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<Paper uid="P93-1026">
  <Title>A COMPLETE AND RECURSIVE FEATURE THEORY*</Title>
  <Section position="3" start_page="194" end_page="194" type="intro">
    <SectionTitle>
2 Preliminaries
</SectionTitle>
    <Paragraph position="0"> Throughout this paper we assume a signature SOR~ FEA consisting of an infinite set SOR of unary predicate symbols called sorts and an infinite set FEA of binary predicate symbols called features. For the completeness of our axiomatization it is essential that there are both infinitely many sorts and infinitely many features.The letters A, B, C will always denote sorts, and the letters f, g, h will always denote features.</Paragraph>
    <Paragraph position="1"> A path is a word (i.e., a finite, possibly empty sequence) over the set of all features. The symbol c denotes the empty path, which satisfies cp = p = pc for every path p. A path p is called a prefix of a path q, if there exists a path p' such that pp' = q.</Paragraph>
    <Paragraph position="2"> We also assume an infinite alphabet of variables and adopt the convention that x, y, z always denote variables, and X, Y always denote finite, possibly empty sets of variables. Under our signature SOR ~ FEA, every term is a variable, and an atomic formula is either a feature constraint xfy (f(x,y) in standard notation), a sort constraint Ax (A(x) in standard notation), an equation x - y, _L (&amp;quot;false&amp;quot;), or T (&amp;quot;true&amp;quot;). Compound formulae are obtained as usual with the connectives A, V, --+, ~-+, -~ and the quantifiers 3 and V. We use 3C/ \[VC/\] to denote the existential \[universal\] closure of a formula C/. Moreover, 1)(C/) is taken to denote the set of all variables that occur free in a formula C/. The letters C/ and C/ will always denote formulae.</Paragraph>
    <Paragraph position="3"> We assume that the conjunction of formulae is an associative and commutative operation that has T as neutral element. This means that we identify eA(C/A0) withOA(C/AC/),andeATwithC/(but not, for example, xfy A xfy with xfy). A conjunction of atomic formulae can thus be seen as the finite multiset of these formulae, where conjunction is multiset union, and T (the &amp;quot;empty conjunction&amp;quot;) is the empty multiset. We will write C/ C C/ (or C/ E C/, if C/ is an atomic formula) if there exists a formula C/~ such that C/ A C/1 = C/.</Paragraph>
    <Paragraph position="4"> Moreover, we identify 3x3yC/ with 3y3xC/. If X = {xl,...,xn}, we write 3XC/ for 3xl ...3xnC/. IfX = 0, then 3XC/ stands for C/.</Paragraph>
    <Paragraph position="5"> Structures and satisfaction of formulae are defined as usual. A valuation into a structure `4 is a total function from the set of all variables into the universe 1`4\] of`4. A valuation ~' into,4 is called an x-update \[X-update\] of a valuation a into ,4 if (~' and a a~ree everywhere but possibly on x \[X\]. We use C/~ to denote the set of all valuations c~ such that ,4, c~ ~ C/. We write C/ ~ C/ (&amp;quot;C/ entails C/&amp;quot;) if CA C C/ A for all structures ,4, and C/ ~ C/ (&amp;quot;C/ is equivalent to C/&amp;quot;) if C/.4 = cA for all structures ,4.</Paragraph>
    <Paragraph position="6"> A theory is a set of closed formulae. A model of a theory is a structure that satisfies every formulae of the theory. A formula C/ is a consequence of a theory T (T ~ C/) if VC/ is valid in every model of T. A formula C/ entails a formula C/ in a theory</Paragraph>
    <Paragraph position="8"> Two formulae C/, C/ are equivalent in a theory T (C/ ~T C/) if cA = C/.4 for every model ,4 of T.</Paragraph>
    <Paragraph position="9"> A theory T is complete if for every closed formula either C/ or -,C/ is a consequence of T. A theory is decidable if the set of its consequences is decidable.</Paragraph>
    <Paragraph position="10"> Since the consequences of a recursively enumerable theory are recursively enumerable (completeness of first-order deduction), a complete theory is decidable if and only if it is recursively enumerable.</Paragraph>
    <Paragraph position="11"> Two first-order structures ,4, B are elementarily equivalent if, for every first-order formula C/, C/ is valid in ,4 if and only if C/ is valid in B. Note that all models of a complete theory are elementarily equivalent. null</Paragraph>
  </Section>
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