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<Paper uid="C69-0101">
  <Title>TREE GRAMMARS ( = 4- GRAMMARS )</Title>
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TREE GRAMMARS ( = 4- GRAMMARS )
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    <Paragraph position="0"> - GladkyA.V. (Novosibirsk) ~ Melt~uk I.A. (Moscow) i. This paper suggests a new kind of formal grammar (hereafter called  A- grammarS) which in some respects is closely related to Chomsky's grammars but differs from these in that it is meant to process trees (in the sense of graph theory) and not to process strings as Chomsky's grammars do. More precisely, we aim at a type of grammar with rewriting rules of the'X ~Y&amp;quot; where X and Y are trees (N.B. : with no linear order imposed on their nodes ~).</Paragraph>
    <Paragraph position="1"> Linguistically, the trees under consideration are dependency (not phrase structure) trees representing natural sentences at different levels of &amp;quot;depth&amp;quot;: roughly speaking, &amp;quot;surface&amp;quot; syntax, ,'deep&amp;quot; syntax, semantics. A- grammars are designed to be used not for generating sentences but rather for transforming given trees into other trees; this covers transitions from one abstract representation of a natural sentence to another (deeper or more superficial) representation of the same sentence as well as transitions from an abstract representation of one sentence to a representation on the same level of another sentence, synonymous to the given One. The conversion of a ~ready ~ surface tree into an actual sentence - a conversion consisting of a) inflexion and b) determination of word order - must be carried out by some autonomous device not included in the conception of A- grammar.</Paragraph>
    <Paragraph position="2"> f~l&amp;quot; _ _ *. ~ from the Greek cJPSv~@poV(tree).</Paragraph>
    <Paragraph position="3"> ~'~. The limitatlons of place and time prevent us from comparing tree grammars with those of Chomsky as well as from referring to other works dealing with more or less analogous matters, such as studies byM. Arapow and V. Borschtschow; G. Veillon, J. Veyrunes and B. Vauquois; Ch, Hockett; and other s.</Paragraph>
    <Paragraph position="4"> The authors are glad to acknowledge here the friendly help and useful suggestions byO.S. Kulagina and A.Y. Dikovsky.</Paragraph>
    <Paragraph position="5"> All shortcomings in the paper are, of course, ours.</Paragraph>
    <Paragraph position="6"> The A-grammar embodies an attempt to formalize the linguistic &amp;quot;Meaning ~Text Model&amp;quot; described, e.g., in Ill. In this model, the starting point for producing a sentence is a detailed semantic description of its meaning conceived as a rather involved graph (not merely a tree) consisting of &amp;quot;semantic atoms&amp;quot; and &amp;quot;semantic links&amp;quot; connecting them. The semantic description is generated outside of the linguistic model and constitutes the input of that model; it is then subsequently &amp;quot;lingualized&amp;quot; (anglicized, russianized etc.) by means of formally specified transformations: i) extracting from the given semantic description (of a family of synonymous sentences conveying the meaning repret sented by that description) the deepest admissible tree-like structures; 2) proceeding in a multi-step fashion from the deeper trees to the more superficial ones; 3) linearizing the most superficial syntactic trees (with simultaneous inflexion where needed) to produce actual sentences. The n-grammars deal with the second phase of this . process only.</Paragraph>
    <Paragraph position="7"> Z. We shall consider trees with labelled branches; nodes are not labelled.</Paragraph>
    <Paragraph position="8"> The labels can be interpreted as names of the types of syntactic link at the corresponding level. For brevity~s sake such trees will here be referred to just as &amp;quot;trees&amp;quot;.</Paragraph>
    <Paragraph position="9"> A tree is called minimal if all its nodes, except the root, are terminal (i. e., with no branches growing out of them). A tree with but one node is called an empty tree and is denoted as C/. The composition of trees is defined as follows: let to, tl, t2, .... tnbe trees, and let in t o some nodesa I, a~, .... a n (not necessarily pairwise different) be marked. Then the result of the composition of the tree t o with the trees tl, t2, . .., t o will be any tree isomorphic to the tree which can be obtained from t o by identifying the roots of the trees tl, t2, ..., tn with the nodes c~ I, a2, ..., an, respectively in t o .</Paragraph>
    <Paragraph position="10"> The composition of t o in which the nodes al, a 2, .... a n are marked with tl, t 2 ..... t n isdendegted</Paragraph>
    <Paragraph position="12"> where a 0 is a terminal node of TO, and ce~, c~ 2 ..... a n a repetitionless enumeration of all nodes of t.</Paragraph>
    <Paragraph position="13"> Now, a n elementary transformatign (ET) of trees is an ordered triple &lt;tl, ~, f&gt;, where t 1 and t2are trees and fis a mapping of the set of all nodes of t 1 into the set of all nodes of t 2 . Instead of &lt;tl, t2, f &gt;, we shall write t 1 ~ t21 f. The tree T f is said to be the result of the application of the ET tl=t 2 I f to the tree T if T and T t can be represented in the form:</Paragraph>
    <Paragraph position="15"> where cr 0is a terminal node of T o , and c~1, cr 2 ..... ~n a repetionless enumeration of all nodes of t 1 . Informally, an application of certain ET to a tree T consists of the substituting of t 2 for an occurence of t 1 in T ; if ~(a node of t I ) is mapped on 8( a node of t 2), i.e., B= f(d), then all &amp;quot;untouched&amp;quot; nodes of T &amp;quot;pending&amp;quot; from ceare transferred to B with the same labels on corresponding branches.</Paragraph>
    <Paragraph position="16"> Example: E Let tl = ~//~ , t2= a/~d G and let f be specified H I as follows: f(A) = E, f(B) = H, f(D) = F. Then, applying the ET t~ ~1 f to the</Paragraph>
    <Paragraph position="18"> T contains three occurences of t ; the replaced one is the subtree of T with the nodes M, N, O, Q.</Paragraph>
    <Paragraph position="19"> 3. A syntactic A-grammar is an ordered pairP =&lt;V,n &gt; where V is a finite set of symbols (branch labels) and a finite set of ETIsscalled rules of grammarr . A derivation in a syntactic A-grammar is a finite sequence of trees where each subsequent tree is obtainable from the preceding one by application of an ET ofn. A tree t I is derivable from T inF' if there exists a derivation in~ beginning with T and ending with T I For linguistic applications, it may prove to be of interest to define some specific types of syntactic A-grammars.</Paragraph>
    <Paragraph position="20"> A syntactic A -grammar will be called exp~nding if each rule it contains has in its left side no more nodes than in its right.</Paragraph>
    <Paragraph position="21"> An expanding syntactic h -grammar will be called minimal if in each of its rules of the form&amp;quot;t 1 ~ t 2 \[ f&amp;quot; the trees t I and t 2 can be represented in the form</Paragraph>
    <Paragraph position="23"> where 1) ~ is a minimal tree, Z) ~1, ~2 ..... ~n is a repetitionless enumeration of all nodes of ~, 3) a~ is the root of ~ , 4) f(et I ), f(~2) ..... f(Ctn) are pairwise different, 5) f(cC/2), f(a~3) ..... f(O/n) are terminal nodes of , 6) for every i = Z, 3 ..... n the label on the branch of ~ending in coincides with the label on the branch of V ending in f(cC/ i ) , 7) for all nodes of t differing from ~I , c~2 ..... cl n, the mapping is identical.</Paragraph>
    <Paragraph position="24"> A minimal expanding syntactic A -grammar will be called context-free if in the expressions (5) and (6) the trees ~r0, ~ , ,%r 2 ..... ~n are unity trees.</Paragraph>
    <Paragraph position="25">  4. Linguistic considerations dealt with in &amp;quot;Meaning~-~Text Model&amp;quot; (see, e.g., \[ 1 \]) imply the introduction of a subset of ETIs, ~2ecial elementary transformations (SETfs). A SET is an ET of one of the following three types:</Paragraph>
    <Paragraph position="27"> Let t~ ~ t 2 \[ f be an ET and let M be a set of ETIs. Then the statement &amp;quot;The ET t 1 = t 2 I f can be simulated byETIs of M&amp;quot; means that there exists some finite sequence m 1 , m 2 , .... m n of ET's in M such that for any trees T and T I where T I can be obtained from T by apptlcation of the ET tl ~t 2 \[ f the tree TI can be obtained from T byapplying m 1 , m 2 ..... mn in tandem.</Paragraph>
    <Paragraph position="28"> Theorem t. Any elementary transformation can be simulated by special elementary transformations.</Paragraph>
    <Paragraph position="29"> 5. For the representation of natural sequences it is reasonable to assume not arbitrary syntactic trees but rather a subset of those - namely, those with limited branching. The precise meaning of limited branching is as follows: for each branch label a~ there is fixed an integer n i such that each node can be a starting ppint at moat for ni branches labelled a i . The trees meeting this restriction are called (n 1 , n 2 ..... n k ).-regular (k being the number of different branch labels); for brevity we shall call these trees simply regular trees.</Paragraph>
    <Paragraph position="30"> Now, a slight modification of the notion of the application of an ET suggests itself: if we suppose that the trees T and T I in (3) - (4) are regular, we need consider onlyET~s with regular left and right sides; such ETIs will also be called regular.</Paragraph>
    <Paragraph position="31"> A regular syntactic h-grammar is an ordered triple &lt; V, ~,\[1 &gt;, where V =~ a 1 , a 2 ..... a k \] is a finite set of symbols (branch labels), ~ is a mapping of V into the set of positive integers (for every a EV the integer C (a) being the maximum number of branches labelled a which can grow out of any single node) ands is a finite set of (~(a 1 ),~ (a 2 ) ..... ~ (a k ))-regular ETWs.</Paragraph>
    <Paragraph position="32"> The set of ali regular syntactic A-grammars may be divided into hierarchical subsets which are fully analogous to the corresponding subsets of the syntactic A-grammars as defined above. Special elementary trans.</Paragraph>
    <Paragraph position="33">  formations (SET r s) can be defined here too.</Paragraph>
    <Paragraph position="34"> Theorem I to Any (r~ , n 2 .... , n k)-regular elementary transformation can be simulated by (n 1 , n 2 ..... n k , 1)-regular SETIs '. ..Theorem 2. a) Ifnl +...+nk~3 orif n 1 +...+nk =I, thenany (nl, n2, ..., nk)-regular ET can be simulated by (nt, n 2 ..... nk )_ regular SET i s.</Paragraph>
    <Paragraph position="35">  b) There exists (1, l)-regular and (2)-regular ETIs which cannot be simulated by (i, l)-regular and (Z)-regular SET~s respectively. 6. If a regularity characteristics (n 1 , n 2 ..... n k ) is fixed on the basis of Some empirical (linguistic) evidence, then a &amp;quot;universal syntax&amp;quot; can be constructed as an abstract calculus of all possible syntactic structures and all possible transformations of these. Choosing (1, 1, l, 1, 10, 1)- regula- null rity* as a first approximation to the deep syntactic description of natural languages, we obtain auniversal (t, 1, 1, 1, 10, 1)-regular A-grammar, &lt;%, ~u, il~&gt;, where V~= In1, a2 ..... a~ is the set of types of deep syntactic connections and where</Paragraph>
    <Paragraph position="37"> consists of the following 80 rules: l) 12 &amp;quot;splitting&amp;quot; rules of the formAT ai(B ' C) \[ f(A) =B and A~ ai(B, C) \[ f(A) =C (i = i ..... 6) 2) 62 &amp;quot;transfer&amp;quot; rules of the form ai(A ' B). aj(B, C)~ai(D , E). ai(D , F) and a i(A, B) . a i(A, C) = a i(D, E) . a i(E, Iv) ; *)The description of deep syntax suggested in \[ i\] is meant here. 6 types of syntactic connections are differentiated and interpreted as follows: connections 1 through 4 link a predicate with its arguments (only predicates with no more than 4 places are considered), connection 5 formalizes the general attributive relation, and connection 6 expresses coordination; a node can be a starting point for only one branch of each of types t, 2, 3, 4, 6 and for several branches of type 5 (we have set the number of the latter at 10 as a sufficient upper limit). The set of all regular syntactic A-grammars may be divided into hierarchical subsets which are fully analogous to the corresponding subsets of the syntactic A -grammars as defined above. Special elementary transformations (SET Is) can be defined here too.</Paragraph>
    <Paragraph position="38"> Theorem 1', Any (r~ , n 2 , ..., n k)-regular elementary transformation can be simulated by (n 1 , n 2 ..... n k, l)-regular SET's'.</Paragraph>
    <Paragraph position="39">  Theorem Z. a) Ifn 1 +...+nk~3 orif n 1 +...+n k =1, thenany (nl, n2, ..., nk)-regular ET can be simulated by (n 1, n 2 ..... nk)-regular SET ' s.</Paragraph>
    <Paragraph position="40">  b) There exists (I, l)-regular and (2)-regular E'T's which cannot be simulated by (i, i)-regular and (2). regular SETIs respectively. 6. If a regularity characteristics (n! , n 2 .... , n k ) is fixed on the basis  of some empirical (linguistic) evidence, then a &amp;quot;universal syntax&amp;quot; can be constructed as an abstract calculus of all possible syntactic structures and all possible transformations of these. Choosing (1, 1, 1, l, i0, 1)- regularity* as a first approximation to the deep syntactic description of natural languages, we obtain a universal (1, l, 1, 1, 10, 1)-regular A-grammar, &lt;V~, ~u, il~&gt;, where Vu= \[a 1, a 2 ..... a~ is the set of types of deep syntactic connections and where consists of the following 80 rules: I) IZ &amp;quot;splitting&amp;quot; rules of the form A ~ ai(B, C) \[ f(A) = B an__~d A~ a~(B, C) l f(A):C (i=1 ..... 6) 2) 62 &amp;quot;transfer&amp;quot; rules of the form ai(A , B) . aj(B, C) ~ai(D, E) . ai(D, F) an__! a~ (A, B) . ai (A, C) ~ a, (D, E) . aj (E, ~) ; *)The description of deep syntax suggested in \[ i\] is meant here. 6 types of syntactic connections are differentiated and interpreted as follows: connections 1 through 4 link a predicate with its arguments (only predicates with no more than 4 places are considered), connection 5 formalizes the general attributive relation, and connection 6 expresses coordination; a node can be a starting point for only one branch of each of types 1, 2, 3, 4, 6 and for several branches of type 5 (we have set the number of the latter at tO as a sufficient upper limit). here i, j = I ..... 6 and either i ~j or i =j = 5.</Paragraph>
    <Paragraph position="41"> 3) 6 &amp;quot;lumping&amp;quot; rules of the form ai(A.B) ~ C (i = I ..... 6). 7. It may be useful, in view of possible linguistic applications, to consider also such regular trees where the branches as well as the nodes are labelled filled regular trees. The node labels may be interpreted as characterized lexeme~ i.e., symbols denoting words, idioms and so-called lexical functions with morphological subscripts attached to them (~ i\], p. 186) The notion of regular ET and that of regular syntactic A-grammar can in an obvious manner be modified accordingly. As a result, we obtain regular lexico-syntactic ~grammars. For these grammars (see p. 6-7) we can define SETts of the types '~splitting','transfer&amp;quot; and &amp;quot;lumping H in a manner analogous to the one above; in addition another type of SET must be introduced: 4) Hrenarning&amp;quot; of a node - a transformation of the form * ~ * , where w i and v~ are node labels.</Paragraph>
    <Paragraph position="42"> If SETIs are understood as transformations of the types I-4, the theorems i I and Z will hold also for this case.</Paragraph>
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