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<Paper uid="E93-1005">
  <Title>Decidability and Undecidability in stand-alone Feature Logics</Title>
  <Section position="3" start_page="0" end_page="31" type="intro">
    <SectionTitle>
2 Preliminaries
</SectionTitle>
    <Paragraph position="0"> Feature logics are abstractions from the unification based formalisms of computational linguistics.</Paragraph>
    <Paragraph position="1"> Originally feature logics embodied just one component of unification based formalisms. Early unification formalisms such as GPSG \[Gazdar et al. 1985\] and LFG \[Kaplan and Bresnan 1982\] have important phrase structure components in addition to their feature passing mechanisms, and the study of feature logic was originally intended to throw light only on the latter. These early unification formMisms are thus highly heterogeneous: they are architectures with roots in both formal language theory and logic.</Paragraph>
    <Paragraph position="2"> In recent years this picture has changed. For example, in HPSG \[Pollard and Sag 1987\] the feature machinery has largely displaced the phrase structure component. Indeed in HPSG the residue of the phrase structure component is coded up as part of the feature system. Logic has swallowed formal language theory, and in effect the entire HPSG formal- null ism is a powerful feature logic, a stand-alone formalism, capable of encoding complex grammars without the help of any other component. 1 In this paper we are going to investigate the computational complexity of the satisfiability problem for such stand-alone feature logics. This is an important problem to investigate. Natural language grammars are expressed as logical theories in stand-alone formalisms, and sentences are represented as wffs.</Paragraph>
    <Paragraph position="3"> This means that the problem of deciding whether or not a sentence is grammatical reduces to the problem of building a model of the sentence's logical representation that conforms to all the constraints imposed by the logical encoding of the grammar. In short, the complexity of the satisfiability problem is essentially the worst case complexity of the recognition problem for grammars expressed in the stand-alone formalism.</Paragraph>
    <Paragraph position="4"> We will tackle this issue by investigating the complexity of the satisfiability problem for one particular stand-alone formalism, namely L KRo. This is the language of Kasper Rounds logic augmented with the strict implication operator. L KRm possesses two of the most fundamental properties of stand-alone formalisms: the ability to express re-entrancy, and the ability to express generalisations about feature structures. It is important to note that L KR~ is actually a fairly minimal language with these properties; many feature logics can express a lot more besides (for example, set values), thus the negative results for L KR=~ we present are rather strong: they extend straightforwardly to richer formalisms.</Paragraph>
    <Paragraph position="5"> So let's begin by defining L KR~. By a signature (PS, S) is meant a pair of non-empty sets L: and S, the set of arc labels and the set of sorts respectively. Syntactically, the language L KR=~ (of signature (PS:, S)) contains the following items: an S indexed collection of propositional symbols (or sort symbols); all the standard Boolean operators; 2 an PS indexed collection of distinct unary modalities (that is, features);  a binary modality ==~; and two special symbols 0 and ,~. We use ~., to make path equations: given any non-empty sequences A and B consisting of only unary modalities and O, then A ~ B is a path equation.</Paragraph>
    <Paragraph position="6"> 1 See \[Johnson 1992\] for further discussion of the distinction between stand-alone formalisms and formalisms with a phrase structure backbone.</Paragraph>
    <Paragraph position="7"> 2That is, we have the symbols True (constant true), False  (constant false), ~ (negation), v (disjunction), A (conjunction), --* (material implication) and 4--* (material equivalence). For the purposes of the present paper it is sensible to assume that all these operators are primitives, as in general we will be working with various subsets of the full language and it would be tedious to have to pay attention to trivial issues involving the interdefinability of the Boolean operators in these weaker fragments.</Paragraph>
    <Paragraph position="8"> Intuitively A ~ B says that making the sequence of feature transitions encoded by A leads to the same node as making the transition sequence coded by B.</Paragraph>
    <Paragraph position="9"> The symbol 0 is a name for the null transition. The strict implication operator =~ will enable us to express generalisations about feature structures.</Paragraph>
    <Paragraph position="10"> We make the wffs of L KR~ as follows. First, all propositional symbols, all path equations and True and False are wffs. Second, if C/ and C/ are wits then so are all Boolean combinations of C/ and C/~ so is (1)C/ (for alll E PS) and so is C/ =~ C/. Third, nothing else is a wff. If a wff of L KR:* does not contain any occurrences of ==~ then we say it is an L KR wff.</Paragraph>
    <Paragraph position="11"> Apart from trivial notational changes, the negation free fragment of L KR is the language defined and studied by Kasper and Rounds. 3 That is, the L KR wffs are essentially a way of writing the familiar Attribute Value Matrices (AVMs) in linear format. For example, the following L KR wff: &lt;NUMBER)pluralA (CASE)(nom V gen V acc) is essentially the following AVM: CASE nora or gen or acc To interpret L KRo we use feature structures M of signature (/~,S). A feature structure is a triple (W, {Rt}tez:, V), where W is a non-empty set (the set of nodes); each Rz is a binary relation on W that is also a partial function; and V (the valuation) is a function which assigns each propositional symbol p E S a subset of W. Note that as we have defned them features structures are merely multi-modal Kripke models, 4 and we often refer to feature structures as models in what follows.</Paragraph>
    <Paragraph position="12"> Now for the satisfaction definition. As the symbol 0 is to act as a name for the null transition, in what follows we shall assume without loss of generality that 0 C/ PS, and we will denote the identity relation on any set of nodes W by R0. This convention somewhat simplifies the statement of the satisfaction 3Computer scientists may have met L KR in another guise. The language of Kasper Rounds logic is a fragment of (deterministic) Propositional Dynamic Logic (PDL) with intel~ section (see \[Harel 1984\]). An L lea path equation A ,~ B is written as (An B) True in PDL with intersection.</Paragraph>
    <Paragraph position="13"> 4 For further discussion of the modal perspective ol* feature logic, see \[Blackburn and Spaan 1991, 1992\].</Paragraph>
    <Paragraph position="14">  definition:</Paragraph>
    <Paragraph position="16"> The satisfaction clauses for True, False, A, --. and *-* have been omitted; these symbols receive their standard Boolean interpretations. If M ~ C/\[w\] then we say that M satisfies C/ at w, or C/ is true in M at w (where w E W).</Paragraph>
    <Paragraph position="17"> The key things to note about this language is that it has both the ability to express re-entrancy (the Kasper Rounds path equality ~ achieves this) and the ability to express generalisations about feature structures (note that C/ ::~ C/ means that at every node where C/ is true, ~b must also be true). Thus L KR~ can certainly express many of the conditions we might want to impose on feature structures. For instance, we might want to impose a sort hierarchy.</Paragraph>
    <Paragraph position="18"> As a simple example, given sorts list and nelist (nonempty list) we might wish to insist that every node of sort nelist is also of sort list. The wff nelist ~ list forces this. As a second example, we might want to insist that any node from which it is possible to make a CONSTITUENT-STRUCTURE transition must be of sort phrasal. That is, if a node has constituent structure, it is a phrasal node. The wff (CONSTITUENT-STRUCTURE) True ~ phrasal forces this. Indeed quite complex demands can be imposed using L KR. For example the following wff embodies the essence of the constraint known as the head feature convention in HPSG: phrasal ~ (HEAD) ~,~ (HEAD-DTR)(HEAD).</Paragraph>
    <Paragraph position="19"> This wff says that at any node of sort phrasal in a feature structure, it is possible to make a ilEAl) transition and it is also possible to make a HEAD-DTR transition followed by a HEAD transition, and furthermore both transition sequences lead to the same node. In view of such examples it doesn't seem Wholly unrealistic to claim that L KR has the kind of expressive power a stand-alone feature logic needs.</Paragraph>
    <Paragraph position="20"> However L TM has crossed a significant complexity boundary: it has an undecidable satisfiability problem. This was proved in \[Blackburn and Spaan 1991, 1992\] using a tiling argument, s Now, the result for the full L t':n=~ language is not particularly surprising (undecidability results for related feature logics, can be found in the literature; see \[Carpenter 1992\] for discussion) but it does lead to an important question: what can be salvaged? To put it another way, are there decidable fragments of L KR that are capable of functioning as stand-alone feature logics? The pages that follow explore this question and yield a largely negative response.</Paragraph>
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