eds.): The Structure of 
Language, Englewood Cliffs, N.J., 1964. 
i: ¸ 
