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<Paper uid="P01-1019">
  <Title>An Algebra for Semantic Construction in Constraint-based Grammars</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 A simple semantic algebra
</SectionTitle>
    <Paragraph position="0"> The following shows the equivalents of the structures in Figure 1 in our algebra: Kim: [x2]{[]subj,[]comp}[r name(x2,Kim)]{} sleeps: [e1]{[x1]subj,[]comp}[sleep(e1,x1)]{}</Paragraph>
    <Paragraph position="2"> The last structure is semantically equivalent to: [sleep(e1,x1),r name(x1,Kim)].</Paragraph>
    <Paragraph position="3"> In the structure for sleeps, the first part, [e1], is a hook and the second part ([x1]subj and []comp) is the holes. The third element (the lzt) is a bag of elementary predications (EPs).2 Intuitively, the hook is a record of the value in the semantic entity that can be used to fill a hole in another entity during composition. The holes record gaps in the semantic form which occur because it represents a syntactically unsaturated structure. Some structures have no holes, such as that for Kim. When structures are composed, a hole in one structure (the semantic head) is filled with the hook of the other (by equating the variables) and their lzts are appended. It should be intuitively obvious that there is a straightforward relationship between this algebra and the TFSs shown in Figure 1, although there are other TFS architectures which would share the same encoding.</Paragraph>
    <Paragraph position="4"> We now give a formal description of the algebra. In this section, we simplify by assuming that each entity has only one hole, which is unlabelled, and only consider two sorts of variables: events and individuals. The set of semantic entities is built from the following vocabulary:  2As usual in MRS, this is a bag rather than a set because we do not want to have to check for/disallow repeated EPs; e.g., big big car.</Paragraph>
    <Paragraph position="5"> 1. The absurdity symbol [?].</Paragraph>
    <Paragraph position="6"> 2. indices i1,i2,..., consisting of two subtypes of indices: events e1,e2,... and individuals x1,x2,....</Paragraph>
    <Paragraph position="7"> 3. n-place predicates, which take indices as arguments null 4. =.</Paragraph>
    <Paragraph position="8">  Equality can only be used to identify variables of compatible sorts: e.g., x1 = x2 is well formed, but e = x is not. Sort compatibility corresponds to unifiability in the TFS logic.</Paragraph>
    <Paragraph position="9"> Definition 1 Simple Elementary Predications  (SEP) An SEP contains two components: 1. A relation symbol 2. A list of zero or more ordinary variable ar- null guments of the relation (i.e., indices) This is written relation(arg1,...,argn). For in- null stance, like(e,x,y) is a well-formed SEP.</Paragraph>
    <Paragraph position="10"> Equality Conditions: Where i1 and i2 are indices, i1 = i2 is an equality condition.</Paragraph>
    <Paragraph position="11"> Definition 2 The Set S of Simple semantic Entities (SSEMENT) s [?] S if and only if s = [?] or s = &lt;s1,s2,s3,s4&gt; such that: * s1 = {[i]} is a hook; * s2 = [?] or {[iprime]} is a hole; * s3 is a bag of SEPs(the lzt) * s4 is a set of equalities between variables  (the eqs).</Paragraph>
    <Paragraph position="12"> We write a SSEMENT as: [i1][i2][SEPs]{EQs}. Note for convenience we omit the set markers {} from the hook and hole when there is no possible confusion. The SEPs, and EQs are (partial) descriptions of the fully specified formulae of first  order logic.</Paragraph>
    <Paragraph position="13"> Definition 3 The Semantic Algebra A Semantic Algebra defined on vocabulary V is the algebra &lt;S,op&gt; where: * S is the set of SSEMENTs defined on the vocabulary V, as given above; * op : S x S [?]- S is the operation of semantic composition. It satisfies the following conditions. If a1 = [?] or a2 = [?] or</Paragraph>
    <Paragraph position="15"> where Tr stands for transitive closure</Paragraph>
    <Paragraph position="17"> This definition makes a2 the equivalent of a semantic functor and a1 its argument.</Paragraph>
    <Paragraph position="18"> Theorem 1 op is a function If a1 = a3 and a2 = a4, then a5 = op(a1,a2) = op(a3,a4) = a6. Thus op is a function. Furthermore, the range of op is within S. So &lt;S,op&gt; is an algebra.</Paragraph>
    <Paragraph position="19"> We can assume that semantic composition always involves two arguments, since we can define composition in ternary rules etc as a sequence of binary operations. Grammar rules (i.e., constructions) may contribute semantic information, but we assume that this information obeys all the same constraints as the semantics for a sign, so in effect such a rule is semantically equivalent to having null elements in the grammar. The correspondence between the order of the arguments to op and linear order is specified by syntax.</Paragraph>
    <Paragraph position="20"> We use variables and equality statements to achieve the same effect as coindexation in TFSs.</Paragraph>
    <Paragraph position="21"> This raises one problem, which is the need to avoid accidental variable equivalences (e.g., accidentally using x in both the signs for cat and dog when building the logical form of A dog chased a cat). We avoid this by adopting a convention that each instance of a lexical sign comes from a set of basic sements that have pairwise distinct variables. The equivalent of coindexation within a lexical sign is represented by repeating the same variable but the equivalent of coindexation that occurs during semantic composition is an equality condition which identifies two different variables.</Paragraph>
    <Paragraph position="22"> Stating this formally is straightforward but a little long-winded, so we omit it here.</Paragraph>
  </Section>
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