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<Paper uid="C82-1050">
  <Title>North-Holland Pab~hing Company Composition of Translation Schemes with D-Trees</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
COMPOSITION OF TRANSLATION SCHEMES WITH D-TREES 315
</SectionTitle>
    <Paragraph position="0"> Remark. Analogically as a translation scheme of the type string - ~-~ree was defined, also definitions of the type string- string (TS \[S,SI ) or of the type D-tree - D-tree (TS \[D,D3 ) can be given.</Paragraph>
    <Paragraph position="1"> By TS \[S, Sj the lefthand side and righthand side of a rule is always a string. By TS \[D,D\] both sides of a rule are always D-trees.</Paragraph>
    <Paragraph position="2"> As TS we denote the set of all translation schemes of all the three types.</Paragraph>
    <Paragraph position="3"> Def. Let TI,... ,Tn be a sequence of TS. We denote as TR(T~,...,Tn)=TR(TI).TR(T2)...TR(Tn). The main definition of this paper is the following: Def. A generative system (GS) is a sequence TI,... ,Tn of TS, whe~'~-TR(Tl,... ,Tn) is a relation between strings and D-trees and for every \[dl,d2~ ~ TR(Tn) there exists a. sl,so,such tha_%</Paragraph>
    <Paragraph position="5"> We call this property of GSI an asynetric property of GS.</Paragraph>
    <Paragraph position="6"> Def. Let GSI be a GS. We say that the function MS is a function of ~ minimal synthesis of GSI, if the following conditions are fulfiled:</Paragraph>
    <Paragraph position="8"> Def. D-grammar (DG) is a T &amp; TS \[S,D~, where T=(VN,VT,S,P) and forgery p PS p,p=LS~--A-~RS there holds, that LS=S(RS).</Paragraph>
    <Paragraph position="9"> De__ff. We denote DRo= ~ TR(T);T 6 DG\]and DRj= ~TR(TI,...sT~) I TI,...,Tj 6 GSS for j 6 N. Por ~ ~ N ~/ ~0} we write IDRj= IF6 DRj; F is a function } .</Paragraph>
    <Paragraph position="10"> Note. We need also one more concept. It is %he concept of an B'~phic generative system for another one.</Paragraph>
    <Paragraph position="11"> Def. Let ~I,V2 be two alphabets and h:Vl--,V2. Let SI=~qTI,LRI,ol), S2=(U2,LR2,o2) be two strings, where oI:UI---~VI, o2:U2-@ V2. We say that a tuple (f,h) is an h-morphia,, from SI to $2, when f:UI--&gt;U2 is a one-to-one mapping which preserves the ordering on nodes and for every u ~ UI there holds that h(ol(u))=c2(f(u)). We say that SI is h-morphic for $2, if there exists an h-morphism from SI to $2.</Paragraph>
    <Paragraph position="12"> Def. Let DI=(UI,LRI,BI,rl,ol) and D2=(U2,LR2,B2,r2,o2) be D-trees. Let-T~,h) be an h-morphism S(DI) to S(D2). Let there hold that \[u,vl PS BI iff It(u), t(v)3 ~ B2 and t(rl)=r2.</Paragraph>
    <Paragraph position="13"> We say that (t,h) is a h-morphi~ from DI %o D2.</Paragraph>
    <Paragraph position="14"> We say that DI is h-morphic to D2, when there exists an h-morphism from DI for D2.</Paragraph>
    <Paragraph position="15"> Def. Let TI=(VNI,VTI,SI,PI) and T2=(VN2,VT2,S2,P2) be TS.</Paragraph>
    <Paragraph position="16"> Let~..V~ A2 VTI---~VN2 t/ VT2, where h(VNI)=VN2, h(VTI)=VT2. Let there exist a one-to-one mapping MP from PI on P2 such, that if p=LSI ~--- AI---&gt; RSI and M~P(p)=LS24---A2---~ RS2, then LSI is h-morphic to LS2, RSI is h-morphie to RS2 and h(AI)=A2.</Paragraph>
    <Paragraph position="17"> We then say, that TI is h-m~rphic for T2.</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
316 M. PI~TEK
</SectionTitle>
    <Paragraph position="0"> Def. Let GSI=TII,... ,T~ and GS2=T21,... ,T2 n both be GS.</Paragraph>
    <Paragraph position="1"> Let T11 be hl-morphic to T2 l, T12 h2-morphic to T22,... and so on to n! we say then, that GS1 is h-morphic for GS2, where h=(hl,... ,~).</Paragraph>
    <Paragraph position="3"> It holds that: -c-c-o.</Paragraph>
    <Paragraph position="4"> We can sea that Dom(TR(T3))= . a n, n,j~ ~, a n c &amp;quot; j,n ~N~ and that TR(T3) is a function.  Notation. I sJ denotes the length of the string s, which is the where U is the set of nodes of s. Assertion 2. Let&amp;quot;GS1 be a generative system, a--~Then there exist an algorithm that computes for every string v the set AN(GSI,v) ~(analysis) with the time complexity bound by a function K1. Ivl~o max ~card (AN (T~,...,Ti;v) j ~-I , where El depends only on GSI. - ~ ~-b) Then there exists an algorithm that computes for every D~tree d the set ST(GSI,d) (full synthesis) with the time complexity bound by function K2. I S(d)l ~ . card (ST(GSI)), where I~2 depends only on GSI.</Paragraph>
    <Paragraph position="5"> Assertion ~. Let GS1 be a GS. Then there exists an h-morphic generative system GS2 for GS1 and an algorithm that for every D-tree d com~utes ST(GS2,d) with m time complexity bound by function</Paragraph>
    <Paragraph position="7"> Assertion 4. Let GSI be a GS. Then there exists an h-morphic gene r~tive sys%'em GS2 for GSI and an algorithm such that for every D-tree d computes MS(d) with a time complexity bound by function K . ~ S(d) I , where MS is the function of minimal syntesis of GS2, Dom(TR(GS1))=Dom(TR(GS2)) and K depends only on GS2.</Paragraph>
    <Paragraph position="8"> Remarks.</Paragraph>
    <Paragraph position="9"> Remark to Assertion I.</Paragraph>
    <Paragraph position="10"> We sketch here a proo'~ of Ass. i. We see that DR o ~ DR I an~ IDR o d IDR I. DLkovskij an~ Medina have shown in \[ 2~ , that TR (T3) from Example I cannot be in DRo~ We see that T3 is a TS. Thus DR O~DR I. Since ~R(T3) is a function , we see that IDR o ~ IDR I. 318 M. PLATEK In the Example 2 we have shown that IT(k) ~ IDR~. Prom the results on composition of pushdown transducers (PST) in ~ &amp;quot;4~ and from the equivalence theorem between TS's and PST's from _i it follows, that IT(k+l) ~ DRk . Thus DRj ~ DRj+ 1 and IDRj - 1DRj+ 1.</Paragraph>
    <Paragraph position="11"> Remark to Assertion 2.</Paragraph>
    <Paragraph position="12"> The algorithm for analysis and synthesis for a GS is based on the idea of Cocke-You~ger-Kasami algorithm. Por a seguence of simple translation schemes of the type string-string the algorithm is presented in Suchomel E7J . The difference between the upper boundary of the time complexity of the full synthesis and analysis is given by the asymmetric property of a GS.</Paragraph>
    <Paragraph position="13"> Remark t 9 Assertion 3.</Paragraph>
    <Paragraph position="14"> The basic idea of the proof is a construction of a new GS to GSI. The new GS, denoted GS2, has full information in the alphabets for a straightforward algorithm for a full synthesis.</Paragraph>
    <Paragraph position="15"> Remark to Assertion 4.</Paragraph>
    <Paragraph position="16"> The idea of the proof is a~alogous to that of Assertion 3. When we have a partition of Dom(TR(GSI)) in the clases of synonymous sentences, the function of minimal synthesis chooses always only one representant of his class. Therefore the algorithm can be so fast. Conclusion remarks.</Paragraph>
    <Paragraph position="17"> When formulating a grammar for natural language, we can use with advantage the modularity of GS. We have shown that the time complexity of the analysis and synthesis for DRj, j &gt;_ 2 is independent on ~. Otherwise the explicative power of DR.. is increasing with j. We have also shown, that to any generative ~system there can be con structed an h-morphic generative system with the full information for a fast algorithm of the minimal synthesis.</Paragraph>
  </Section>
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