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<Paper uid="C88-1009">
  <Title>Feature Graphs and Abstract Data Types: A Unifying Approach</Title>
  <Section position="2" start_page="0" end_page="40" type="intro">
    <SectionTitle>
1. Introduction
</SectionTitle>
    <Paragraph position="0"> Unification-based grammar formalisms have become a popular field o'f research. The subject has attracted the interest not only of (computer) linguists but also of computer scientists, especially in the area of logic programming, see e.g. \[Pe 87\]. Due to the formality of grammars we also observe activities developing foundations for the formal semantics of various approaches on unification grammars, e.g. \[PS 84\], \[KR 86\], \[RK 86\], \[Pe 87\], \[Jo 87\].</Paragraph>
    <Paragraph position="1"> In this paper we investigate the relationship between feature graphs on the one hand and algebraic specifications of abstract data types (ADTs) on the other hand. There is a natural correlation between both these areas since a feature graph as well as an abstract data type defines a collection of objects together with functions relating the objects.</Paragraph>
    <Paragraph position="2"> We present a formal semantics for feature graphs by assigning to each feature graph G an equational ADT specification ~(G), called fg--specification. This opens the rich world of ma'thematical foundations of ADT specifications (e.g \[GTW 78\], \[EM 85\]) in order to obtain a better - not only - formal understanding of the nature of feature graphs.</Paragraph>
    <Paragraph position="3"> In particular, we provide a model-theoretic characterization of various consistency conditions for feature graph specifications reflecting the consistency concepts usually imposed on feature graphs such as clash-freeness and acyclicity. These  model-theoretic characterizations have proof-theoretic counterparts in terms of syntactic conditions on the deductive closure of the set of equations of r(G)o Although the proof-theoretic consistency characterizations are of syntactic nature, a test of their validity requires to examine the deductive closure of the set of equations of a fg-specification. Our objective is to restrict consistency checks to equations explicitly mentioned in a fg-specificationo In the ADT-world there is a well-known tool for such tasks: the Knuth-Bendix algorithm (\[KB 70\]). We present a Knuth-Bendix like completion procedure transforming any fg-specification into a reduced normal form. We show that the model-theoretic consistency characterizations for G are equivalent to the presence resp. absence of certain types of equations in this reduced normal form.</Paragraph>
    <Paragraph position="4"> These results are used for defining the semantics of the unification of two feature graphs GI and G2 as the (set-theoretic componentwise) union of ~(GI) and x(G2 ) followed by the normalization process using the completion algorithm and the consistency check on the resulting set of equationsdeg</Paragraph>
  </Section>
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