I. Introduction 
A study of formal properties of different styles of formal 
grammars is of great interest because each style (i.e., formal char- 
acter of rules) is well suited for characterizing certain aspects of 
natural language structure and is awkward for characterizing certain 
other aspects. The awkwardness can be due to either an inherent 
difficulty in characterizing a certain aspect (e.g., the characteriza- 
tion of the notion of the 'head' of a constituent in a PSG) or an 
unnecessary complexity in characterizing a certain aspect (e.g., the 
statements concerning the relational aspects in a PSG) or actually a 
counterintuitive characterization (e.g., this often happens in a PSG, 
especiai\].7 in the context of transformational gra~nnars, because a 
PSG allows an 'uncontrolled' introduction of new 'nonterminals' ). 
This naturally suggests a study of formal gramnars of mixed types in 
order to take advantage of different styles.* Thus we try to see how 
far we can succeed in setting up a class of granmmrs which has no 
more power than necessary and which also can characterize different 
aspects of natural language structure in a natural way. 
The class of grammars studied here and in Joshi (1969) have been 
motivated by the type of granmmr proposed by Harris (1962, 1968). 
These gra,~ars also arose out of an attempt to formalize certain aspects 
of the type of grammar considered by Joshi (1966) where it was used 
for defining structures for the purpose of defining transformations 
and ultimately for constructing a transformational decomposition pro- 
cedttre. 
First, in Section 2, we will introduce a new style of •formal 
grammars called String Adjunct Grammars (AG). The only purpose of 
Section 2 is to state some of the basic concepts and results concerning 
AG's (including a brief discussion of their linguistic relevance) which 
are needed for the presentation of the material in Section 3 (for a 
detailed treatment of AG's, see Joshi, Kosaraju, Yamada (1968)). 
In Section 3, we will introduce a class of grammars called Mixed String 
Adjunct Gra,~ars (~G) which use two different types of rules - adjunc- 
tion rules and a special type of rewrite rules. After studying some 
properties of ~AG's we introduce Generation Schemes G s = (G, A). A G S 
maps strings in the language, L(G), corresponding to an MAG, G, into 
strings in the language, L(G'), corresponding to another MAG, G'. 
* See also Robinson (1968) for a similarly motivated work. 
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Strings in L(G) and ~(G') are both 'well-formed'. In Section 3.~ 
we discuss briefly the linguistic relevance of the material in the 
earlier sections. A detailed development of the various ideas intro- 
duced here will be reported in Joshi (1969). 
Fig. A. at the end summarizes the hierarchy of some subclasses 
of AL's and MAL's in relation to the phrase structure hierarchy. 
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2. String Adjunct Granmmrs (AG) 
Briefly an AG consists of a finite alphabet, a finite set of 
strings on this alphabet and a finite set of adJunction rules which 
state how certain adjunct strings are adjoined to certain host 
strings. The corresponding language called a String Adjunct Language 
(AL) is then defined as the set of all strings derived from a cer- 
tain specified subset of the given set of finite strings. The rules 
in an AG have a considerably different formal character as compared 
to the 'rewrite rule' in a general phrase structure grmmmar (PSG). 
The language hierarchies of AG's and PSG's cut across in many 
interesting ways. 
2 .I Local String Ad,iunct Grammar 
We will define a local string adjunct grammar (LAG) as follows. 
Let A = ~al, a2, ..., am) be a finite alphabet. Let Z be a finite 
set of finite strings on A and let E c C E be a distinguished set of 
strings on A. We will call Z the set of basic strings and F e the 
set of basic center strings. We will define a local left ad,~unction 
rule, ~i~k as a 3-tuple (ai, ~, £k) where a i e Z, a i • E and ~k is a 
p-~t 0f~ad,iunction in a i. WeUwill call a i as the (~esic) host of 
~iJk and aj as the (basic) ~ of ~ijk" The point of ad-j~ction 
of Ai~k'k refers to the point of adjunction which is to the left of 
t~ekth ~ymbol of the host a i where we associate with each string 
ai • ~, ~i = ail ai2 ... ani ~ aij e A, and J = i~ 2, ...~ ni~ 2n i 
points of adjunction, one to the left and one to the right of each 
a i.. Note that a i # 8 ~ the null string. A local right adjunction 
rule ri~k is similarly defined as a 3-tuple ~i, oJ, rk), ~i e ~, 
~d ¢ ~ ~ rk is the point of adjunction of rij~o and. refers to the 
p6int of adjunction to the right of the kth symbol in the host ~i. 
In general, (~i, ~ ~k ) will denote a local ad,iunctio~rule , uij k- 
If ui.ik is a local~left ad~unction rule then ~k = ~k and if Uijk~is 
a loc~l right adjunction rule then ~k = rk" Finally, we have the 
f on~ng 
Definition 2.1.1 A local string adjunct grammar (LAG), G is a 6- 
tuple, G = (A, ~, ~c' ~' ~a' J) where A is the alphabet, ~ is the 
set of basic strings, L~ is the set of basic center strings, ~h is 
the set of basic host sErings, ~ is the sat of basic adjunct strings, 
and J is a finite set of local adjunction rules. ~ = ~uil(ui, uj, 
~k ) ¢ J\]' ~a = ~ajl(~i' ~j' ~k ) e J}, and ~ = ~cU~4J~a . Further 
Z c may contain ~ but ~ ~ Z h. 
-S- 
/ / 
Given J, S h and ~ are completely specified and S = ZcUSh~ 
re. Further the alphabet need not be explicitly stated. Hence, 
unless otherwise necessary we will write G as a pair (Sc, J) 
instead of a 6-tuple as in the definitic~ above. 
Example 2.1.1 Let ~ = (~c, J) where ~c -- rabcl, and 
J = {u I = (abc, pq, rl) , u 2 =.(pq, pq, ~2)}" \[We will write uij k 
as Just u. The indexing of u's in J is arbitrary and is merely for 
convenience.\] Here Z c = rabc\], Z h = ~abe, pq}, re = fpq\], E = 
{abe, pq\]. Note that abe is a basic center string but pq is not. 
u 1 is a local right adjunction rule and t~ 2 is a local left adjunc- 
t~on rule. Here A = ~a, b, c, p, q}. 
2.2 Local String Adjunct Language (LAL) 
The meaning of an adjunction rule, say, u = (ui, a i, ~k) is 
that fr~n u i we can derive a new string by adjoining u~Vto the left 
of the kth symbol in u i. Thus, for example if u = (abe, t, ~2) we 
can derive a string atbc. However, in order to define the language 
L(G) corresponding to a given LAG, Gjwe must first define how the 
rules of adjunction are extended to derived (i.e. non-basic) host 
strings and adjunct strings. Here we will give an example to 
illustrate the main idea and c~it the precise definition (see Joshi, 
~)saraju, Yamada (1968)). 
Example 2.2.1 Consider the LAG, G,in Example 2.1.1. P = ~abc, 
pq), Z c = {abe\], J = \[u 1 = (abe, pq, rl) , u 2 = (pq, pq, ~p)\]. From 
abe by one application of u I we obtain apqbe. We regard~he points 
of adjunction of apqbc to be the same as abc, i.e., the positions to 
the left and right of the symbols a, b and c. This apqbc is a de- 
rived host and we can apply u I again, obtaining apqpqbc where the 
newly adjoined pq is i,,,ediately to the right of a. 
Again, starting with pq, by one application of u 2 we obtain ppqq. 
Since pq is both a basic host and a basic adjunct (in the same rule, 
in this example), ppqq is a derived host as well as a derived adjunct 
and hence it can be used as a host or as an adjunct or both in the 
rule u 2 . This allows us to derive strings pppqqq, ppqpqq, ppqppqqq, 
etc. Since all of these are derived fr~n pq they can be used as 
adjuncts in Ul, allowing us to derive apppqqqbc, appqpqqbc, 
appqppqqqbc, etc. If we use apqbc as a host in Ul, we can also de- 
rive apppqqqpqbc, appqpqqpqbc, etc. Thus we can derive, for example, 
fromthe string abc ¢ r c the strings apppqqqbc, appqpqqbc, 
apppqqqpqbc, apqpqbc, apqpqpppqqqbc, etc. All these strings will be 
included in the language L(G) corresponding to G. 
Example 2.2.2 Let G = (E c, J), F~ = \[ab\],'J = ~u I = (ab, ab, rl) \]. 
This gra~nar generates the language L(G) = ~ w / w is a string on 
A; "the number of a's in w" = "the number of b's in w" and for any 
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\. 
initial proper substring of w, the number of a's is greater than the 
number of b's). This language is context-free and is known to be 
non-linear (Schatzenberger (1961)). 
Remarks 2.2.1 
i. In the generation of a string in L(G) we observe that once 
a string is adjoined to a host then the adjunct string cannot receive 
any further adjuncts. In other words a string which is to become an 
adjunct string must acquire all its adjuncts prior to its being used 
as an adjunct string. 
2. Let w be a string in L(G) derived from some string ui ~ ~c" 
The generation of w does not begin, however, with the basic center 
string unless, of course, w is just a basic center string itself or 
a center string with adjuncts which themselves do not receive any 
other adjuncts. We have to start from the "innermost" adjunct 
(adjuncts) and work our way "inside out" and finally use the basic 
string which is to become the center string of w. 
3. During the generation if a host string receives two (or more) 
adjuncts then we have the two following situations. If the two ad- 
juncts are adjoined at distinct points of adjunction of the host, then 
clearly those adjuncts can be adjoined in any order. However, if the 
two adjuncts are adjoined at the same point of adjunction of the host 
the order is significant. Let u I = (ai, oj, ~k ) and u 2 = (ci, am, ~k ) 
be two rules. Let ~k -- ~i for example. If u I is used before u 2 then 
we obtain OjamOi; but if u~ is used before ul-then we obtain amajO i. 
In other words, the adjunct adjoined later in the derivation is 
closer to the point of adjunction in the host (~6~which it was adjoined) 
than the adjunct adjoined earlier in the derivation. 
2.3 'Tree representation for a derivation in LAG 
Let G = (~c, J) be an LAG. Let the rules in J be arbitrarily 
numbered u_, u_, ...,u • The generation tree is constructed as 
follows, ll)~If u~ =n(oi3 oj, ~k ) is used in the derivation then 
we represent this as in Fig. 2.3.1a. Here we have two labeled nodes 
aj and °i and a directed branch from oj to °i with the label u~: ~k" 
2~ Let a host oi receive more than one adjunct, say, ajl , oJ2''"' 
ejmat points of adjunction gk I'~ gk o'~ "''' gkm' i.e., we use rules 
ui~ = (oi' aj2, ~k ) ¢ J, ~ = l, 2, ..., m. We represent this as 
in Fig. 2.3.1b. Now, in view of Remark 2.2.1-3 we impose a right to 
left ordering on the points of adjunction of a host and thus in effect 
define an equivalence relation on the set of derivations of a string 
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f,c,. 2.3.1 ~. 
bq Pq 
(,I) (9) 
:i I: rz 
~:~ X~.:r~ 
t~t~) a-i>c , 
~4 ,',, Ex~,~,,b, le. 3... 3. i. 
-6- 
~ler';vo, 4;i'or~ o: 
in L(G). The tree representation of a derivation with the above 
conventions will be called a ri6ht to left (r-~) tree representation. 
Note that the tree representation of a derivation of a string in 
L(G) is a rooted tree and the string labeling the root is in Ec" 
2..1 Example = labc, Let G = (E c, J) be an LAG where Zc -- \[abc~, and 
j = KUl pq, r2) , u 2 = (abc, rs, r2) , u S = (pq, pq, rl) , 
u~ =(abc, t, ~l ), ~ = (pq, t, ~)3. The following is a string in 
L(G). w = ttabrsppqptqqrspqc 
Fig. 2.3.2 show an r-~ tree representation of a derivation of w. 
We have numbered the nodes for convenience. Nodes l, 3, 5, 6, 7, 
8, and 9 are terminal nodes. Node lO is the root node. 
The derived strings corresponding to the nodes of the tree in 
Fig. 2.3.2 are as follows: 1. t; 2. ptq; 3. Pq; 4. ppqptqq; 5. Pq; 
6. rs; 7. rs; 8. t; 9. t; 10. w = ttabrsppqptqqrspqc. 
2.4 
Theorem 2.~.i Every LAL is a CFL (context free language). 
class of LAL's is properly contained in the class of CFL's. 
(L = ~anbn I n ~ i\] is not an LAL. 
The 
2.5 Distributed Strin~ Adjunct Granmmr (DAG) and Lan6ua~e (DAL) 
We will generalize the local adjunction rule as follows. 
Definition 2.~ A distributed ad~unction rule, dij k is a 3- 
tuple, (ai, (aj), ~k ) where a i ¢ E; aj ¢ E; (aj) denotes a specified 
segmentation of aj; {k is an adjunction 'vector', ~kl, %,..., ~kn; 
Ek i's are the points of adjunction of ai," and ~k i = ~ki or rki , 
1 ~ k i ~ m, k i ~ ki+l, sad if k i = ki+ 1 then ~ki = ~k i sad ~ki+ l- rki+l. 
The meaning of ~i-'k is that from the host oi 
string, say, ap, by adjoining the segments of aj, 
ajn at the points of adjunction ~kl, ~, ..., ~kn 
ly. That is, we 'distribute' the specified segments of a~ over a i 
at the points of adjunction ~, ~, ..., and ~k ; the j~h segment 
n 
is adjoined at ~kj" The Condition on ~k prevents permutation of 
the segments. 
we can derive a 
J2 " ' Jl 
of ai, respective- 
-7- 
The language L(G) corresponding to a given DAG, G, can be 
defined by generalizing the definition in Section 2.2. The main 
idea is that when a derived string is segmented each segment contains 
all the adjuncts (and adjuncts of adjuncts etc.) of all the symbols 
in that segment. The tree representation in Section 2.3 can also he 
generalized to DAG ' s. 
Example 3.1.1 Let G = (Z n, J) he a BAG where Z~ = \[abc}, and 
= {u l = Cabc, (p) (qr), rl~3) , u 2 = (pqr, (p) Cq~ (r), ~i~2~3) \]. 
Here Z = \[abc, pqr}, andA = \[a, b, c, p, q, r}. Note that in t h 
in u 1 and u 2 is not the same. Then 
n n ~(G)-\[ap 
b 5 r ~...q r ¢In i~0, 
for i = i, 2, ..., m; m ~ i}. 
Example 3.1.~ Some other DAL's are: LI = \[a~omln ~ 11, 
L, 2 = \[anbncnln ~ i}, L 3 = \[x xRI x ¢ AA*, "~Rx = reversal of x}, 
% = \[XXlX C AA*\], etc. 
Theorem 2.~.1" The class of LAL's is properly contained in the 
class of DAL's. Every DAL is a CSL (context sensitive laaguage). 
The clas~ of DAL's is properly contained in the class of CSL's. 
(L = \[a n In ~ l} is a CSL but not a DAL). 
Theorem 2.~.i There are languages which are inherently ambiguous 
in the class of CFL's but which are unambiguous in the class of DAL's. 
(Ex,=;le: ~. -~ {a%Jckl i, j, k :. i; i : j or j : k) 
2.6 String Adjunct Grammars with Null Symbols (AGN) andLanguage 
We will now introduce a somewhat modified form of AG's (IAG's 
or DAG's) called string adjuncts grammars with null symbols (AGN). 
The modification consists of allowing in the alphabet a very special 
type of "non terminal" symbols called "nul___!l symbols". The main idea 
is to use the null symbols to tag the strings in Z. The null sym- 
* It is possible to generalize the local adjunction rule in the fol- 
lowing manner also. This generalization permits one to adjoin to the 
host a set of local adjuncts simultaneously, i.e. (ai, ajl , ~J2' "'" 
ajn , ~k ) where a i c Z, k = l, 2, ..., n, and ~k is the adjunction 
'vector' as before. We will call these IAG's with simultaneous 
(Continued on Page 9). 
-8- 
bols have no points of adjunction associated with them and they do 
not receive any adjuncts. The null symbols are ulitmately erased 
(i.e., rewritten as a null string E ). 
Definition 2.6.1 An IAGN (or DAGN), G, is a 7-tuple (A, N, ~, 
Zc, Dn, Ea, J) where A is a finite alphabet, N is a finite set (pos- 
sibly empty) of null symbols, ~ is a finite set of basic strings, 
Zc c Z is the set of basic center strings, ~h is the set of basic 
host strings and ~a is the set of basic adjunct strings, 
E = Z c U E h U E a and J is a finite set of adjunction rules. Further 
a. A~N =~; b. If ~ ¢ Zthen s c (A~N) (AUN)*; c. There is 
no rule in J which adjoins adjuncts to the left or right of a null 
symbol, i.e., null symbols have no points of adjunction. Thus for 
a oi c Z the adjunction 'vectors' are the same as those that can be 
defined for the same o i without the null symbols, i.e., as far as 
adjunctions are concerned we ignore the null symbols. We will use 
Greek symbols for the null symbols, and unless otherwise necessary, 
we will write an LAGN (or DAGN), G, as just the pair (Ec, J). 
Theorem 2.6.1 The class of LAL's C the class of LALN's and the 
class of DAL's c the class of DALN'a. (We conjecture, however, that 
we have '"' ~.ns~ead of " _C "). 
2.7 
An adjunction rule u, if applicable, can be applied arbitra- 
rily many times. In this sense it is repeatable. We can also 
consider nonrepeatable rules, (nr-rule)~ in the sense that if a rule 
u = (ui, aj, ~k) is nonrep~atable then for each occurence of the 
host o i in a derivation, u can be applied no more than once. Let 
nr-AG and nr-AL be the correspondin~ grammars (i.e. AG's which have 
at least one nr-rule) and languages. The class of LAL's the class 
of LAI/Y'S (L = \[anbnln ~ I\] is an nr-LAL but not anIAL). ~ 
2.8 
We say that a local a~junction rule is uniform if in a rule u 
the adjunct aj adjoins to the left (or right) of some symbol ag ~ A 
in the host of u, then a~ adjoins to the left (or right) Of ag 
wherever ag occurs in 8n~ string in ~. An LAG, G, is uniform if all 
its rules are uniform. 
(,Continued fremPage 8.) 
ad.iunction rules (LsAG). It can be shown that the class of LAL's 
C the class of LsAL's C the class of DAL's. This observation is 
due to Leon Levy. 
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2.9 
AG's with the condition ~ = ~_ are of special interest. Strings 
in  ean be considered as ele ot ry sentences (or sentence fo ) 
in L~G)-- If ? = ~c then every string in L(G) Can be decomposed into 
a set of elementary sentences (or sentence forms). Note also that 
if ~ = ~c then in the r-~ tree representation of the derivation of 
string in L(G) every node is either a sentence or a derived sentence. 
2 .iO 
In an LAG (or DAG) we do not have nontermlnals in the sense of 
the nonterminal alphabet of a PHG. We have, however, auxiliary 
symbols used implicitly such as the ~k's corresponding to the points 
of adjunctions. But these auxiliary symbols are used purely as 
position markers and do not have the same interpretation as the 
nonterminals in a PSG (i.e., the auxiliary symbols ~k's do not cor- 
respond to phrase types) • If we consider the marking symbol, A, 
used in making precise the definition in Section 2.2, (see Joshi, 
Kosaraju, yamada (1968)), also as an auxiliary symbol then one can 
posslbly consider a i (a i e A) as a nonterminal which can be inter- 
preted as a phrase ^type but with the added interpretation that a 
rhrase type a i has a i as the 'heed' (or 'center') of the phrase. 
Each sentence in L(G) has also a 'center' namely the center string 
of W. 
In an IAGN (or DAGN) the null symbols are, however, like the 
nonterminals in the PSG although highly restricted. The null symbols 
are used to tag basic strings and therefore they are not used as 
position markers; in fact, they have no positional interpretation. 
The null symbols as nontcrmlnals are highly restricted because they 
are never 'rewritten' (in the sense of a PSG) into any other string 
except the null string, i.e., the only 'rewrite rule' associated 
with a null symbol, say~, is ~ ~ £. 
IAGN (or DAGN) can be considered as gra~nars of a mixed style 
as we use rules of two different styles - adjunction rules and 
'rewrite rules' of a special type. This is a very simple example of 
a mixed ~rammar. In Section 3 we will be considering some more 
interesting classes of mixed grammars. 
2.11 
In the linguistic context the alphabet A in an AG, G, will 
consist of symbols which denote major dictionary classes (lexical 
classes) such as N (nouns), t (tense, auxiliaries), % (adjectives), 
V (verbs), P (prepositions), wh (who, which, whom), R (pronouns), 
D (adverbs), Q (quantifiers), etc. N, t, A, V, etc. are thus 
-i0- 
preterminal symbols. The basic center strings thus correspond to 
basic (elementary) sentence forms, e.g., N t V (John came), N t V N 
(Jipb~ht books), N t V P N (people rely on John) , etc. (A 
subcategorization of V's is implied here and is not explicitly 
shown). Basic adjunct strings are basic adjunct forms, e.g., P N 
(from Philadelphia), A (old), wh N t V (whom John saw), wh t V N 
(who saw Jim), D (~--~, etc. Each de-~Ive'~-r~-~ in L(G) is 
thus a derived sentence form, e.g., (assuming suitable adjunction 
rules), N P N t V N (a man from Philadelphia bought books), A N t V 
(an 61d man came), N wh ~ t V t V D (the man whom Bill saw ran 
~), N wh N wh t V ~ t V t V D (~he books (which) the man who 
met Jim boushtwill arrive soon) , etc. (ignoring articles for 
simplicity). 
In an AG, lexleal insertion takes place as each basic string 
is brot~ht into the generation of a sentence. Let a i = ailal2 .... aim ' 
a i e A be a basic string. As o i is brought into the generation of 
J , , a sentence, each aij can be rewritten as s set, say, ~, of syn- 
tactic features and-dictionary items can be inserted immediately. 
The verification of selectional restrictions that hold within the 
domain of a basic string can be inmediately carried out as any pair 
of adjacent symbols of ~i are contiguous at this stage. If the 
basic strings are properly chosen then most selectional restrictions 
are brought to bear within the domain of some basic string, and in- 
deed it turns out that basic strings (with reasonable linguistic 
interpretations) can be set up in this way. 
There are some restrictions which hold between a host and an 
adjtmct string; e.g., in N wh N t V t V (the man whom John met 
arrived), wh N t V is an adjunct of N t V and the N in N t V is 
really the 'object' of V in wh N t V. Some other examples are: 
Zeroing in conjoined sentences, e.g., everyday, he runs and swims; 
he plsyed tennis hut she didn't, etc. Restrictions between suc- 
cessive adjuncts at ih6 same p0int of adjunction of the host 
(ordering restrictions) as in I sm lqokin~ for a book with a green 
eoverwhish was lying here somewhere. Restrictions between a ~ost 
and two or more adjuncts at different points of edjunctlon of the 
host as in boMs who ca w swim distrust boys who 9an't. All these can 
be easily veriffed. 
• AG's are well suited for formulating the 'endocentric' proper- 
ties in the sense that this aspect of a constituent can be explicitly 
brought out in the structural description. There are, however, 
constituents which are not 'endocentrlc'. These are 'exocentric' 
in the sense that we cannot replace them by any word of a character- 
iziug category contained in them such that the constituents can be 
considered as constituent expansionsof the characterizing category; 
e.g., who willrepresent us at the meetin ~ in who will represent us 
-Ii- 
at the meetln~ is unclear, etc. AG's are not well suited for 
formulating the exocentric properties. These properties are better 
characterized bythe use of a 'nonterminal' and 'rewrite rules' in 
the sense of a PSG (see Section 3)- 
Sentence adjuncts (e.g., in general, today) can be handled well 
in an AG; in particular, that these adjuncts can occupy various 
sentence positions can be easily characterized in an AG. This is 
awkward to characterize in a PSG. However, the property that these 
adjuncts ere adjuncts of a sentence is better characterized by the 
use of a nonterminal. 
Distributed adJunction rules are required to handle eases such 
as two and three are even and odd numbers respectively which is a 
case of an intercalated structure. Such structures are not too 
• frequent. However, if one tries to construct AG type gran~ars as 
base for transformational grammars then the need for intercalated 
structures is not so marginal. This is primarily because one tries 
to relate each adjunct to an elementary sentence (i.e., one tries 
to constitute the adjunct and host strings in such a way that the 
underlying elementary sentence (s) could be reconstructed frcan them). 
Some examples are: the man who came ... (double underline indicates 
the distributed adjunct); John's proof of the theorem, etc. (see 
Hi~ and Joshi (1967), Joshi (1906, 1969), for further details). 
The kinds of intercalated structures possible in a DAG apparently 
are adequate for this purpose. 
If E = ~c then each string w c L(G) has a representation in 
terms of basic 'sentences' (or basic center strings). In general, 
adjuncts are not strings in ?~ and hence ~ $ E c. But what seems 
to be true of a natural language is that one can 'almost' construct 
an AG, G, (actually, a mixed AG, see Section 3) for which ~ = ~c 
and define a set of operations (these consist of permutations, 
deletions, and additions of constants; these operations can be 
related to transformations in a given language) for each ~ c E 
and for strings derived from ~ such that we can construct a new AG, 
G', with basic strings E', ~c' where E' $ ~c' and strings in E' 
are transformationally derived from strings in ~. Strings in L(G') 
except for morphophonemic operations are the strings (sentences) 
in the language. In transformational analysis we go in the reverse 
direction, i.e., from G' to G and reconstruct the set of basic 
'sentences' (together with the derivation in G) underlying a given 
sentence generated by G'. (See Joshi (1969) for further results 
and details. ) 
t 
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3- Mixed Grammars 
3.1 
In AGN's (i.e., AG's with null symbols) in Section 2.6 we use 
a very special type of null symbols which are ultimately erased.* 
The only rewrite rule associated with a null symbol is ~ - 
where ~ is a null symbol and ~ is the null string. The AGN's are 
thus a class of mixed grammars as rules of more than one style are 
used. It is, huwever, a rather simple class of mixed granmmrs. 
3.2 Mixed Strin~ Adjunct Granmmr (M~G) 
We will now consider a more interesting class of mixed grammars. 
The main idea is to allow a single 'nonterminal' (in the sense of 
a PSG) in an AG and a special type of 'rewrite rules' associated 
with this nonterminal. We will, however, call them 'replacement 
rules'. The reason for adopting this new terminology will become 
clear later. More specifically, we will define a Mixed Strin~ Adjunct 
r~ (with replacement rules) (MAG), G, as follows. 
First, in addition to the 'terminal' alphabet A we will have 
a 'nonterminal' S. The set of basic strings, E, and the set of 
basic center strings, Zc, will now be strings on (A U \[S}). Thus 
a string a i c ~ may now contain one or more S's in it. A oi c 
which does not contain S will be called an elementary basic string 
and a ui ¢ E which contains one or more S's will be called a 
basic string. The adjunction rules (local or distributed) are 
defined as before and we adopt the same convention as in the case 
of the null symbols, i.e., in an adjunction rule if the host is a 
complex basic string we disregard the S symbols in it as far as 
points of adjunction are concerned. Thus the points of adjtu%ction 
of a string, say, a i = aS bS are the points of adjunction which are 
to the left and right of a and b. Further, the S symbols have no 
points of adjunction. Of course, ~f an adjunct string, say, ~ is 
a complex basic string and has a specified segmentation, then ~ach 
symbol S in oj must be included in some segment of aj. 
* Actually, it is possible to define the recursive extension of 
the adjunction rules such that the null symbol associated with any 
basic string is erased at the time the string is ad.ioined to some 
host string. The null symbol associated with the center string 
is then erased at the end. 
- 13 - 
We now define a replacement a rule Pij (often written Just 
as p) as a pair < ai, oj >where a i c ~, a i is a complex basic 
string, and a S ¢ ~c" The w~aning of a replacement rule p = 
< ai, aj > is that ~r~ a i one can derive a new string by replacing 
some occurenee of S in a i by a.. Thus we have the following S 
Definition ~.2.1 An MAG, G, is a 9-tuple, G = (A, S, E, E c, ~, 
• Ea' E~' J' R) where A is the alphabet (terminal), S is a 'nonterminal' 
s~m~ol (S ~ A), Z is the set of basic strings, ~c is the set of basic 
center strings, E h is the set of basic host strings, E a is the set 
of basic adjunct strings, Z r is the set of basic replacer strings, 
J is a finite set of adjunction rules (local or distributed), and 
R is a finite set of replacement rules. ~ = \[oi!(ai, aj, ~k ) ¢ J) 
u (%1 < 5, aj > ~ RL z a = {ojl(oi, ~j, ~k) ~ J}, and 
Z r =\[ajI< ai, aj > ¢ R\]. Z = ~e U Z h U Z a U Z r. Z c may contain 8 
h~ ~ 
Given J and R, Zb, Za, and ~r are completely specified and 
= ~c U Z b U ~a U Z r. A need not be explicitly stated. Since S 
is the only 'nonterminal' it need not be explicitly stated also. 
}~nce, we will write an MAG, G, as a triple (Ec, J, R) instead of 
a 9-tuple as in the definition above. 
Example ~.2.1 Let G = (~c' J' R) be an MAG where Ec = \[abc, 
pq, ahSc}, J = {u I = (abSc, (a)(b)(c), ~1~2~3), ~ = (abe, (a)(b)(c), 
~I~2~3)\], and R = {Pl = < ahSc, abSc >, P2 = <abSc, pq ~\]. 
Example ~.2.2 Let G = (7~c, J, R) be an MAG where ~c = lash, C\], 
J = {u I = (aSh, (a) (Sb), rlr2)\], and R = \[Pl = <aSb, aSh >, 
P2 = < ash, c >}. 
3-3 Mixed String Adjunct Language (MAL) 
We now have two different styles of rules in G, namely, the 
adjunction rules J and the replacement rules R. If R is empty then 
we have an AG and we know how the generation proceeds in this case. 
In particular, we note the 'inside out' characteristic of the 
generation. If R is not empty then we have replacement rules 
associated with the symbol S. The generation still proceeds in an 
'inside out' manner. In order to define the language corresponding 
to an FAG, we must state how the rules in J and R are extended to 
derived strings. Rather than giving a precise definition, we will 
illustrate the main idea by the following example. 
Example 3.3.i Consider the MAG, G, in Example 3.2.1. Starting 
with the complex basic string abSc and using it as a host in u I 
and the string abc as an adjunct in Ul, we obtain aabbScc. Usxng 
this as a derived host in the replacement rule p~, we obtain 
aabbpqcc. But this is a string derived from abS~ and therefore 
it can be used as a replacer string in p~. Thus we can obtain 
aabbaabbpqcccc (see Fig. 3.3.1). The language L(G) = L I is 
As far as rules in J are concerned we require 'inside out' 
generation. In order to define consistently the recursive exten- 
sion of rules in J and R together it is necessary that once a 
replacer string replaces an S no further adjunctions or replacements 
can be made on it. Thus before a replacer string is used it must 
be completely built up (i.e., it must have received all its adjuncts 
and adjuncts of adjuncts etc., and all occurences of S must have 
received their replacements).* This was the reason for calling 
the rules in R as replacement rules rather than rewrite rules. 
Thus the generation begins from either (a) the 'innermost' 
host - adjunct pair(s) or (b) the 'innermost' complex basic host - 
replacer pair(s) where the replacer is an elementary basic center 
string, or (e) both (a) and (b). 
The reader may amuse himself by working out the language (1%) 
corresponding to the MAG in Example 3.2.2. It is rather complicBted 
to write it down in a parametric form. 
* It is assumed that for every complex basic string, say, ~i' either 
there is a rule < ~i, a.S > where ~j is an elementary basic string 
or there is a sequence 6f rules < Oil , ajl >, < si2 , ~J2 >' "''' 
< gin' ~Jn >where = = • k = l, 2, ... n-l, and si I ~i, SJk ~Ik+l' 
~Jn is an elementary basic string. Otherwise, ~i can be removed 
from G without affecting L(G). 
- 19 - 
Both L I and L 2 are CSL's (Context sensitive languages). They 
are both DA~'s also. This can be shown by constructing the equi- 
valent'DAG's. Let G I = (~c' J) be a DAGwhere ~c = {abpqc, pq\], 
and J = {u I = (abpqc, (ab)(c), r2rg) , u 2 = (abpqc, (a)(b)(e), 
~i~2~5) \]. Then L(GI) = L I. Similarly let G 2 = (Ec, J) be a DAG 
where Z c = {acb, c\], and J = \[u 1 = (acb, (a)(cb), rlr3) , 
~2 = (acb, (a)(b), rlr3) , u 3 = (acb, (a)(b), ~2~3) , u~ = (ab, 
(a)(b), rlr2), u~ = (ab, (a)(cb), rlr2) }. It can be shown that 
L(G2) = L 2. In fact, we have the following 
Theorem ~.~.I For every NAG, G, there is an equivalent DAG, G', 
(i.e., L~G) = L(G')) which can be effectively found. 
We will omit the proof here as it is rather long. An examina- 
tion of MAG's in Examples 3.2.1 and 3.2.2 and their corresponding 
DAG's, G I and G 2 above will give the reader some indication of hew 
the proof can be constructed. This is an interesting result because 
it shows that as far as weak generative power is concerned, we can 
eliminate S, the only 'nonterminal' we have used. It can also be 
shown (this is easily seen from G I and G 2 above) that if the NAG, 
G, satisfies the condition that E = ~c, then the equivalent DAG, G~ 
will not necessarily satisfy the corresponding condition E' = Z ' 
In fact, for some NAG there will be no equivalent DAG satisfyin~ 
this condition (see Section 3.9.6 for linguistic relevance). 
Remarks 3.3.1 
i. In an MAG, G, not every basic string is a string on A (e.g., 
the complex basic strings). However, in the tree representation of 
derivation of a string in L(G), the derived strings at each node 
are strings on A, just as in the case of an AG. In fact, if this 
condition is not satisfied the tree will not correspond to a tree 
for some string in L(G). 
2. The symbol(s) S in a complex string, say, ~i' will be referred 
to as a contained S. ~. will also be called a container strin~ and 
the repla--c-~s-t~-~ng(s) ~or S will be called contained string(s~. 
Let•~i = abSc, ~j = dSe, a k = gSh, and ~ = pq. Let u I = (abSc, 
dSe, rl) be an adjunction rule and Pl = <abSc, gSh >, P2 = <abSc, 
Pq >' P3 = < dSe, pq >, and P4 = < gSh, pq > be some replacement 
rules. Consider the following tree representation of a derivation 
(Fig. 3.3.2: the superscripts on ~ are used to distinguish the two 
i is contained distinct occurences of the string pq). Note that a~ 
-16- 
o.k, c~c 
C (~) c~b g~ 
(4) a.bc 
-6-Ae ¢mode$ 
"~ 3~4-s 
~.. 0.be 
z. F9 
q. ~c 
,5, o_o,%b<xo, bb ~Iccc~ 
FI&. 3. 3, I. 
W= ~.bloo.~bh~%cccc ='~n MAGr G 
~ E~"~'~ ~.2. "f. 
= aSe 
,l 2. 
o.~Sc = o-,: 
-17- 
FiG "~.%. 2.. 
A 
in uj and aj is adjoined to o., and u~ is contained in and 
Uk ° k 
'depth' 2) with respect to ~i and this is so both in the sense of 
a PSG and an MAG. Now ul is two levels down where the first level 
is due to an adjunction ~ud the second due to containment, but 
is two levels down where both levels are due to containment. 
Thus in an MAG the depth of embedding of string can be characterized 
not only by the number of levels involved but also by stating for 
each level whether it is due to adjunction or containment. There 
is also the possibility of characterizing an arbitrary depth of 
embedding in terms recurrent patterns of adJunction and/or con- 
tainment levels. 
3.4 Deformations and Generation Scheme 
In this section we will be concerned with the construction 
of an MAG, G, with ~ = ~c for an MAG, G', (for which ?2 ~ ~c', 
in general) such that G is related to G' by means of certain 
operations (see Section 2.11 - last paragraph). 
Let A = Ill' 12' "'" ~n \] be a finite set of deformations 
(to be defined later). 
Definition 3.4.1 A Generation Scheme, G~ is a pair (G, A) 
where G is an NAG, G = (~c, J, R), A is a f~nite set of deformations, 
and with each rule u e J and each rule p ¢ R, we associate unique 
subsets of A, say, ~u and ~ respectively. 
Let C be a finite set of constants. We say that a string s' 
is a deformed a if every symbol of ~' is either a constant (i.e., 
is in C^) or is a symbol of a or both, That is, a' is obtained 
from a Ey one or more of the operations of permuting, deleting, 
or repeating symbols of a or adding one or more symbols from C . 
0 
Definition ~.4.2 Let u = (~i, a~, ~k ) c J and let d u be the • ' v 
assoclated subset of deformations. Then a Ji ¢ Au maps the rule 
u = (ai, aj, ~k) into a 3-tuple (ui', aj', ~k') where a i' = ai, 
aj' is a deformed aj (the specific choice of operations of permuting, 
deleting, or repeating symbols Of a j or adding constants is deter- 
mined by ori ) , and ~k' is an adjunction 'vector' of ui not necessarily 
- 18 - 
L 
the same as ~k" We write this as Ji(u) = u' where u' = (ai' , 
uj', ~k').* Similarly, a li¢ Ap, P = < a i, aj >, maps p into 
a pair < oi' , oj' >where a i' = oi and aj' is a deformed aj 
( h i determines the specific deformation as before). We write 
this as li(p) = p', where p' = < ai' , oj' >. 
Note that li deforms uj and also specifies a new adjonction 
'vector'. Note also that u and p are rules in the MAG, G. u' 
and p' are not rules in G. However, they will be later interpreted 
as rules in another MAG, G'. 
Each oci can then be extended to the derived hosts and derived 
adjuncts in u, and to derived hosts and derived replacers in p in 
the obvious way (i.e., if a symbol of ~ is permuted it carries 
with it its adjuncts, and their adjunct~ etc.~ if a symbol of G~ 
is deleted then we delete its adjuncts, and their adjuncts etc., 
addition of constants is not affected). More complicated extensions 
have to defined however for the more complicated li's. (See Section 
3-5-5, and for further details, Joshi (1969). 
For a given Generation Scheme, G s = (G, 4), we will define the 
l a~ngu~e__ ~c°rrespondi~ _to Gs,~L G~, as follows. We will give her--~ 
only an informal definition. We carry out the generation in the 
MAG, G, as described in Section 3.3, with the following modification. 
If during the generation we plan to use a rule u then instead of 
i~n ~ ~ a~ ~. Similarly,p weif using u we use the rule u' = ~ (u) where ~e we plan to use a replacement ~e p then s of using use 
the rule p' = ~m(P) where 6 m c ~p. The choice of or~ from ~u and 
~m from ~ is nondeterministic. 
* This definition is not quite precise as stated. First, note that 
~k' is an adJunetion 'vector' and hence its components must satis- 
fy certain conditions (see Section 2.5). Secondly, if ~k' has 
more than one component then 6 i must also specify the appropriate 
segmentation of ~j '. This definition is also weaker than required. 
More c~nplicated ~.'s can be defined and are required (see Section 
3-5-9) - i 
- 19 - 
Note that here we use the word 'language' in the usual sense, 
i.e., a set of strings on A. Ultimately, however, we are inter- 
ested in the corresponding strings of lexical items. The lexical 
insertion proceeds in G in the same general manner as in Section 
2.11 (paragraph 2), (see also Section 3.~). In thls case the 
choice of a J~ from ~ and of Jm from ~ may depend on the 
lexical entries for a i and a. which are, of course, available 
at this point in the generation. 
The 'language' derived in this way is L(Gs). The derivations 
are not in G but in G s. We could, of course, allow the generation 
in G to proceed independently but concurrently with the generation 
in G . In this case, we would derive a pair of 'corresponding' 
strips, say w and w s where w c L(G) and w S ~ L(Gs). Note that 
G S is not an MAG; however, we have the following 
Theorem ~.4.1 For every Generation Scheme, G S = (G, 4) there 
is an equivalent MAG, G', (i.e. L(G') = L(Gs)) , and G' can he 
effectively found. 
The proof is rather involved and we will omit it. At least 
for the 6's in Definition 3.~.2 one can state the main idea as 
follows.* Let G = (~ , J, R) and G' = (~ ', J', R'). Then 
i. ~c' = ~c ° 2. j'Cobviously contains ~ll the u' = 6i(u), u ~ J, 
as adjunction rules. But J' also contains some additional rules 
which are needed for the following reason. Let a~be an adjunct 
string in G and let some ~deform aS into aj'. N~w, if a.j is also 
a host string in some adjunction rule, say, u in J then we must 
add a new adjunction rule(s) in G' which in e~fect allows one 
to adjoin the adjunct in u (actually, its deformations under all 
possibleS's) to a~', withrthe adjunction 'vector(s)' appropriately 
specified. 3. R'Uconsists of all the p' = ~i(p), p ¢ R, as 
replacement rules. R' also contains some addltional rules which 
are needed for the same reason as in 2 above. 
We can impose the condition ~ = ~c on the MAG, G. But then 
G' (equivalent to G S = (G, 4)) need not satisfy a similar condition, 
i.e., ~' need not be equal to ~'. This is because ~' contains, 
besides strings in ~c' (= ~c )' the deformed adjuncts and deformed 
replacer strings. 
I 
* The proof extends to s~ne of the more complicated J's (see 
Section 3.~-~)- 
- 20 - 
From Theorems 3.~.i and 3.3.1we then have 
Corollary~.~.% For every G s = (G, A) there is an equivalent 
DAG, G", (i.e., L(Gs) = L(G")) which can be effectively found. 
3.9 Linguistic Relevance 
In this section, we will briefly discuss the various results 
in Sections 3.1 - 3.4 in the linguistic context and provide some 
Justifications and interpretations for these. 
3.9.1 
As is evident from the discussion in Section 2.11, the main 
motivation for considering FAG's is to provide suitable representa- 
tions for certain structures (e.g., that he went home surprised 
me, I told him to 5o home, t.hat John will come is Certain, I tried 
to read the book, etc.). The purpose for considering MAG's with 
= ~c is the same as in Section 2.11 (last paragraph). 
3.5.2 
In Section 2.11 we have seen that many restrictions have as 
their domain a basic string or a basic string and its adjuncts; 
and these can be easily stated and, at the time of adjunction, 
easily verified. These remarks obviously hold for an FAG as far 
as adj~netions are concerned. However, in addition to these, in 
an FAG there are many restrictions which have as their domain a 
complex string and its replacer(s) string (i.e~., a container string 
and the contained string(s)). These also can be easily stated and, 
at the time of replacement, easily verified. Apart from selectional 
restrictions, some of these restrictions are: (a) Identity of one 
of the N's in the container string and one of the N's in the con- 
tained string (identity of the 'subject' or 'object' noun in the 
container string and the 'subject' or 'object' noun of the con- 
tained string), e.g., I told. John to go home, I promised Bill to 
purchase books, John deserves promotion, He suffered defeat, I 
forced him to s~rim, He is uninspirin 5 as a teache__~r, I saw the bo~ 
bein~ beaten by the policeman, etc. Actually, since we are 
considering derivations in an MAG with ~ = ~c, we should have 
written these somewhat as follows: I told John" (John" should* 5o 
home), I' promised Bill (l'would purchase books ), John'dese'rves 
NV~-~should* pror~ote John'l, He '~ suffered (N defeated hin0, He ~ is 
unlinspiring (He'1% teac h N to N), etc. ( , marks the eleme-~ with the 
same reference; %: untensed (or tenseless) 6). (b) Certain 
conditions on replacing a noun by a pronoun, e.g., Joh~ hoped (he" 
perhaps %. 
- 21 - 
/ 
J 
will wln~ but not Heehoped (John'will win), etc. (c) Possible 
correlations between tenses in the container and contained strings 
(see examples in (a) above). 
Later, in the context of generation in G s = (G, A) we will 
discuss some additional restrictions. These do not affect the 
'well-formedness' (with respect to the satisfaction of restrictions) 
in G. Thus the strings of lexical items corresponding to strings 
in L(G) are also 'well-formed'. 
3.5-3 
We now consider the derivations in G s = (G, A). Obviously, 
the purpose of 6's is to obtain the corresponding strings in 
G s (i.e., also in G'), e.g., Ltold John to gq home --I told Jo_~ 
~d go h °m-e), etc. Note that for each rule u or p in G, 
the 6's are selected from A u or ~ respectively. Some examples 
of restrictions on ~'s are: (a) The choice of a ~ from A u (or Ap) 
may be affected by the lexical entries for the container and 
contained strings, mostly by the verb (including is___~A and is___NN) 
of the container string, e.g., I tried to drive a car, I tried 
drivin~ a .car, but ~nly I stopped drivin~ a car; That he answered 
the .letters is true, His answering the letters is strange, but 
not His answerin5 the letters is true, etc. (b) Choice of a 
particular preposition in a deformation may depend on the lexical 
entries for the container and contained (?) strings, e.g., I 
know of John's coming, I believe in (my) leaving early, etc. 
(see Section 3.~.5 and Joshi (1969) for further details). 
3.9.4 A Simple Example of G, G s and G' 
Let G = (Ze, J, R) be an NAG (with Z = Z c) where A = \[N, t, 
V,A\];Ze=Z=\[~i=NtV, a 2=~tvN,o3=~tvS,o4= 
N t VN S, a9 = S t VN , a6 = S t VA, c 7 = S t V S}*; 
J = \[(al, aj, rl\] U ~(o 2, ~j, r I) O \[(a 2, oj, r~¢)} O ~(o 3, oj, rl) 3 
U {(a\]¢, oj, rl)~} U {(ab, o~i' rl~ \] U \[(aS, (~j, r3)}, J = I, 2, 3, b~) 
* A subcategorization of N's, V's, and A's is implicit here and 
is not shown explicitly in the notation. 
- 22 - 
R= {<al, aj >I i=3, ~, 9, 6, 7; J =I, z, ..., 7}. 
Let G s = (G, A)be a Generation Scheme where A = \[ or i} is 
a set of deformations ~d the ~'s are defined as follows. 
(Set of Constants, Cq = \[wh, __that, ~t°' __,'sing\]) 
~l: (ai' aj, ~k) ~ (ai, aSl , ~k) where i = i, 2, 3, ~, 9; J = i, 
i =wh t V, I z, 3,~; (a i, aj, ~k) c J; and% ~2 =whtv~' 
aS =wht V S, a =wht VN S.* 
2 62: (ai, aj, ~k ) --~ (ai, aj, ~k ) where i = 1, 2, 3, ~, 5; 
2 =whN t V, j = z, ~, 5; ("i' aj, %) c J; and a z 
2 =wh S t V. =whNt V S, o 9 
 S: < ai' 
S= ~ 
: < a i, 
< a i, 
< al, aj S > where i = S, ~, 5, 6, 7; a S > ..@ 
i, 2, ..., 7; < a i, aj > c R; and aSS = that aj. 
4 aS > ~< ai, a S >where i = S, ~; J = i, 2, 3, 4; 
a S > ~ R; and af' =to V, a2~ = toVN, a34 = toV S, 
toVN S. 
~9: < ai' ~J > ~ < ~i' ~j9 >where i = 3, 5, 6, 7; J = I, 2, S, 4; 
< ai, aj > e R; a19 = (N's) V ing, a25 -- (N's) Ving N, 
a35 = (N's) Ving S, ~45 = (N's) Ving N S. 
We have not shown explicitly the various subsets of A 
associated with the rules in J and R in G. But, these can be 
worked out fr~n the specification of the o~'s above. 
\ 
\ 
Actually, we should use here a distributed adjunction rule to 
account for the definite article to the left of N in the host. 
We leave this out in order not to complicate the example. 
- 23 - 
/ 
An NAG, G', equivalent to G s =(G, A) can now be easily 
constructed (see the discussion under Theorem 3~4.1). We will 
not write G' here as it is too long. It is easily seen that we 
can derive in G s =(G, A) and therefore in G' sentences such as, 
John wants to ~o home, I prefer walking, The man who came ordered 
Jim to shut the door, ! promised Bill to tell John that he should 
visit Fred, Bill's forcin 6 John to resign annoyed him, 
the doors blue was the custom, My asking him towrite a paper 
caused his leavin 6 the job, etc. 
3.~.~ Some Complex Deformations* 
Some examples of deformations more complicated than those in 
Definition 3.4.2 ~re as follows: 
a. A J may be defined such that it requires the adjunct string, aj, 
~n the rule u = (~i' ~j' ~k ) to be a derived string. ~ then refers 
to not only ~ but string(s) which maybe at most a fixed finite 
depth (counted in terms of adjunction and containment levels) rela- 
tive to o.. Mostly depth i (and occasionally depth 2) is adequate, 
e.g., The man who had the keys finally came ~The man finall~ 
came who had the keys: The man who finall~ came who had the keys ... 
b. A J may be defined as above but with the possibility of 
--o r referring to not only o. but a string which is at an arbitrary 
depth relative to ~. where ~he arbitrariness of the depth is so 
constrained that itScanbe specified in terms certain recurrent 
patterns of adjunction and containment levels, e.g., The meetin 6 
(which) I selected John to represent us at ..., The people we hope 
that John told to water the plants ..., etc. \[Although slightly 
out of place, it might be worth mentioning here that the distinction 
between adjunetion and containment levels also helps in stating 
certain pronouning restrictions to some extent, e.g., the pronoun 
is in the contained string and not in the container string: John" 
thou6ht he'will win but not He'thou6ht John°willwin; but if-~ 
have an adjunction level then we have both: People who know Bill" 
like him~ and People who knowhim~like Bill~ if we have an adjunction 
level and a containment level, we again have both: People who know 
Bil~ want to help him°and People who knowhim~want to help Bill~ 
and if we have two successive containment levels, we have John'asked 
Bill to tell Mary to see hi'but not He" asked Bill to tell Mary to 
see John5 etc. S 
* For further details see Joshi (1969). 
- 24 - 
! 
/ / 
/ 
C 
c_. A ~ may be defined such that it not only deforms the adjunct 
string aj but also deforms ~he host string a i. Since the host a i 
can also-be deformed by such a or, the precise definition of how 
generation proceeds in the Generation Scheme, Gs, becomes complicated. 
Such ~ 's can be used to obtain from the same* container string 
--contained string pair~two related sentences such as, e.g., Ths__ tt 
he came surprised me and it surprised me that he came, etc. 
d_. Sets of related l's to cover certain zeroings which have as 
their d~nain the container and the contained strings, e.g., I 
promised him to come ~ I promised him that I would come, etc. One 
may also include here zeroing of 'appropriate' verbs, Vamp, e.g., 
I expect him ~ I expect him to Vap p where Vap p = ~_, ~rrive, etc.\]; 
perhaps also I shall ~o ~ I promise you that I shall ~o (Harris (1968). 
3.9.6 
From Theorem 3.3.1 we know that for every NAG there is an 
equivalent DAG which can be effectively found. This means that we 
can eliminate the nonterminal S as far as weak generative capacity 
is concerned. Of course, we don't choose to eliminate S, but it is 
interesing to see the implications of this theorem. If one examines 
the proof of this theorem, we notice that in effect for every complex 
basic string, say, a i = abSc and for every elementary basic string, 
say, pq, which is a replacer for ai~,@we set up an 8djunction rule 
(in this case a distributed rule) such as (pq, (ab)(c), ~ir2 ). Thus 
we will have to consider I know that in I know that John went h~ne 
as an adjunct of John went home. Now (in the spirit of the discus- 
sion in Section 3.4) adjuncts are obtained by deforming a string in 
Fe; also adjuncts have a certain degree of mobility within the host. 
This is perhaps the reason why in some cases we come close to reali- 
zing this, e.g., (i) I hope that John will win: we can obtain a 
sentence and a semi-sentence, John will win} I hope so or a sentence 
and sentence adjunct, John, I hope, will win, (2) That John passed 
the examination sur2rised me: John passed the examination} it sur- 
~rised me, John passed the examination t to mY surprise, etc. 
* This avoids having two distinct strings in G generating strings 
which are paraphrases of each other. If, however, we allow this 
possibility the structure of G s can be considerably simplified. We 
do not follow this approach but the nature of these simplifications 
is discussed in Joshi (1969). In a different context and in a 
different framework, Keenan (1969) has made a similar comment. 
- 25 - 
d 
3.9.7 
In section 3.~, in G S = (G, A) we imposed on G the condition 
that ~ = ~. Then in G' (equivalent to Gs) every adjunct string 
is obtained by deforming some string in Ec(= E) in G. However, 
adjuncts such as, e.g., quite in quite forgot, ~ry in very long, 
sc~e quantifiers (all, sc~e, etc.), sc~e occurences of articles, 
sc~e time and manner adverbials, etc. pose a problem here. There 
are a couple of ways around this problem. 
One solution is to consider these adjuncts as primitively 
adjoined in G (i.e., regard them as a sort of primitive adjuncts 
in G)*. G, of course, will no longer quite satisfy the condition 
E=E . c 
Another more attractive solution (certainly, motivated by some 
current trends in transformational theory) whichwill maintain the 
condition that every basic string is also a basic center string is 
to construct (i) another NAG, G", by retaining all strings in ~e 
in G, excluding the primitive adjuncts in G, but adding new complex 
basic center strings (these will now more and more become infra- 
~entence forms), and also adding new adjunction and replacement 
rules, and (2) a new Generation Scheme Gs" = (G", A"), where A" 
is a new deformation set, and G" satisfies the condition ~" = Zc" , 
such that Gs" is equivalent to G. (At this point, we may also 
remove the tenses, auxiliaries, and prepositions. Basic strings 
in G" will then be strings of N's, V's (including isA and isN) 
and S's). Thus we have the alternating sequence of MAG's and 
Generation Schemes**. 
G ...... G s = (G", ~") ~G: Gs(G , ~) ~G' 
and 
T,(G") : L(%") = L(G) : LCGs) = T,(G') 
* Note that there are very strong restrictions on the repeatability 
of these primitive adjuncts. 
~* In principle, we could consider arbitrarily many intermediate 
stages, between the first and the last MAG's. However, there would 
not be much point in considering such sequences, unless each inter- . 
. L • mediate stage has some reasonable llnguistlc interpretatlon. 
- 26 - 
where G" underlies G and G underlies G'. Further development 
of these ideas in so~e detail will be reported in Joshi (1969). 
Ap~ndix: Fig. A, su~narizes the hierarchy of certain subclasses 
of AL's and MAL's in relation to the phrase structure hierarchy. 
(The replacement rule in an NAG can be generalized in such a way 
that all occurences of S in a complex basic string are simultaneously 
replaced by a specified set of replacer strings. We call such a 
grammar an MAG with simultaneot~ replaceme~rul_ess, MsAG , and the 
corresponding language, MsAL. It can be shown that MAL ~ MsAL 
CSL. An MsAL has the property that the lengths of the strings 
in it (assuming an ordering in terms of increasing lengths) grow 
no faster than an exponential. The whole class of MsAG's as such 
does not appear to be linguistically relevant. ) 
~- - 27 - 
/ r 
CSL 
CFU % ~nr- L 
P,~- 
// LsI~ 
L ALAI~ 
/. e n9 ~.e.Dg~.) 
I,c|,~;.,, w~/, Co,dec'~",e'~ ET'~'lc*..y 
F,~. /~. 
- 28 - 

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