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<Paper uid="P84-1015">
  <Title>USES OF C-GP.APHS lil A PROTOTYPE FOR ALrFC~ATIC TRNLSLATION,</Title>
  <Section position="3" start_page="0" end_page="61" type="intro">
    <SectionTitle>
2. DEFINITIONS
</SectionTitle>
    <Paragraph position="0"> C-graph. A c-graph G is a cycle free,labelled graph \[1,9\] without isolated nodes and with exactly one entry node and one exit node. It is completely determined by a 7-tupie: G=(A,S,p,I,O,E,C/), where A is a set of arcs, S a set of nodes, p a mapping of A into SxS, I the input node, 0 the output node, E a set of labels (c-trees, c-graphs) and E a mapping of A into E. For the sake of simplicity, arcs and labels will be merged in the representation of G (cf. Fig.1 . Interesting c-graphs are sequential c-graphs (cf. Fig.2a) and bundles (cf. Fig.2b).</Paragraph>
    <Paragraph position="2"> Fig.2. A seq. c-graph (a) and a bundle (b).</Paragraph>
    <Paragraph position="3"> C-trees. A c-tree or a tree with decorations is an ordered tree, with nodes labelled by a label and a decoration that is itself a decorated tree, possibly empty.</Paragraph>
    <Paragraph position="4"> Classes of c-graphs. There are three major classes: (1) recursive c-graphs (cf. Fig.3a) where each arc is labelled by a c-graph; (2) simple c-graphs (cf. Fig.l) where each arc is labelled by a c-tree and (3) regular c-graphs, a proper sub-class of the second that is obtained by concatenation and alternation of simple arcs (cf. Fig.3b). By denoting concatenation by &amp;quot;.&amp;quot; and alternation by &amp;quot;+&amp;quot;, we have an evident linear representation. For example, G4=g+i.(j+k). Note that not every c-graph may be obtained by these operations, e.g.G.</Paragraph>
    <Paragraph position="5"> Substructures. For the sake of homogeneity, the only substructures allowed are those that are themselves c-graphs. They will be called sub- null -c-graphs or seg's. For example, G1 and G2 are seg's of G.</Paragraph>
    <Paragraph position="6">  a) A recursive c-graph.</Paragraph>
    <Paragraph position="7"> b) A regular c-graph. G4= Fig.3. Two classes of c-graphs.</Paragraph>
    <Paragraph position="8">  Isolatability. It is a feature that determines, for each c-graph G, several classes of seg's An isolated seg G' is intuitively a seg that has no arcs that &amp;quot;enter&amp;quot; or that &amp;quot;leave&amp;quot; G'. Depending on the relation that each isolated seg keeps with the rest of the c-graph, several classes of isolatability can be defined.</Paragraph>
    <Paragraph position="9"> a) Weak isolatability. A seg G' of G is weakly isolatable (segif) if and only if for every node x of G' (except I' and 0'), all of the arcs that leave or enter x are in G ~. E.g.: G5=i is a segif of G.</Paragraph>
    <Paragraph position="10"> b) Normal isolatability. A seg G' of G is normaly isolatable (segmi) if and only if it is a segif and there is a path, not in G', such that it leaves I' and enters 0'. Example: G6=k is a segmi of G.</Paragraph>
    <Paragraph position="11"> c) Strong isolatability. A seg G' of G is strongly isolatable (segfi) if and only if the only node that has entering arcs not in G' is I' and the only node that has leaving arcs not in G' is 0'. When G' is not an arc and there is no segfi contained strictly in G', then G' is an &amp;quot;elementary segfi&amp;quot;; if G contains no segfi, then G. is elementary. E.g. G4 is a segfi of G.</Paragraph>
    <Paragraph position="12"> Order and roads. Two order relations are considered: (l) a &amp;quot;vertical&amp;quot; order or linear order of the arcs having the same initial node and (2) a &amp;quot;horizontal&amp;quot; order or partial order between two arcs on the same path. A road is a path from I to 0 Vertical order induces a linear order on roads.</Paragraph>
  </Section>
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