l string p(r.s) l and P(ris) p(rs) ~nerefore, That is C(ris if and only if p(rs) is neighborhoods as the complete Generally, I % we put J(y) and J(y) may be real or .... s--j) = ---ri---sj--and s. in it. Any such string ix acceptable if and j : ---r.---s--- l is acceptable if and only if .... r---s--P(ris j) is acceptable j) = C(rs). then xy while xy does not always result Sakai 18 = C(rs), = C(rs) in x : C(r) or y = C(s). We have generalized and transferred the concatenation of strings to concatenated sets of strings and then to concatenated complete neighborhoods. The complete neighborhood representation provides us with a less complicated approach, especially when the strings are syntactically ambiguous. The distribution class J(x) means the narrowest classification of strings and no further subclassification is possible, while its complete neighborhood x can be subclassified if x is not an elementary neighborhood. If rg J(x) and x = y Uz, then we can talk about imaginary strings r' and r", such that C(r') = y and C(r") = z. These imaginary strings, always referred to implicitly in terms of distribution classes, can be discussed explicitly in terms of complete neighborhoods. 9.2. We make distinction between the concatenation xy : c(r)c(s) of complete neighborhoods and the complete neighborhood z : C(rs). ~%e former means a set consisting of concatenated contexts. The properties of the language is introduced when it is written in the form or where the property z of rs. Thus, z ambiguous even if 9.3. We find it neighborhood in a unified way. We saw that a complete neighborhood x can be represented by a union of elementary x = Oe(i) Let us introduce coefficients x(i) : = and no other cases possibly x(i)e(i)

xy= z

C(r)C(s) : C(rs),

x of r and the property y of s result in another property

can be an empty set even if neither x nor y is empty, and

neither x nor y is ambiguous. advantageous to have a system which represents every complete x(i), o i

if

if occur.

= e(i)

=0

neighborhoods e(i): with x ~ e(i) ~ O.

such that

e(i),Ox =o,

e(i) -- x;

We put

if x¢i) : l,

if x(i) : o. Sakai 19 coefficients, we can write x = Dx(i)e(i), y = ~y(j)e(j), z = Uz(k)e(k). z=xOy, x U Y = (U x(i)e(i)) U (Oy(j)e(j)) = U(x(k) + y(k))e(k) = U z(k)e(k). e(k) c_ x or e(k) ~ y, e(k) ~_ z. x(k) + y(k) = z(k), 0~-0=0, l+O =0 +l= 1 +l= 1. z = xy, z = (Ux(i)e(i))(~y(j)e(j)) = DU x(i)y(j)e(i)e(j) = UU z(i,j)e(i)e(j). of concatenation, e(i)e(j) ~ xy e(i) C x and e(j) c y. z(i, j) = 1 x(i) = y(j) = 1. x(i)y(j) = z(i,j), 1Xl=l, 0 XO =0 Xl= IXO = O. In virtue of these (1) If then If then ~\]erefore, for we have (2) If then By the definition if and only if That is, if and only if Therefore, for we have (3) A concatenation of two elementary neighborhoods is a complete neighborhood, and it is also a union of elementary neighborhoods:

e(i)e(j) = Ua(i,j,k)e(k)"

Z = xy

= U~ z(i,j)e(i)e(j)

: UUU z(i,j)a(i,j,k)e(k)

= Uz(k)e(k), we have e(k) ~ z if and only if e(i)e(j) ~ z and e(k) ~ e(i)e(j). Therefore, for the expression

z(i,j)a(i,j,k) we have 1 X 1 = l, Writing = z(k), 10. Concatenation i0. i. be cause if and only if Sakai 20

0 XO =OXI= 1XO =0. of Distribution Classes.

G(u)G(v) ~G(uv),

r~ £ G(u)~(v)

r £ G(u) s £ G(v)

C(r) N U / 0 and C(s) ~ V / 0

(C(r) ~ u)(C(s) ~ v) = C(r)C(s) N uv / o and then ,if and only if l0.2. because if and only if II It C(rs) ~uvJ0 rs £G(uv). H(u)H(v) c ~(uv), rs E ~(u)~(v) r ~ H(u) and s g H(v) C(r) D u and C(s) ~ v C(r) ~ u =u and ~C(s) ~ v = v II I! then if and only if lo._.._5.5. because if and only if II It II I1 then if and only if 10.4. because if and only if then if and only if (C(r) ~ u)(C(s) N v) C(r)C(s) ~ uv C(rs) 2 uv rs ~ H(uv). I(u)I(v) ~ I(uv), rs £ I(u)I(v) r ~ I(u) and C(r) c u and C(r) ~ u = C(r) (C(r) ~ u)(C(s) D v) C(r)C(s) c uv m C(rs) c uv N rs 6 I(uv). J(u)J(v) C J(uv), rs ~ J(u)J(v) r ~ J(u) C(r) = u C(r)C(s) = uv

C(rs) : uv

rs £ J(uv). and and = C(r)C(s) N uv : uv

s ~ I(v)

C(s) ~ v and C(s) ~ v = C(s)

: C(r)C(s) D uv : C(r)C(s) s 6 J(v) C(s) = v Sakai 21 i!. Rules for Recognition and Generation.

Each rule of a grammar indicates the arrangement of a few items to be concatenated, accompanied by some other necessary informations. We assume the items arranged in a rule are either complete neighborhoods or distribution classes. Let us see what happens during the generation and recognition of a string of symbols.

In case a grammar is given in terms of complete neighborhoods, the input text is converted to a string of complete neighborhoods before the syntactic analysis begins. At the very end of generation, a terminal node accompanied by a complete neighborhood x is replaced by a string s whose complete neighborhood C(s) shares at least one elementary neighborhood with x.

~nen the syntactic rules are expressed in terms of sets of strings, input text to be analyzed is replaced by a string of distribution classes. If a symbol string belongs to more than two sets of strings, their meet replaces the symbol string. At the end of a generation, the synthesized output string is obtained by replacing the set of strings on @ach terminal node by a string which is a member of the' set. ll.1. An acceptable string can be generated and analyzed making use of a tree with its nodes marked by complete neighborhoods. The expansion of a node z to a concatenation xy of nodes x and y implies z ~ xy, because otherwise further expansion of x and y may yield a structure which can not be accepted by z. Transformational rules can be a}?plied more freely because a transformation does not imply such a restriction. However, attention ahould be paid not to add any other contexts to the complete neighborhoods attached to the nodes already generated. Finally, each terminal node is replaced by a lexical element. ~%e string obtained after applying all the obligatory rules must be an acceptable string.

~ne analysis is carried out by testing all the possible transformations and trying all the possible contractions. At any rate, both generation and analysis can be carried out if we have a set of rules which gives concatenation z = x---y for any x, ---,y of the language, and the transform y(1)y(2)---y(n) of any string x(1)x(2)---X(m) of complete neighborhoods. ll.2. Acceptable strings are also generated by starting from the node P(O) which is the set of all acceptable strings. It is replaced by its subset

P(1)P(2)---P(i)---P(m) ~ P(O) which is a concatenation of nodes P(i)'s. Each node P(i) also represents a set of strings, and it may or may not be replaced again by

P(il)---P(ij)---P(in) ~ P(i). the

Sakai 22 On each step of expansion, a choice is made by taking a subset of strings. ~e possible choice becomes narrower and narrower. It is expected that the string obtained by applying obligatory rules and by replacing each terminal node by a lexical element is an acceptable string.

~is is not always true if the replacement of a node is independent of the other nodes already generated. %his difficulty is overcome by executing a syntactic analysis after every step of expansion. If the analysis does not prove the possibility of obtaining an acceotable string, another subset should be chosen as a candidate. ~ne check by analysis should be tried after a transformation if it is a local or a generalized one. All the nodes, terminal and non-terminal, are sets of symbol strings. A generated string of nodes is analyzed by tracing back the P(O) which covers the whole acceptable if otherwise.

Any given string can be this case, however, the tree on every possible combination whole string may be covered history may be accounted for 11.3. ~ihe Rules for generation and those for recognition are essentially the same. They may be prepared in terms of complete neighborhoods or distribution classes. ~le rules will be prepared without any formal ambiguity if their definitions are carefully observed. Some formal systems are given in the following pages as examples of sin:pie types of grammar. . !2. . . . Conu~lete Neighborhood Re,,resen~.~<~on ~ - c ~ ~ Concatenation Rklles ,.

We say a set of concatenation rules is con~plete if it gives the concatenation

Z = xy of any complete neighborhoods x and y of the language. It is not necessary, however, to list all the ioossible x's and y's. Much less number of rules can cover all ~he ~ossible com!iete nei~3hborhoods if their use is y rcper!y pro gramme d.

We consider a rule f(uv;w) represents a relation between the concatenated complete neighborhoods uv and another complete neighborhood w. Each rule

path of generation. If the analysis string, the generation is acceptable, goes back to and not

analyzed by applying rules to the string, in

structure is not known. Rules should be tested

of terminal and non-terminal nodes, so that the by a single node and the possible derivational

by the concatenationai and transformational rules. Sakai 23 will give information to xy if x (=) u and y (=) v: (xN u)(yNv) : xy uv; which is a part of xy = z. In order to obtain th~ given concatenation xy, we determine a set R(xy) of rules applicable to xy. Each rule is decided whether or not it is applicable to xy by the condition g, so that f(uv;w) 6 R(xy) if and only if g(x;u) and g(y;v). ~%e term w is read out of the rules in R(xy) so that z = xy may be determined, it is obvious that there exist certain restrictions in choosing the type f of rules, the condition g for determining R(xy), and the procedure of finding z. We have to specify these three for the grammar to be written. When the complete neighborhood z is given and its expansion xy is to be found, the set E(z) of applicable rules is determined by the condition h(z;w): R(z) = set(f(uv;w): h(z;w)). The situation is a little complicated in this case. We can possibly expect a case where both z = xlY 1 and z = x2Y 2 are true under the condition x I ~ x 2 = 0 and/or Yl ~ Y2 = O. Note that this is not the case of formal concatenation of sets N CO = (A N C)(BN O). The concatenations xiY I and x2Y 2 happened to be z by the syntactic reason of the language being studied. A storage as any rule in R(z) proves a possibility, rule is applied to it. However, if x. i ~ - x. ~ and then either xiY i or xjyj is just trivial. rules and the program which applies the a set of x i~ accompanied by the subset rules for this purpose will not be discussed here, because the principle is similar to the case of finding z from x and y. In order to see some properties of rules, we assume simple forms of f(uv;w): uv (=) w, UV :D W, UV ~ W, space is assigned to each xiY i as soon and xiY i is modified every time a Yi ~Yj'

The choice depends upon the type of rules to the text. Finally, we have R(z;i) of R(z). Possible types of Sakai 24 UV = W. The condition g will be assumed simply as (=), o_, c_, or =.

The condition of constituents can be replaced by a condition imposed on the whole concatenation: (i) xy (=) uv if and only if xyNuv = (x~ u)(yDv) /0

T! x (=) u and y (=) v; (a) xy E uv if and only if xyZ uv = (x~ u)(y Nv) = uv I! u=xNu and v = y ~ v I'I (3) similarly, if and only if (4) if and only if uv (=) applicable to xy if x (=) Then, for such a rule, we xy (=) We can also assume the rules are xy ~ uv, xy ~_ uv, xy = UV. We can not decide which part of w belongs to uv, unless some other information is available. then be applicable ~en This is true xDu xy ~_ uv X~ U xy = UV X = U and and and rules w, u have uv (=) and y~ v; y y of the y w. applicable if ~ v; = v. form (=) v. the

to for

uv~ w u

(x~ u)(y~v) rules of the form UV ~ W xy if and only if

x ~ u and

xy ~ uv~w. any rule in R(xy) = set(uv relation = xy~ uv~xyn w. y ~_ v. ~ w: xy~uv). Sakai 25 If the set R(x~ has sufficient rules to give xy : Uw, we can 12~2,2. then doom to e(i)e(j) If for instance, and if we have the rules

e(1)e(2) ~ e(5) 0 e(6),

e(1)e(3) ~ e(5) and e(1)e(4) ~ e(6), then xy ~ e(5) U e(6). These rules will be broken

e(1)e(2)

e(1)e(2)

e(1)e(3)

e(i)e(4) and then contracted as

e(l)(e(2)

e(1)(e(2) (t) e(4)) where the symbol (+) means an alternative

~e number of elementary neighborhoods analysis becomes more precise, and hence a e~ementary neighborhoods comprises a great type of rules is preferred when a particular (Opler et al., 1963). 12_~.. Let us consider a set of rules of the form

uv ~ W. We assume a rule is applicable to xy if

x (:) u and ~ e(6),

choice.

increases rapidly as the linguistic

grammar prepared in terms of

number of entries. However, this

technique is available on machine y (:) v. We have, then, u)(y v) :xyOuv xy w. find xy by simply taking the union of all the w's in R(xy).

If the rule~ are applicable to xy when

x ~_ u and y c_ v,

xy c uv = w.

We le%ow that a concatenation xy of any two neighborhoods is broken

the concatenations of elementary neighborhoods e(i)e(j) and that each

is represented as a union of elementary neighborhoeds.

x = e(1),

y = e(2) 0 e(3) ~ e(4), down as

~ e(5)

~ e(6)

~ e(5)

~ e(6), (+) e(3)) D e(5) uv _~w are applicable to xy if and only if x ~ u and y ~ v. For all the rules in the set R(xy) of applicable rules, xy ~_ uv c w. R(xy) = set(uv~ w: x ~ u, y~ v). Then, for each rule in R(xy), we have xy ~ uv ~ w. Taking all the rules in R(xy), we can expect xy = Dw, and, if the set of rules is prepared so as to xy by taking the intersection of w's in R(xy). and let R(xy) be then we have for all the rules where the union so that then we can find rules in R(xy). then If is true for all by UV = W~ the set of rules such that

x (=) u and y (=) is the set of all the rules

x o u and y ~_ v,

xy ~ uv = w in R(xy). Then,

xy ~_ Uuv = U w,

to cover all the rules in xy = U uv, meet this v. satisfying R(xy); if taking the Sakai 26 we have condition, we can find the condition the rules are prepared union of w's of the applied to xy when is the concatenation simD1y by rules are pre~fared so that they may be x ~_ u and y C v, xy ~_ uv = w. xy = ~ uv the rules in R(xy), then we can find the desired concatenation xy = Nw. rules are represented in terms of elementary neighborhoods in Sakai 27 e(i)e(j) = w(i,j), virtue of the coefficients x(i) and y(j), we have x ~ U = X y ~ v = y (X D u)(y~ a rule is applicable x(i) = y(j) z = xy is obtained rules: z = Uw = the form then, in Therefore, The result applicable

12.5. The rules are prepared

condition and requirement. In plete neighborhood is represented by a code consisting of a number of digits and each digit is checked, modifiedand transferred independently.

Suppose x and y are given and their concatenation z = xy is required. Both x and y can be syntactically ambiguous and their ambiguity is to be reduced in the course of finding z. Initially, z is assumed to be the set of all the possible contexts, x, y and z are transferred to a temporary storage space (xl,Yl,Zl). A rule is applicable if

x (=) u, y (=) v and z (=) w, and the set (xl,Yl,Z l) is modified everytime a rule is applied. If a rule proves x I (=) u, Yl (=) v, z I0 w = O, then the rule is not applied to this set, and another set (x2,Y2,Z2) is stored in another storage space as another possible result. able rules are applied one after another to all the possible (xi,Yi,Zi). Similar procedure is repeated over again on two All the applicsets of languages simultaneously, so that the syntactic structure can be transferred from the tree structure in one language to that of another language. ~he form of the tree is preserved but their nodes are marked by the labels specific to each language, input, intermediate or output language. 13. Distribution Class Representation of Concatenation Rules.

Possible concatenation of a language can be formulated as concatenated

of strings. Let sets R = set(r: h(r)) ~ e(i) = x(i)e(i), S e(j) = ykj ' )e(j) ,

V) = x(i)y(j)e(i)e(j).

to ~y if

= i. as the union of all the w(i,j)'s of the ~x(i)y(j)w(i4j). and used more

the following freely according to the given

scheme (S'akai, 1961), a comSakai 28 and S = set(s: h(s)) be sets of strings satisfying the conditions h(r) and h(s), respectively, and let their concatenation have the property k(rs), so that

rs E T = set(t: k(t)). We consider the concatenation rules of the form

RS ~ T,

which reads : if r 6 R and s £ S, • then rs K T. The point of this representation is that, and s 6 S h~ S i~--- OS k, then as many rules are applicable to rs and they give rs E The intersection T' has less the character of the strings course, these procedures are the sets. Each set in the rules is represented by a code. Every entry of the lexicon has a code and it can be determined whether or not the string belongs to any given set. These codes are to be generated and attached to rs to indicate that it belongs to the set T'.

Practically, it is convenient to classify the strings in terms of their complete neighborhoods:

R = set(r:

S = set(s:

T = set(t: A grammar of concatenation will

R(u)S(v) c with a relation f(uv;w), and the rules can be described choice of R(u)S(v), T(w) and f(uv;w). In order to see simplify the situation by making use of the distribution J, and by assuming the relation f(uv;w) as uv(=)w, UV~ W~ the principle, we

classes G, H, I and N --- S% : number of elements and, if the rules are precise, in it is determined as precisely as required. not to be done by listing up all the members of Of h(C(r);u)) = R(u), h(C(s);v)) = S(v), k(C(t);w)) = T(w).

be given as a set of rules of the form in a number of different ways according to the Sakai 29 or EV ~ w, uv = w. lhe type of T(w) is chosen so that the grammar may describe the language adequately. 13.___~1. G Renresentation.

Put

~(u) = G(u), s(v) = G(v). If then r E G(u), s E G(v), uv (=) w, rs E G(u)G(v) ~ G(uv), then C(rs) (=) uv (=) w. If then r ~ G(u), s ~ G(v), uv ~ w, rs E G(u)G(v) c G(uv), then C(rs) (:) uv ~ w. If then If then

r E G(u), s ~ ~(v), uv ~ w,

rs 6 S(u)G(v) C G(uv) c G(w).

r E G(u), s g G(v), uv = w,

rs ~G(u)G(v) ~ G(UV) = G(w). Even if a few rules are applicable to rs in these cases, that is,

rs E G(w~) ~ G(w i) ~ --- ~ G(wx), we have no simple way to find C(rs) from w's. We can not specify a set of less members which adequately indic'ates the property of rs, unless more specific information is available. 13.1.1. Suppose, however, u and v are elementary. If C(r) (=) u and C(s) (=) v, then C(r) ~ u, C(s)~v. That is, r ~ G(u) = H(u), s ~G(v) = H(v). For further discussion, see "H Representation", where u or v is not necessarily elementary. 13.1.2. Assume C(r) and C(s) are elementary. If C(r) (=) u and C(s) (=) v, then C(r) ~ u and C(s) ~ v. That is, r ~ l(u) and s E I(v). For further discussion, see " I Representation", necessarily elementary. 13.2. H Re-oresentation.

Put

~(u) = :~(u), S(v) = ~(v). where no neighborhoods are S~<ai 30 if then r ~ H(u), s 6H(v), rs 6 H(u)~(v) _~ H(uv). .z,3.2..l. If uv (=) w,

then rs ~ H(u)H(v) ~ H(uv), then C(r~) o_ uv (=) w, then C(rs) (=) w, then rs 6 G(w). We put ~(w) = Q(w). However, there is no simple procedure of finding the intersection of G(w)'s. We can not specify the features of the strings by finding more rules applicable to rs, unless more specific informa{ion is available. 13.2.2. If uv ~ w, then rs ~ H(u)H(v) ~ H(uv) ~ H(w), because H(uv) = H(w U wi) = H(w) ~ H(w') ~ H(w). We put T(w) = H(w) to have the rules of the form If a number of rules are applicable and then then

The complete although 13.2. 7 . then then 13.2.4. Then rs £ H(uh)~(v h) c_ X(w,n) rs 6 H(ui)H(v i) ~-H(w i) rs ~ X(uk)~:(v k) c_ ~(wk), rs £ H(w h) ~ H(w i) ~ = :~(wh C wi U --C(rs) O_ w h U w iU --rules of this type are essentially neighborhoods

xy ~ uv ~ w, they are encoded as the sets of If Put uv ~_ w, C(rs) ~_ uv~_ w, rs 6 G(w). UV = W. rs ~ H(u)H(v) ~ H(uv) - - - OH(w k) Owk )' Ow k" the same as the rules of strings; = H(w). Sakai 31 The situation is the same as the case above, where uv ~ w. 13.3. ,I, Re, presentation.,

Put R(u) = i(u), S(v) = l(v). If r £ i(u), s 6 i(v), then !3.3.1. If then No relationship is 13.3.2. If then No definite T(w) is 13._~3.. We consider

rs ~ I(u)I(v) ~ l(uv).

uv (=) w,

C(rs) c uv (=) w. relevant between C(rs) and w.

urn_w,

C(rs) c uv ~ w. m

available, such that I(u)l(v) ~_ T(w).

the rules of tie type i(u)I(v) with uv ~ If r£ then rs { If a nmmber of rules are then rs £ = I(w h ~ w i~ --- ~Wk). %herefore, the rules of this type are equivalent to those of the type

xy C uv C w. 13.3.4. Put UV = W, • Then rs ~ I(u)i(v) ~ I(uv) = I(w).

This is the same to the case mentioned above.

13.4. J Reoresentation. Put ~(u) = J(u), S(v) = J(v). This type of grammar is not practical because every real distribution class J of the language must be listed in the rules, k~:is condition corresponds to the com}~lete neighborhood representation of rules f(uv;w) applicable to xy only if 13.5. Practically, can be more flexible x = u and y = v. the rules can be written more freely and the program

and efficient, provided that a more sophisticated w. i(u), s£1<v), I(uv) = i(w). applicable to rs, I(w h) N I(wi)~ --- O I(wk)

Sakai 32 scheme is introduced to the G Representation and the condition f(uv;w). ~is is realized by representing the sets of strings by codes, so that the union and the intersection of any two sets are determined by the operation on the codes. 14. Some Remarks on Transformation. 14.1. It is generally agreed that we generate acceptable strings by starting with an axiom and expanding it repeatedly into a string of constituents. This procedure is taken care of by concatenation rules. After generating one or more strings by this procedure, they are transformed to yield another string.

Let us imagine another function of our normative device. We give it a pair r = (r',r") of acceptable strings

r' = r'(1)r'(2)---r'(i')---r'(m') and r" = r"(1)r"(2)---r"(i")---r"(m"). The pair r will be referred to as a string

r = r(1)r(2)---r(i)---r(m) with m = m' + m". We put m" = 0 if the string r" is absent. We then give it another acceptable string and ask it whether and r" are true. rated from r by a orano_o~.n. If it ask it whether or T~Xr~f ~ , we consider by r' and r". We transformation but posed to have been unless appropriate, possibly non-linguistic, situation is beyond the scope of Syntaetics.

A transformation or an inverse transformation is called singularly if r" in r is absent, and it is a generalized one if both r' and r" are present. If it is an embedding transformation, r' and r" are called matrix and constituent strings, respectively.

If we understand the transformation in the sense mentioned above, the transfer of syntactic structure from one language to another is also a transinformation is supplied. ~%is

s = s(1)s(2)---s(j)---s(n),

or not the string s as an expression is true if both r' If the device says "yes", we consider the string s is genetransformation. We call r the original string and s its says ~'no", no such transformation exists. Conversely, not r' and r" are true if s is true. If the device says

an inverse transformation exists, such that s is expressed can find many cases in which the device would say "yes" for

"no" for inverse transformation. Some information is sup-

lost in generating the string s, which can not be retrieved we Sakai 33 formation (Gross, 1962). 14.2. If it is known that r is transfor~ed to s, then ..... ~n~o fact ms used to generate a particular string. If r is known to be an inverse transform of s, then this is used to recognize s, giving a possible derivational history. if no other such transformations are found, r is the only nearest history. Otherwise, the ambiguous history is to be accounted for by other rules.

If we find r and s such that r is true if and only if s is true, then we say r and s are equivalent and write

r eqv s. Obviously, this equivalence is symmetric, reflexive, and transitive. A transformation that transforms a string into an equivalent string is called an equivalence transformation, if we have a grammar consisting of equivalence transformations only, it can be used for both synthesis and analysis.

Let us confine ourselves to the equivalence transformations in order to simplify the discussion, and assume we have a set of rules or a normative device. A generalized transformation transforms a oair r = (r' r") of strings into one strin~ s. ~e inverse transformation by the same rule dissolves a string s into a pair of strings (r',r"). ~en, r' or r" is regarded as an s, and, if we find an appropriate rule, it is again dissolved into strings. By repeating the same, we have a number of equivalence which can be arranged as a tree:

s eqv (r(1),r(2));

r(1) eqv (r(ll),r(12));

r(2) eqv (r(21),r(22));

r(ll) eqv (r(lll),r(ll2));

r(12) eqv (r(121),r(122));

If an acceptable string t can no longer be dissolved into strings, we call t a terminal or an atomic acceptable string. this procedure, the strings are expected to become shorter and

equivalent information is expressed by many separate strings. still possible to transform an atomic string to another atomic of a singulary transformation. We have different atomic strings mutually equivalent. We may pick up one of them and call it a

1~e sequence of inverse transformations is not always uniquely determined. There can be other orders of dissolving a given string into atomic strings. We can make the grammar less redundant by studying the possible sequences of two acceptable

relations two acceptable

~nroughout simpler, because

It will be string by means

which are kernel string. Sakai 34 inverse transformations. If the rules are all equivalence rules, there is no theoretical problem of ambiguity• ~ne investigation of these problems requires quite a different treatment, and will not be included in this paper. 14.3. Sometimes, it is considered more linguistically reasonable to assume that a string is not acceptable but its transform is " an acceptable S ~ ~rln~ " ~ or

~. a constituent of an acceptable string, in some other cases, a s~ing may be an acceptable string and its transform may not be an acceptable string or a constituent thereof In other wo~as, a transformation is applied to an unacceptable string or a transformation results in an m%acceptable string. may prepare the rules in such a way that a sequence of obligatory transformations is contracted to a single ~ale. This seems formally simpler and consistent. However~ it will result in a more entangled system of grammar. We admit some of such strings as potentially acceptable and indicate it by a marker, This convention is somet~nes useful not merely as a technique but also as a consistent and more plausible derivation of acceptable strings. It is known that a string of a Chinese dialect marked potentially acceptable for the derivation of apparently inconsistent strings is quite acceptable in another dialect (Wang, 1964). 14.4. A generalized We

transformational rule consists of terms u and v, where u = (u',u")

= u(1)u(2)---u(i)---u(m), u' = u,(1)u'(2)---u'(i')---u'(m'), u" = u,(1)u"(2)---u"(i")---u"(m"), m = m r. ~ m ~r,

u becomes v~

v = v(!)v(2)---v(j)---v(n).

Most rules are accompanied by a number of restrictions imposed on the original strings and their transforms as well as some manipulations of strings. ~ese are classified into a few types and subroutines are to be prepared for them. Some of the operations are listed below, which have been picked up sporadically from the rules for generating Chinese strings (Hasimoto, 1964). (0) A routine supervising the subroutines takes care of the whole procedure

of applying the rules to a string, if the rules are prepared in a defin-

ite format, they are automatically checked and applied to the given string.

Certain segments r(h) and r(i) in the original string must or must not

share a certain feature in common and/or a segment r(j) must or must not (I) have a certain feature.

string and same feature must satisfy Sakai 35 the segment ~(o) of the trans-

specified by the rule.

the condition similar to (I).

The segment r(i) of the original

form must or must not have the

Some segments in the transform

Absence and/or presence of particular segments must be cne ~ c~ed.

Positions of certain segments in the string must be found.

A check of the derivational history somet~les decides the recursive

application of the rule~

The tree structure must or must not be changed by the final procedure of

a transformation. ~ . No rule describes a transformation of an individual string r into an individual string s. The rule says, if the string r has the feature (2) (3) (4) (5) (6) (7) u : u(1)u(2)---u(i)---u(m), then it is transformed to another string s which has the feature

v : v(1)v(2)---v(j)---v(n).

What are these features? They must be defined on the basis of the answers of our normative device. The program must be consistent with the features defined. Once a program is written and decided to be used, the program is the definition. If the program is modified, the rules and the lexicon are to be modified.

Since the transformations are applied to P-markers, a string is considered to be a tree-like string, if it is a linear string of terminal nodes, the other non-terminal nodes and the branches are to be determined by virtue of the concatenation rules. We consider the labels u(i) and v(j) are complete neighborhoods, if the concatenation rules are written in terms of complete neighborhoods. If the concatenation rules are written in terms of distribution classes, u(i)'s and v(j)'s are considered to be distribution classes. 14.6. The complete neighborhoods are defined on the basis of concatenated strings and we have to associate them with the labels given to the nodes of our transformational rules in order that the kernel strings can be transformed. Let us see what happens when the nodes are assumed to be complete neighborhoods. Let

p = (p',p") be a pair of acceptable strings p' and p", and let

r = r(1)---r(i)---r(m) be a segment of p. The pair p is transformed by T into Sakai 36 q = T(p), and the segment appears in q as

s = s(1)---s(j)---s(n). Some strings may have been added and some others may have been deleted. Put x(i) : C(r(i)), x : C(r)~ y(j) = C(s(j)), y : C(s). By definition, x : x(1)---x(i)---x(m), y = y(1)---y(j)---y(n). Any string belongs to one and only one distribution class J. Therefore, instead of T(r(1)---r(i)---r(n)) = s(1)---s(j)---s(n), we write T(J(x(1))---J(x(i))---J(x(m))) = J(y(1))---J(y(j))---J(y(n)). Since all the elements in a J has the same complete neighborhood, we rewrite the above as T(x(1)---x(i)---x(m)) = y(1)---y(j)---y(n). This is rewritten again by breaking down in the form X = x(1)---x(i)---x(m), y : T(x) = y(1)---y(j)---y(n). If we have a complete set complete neighborhoods of the borhood x. The transformation y is to be generated in virtue the structural requirement of as: ~ne complete neighborhood x of x(!)---x(i)---x(m) of complete neighborhoods is transformed to another complete neighborhood y of the node dominating the string y(1)---y(j)---y(n).

of rules which gives the concatenation of any language, then we can find the complete neigh-

takes place when x is changed to y. The string

of the information brought forward from x and y itself. A transformation is then interpreted the node dominating the string Sakai 37 This interpretation, however, suggests a few problems, 14_~. We know that

J(x(1))---J(x(i))---J(x(m)) (J(x),

J(y(1))---J(y(j))---J(y(n)) m J(Y)"

The statement "x is transformed to y" is a generalization of the original

fact, and this generalization is not always true. The text should be checked before a transformational rule is applied to it. Some separate steps for this purpose will save the machine time.

(1) A text to be parsed must consist of segments specified by the

correct segmentation can be done by finding the tree structure

text. Therefore, the concatenation rules must be prepared so

account for the structure of any acceptable strinG.

Not all the trees of the specified form undergo the inverse transformation

so that the derivational history may be traced back. The nodes are

labeled. A tree of a form can correspond to a number of trees whose nodes

have different labels.

When a string is being synthesized, the text is given as a pair of P-

markers. A rule can be applied only if the P-markers meet the condition

specified by the rule. 14.8. We may regard the structure mentioned above as a representation of derivational history. The history can be recorded by listing all the derivational steps the string has experienced. This representation, however, will be redundant and inefficient, because it is likely to occur that an identical series of transformations is applied to strings of different history. On the other hand, it is also possible that the strings p and q of different histories result in an identical string s by a transformation and the string s is ambiguous in that the s from p can undergo a sequence of transformations and the s from q another; thus the structure itself can not be an absolutely reliable marker.

We think it more practical to associate the rules with the features in the P-marker to which the rules are applied. '~lese features should correspond to the series of transformations applicable to the P-marker in case of synthesis and the series of inverse transformations in case of analysis. We have some rules with notes on the type of transformations to which the resultant strings may be exposed (Hasimoto, 1964). 15. Complete Neighborhppds and Transformational. Rules.

Let us assume u(i)'s and v(j)'s are complete neighborhoods. rule. The

of the as to ! ~ (2) (3) Saka± 38 ~ . Two strings r and s may replace th~ same non-terminal node to yield a longer acceptable string. However, when a transformation T is to be applied, they must hav~ the specified structure; thu~ the str!n~ p with r a~ a ~e~ment in it may be transformed by T, while the string q which differs from p only in that it has the segment s in the place of r may not. The lack of q by T means C(r) / C(s). 1_~,2. Because of this complexity involved in natural languages, we encounter a difficulty when we try to prepare a set of syntactic data for practical purposes. We refine the definition of complete neighborhood in such a way that C(r) of a string r is the set of all contexts of r which appear in the strings to which no transformations have ever been applied during their derivation. The difference between r and s is found in their internal structure, if the machine is given only the input string to be parsed. difference, where

Let strings, Let d(i;j) possible possible we have the elementary neighborhood e(i;j) defined over the set of kernel strings and transforms. These e(i;j)'s are no longer necessarily disjoint:

e(i;j) ~e(i;j') ~ c(i). l_l_l~. ~e separation of kernel strings and transforms siderable complexity. Let q be a transform. It is a a transformation in a sequence of transformations and string to be transformed by the following transformation.

A transformation is accompanied by the set P of original strings and the set Q of transforms: still involves a contransform generated by it can be an original P = set(p: T is applicable to p), In order to indicate this kernel strings, transforms, kernel strings and over the set of kernel we put

U O(r) =

is defined

is defined

is defined

transforms. c(i) be an elementary neighborhood defined and let r be a real or imaginary string such that

C(r) = o(i).

be the elementary neighborhood defined over the set of all the transforms of which r is a segment, where j corresponds to the sequence of transformations. Putting

c(i)~ d(i;j) = e(i;j), c(r) C(r) D(r) E(r) over over over ~.,e sez the set the set of of of Sakai 39

Q = set(q: q = T(p), p in P). We simplify the situation by defining the complete neighborhoods over P and over Q. The feature of T is shown more explicitly in this way. Let A be a node and imagine a derivation by the context sensitive rules

A ~> BC

B F / ---C

c--> G / B--where the s~nbols are assumed to be complete neighborhoods. Let B be replaced by F first to yield FC, and the third rule can no longer be applied because of the lack of its necessary environment B---. When these rules are to be used in analysis, none of the contexts ---C or B--- is relevant in the given string FG of complete neighborhoods. We can get rid of this difficulty by defining B and C over a set of strings and F and G over another, and by considering a transformation from BC to FG, prohibiting the operations on the strings FCand BG. l~t

p = p(1)---p(i)---p(m) be a string in P, and let

q = q(1)---q(j)---q(n)

= T(p) be the transform of p by T. We define the complete neighborhood of ~(i) over P and that of q(j) over Q. By modifying the meaning of the notation, we put

x(i) = C(p(i)) over

y(j) = D(q(j)) over The requirement that p(i) should appear as

p(i) = q(j), c(p(i)) # o, O(q(j)) 0; if p(i) does not occur in

x(i) if q(j) does not occur in

y(j) The relational conditions

P,

Q. q(j) in Q gives P P ~ Q; Q P ~ Q.

Q, then = C(p(i)) = E(p(i))

P, then = D(q(j)) over = E(q(j)) over

imposed on the segments p(i) of the original string over over Sakai 40 and q(j) of the transform are indicated in terms of E(p(i)) and E(q(j)), or by a relation between C(p(i)) and D(q(j)).

be set Q can include a part of the set P' of original strings to which another transformation T' can be applied. ~hus, we can classify the strings with respect to possible transformations. We have no positive grounds to assume any natural language has a stratified system of layers arranged one over another. • -~ 15.4. Let u = (u' u") = u(i)---u(i)---u(z) be a pair of concatenations u' = u'(1)---u'(i')---u'(m') . and u" = u"(1)---u"(i")---u"(m") of complete neighborhoods u'(i')'s is linear, the non-terminal nodes rules. We assume the rules of the form f(T(u);v) mean, over Q, a relation between T(u) and v. We assume further a rule is applicable to the given pair of concatenated complete neighborhoods x = (x',x")

= x(1)---x(i)---x(m) if the condition g(x;u) holds. That is, if g(x;u) over P, then f(T(u);v) over Q. We expect to find the transform T(x) in terms of v of the rules in the set

R(x) = set(f(T(u);v): g(x;u)) of the applicable rules.

Given the rules of the same form and a string represented by a concateand u"(i")'s defined over P. If the string

are to be determined by concatenation nation

y = y(1)---y(j)---y(n) of complete neighborhoods, an inverse transformation is to be carried out by finding the set R(y) = set(f(T(u);v): h(y;v)) of applicable rules.

With all the linguistic difference transformational rules, they exhibit

between the concatenation rules and formal similarities when the labels are Sakai 41 assumed to be the sets of contexts. We will not repeat a similar discussion on the choice of f(T(u);v), g(x;u), h(y;v) or the algorithm for finding x or y. 16. Distribution Classes and Transformational

Let p be a string and T(p) its transfo~n be a set of strings p to which T is applicable. of P as the set of all T(p)'s:

T(P) = set(T(p): p in P). A rule will be written in the form

f(T(P);Q) to indicate a relation between the sets T(P) and Q.

In order to specify the sets a little closer to the form of rules usually prepared by linguists, we put

Rules. by the transformation T. Let P

We defined the transform T(P)

p = p(1)p(a)---p(i)---p(m)

q = q(1)q(2)---q(j)---q(n), where p(i)'s and q(j)'s are segments in p and q, respectively.

P : P(1)---P(i)---P(m)

Q = Q(1)---Q(j)---Q(n), which are to be understood as concatenated sets if strings.

A rule of the form f(T(P);Q) is applicable to the string

p(i) ~ P(i) for giving T(p) 6 T(P), so that f(T(P);Q) provides us with the information governed lexicon and each constituent in the string given a marker which indicates whether or provided that the sets are established systematically. Because of the ambiguous property of real strings, the markers will be given interms of complete neighborhoods defined over the set of (potentially) acceptable strings. 17. Establish!~ent and Representation of Complete Neighborhoods.

A syntactic function is called a complete neighborhood if it is defined as a set of contexts. We use conventional terms and redefine them assigned to complete neighborhoods.. 17.1. In establishing a set of complete neighborhoods of a natural we assize a few of them as undefined terms and derive the others by concatenation rules. Sometimes, there will be a choice among a few i = i, 2, ---, by this rule.

under analysis not it belongs Then we put p, if

string in the synthesis is m, Each

or to any set of strings, as symbols language, hypothetical hypothetical Sakai 42 rules. We take one of them to define a complete neighgorhood and regard the others as the property of the complete neighborhood defined by the former. Thus, we distinguish two kinds of rules: definition rules and property rules. Let

axb = c and xd = f be hypothetical rules. If one decides to regard the former as the definition of x, the latter is a property of x. ~%is method is applied not only to phrase structure grammar but also to transformational o~ammar, because both transformations and inverse transformations are applied to a (pair of) P-marker(s) to yield another (pair of) P-marker(s).

Every time a definition rule is established as a hypothesis, it must be tested as to whether or not it contradicts any other definition rules. ~,o "~ property rules should contradict any other rules. %~nenever a contradiction is found, the source of trouble must be found out by tracing back the definition rules, and the hypothesis that has given rise to the trouble must be modified. 17.___~2. The complete neighborhoods of all the acceptable strings (as distinguished from the other ambiguous interpretations of the same string) are identical to each other and consist of one element indicating that the strings are acceptable. It seems adequate, for most of the natural languages, to admit two complete neighborhoods, nominals and verbals, although there are no rigid grounds. Many others are derived from hypothetical concatenations that can occur in acceptable strings.

The prepositions in many European languages are subclassified by the case of thenominals they govern, and the nominals by their case, gender and number. A rule for yielding prepositional phrases will be stated as follows: a preposition that governs nominals of case c, followed by a nominal of case c', of any gender and of any number, results in a prepositional phrase, provided the cases c and c' are the same. As suggested in this example, subclassification and desubclassification are useful to describe syntax. A number of indices are made use of in subclassifying a broadly defined complete neighborhood. The example above will be rewritten, by introducing the indices c for case, g for gender and n for nu~nber, and a coefficient d(c,c'), in the form

prep(c) n(c';g;n) = d(c,c') prep-n,

d(c,c') = 1 c = c',

= 0 c ~ c'. where if if Sakai 43 %he indices g and n are arbitrary if the preposition in question takes nominals of any gender and of any number.

Usually, a linguist will define complete n,~g~noornoocs broadly so that the majority of acceptable ~rmn~ ..... .... may be generated and recognized correctly. As his analysis proceeds further in c~eoa1~, he ~ill take an exa~mT~le that is not generated or recognized correctly by his broadly defined complete neighborhoods: generation may give him some unacceptable strings or the syntactic analysis may give him erroneous or unnecessarily ambiguous interpretations. He will then trace back the definitions and find out some of his rules hold in his example with respect to a subset of one of his complete neighborhoods. Suppose he has a set R(xy) of rules to concatenate x and y. His new example will indicate that the rules are not always true. He may then establish the subsets x', x", y', y", and a new set of rules which allows x'y' and x"y ~', for instance, but not x'y" or x"y'. 17.3. Let a broadly classified complete neighborhood be shown by a symbol, say, v. If a subclassification thereof is desired, we introduce an index p, such that v : v(p l) U v(p 2) U--- Uv(pn). When the subclassification v(o) : is not necessary, we put p = O; Uv(pi ), i : l, e, ---, n. The union of a few subsets

are written as v<Pl,P3,P 5) : v(P I) U v(P 3) U V(Ps),

etc. If a complete neighborhood is to be subclassified from a few different points of view, ~s many indices are introduced:

v(p;q), v(p;q;r), etc;

v(Pl,P2; q) = v(Pl; q) ~ v(P2;q), v(p;ql,q a) : v(~;q l) U v(p;qa), v(p;o) n v<o;q) : (Uv(p~qj)) ~ (Uv(pi~q)) = v(p;q),

etc. Hence, for the distribution classes ~(V(Pl,pa;~)) : H(V(~l;q)) N X(v(Pa;q)), I(v(p;q)) = !(v<p;o)) ~\]i(v(o;q)), Sakai 44

etc. Sometimes, an index depends upon other indices:

v(p;q(r;~;t)),

~%e meanings of r, s and t depend upon the meaning of q. where the broad a classification

It will be "razbijenije", kind of representation, used by many research groups, enables us to describe the syntax of a language systematically. Each digit can be regarded as an indication of a certain feature common to some elementary neighborhoods, and classifies them according to their specific 1_~.4. Suppose a concatenation rule f(uv;w) complete neighborhoods to determine z = xy, represented by the indices in the form

x = (a(x);b(x);---;n(x)),

y = (a(y);b(y);---;n(y)),

z = (a(z);b(z);---;n(z)),

u : (a(u);b(u);---;n(u)),

v = (a(v);b(v);---;n(v)),

w = (a(w);b(w);v--;n(w)). If a rule indicates the relation between the an index k(w), and if all the others are independent u = (o;---;o;i(u);O;---;o), v : (o;---;O;j(v);O;---;O),

w = (O;---;O;k~w);O;---;O). F If the pairs (i(x),i(u)), (j(y),j(v)) and (k(z),k(w)) specified by the grammar system being used, the rule gives a z modified by this rule. ~ne rule gives no other indices. This information should not be lost We have to indicate in the rule how to transfer the or y. A simple method was used in a translation program (Sakai, 1961).

A transformational rule requires that certain features of the original for example.

The above be represented scheme may be further generalized. by a number of indices

(a;b;c;---;n),

class symbol is one of the indices and each index represents

from a certain point Of view.

of interest to compare these indices with the concept of "okrjestnostj" (Kulagina, 1958) or "sememe" (lamb, 1962). Let a complete neighborhood ~nis features. is to be applied to a text xyof and the complete neighborhoods are pair (i(u),j(v)) of indices and of these, we have

satisfy the condition

is applied to xy and information as for the if it is in x or y. information to z from x Sakai 45 string are carried forward to its transform. ~lis requirement is usually indicated by the identity of features of certain segments in the original string and its transform. The use of rules is to be programmed in such a way that, if the rules are applicable to the string regardless of a certain index, the value of the index in the original string is transferred to the corresponding index of the transform, and vice versa in case of an inverse transformation. 17_~.. An extremely simplified example is given. %~ne complete neighborhoods are no longer treated as sets. The symbol ~'+" means "or". The symbol "=" does not necessarily mean an identity: it can be replaced by an arrow. The segments of the string

~hey are red ~ianes

1 2 3 4 are rePresented in the form (h,k):

(i,i)

(1,3)

(2,3)

etc. Both (h,i)(j,k) and (h,i) and (j,k). The following

adj:

adj-pred: adjectival predicate

anim: animate

compl: complement

inanim: inanimate

m: masculine

n: nominal

n/n: modifier of nominal

nom: nominative

pl: plural

pn: pronoun

s: sentence

v: verbal

-k: ends with k

-t: ends with t = they, = they are red = are red,

* (j,k) mean the concatenation of the strings (h,i)

abbreviations are used. adjective Sakai 46 Input Langua ~e ..... (l,4)(v;s) = (l,1)(pn;3rd "'~),~.- ~ (2,2)(v;be;pres;3rd;pl) * (3,4)(n;p!) (3,4)(n;pl) : (~,3)(ac, j) ~ (4,~)(:~,;'?l) Intermediate Renresentaion. (l,4)(v) = (i,l)(pn;3rd',p-,;'nom) '~ (2,2)(conula;pres). * (3,&)(n;compl;p i) (3,4)(n;comp!;pl) = (3~3)(n/n) * (4,4)(n;compi;pi) Output Lan~ua{e (Russian) (i,l)(pn;3rd;pl;nom) = on(Dl;nom) = oni (2,2)(copula;pres) = () (3,3)(red)(n/n) = krasn(adj;hard) (4,4) (plane)(n;comp!;}l) = (rubank(-k) ~ samoljet(-t))(n;m;pl;nom)

= (rubanki + samcijety)(n;m;pi;nom) (3,4)(n;compl;pl) = (b,b)ta~:;na;~J (i,4)(v) = (i,i)~,~,)<~ ' . . . . . . ' " ,') . = onto " Output L~ n , ' : u a : : e :- " " (1,1)(pn;Srd;pl;nom) = (i~are(anim) ~ sore(inanim))(pn;pl;nom) (2,2)(copula;pres) = ar(v;%;pres:final) = ar-u (3,3) (red)(n/n) = aka(adj-pred~n/n) (4,4) (plane) (n;compl;p!) = (keimen ¢ hikooki) (n;inanim;compl)

~ (4, ~)(n;m;pl;nom) = (3,3)-yje(4,4) k~'asnvje ~ ( I"UO~'.-~ ....... -i- samoljety) (3,4)(n;compl;pl) = ((3,.-5)-i(4,4))(n;inanim)-de (l,4)(V) = (l,!)(anim,:inanim~pn;pi;nom)

( 2,2 ) (v; %; lores ; ~inai)

= (l,i)tlnan~m;pn;ip;nomj * <p,4)(u;mz~onmm)-~e * (2,2)(v;4;pres;final)

= sorera (ga ~ wa) akai (heimen + hikooki) de aru 17.6. We observe in ~he above example ~hat the index of an animate or an inanimate object affects the choice of a lexicai element in Japanese while it is not relevant in ~zlzsn. if'his phenomenon may be considered syntactic in one lauguage and semantic in another. Take two languages A and B, and suppose A has a syntactic marker o '~ qender and '5 does not. The gender is considered syntactic in A and sema:r~ic iu S. The syntactic genders are trary and can not be al'.~-,?/~ nrcse::'vec i'a the ~ranszer process to another. We will ,:~v~ -~= t;o --'~*e.,~ar<.~~: - ~ two se;oarate_ procedures gender. Si::;ilar ~-" .~r;.~ - ~ ....... ... : armse ~.~.~ res::~ec~ to ozher indices

~e choice of iexical elements de\]cends greatly upon the of language, k~ne ~.~on =-' ~ ....... is si;t;iiar when we observe some longer constituents.. The ch,:,.ice of constituents is limited semantic or habitual reasons as indicated by the branches of' the second kind sometimes arbi-

from one language for handling

habitual usage combinations of by logical, Sakai 47 in the net strings. Sometimes the choice is quite capricious. It seems more practical to handle this kind of information separately (Matthews, 1965), corresponding to the separate normative devices the lin&~ist has conjectured. Acknowledgment.

The need of defining distribution classes was recognized when I was with the Machine Translation Project, bniversity of California. The basic approach was worked out at the First Research Center, Defense Agency of Japan, and was refined and finished at the Project on Linguistic Anaiysis, Ohio State University. I appreciate the encouragement of these organizations. References. Gross, M.: On the Equivalence of Models of Languages Used in the Fields of

Mechanical Translation and Information Retrieval, NATO Advanced Study

Institute on Automatic Translation of Languages, Venice, 1962. Hasimoto, A. Y.: Revised Rules of Mandarin Grammar, Project on Linguistic

Analysis, Ohio State University, Columbus, Ohio, 1964. Kulagina, O. S.: Ob Odnom Sposobje Oprjedjeljenija Grammaticeskix Ponjatij

na Bazje Tjeorii ~ho~estv, Probljemy Kibjernjetiki, Vypusk i, Moskva,

1958. Lamb, S. M.: Outline of Stratificational Grammar, University of California,

Berkeley, California, 1962. ~tthews, P. H.: Problems of Selection in Transformational Grammar, private

circulation, indiana University, to appear in the Journal of Linguistics,

No. l, 1965. Opler, A.; Silverstone, R.; Saleh, Y.; Hildebran, M.; Slutzky, I.: The Applic-

ation of Table Processing Concept to the Sakai Translation Technique,

Mechanical Translation, vol. 7, No.2, 1963. Parker-Rhodes, A. F.: A New Model of Syntactic Description, 1961 International

Conference on Machine Translation of Languages and Applied Language

~alysis, Her Majesty's Stationary Office, London. Sakai, I.: Syntax in Universal Translation, 1961 International Conference

(See above). Wang, W. S.: Two Aspect Markers in Mandarin, Project on Linguistic Analysis

(See above), Report No. 8, 1964. Sakai 48 Appendices. A-I. Sets. a ~ A; ~ in A: ~ is an element of the set A; ~ belongs to A; ~ is in A. a~A; ~ not in A: a~A is not true. A (=) B: there is at least one element which belongs to both A and B. A~ B; B~ A: if a ~ A, then a~ B; A is a subset of B; B is a superset of A. A = B: a ~ A if and only if a ~ B; A~B and A~B. A # B: A = B is not true. A = O: there is no element in the set A; the set A is empty. A = set(a,b,c,d): A is a set whose elements are a,b,c and d. A = set(ai: i = 1,2,---): A = set(al,a2,---). A = set(a: f(a)): a~ A = B ~ C: A = set(a: A = Us i, i = l,a,---: A = U B for f(B): A A = B~ C: A = set(a: A = D i' i = 1,2,---: A = ~B for f(B): A A-2. Boolean Coefficients. We introduce coefficients which indicate Let a, b, etc. be the coefficients and x, y, cient is either 0 or i: ax = 0 = empty set, =x, The sum a ÷ b and the product ab = a axU bx = (a + b)x and ax ~ by = ab(x D Y) Therefore, the coefficients are Boolean: Consequently, for concatenation, we have (ax)(by) = abxy. A if and only if f(a) is true. a ~ B or a ~ C); A is the union of B and C.

~ : BIU B 2U--- • is the union of all B's a ~ B and a ~ C); A is

, = n B 2 n -is the intersection of satisfying f(B). the intersection or meet of B and C. all B's satisjying f(B). presence or absence of sets. etc. = O, =!. are if if X b)(x ~ y) = x D Y, if a = b : i, = O, if a = 0 or b = O. sets. The value of a coeffidetermined by

a = I or b : l,

a = b = O,

if

if

X = x, = O, = (a a a b = i + 0 = i + i = i, Sakai 49 Table of Contents 1. Introduction.

2. Symbol; String; Language.

3. Context: Neighborhood.

4. Equivalence of Contexts.

5. Complete Neighborhood.

Elementary Neighborhood.

7. Distribution Class.

8. Concatenation.

9. Concatenation of Complete Neighborhoods. lO. Concatenation of Distribution Classes. ll. Rules for Recognition and Generation. 12. Complete Neighborhood Representaticn of Concatenation Rules. 13. Distribution Class Representation of Concatenation Rules. 14. Some Remarks on Transformation. 15. Complete Neighborhoods and Transformational Rules. 16. Distribution Classes and Transformational Rules. 17. Establishment and Representation of Complete Neighborhoods. Acknowledgment. References. Appendices. Table of Contents. .