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<Paper uid="P92-1010">
  <Title>Reasoning with Descriptions of Trees *</Title>
  <Section position="3" start_page="73" end_page="74" type="intro">
    <SectionTitle>
2 Quasi-Trees
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    <Paragraph position="0"> In this section, we use the term relationship to informally refer to any positive relationship between individuals which can occur in a tree, &amp;quot;a is the parent of b&amp;quot; for example. We will say that a tree satisfies a relationship if that relationship is true of the individuals it names in that tree.</Paragraph>
    <Paragraph position="1">  It's clear, from our discussion of their applications, that quasi-trees have a dual nature -- as a set of trees and as a set of relationships. In formalizing them, our fundamental idea is to identify those natures. We will say that a tree is (partially) described by a set of relationships if every relationship in the set is true in the tree. A set of trees is then described by a set of relationships if each tree in the set is described by the set of relationships.</Paragraph>
    <Paragraph position="2"> On the other hand, a set of trees is characterized by a set of relationships if it is described by that set and if every relationship that is common to all of the trees is included in the set of relationships. This is the identity we seek; the quasi-tree viewed as a set of relationships characterizes the same quasi-tree when viewed as a set of trees.</Paragraph>
    <Paragraph position="3"> Clearly we cannot easily characterize arbitrary sets of trees. As an example, our sets of trees will be upward-closed in the sense that, it will contain every tree that extends some tree in the set, ie: that contains one of the trees as an initial sub-tree. Similarly quasi-trees viewed as sets of relationships are not arbitrary either. Since the sets they characterize consist of trees, some of the structural properties of trees will be reflected in the quasi-trees. For instance, if the quasi-tree contains both the relationships '% dominates b&amp;quot; and &amp;quot;b dominates c&amp;quot; then every tree it describes will satisfy &amp;quot;a dominates c&amp;quot; and therefore it must contain that relationship as well. Thus many inferences that can be made on the basis of the structure of trees will carry over to quasi-trees. On the other hand, we cannot make all of these inferences and maintain any distinction between quasi-trees and trees. Further, for some inferences we will have the choice of making the inference or not. The choices we make in defining the structure of the quasi-trees as a set of relationships will determine the structure of the sets of trees we can characterize with a single quasi-tree. Thus these choices will be driven by how much expressive power the application needs in describing these sets.</Paragraph>
    <Paragraph position="4"> Our guiding principle is to make the quasi-trees as tree-like as possible consistent with the needs of our applications. We discuss these considerations more fully in (Rogers &amp;5 Vijay-Shanker, 1992).</Paragraph>
    <Paragraph position="5"> One inference we will not make is as follows: from &amp;quot;a dominates b&amp;quot; infer either &amp;quot;a equals b&amp;quot; or, for  some a' and b', &amp;quot;a dominates a', a' is the parent of b', and b' dominates b&amp;quot;. In structures that enforce this condition path lengths cannot be left partially specified. As a result, the set of quasi-trees required to characterize s' viewed as a set of trees, for instance, would be infinite.</Paragraph>
    <Paragraph position="6"> Similarly, we will not make the inference: for all a, b, either &amp;quot;a is left-of b&amp;quot;, &amp;quot;b is left-of a&amp;quot;, &amp;quot;a dominates b&amp;quot;, or &amp;quot;b dominates a&amp;quot;. In these structures the left-of relation is no longer partial, ie: for all pairs a, b either every tree described by the quasi-tree satisfies &amp;quot;a is left-of b&amp;quot; or none of them do. This is not acceptable for D-theory, where both the analyses of &amp;quot;pseudo-passives&amp;quot; and coordinate structures require single structures describing sets including both trees in which some a is left-of b and others in which the same a is either equal to or properly dominates that same b (Marcus, Hindle &amp; Fleck, 1983).</Paragraph>
    <Paragraph position="7"> Finally, we consider the issue of negation. If a tree does not satisfy some relationship then it satisfies the negation of that relationship, and vice versa. For quasi-trees the situation is more subtle.</Paragraph>
    <Paragraph position="8"> Viewing the quasi-tree as a set of trees, if every tree in that set fails to satisfy some relationship, then they all satisfy the negation of that relationship.</Paragraph>
    <Paragraph position="9"> Hence the quasi-tree must satisfy the negated relationship as well. On the other hand, viewing the quasi-tree as a set of relationships, if a particular relationship is not included in the quasi-tree it does not imply that none of the trees it describes satisfies that relationship, only that some of those trees do not. Thus it may be the case that a quasi-tree neither satisfies a relationship nor satisfies its negation. null Since trees are completed objects, when a tree satisfies the negation of a relationship it will always be the case that the tree satisfies some (positive) relationship that is incompatible with the first. For example, in a tree &amp;quot;a does not dominate b&amp;quot; iff &amp;quot;a is left-of b&amp;quot;, &amp;quot;b is left-of a&amp;quot;, or &amp;quot;b properly dominates a&amp;quot;. Thus there are inferences that can be drawn from negated relationships in trees that may be incorporated into the structure of quasi-trees. In making these inferences, we dispense with the need to include negative relationships explicitly in the quasi-trees. They can be defined in terms of the positive relationships. The price we pay is that to characterize the set of all trees in which &amp;quot;a does not dominate b&amp;quot;, for instance, we will need three quasi-trees, one characterizing each of the sets in which &amp;quot;a is left-of b&amp;quot;, &amp;quot;b is left-of a&amp;quot;, and % properly dominates a&amp;quot;.</Paragraph>
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