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<Paper uid="C82-1050">
  <Title>North-Holland Pab~hing Company Composition of Translation Schemes with D-Trees</Title>
  <Section position="2" start_page="0" end_page="0" type="intro">
    <SectionTitle>
INTRODUCTION
</SectionTitle>
    <Paragraph position="0"> A generative system for Czech was presented in Sgall \[~ .</Paragraph>
    <Paragraph position="1"> The concept of a generative system was studied by Pl~tek \[4\] and Pl~tek and Sgall \[~ . In this paper we use a similar approach as that presented by Haji~ov~, Pl~tek and Sgall in ~3~ * We define generative systems as a fundamental device for construction of grammars of natural languages. We give here some mathema tical results to illustrate the usefulness of the new concept.</Paragraph>
    <Paragraph position="2"> We try first to formulate the necessary requirements on a grammar G of a natural language L. The grammar G must determine: a) The set of all correct sentences of the language L.</Paragraph>
    <Paragraph position="3"> The set will be denoted by LC.</Paragraph>
    <Paragraph position="4"> b. ~he set of the correct structural descriptions (SD) of the anguage L. SD represents all meanings of all sentences of LC.</Paragraph>
    <Paragraph position="5"> c) The relation SH between LC and SD. The relation SH describes the ambiguity and the s~11ony~ of L.</Paragraph>
    <Paragraph position="6"> By a structural description we understand a dependency tree (D-tree). The concept of a simple translation scheme from \[I\] is a generalisa tion of context-tree grammar. We introduce here a similar concept of a translation scheme, in this case as a generalisation of dependency grammar (see \[2\] , \[5~}.</Paragraph>
    <Paragraph position="7"> A generative system (GS) is defined as a sequence of translation schemes with a special as~etric property.</Paragraph>
    <Paragraph position="8"> We show that the explicative power of GS increases with the length of GS. We present results concerning on algorithm for the analysis and synthesis of GS and show that its time complexity is independent on the length of GS.</Paragraph>
    <Paragraph position="9"> Moreover for a given GS we can construct a similar GS, for which a fast algorithm for synthesis exists.</Paragraph>
    <Paragraph position="10"> Definitions.</Paragraph>
    <Paragraph position="11"> Notation. The vocabulary, sets of nodes, edges and rules are here nonemDtyand finite sets.</Paragraph>
    <Paragraph position="12">  Let R be a relation. We denote</Paragraph>
    <Paragraph position="14"> Def. A ~ over a vocabulary V is a triple S=(U,LR,o), where U i~-~ set of nodes, LR a linear ordering of U, o:U--*V. Let o(u)=A. We say that A is the value of node u. Let S=(U,LR, o), Sl=(U1,LRl,ol), S2=(U2,LR2,o2) be the str'.~ and u ~ U. We say that $2 is ~ from S by replacin~ u by S1, when the string SI is place~n the predecesor and the succesor o~ nods u and otherwise $2 does not differ from S. We denote as V the set of all nonempty strings over V.</Paragraph>
    <Paragraph position="15"> Def. Let S1 = (U1,LRl,ol), 62 = (U2,LR2,o2) be strings.</Paragraph>
    <Paragraph position="16"> Let U1 = ~Ull,...,Uln~ and U2 = {u21,deg..,U2n~ and ul l, .... ,ul n be in the ordering LR1, and u21,... ,u2 n in the ordering LR2 and ol(uli)= o2(u2 i) for all i between 1 and n. Then we say that S1 and $2 are equivalent.</Paragraph>
    <Paragraph position="17"> We shall not distinguish between equivalent strings.</Paragraph>
    <Paragraph position="18"> Def. A quintuple SR=(U,LR,B,r,o) is called a D-tree over V,when S(SK~(U,LR, o) is a string and o:U--eV, B(SR)=(U,B.r) is a tree with the root r and when the following condition holds: The nodes of every path in B(SR), which begins with a leaf, are nodes of a substrlng of S(SR). We say that S(SR) is a projection of SR.</Paragraph>
    <Paragraph position="19"> Def. Let SRl=(UI,LR1,Bl,rl,ol) and SR2=(U2,LR2,B2,r2,o2) be D-t~'~s. Let strings S(SR1) and S(SR2) be equivalent. Let f be a one-to-one mapping from UI on U2, which preserves the ordering LRI to the ordering LR2. Let f(rl)--r2 and let it hold that o \[u,v\] a B1 iff \[f(u), f(v)3 @ B2. Then we say that SR1 and SR2 are equivalent. We shall not distinquish between equivalent D-trees. Def. Let D=(U, LR.B,r,o), Dl=(U1,LR1,Bl,rl,ol) and D2=~O~,LR2,B2,r2,o2) be D-trees and u ~ U. We say, that D2 is produced from D by replacin~ u by D1, when S(D2) is produced from S(D) by replacing u by S(D1) and the neighbours of rl in B(D2) are the same as neighbours of u in B(D). Otherwise D2 does not differ from D.</Paragraph>
    <Paragraph position="20"> Def. A translation scheme of type string- D-trees (TS IS,D3 ) is ~-~uadruple T=(VN,VT,S,P), where VN is a the vocabulary of nontermlnals, VT the vocabulary of terminals, VN ~ VT=~, S e VN and P is a set of rules of the following type: LS-~-A--~RS, where A ~ VN (the middle of the rule) LS (the lefthand side) is a string over VN u VT, RS(the righthand side) is a D-tree over VN ~ VT and the following condition holds: When all nodes with terminals are erased from S(RS) and LS, then we get two equal strings.</Paragraph>
    <Paragraph position="21"> Let p=LS~--A---~RS. We write \[ LSI,RSI~ p--~LLS2,RS2~ , when (i):the leftmost nonterm/nal node of LS1 is some u with the value A, (ii):the leftmost nonterminal node of RS1 is some v with the value A and (iii):LS2 is produced from LS1 by replacing u by LS and RS2 is produced from RS1 by replacing v by RS.</Paragraph>
    <Paragraph position="22"> ~2 p--~ is denoted as ~ and~ is the transitive closure of_~.</Paragraph>
    <Paragraph position="23"> PEP We denote as TR(T)= i\[\[LS,RS\] ; /S,S\]~\[LS,RS\] , LS,S (RS)PS VT +} .</Paragraph>
  </Section>
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