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<Paper uid="P93-1027">
  <Title>ON THE DECIDABILITY OF FUNCTIONAL UNCERTAINTY*</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Feature logic is the main device of unification grammars, the currently predominant paradigm in computational linguistics. More recently, feature descriptions have been proposed as a constraint system for logic programming (e.g. see \[ll D . They provide for partial descriptions of abstract objects by means of functional attributes called features.</Paragraph>
    <Paragraph position="1"> Formalizations of feature logic have been proposed in various forms (for more details see \[3\] in this volume). We will follow the logical approach introduced by Smolka \[9, 10\], where feature descriptions are standard first order formulae interpreted in first order structures. In this formalization features are considered as functional relations. Atomic formulae (which we will call atomic constraints) are of either the form A(x) or zfy, where x, y are first order variables, A is some sort predicate and f is a feature (written in infix notation). The constraints of the form xfy can be generalized to constraints of the form xwy, where w = fl-.. fn is a finite feature path.</Paragraph>
    <Paragraph position="2"> This does not affect the computational properties.</Paragraph>
    <Paragraph position="3"> In this paper we will be concerned with an extension to feature descriptions, which has been introduced as &amp;quot;functional uncertainty&amp;quot; by Kaplan and Zaenen \[7\] and Kaplan and Maxwell \[5\]. This formal device plays an important role in the framework of LFG in modelling so-called long distance dependencies and constituent coordination. For a detailed linguistic motivation see \[7\], \[6\] and \[5\]; a more general use of functional uncertainty can be found in \[8\]. Functional uncertainty consists of constraints of *This work was supported by a research grant, ITW 9002 0, from the German Bundesministerium ffir Forschung und Technologic to the DFKI project DISCO.</Paragraph>
    <Paragraph position="4"> I would like to thank Jochen Dhrre, Joachim Niehren and Ralf Treinen for reading draft version of this paper. For space limitations most of the proofs are omitted; they can be found in the complete paper \[2\] the form xLy, where L is a finite description of a regular language of feature paths. A constraint xLy holds if there is a path w E L such that zwy holds. Under this existential interpretation, a constraint xLy can be seen as the disjunction = I ,.,, e xLy L}.</Paragraph>
    <Paragraph position="5"> Certainly, this disjunction may be infinite, thus functional uncertainty yields additional expressivity.</Paragraph>
    <Paragraph position="6"> Note that a constraint zwy is a special case of a functional uncertainty constraint.</Paragraph>
    <Paragraph position="7"> To see some possible application of functional uncertainty we briefly recall an example that is given in Kaplan and Maxwell \[5, page 1\]. Consider the topicalized sentence Mary John telephoned yesterday.</Paragraph>
    <Paragraph position="8"> Using s as a variable denoting the whole sentence, the LFG-like clause s topic x A s obj x specifies that in s Mary should be interpreted as the object of the relation telephoned. The sentence could be extended by introducing additional complement predicates, as e.g. in sentences like Mary John claimed thai Bill telephoned; Mary John claimed thai Bill said that ...Henry telephoned yesterday; .... For this family of sentences the clauses s topic x A s comp obj x, s topic xAs comp cornp obj x and so on would be appropriate; specifying all possibilities would yield an infinite disjunction. This changes if we make use of functional uncertainty allowing to specify the above as the single clause s topic x A s comp* obj x.</Paragraph>
    <Paragraph position="9"> Kaplan and Maxwell \[5\] have shown that consistency of feature descriptions is decidable, provided that a certain aeyclicity condition is met. More recently, Bander et hi. \[1\] have proven, that consistency is not decidable if we add negation. But it is an open problem whether consistency of feature descriptions without negation and without additional restrictions (such as acyclicity) is decidable. In the work presented here we show that it indeed is decidable.</Paragraph>
  </Section>
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