<?xml version="1.0" standalone="yes"?>
<Paper uid="C65-1021">
  <Title>1965 International Conference on Computational Linguistics SOME MATHEMATICAL ASPECTS ON SYNTACTIC DISCRIPTION</Title>
  <Section position="1" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1965 International Conference on Computational Linguistics
SOME MATHEMATICAL ASPECTS ON SYNTACTIC DISCRIPTION
</SectionTitle>
    <Paragraph position="0"> Abstract. The purpose of this paper is to help linguists contruct a consistent, sufficient and less redundant syntax of language.</Paragraph>
    <Paragraph position="1"> An acceptable string corresponds to an expression or an utterance: it may be a natural text, a string of morphemes, a tree structure or any kind of representation. A sharp distinction is made between the syntactic function which is an attrib trin s and the distribution class which is a set of strings. Syntactic function of a continuous or discontinuous string is defined as the set of all the acceptable contexts of the string, and is called a complete neighborhood. Two contexts are equivalent if they accept or reject any given string at the same time. An elementary neighborhood is the set of all contexts equivalent to one context.</Paragraph>
    <Paragraph position="2"> Four simple distribution classes are proposed and their properties are discussed. Concatenation rules of a language can be described in terms of concatenated complete neighborhoods or concatenated distribution classes. Some possible representations and their consequences are discussed.</Paragraph>
    <Paragraph position="3"> Transformational rules are also described in a similar way. However, there is another problem of correspondence of original strings to their transforms. It is useful to establish subsets of elementary neighborhoods and this subclassification may contribute to a simplification of the clumsy representation of derivational history.</Paragraph>
    <Paragraph position="4"> Finally, some trivial but practically useful conventions are described.</Paragraph>
  </Section>
  <Section position="2" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1. Introduction.
</SectionTitle>
    <Paragraph position="0"> ~he grammar of a language should be consistent throughout its whole system. No features should be left unformulated in order that the grammar be a complete one. At the same time, it is desirable to prepare the grammar as compact as possible. These are important requirements especially when the grammar is a machine-oriented one. The knowledge on the formal properties of syntax will help us construct an objective system of grammar. Every term used in a description should be rigorously defined and no ambiguous expressions are allowed. If the consequence of grammar rules deviates from the proper usage of the language~ we will be able to trace back the definitions and locate the source of trouble.</Paragraph>
    <Paragraph position="1"> When the grammar rules are given in terms of concatenated symbols, we must know the formal definition of the symbols before writing a program by which the rules are applied to the text. If a grammar rule describes the nature of a P-marker, the label given to each node in the P-marker must have an unambiguous definition which relates the meaning of the symbol to the strings supplied as texts.</Paragraph>
    <Paragraph position="2"> Sahai 2 We need, at least, an objective criterion by which we can specify a language. This criterion will be a dichotomous decision whether or not a given symbol string belongs to the language in question. We leave the decision to native speakers and consider the acceptable strings undefined. A substring of an acceptable string is said to have a syntactic function or a part of speech. The syntactic function of a s~boi string is considered as the set of all acceptable utterances in which the string occurs. We eliminate the string in question and define its syntactic function as the set of all acceptable contexts of the string. The set of all acceptable contexts of a string is called a complete neighborhood.</Paragraph>
    <Paragraph position="3"> A distribution class can be defined as a set of strings whose complete neighborhoods are related to a given set of contexts in a specified way. We propose four simple definitions of distribution classes.</Paragraph>
    <Paragraph position="4"> With these fundamental concepts of parts of speech and distribution classes, we can proceed to a more formal system of syntactic description. However, a few questions may be immediately raised. Is it really possible to construct a grammar in such an elementary way? How can we list the elements of a set picking them up out of a practically infinite nmmber of strings even though each string is assumed to be of finite length? Is it not useless to establish such sets for a natural language, most of which are likely to have only one element? Etc. Etc.</Paragraph>
    <Paragraph position="5"> We should be better off if we were to create a new languaze by preparing a grammar and a lexicon. Unfortunately the situation is quite contrary when we are to handle a natural language. The language exists. We want to find out a grammar that accounts for all and only the acceptable strings of the language. We regard a language L as a set of strings generated by a machine M, whose internal structure is not known to us. We can observe only a part of the set of generated strings in a limited length of time. We want to construct a hypothetical machanism M' that generates all and only the strings in L. The internal structure of M and ~'~ may not be the same. ~%e output of M' is checked if it is an element of L, and strings are supplied to M' to see if M' accepts a string if and only if it is an element of L. To do this, we must have the set L, or a mechanism which tells us whether or not the given string belongs to L. We call this mechanism a normative device. It is a native speaker if a natural language is to be discussed. We simplify the situation by assuming a few separate strata in the mechanism. A string generated is supposed to have been transferred from a stratum to another before it becomes a string of natural language. An utterance has a few different forms corres-Sakai 3 ponding to the strata. Each form has its own grammar. The normative device will be a linguist in this case.</Paragraph>
    <Paragraph position="6"> Since the number of strings is practically infinite, a linguist trying to constuct a grammar will find it advantageous to establish rules that hold for a set of strings or for a set of relevant facts. A linguistic phenomenon may be analyzed from various points of view which will help him avoid listing a tremendous number of phenomena and rules. He will attach certain markers to the stringm according to the way he considers consistent with his usage of language. He will then write down the rules in terms of the markers. He may also establish his rules in terms of sets of strings which share some common features in their mai~ers. The procedure of using these rules consists of two parts. ~%e one is a routine that compares a rule with the text and decides whether or not the rule is to be applied. The other is a transfer routine by which the relevant infon~ation is read out of the applicable rules and transferred to the text. In these procedures, both comparison and transfer are carried out with the coded markers. It is important that the meaning of the codes is unambiguously defined so that the code obtained in the text is exactly what the linguist wants to mean.</Paragraph>
    <Paragraph position="7"> Some of his rules may account for a certain n~mber of texts he has examined but may fail to account for some others or to rule out similar but inconsistent facts. He will test his rules by applying them to a natural text or by generating strings. The normative device will tell him whether or not a string supplied to it is acceptable but not tell him why. It is obvious that these procedures can not be carried out practically on every string that may be supplied to a machine in the future, and that nobody will be able to predict what can occur when an arbitrary string is supplied to the machine. Nevertheless, it is required that a grammar may deal with most of the texts supplied in the future.</Paragraph>
    <Paragraph position="8"> His ~rammar is inevitably affected by the nature of the normative device.</Paragraph>
    <Paragraph position="9"> If the normative device is so strict as to reject every string which fails to meet such requirements as that its style must be just an ordinary one, the statement must be logically correct, the lexical usage must conform with the regular way of the language, etc., etc., then the linguist must prepare a separate rule for almost every string. He can break down the decision procedure into a few separate steps. The first device will accept a string if it finds the internal relationship of the string is acceptable, regardless of the reality the string designates. If the grammar is to be applied to input texts Sakai 4 whose structure is always grammatically correct and unambiguous, a grammar which satisfies the requirement of this device ~ wl~ be enough. However, it will give many unusual strings if it is used in random generation and many ambiguous alternatives if it is used for analysis, ~hC/ second device may reject tl%ose strings whose structure shows an unallowable combination of lexical elements, thus eliminating some of the ambiguous alternatives in analysis and suppressing the output with improper usage of lexical elements in synthesis.</Paragraph>
    <Paragraph position="10"> The third device may reject as unacceptable those strings which are not logically consistent. If one wants to have more rigorous grammar that may be used for random generation of only non-surprising sentences, he may add more devices to the preceding ones, so that the grammar may be tested from such points of view. He will prepare his grammar keeping the characteristics of his normative device in mind. A number of digits will be assigned to the coded form of markers corresponding to each step of decision. ~ne procedure will be programmed so as to handle these digits independently, thus allowing a number of rules to be applied to the same string, if certain digits are related to each other, and a particular combination ,of codes is to obey a particular rule, the rule will be prepared independently and the general procedure will be prohibited. ~nis is done by a simple technique in coding and programming.</Paragraph>
    <Paragraph position="11"> As we see on the following pages, a number of similar but different representaions are possible. If we are not ready to understand the exact meaning of codes and rules and to prepare the right program for the representation chosen, the rules established on the basis of ad hoc definitions will result in a chaos. The formal property is not confined to a certain language, but it is common to many, probably to all, languages. A grammar will not deviate greatly from its proper constuction if its formal property is carefully examined.</Paragraph>
    <Paragraph position="12"> ~. Symbo!~ String; Language.</Paragraph>
    <Paragraph position="13"> 2.__~I. Symbol is an undefined term. Morphs, morphemes, lexes, lexemes, or some other units may be regarded as symbols. Any unit consisting of a number of symbols is called a string. All the strings are possible strings. If a string is considered &amp;quot; ~ ~ meanln~u+-, then it is an acceptable string. Each acceptable string is an undefined term.</Paragraph>
    <Paragraph position="14"> These definitions are quite fon~al. If we confine ourselves to the problems in morphotactics, the symbols are morphs and the acceptable strings are what are called expressions or utterances. A symbol may be a morpheme and a linear arrangement of morphemes is an acceptable string if it is reco~jnized as a mori:,hemio =,j ::'osentaticn of an u-ctu:'-.,&lt;=e'. A string need not always be a linear a,~ra~gemen% of &amp;quot;~ ~-- &amp;quot; a l~mo. We may rega~t.~ labeled tree called a P-marker as a string~ and a labeled node as a re-0resu~rlon of the subsZrin~{ dominated by the node~ al~ouZ~.. ~e term strin~ seems inadequate in this oa~e A node represents a P-marRer consistin/ of all +~.he terminal and non-terminal nodes it dominates. We can regard a P-marker as a L='ee-l/ice strin Z of P-markers dominated by the former. &amp;quot;'- ~'~ .... &amp;quot; ~ ..... ,&amp;quot;~ :~o~e. x~nc of .......... es may be added to the syntactic tree in order Zo indicate the re!ationshi3 a~=on 1%he constituents. We call this renresentation a net~ provisionally. We l:~a y reoard a net as a string co~.&amp;quot;sisting ~: ...... &amp;quot;~ -~ &amp;quot; ........... e. of labeled nouns, w;:ose arrangement is shovm by two kinds of branches.</Paragraph>
    <Paragraph position="15"> We define a langua='e as a see of accei=table .... ihc ~&amp;quot; ,. -'- _ s ~r~_njs. acce.n=a3~e string of a natural -'a,&amp;quot;-:,'&lt;~ is considered =o have as ..... ~ &amp;quot; .... ..ly versions as the nusoer of strata established &amp;quot;bLr linouist. Ear=. v,~-.&gt;;ion of at. accen, table s~cz~ing is an element of the language defined on the st,.~atm= i.n ou=,=~:;.,on. A transfer from one version to another is essentially a translation.</Paragraph>
    <Paragraph position="16"> 2.p. Su~o'.~ose we have a Linear sz, r:Ln:j. !,',e ~,ndegcer'r'a~oD the sLrzng by delet.n~ some of the s~.:ools therein and ..~. ~ ......... ~...n o&amp;quot; -&amp;quot; a s:p~bol of absence &amp;quot;to each point of deletion, if a symbol,, o- absence is foiio',Jed by another .,.,,,e&amp;lauu.y, ~ .... &amp;quot;&amp;quot; ~&amp;quot;~' -~ne,y are contracted to one. A ~&lt; .... . ..... e~z strin~ is continuous if it is not interrupted ,%- ~ ~+-~ the nodes in a syntactic tree are palatially ordered. A node includes a~ot~ ~ .... ...... if the linear str~_ng ..... covered 0y zne latter ms a part of the linear s~l~,a covered by the for='~er. A t:cee-iike strmn\[~\] is continuous~ if and only if (i\] all the nodes of the sLrin~ are included in one node D, and (2) there are no o d:er nodes which are not included ir~ Ddeg  * ~.~ * .~ ~*~ ~ * I A ;%et strln&lt; is continuous, 4~ and only m~ ~.~e s~jntactmo tree is continuous and no branches of ~ne second ' 4 &amp;quot; -&amp;quot;~' ~'&amp;quot; ,~.nQ are broW&lt;ell o.~. Any o. ~ ....... ~ ~. -~ ~,I ........ s 3~ a sLrin Z is called a se&amp;~nent, it may be either continuous or U~CO~uoZnUOGo. ~ discoP.tiZlUOUS sec',',~enL consists of a few nar~s se.narated from each otl.er. Each o~,z~t of se&lt;':::e:&lt;% ::s Ca~__~,~ a fra-~,,lent which is necessarily con'~inuous (~-az-l&lt;er-i.~-.odes~ itdl).</Paragraph>
    <Paragraph position="17"> ~. boll ~el{ ~ : ......... (,~_ ,,.,,, ~,.</Paragraph>
    <Paragraph position="18"> 5.PS=- Context i :2.cce'-'-,&lt;:3_e Conbe:,:=.</Paragraph>
    <Paragraph position="19">  Let r be a strin~ an&amp; ~eL s be a seonenu of r. ~:,e s~.~,~ r may be continuous or discontinuous. ;lhe other :taru~ c of z&amp;quot; _~s called the co~-'~c.~.~ of s ...... ~.~.~u .... ~.~ Lf.eZ:i We ;.sa~, r c ~.s &amp;i% ~c,..,.z.,~,_~u.~,~ OOl%Le\]&lt;L O-&amp;quot; S~ or c</Paragraph>
    <Paragraph position="21"> * f the discussion is confined to a co~.~-:.ee cr.rase scruczurc _an:Cu~je, it seems more convenient to modify the concepts acceptable string and context; any immediate constituent of an acceptable szring is also acceptable, and a context is acceptable to a string if the string, its context and the whole string are all acceptable, if the constituents are continuous, the situation becomes simpler. ~ne context c = r()t is acceptable to s, if r, s, t, aud rst are all acceptable. Either r or t may be absent.</Paragraph>
    <Paragraph position="22"> 3.2. Neighborhood.</Paragraph>
    <Paragraph position="23"> A context is an interrupted string which becomes a continuous string if an appropriate segment is supplied to its points of interruption. Let y = set(cl,c2,---,c n) be a set of contexts and let s be a string. If all the contexts in y become acceptable strings when s is supplied to them, then the set y defines a property of s. We call the set y an acceptable neighborhood of s. If y is an acceptable neighborhood of strings Sl, s 2, s 3, for instance, then we say y is an acceptable neighborhood of</Paragraph>
    <Paragraph position="25"> and we consider the set y represents a syntactic property common to all the strings in S. ~ote that our neighborhood is not the same as the okrjestnostj  (Kulagina, 1958). A set of acceptable strings with a string s is called a paradigm of s (Parker-Rhodes, 1961); our neighborhood is a paradigm in which the string s is lacking.</Paragraph>
    <Paragraph position="26"> 4. Eouivalence of Contexts.</Paragraph>
    <Paragraph position="27"> Let c and c be two contexts. Suppose a string s is acceptable to both z 3 c and c., and another ..... &amp;quot;~ ~ * ~ing t is not acceptable to c. or c.. In this case, ! j i ,\] we can not tell the difference between c. and c as far as the acceptance of l 3 the strings s and t are concerned. We say these contexts are equivalent to each other and write c i eqv c j, if the condition &amp;quot;c is acce~tabie to ~ ~ring s, if and only if c is acceptable to s&amp;quot; is satisfied for every possible string s of the language. ?he relation of equivalence is symmetric, reflexive, and transitive: (i) c. ecv c.; (2) if c i eqv cj, then c~ eqv ci; Sakai 7 (3) if c i eqv c. and c eqv Ck, then c. eQv c k- j .i &amp;quot; ~. Complete Neighborhood.</Paragraph>
    <Paragraph position="28">  ~u~. Let y be an arbitrary set of contexts, it may include contexts which are not equivalent to each other and may not include all the contexts which are equivalent to some context in it. ~he comolete, n-'ei ''~..~o~nooa'~ &amp;quot; N(y) of y is the set of all contexts equivalent to some context c' in y: N(y) = set(c: c eqv c' ~ &amp;quot;~ C' in ~o~ some y;.</Paragraph>
    <Paragraph position="29"> A set of contexts is complete or is a complete neighborhood if and only if it is the complete neighborhood of itself. Take a string s and let C(s) be the set of all the contexts acceptable to it. We show ~u ~ C(s) is ~.a~ complete.  c eqv c' for some c' in C(s), c eqv c' and c' is acceptable to s, c is acceptable to s, o g c(s), N(c(s)) C(s).</Paragraph>
    <Paragraph position="30"> From (i) and (2), we have : c(s).</Paragraph>
    <Paragraph position="31"> Therefore, C(s) is complete. We call C(s) the complete neighborhood of the string s.</Paragraph>
    <Paragraph position="32"> We may pick up an arbitrary segment of an acceptable string, call the other part the context of the segment and establish a complete neighborhood of the segment. This kind of complete neighborhood contributes nothing to a grammar but some redundant rules. These practically nonsensical complete neighborhoods give rise to no trouble, because they never appear in any rule of the language.</Paragraph>
    <Paragraph position="33"> ?he complete neighborhood C(s) of a string s is considered to correspond to the syntactic function or the Dart of speech of the string s. The elements of C(s) shire a common property that every one of them can be an acceptable context of s, while no other context~ which do not belong to C(s) are acceptable to s. ?his property of C(s) leads us to the application of complete neighborhood to a given set of contexts supplied as text. Let S be an arbitrary set of contexts. Some elements of S may be accepted by s and some others may not. The elements accepted by s must, at the same time, belong ~o C(s), that is, Co C(s)~ S. If</Paragraph>
    <Paragraph position="35"> then the string s can not occur under the contextual condition defined by S, and vice versa. If</Paragraph>
    <Paragraph position="37"> then we have no means to distinguish the syntactic function of s and t with respect to the given S. If S is the set of all the possible contexts of the language, then c(s) N s = c(s) for any string s. If</Paragraph>
    <Paragraph position="39"> then we have no means to d~stlngmlsn zne s~tactic function of s and t so far as only the acceptability is concerned.</Paragraph>
    <Paragraph position="40"> 5.2. It occurs very often that a string r behaves like a string s under a certain condition, and like t under another condition. This phenomenon will be restated as follows: for some set S' of contexts, C(r) N s' = c(s) O s,, and for another set S&amp;quot; of contexts,</Paragraph>
    <Paragraph position="42"> ~,~en, xN ~'N s,,= yqs, N s,, and xNs'N s&amp;quot;= ~O s',q s&amp;quot;.</Paragraph>
    <Paragraph position="43"> Taking the union of these two, we have x('l s, .q s,, = (yd ~)Ns'lq s,,.</Paragraph>
    <Paragraph position="44"> This means that r acce~ots every context in ~' ~ S&amp;quot; ~ if it is acceptable to s or t. Now, we will see the behavior of r with respect to the context set</Paragraph>
    <Paragraph position="46"> This result su~'&gt;e~+~oo _~ that the behavior of r may be interpreted in terms of y and z, and that y and z may account for something lacking in x with respect to</Paragraph>
    <Paragraph position="48"> then, for some c in C(s) and some c in C(t), we have c. noz ecv c~.</Paragraph>
    <Paragraph position="49"> \]- o 6.2. k se~ oPS all ....... ~- ,.ut~z~j equivalent contexts, called an elementary neighborhood, leads us to a concept of the ultimate unit of syntactic function. Given the elementary neighborhood e(i) with c. as an element is defined ! a context ci, as</Paragraph>
    <Paragraph position="51"> Since the equivalence is symmetric, reflexive and transitive, any two distinct ele::;e-=~ary neizhborhoods have no elements in common.</Paragraph>
    <Paragraph position="52"> 6.__~. Let x be a co,mi~iete neid_borhood and e(i) an elementary neighborhood. ~ an element c in x ~s a ::,emoer of e(i)~ then</Paragraph>
    <Paragraph position="54"> for all ek~)'s ~=ving at leas~ one element in x. Every elementary Sakai i0 neighborhood is complete. An intersection of complete neighborhoods is complete. Every union of elementary neighborhoods is a complete neighborhood.</Paragraph>
  </Section>
class="xml-element"></Paper>