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<Paper uid="J94-1001">
  <Title>Computing with Features as Formulae</Title>
  <Section position="3" start_page="2" end_page="2" type="metho">
    <SectionTitle>
3. Languages for Expressing Feature Structure Constraints
</SectionTitle>
    <Paragraph position="0"> There are many different possible constraint languages. Specialized languages can be constructed specifically for the task of expressing feature structure constraints (such as Kasper and Rounds's FDL \[Kasper and Rounds 1990\] and Johnson's attribute-value languages \[Johnson 1988\]). Alternatively, the constraints may be able to be expressed in some standard language, so that the satisfiability problem for linguistic constraints is reduced to the satisfiability problem for that language, as is done here. 4 Johnson (1990a), following a suggestion first made in Kaplan and Bresnan (1982), showed how attribute-value constraints could be formalized in the quantifier-free sub-set of first-order logic, while later work (Johnson 1991a, 1991b) proposed a different formalization in the Sch6nfinkel-Bernays' subset of first-order formulae. 5 Roughly speaking, there is a trade-off between the expressive power of a language and its computational tractability. For example, the satisfiability problem for the language consisting of conjunctions of equalities and inequalities of first-order terms can</Paragraph>
  </Section>
  <Section position="4" start_page="2" end_page="4" type="metho">
    <SectionTitle>
4 A third approach, developed by Smolka (1992), is to define a specialized language tailored for
</SectionTitle>
    <Paragraph position="0"> expressing attribute-value constraints and note its translation into some standard language, in this case, also the Sch6nfinkel-Bernays' class. 5 Of course, there is no a priori reason for these subsets of first-order logic to be optimally suited for expressing feature structure constraints. Kasper and Rounds (1990) and more recently Blackburn (1991) and Blackburn and Spaan (1992) have suggested that it may be useful to express feature structure constraints in a special kind of modal logic. Johnson (1991b) also discusses the application of general first-order logic and nonmonotic logics to the specification of more complex constraints on feature structures.</Paragraph>
    <Paragraph position="1">  Mark Johnson Computing with Features as Formulae be decided in quasi-linear time using the congruence-closure algorithm, but this language can only express conjunctions of feature-value equalities and inequalities. If this language is extended to allow disjunctions (so that disjunctive feature-value constraints can be expressed), the satisfiability problem becomes NP-complete (Gallier 1986; Kasper and Rounds 1990; Nelson and Oppen 1980).</Paragraph>
    <Paragraph position="2"> Since disjunctive constraints seem to be a practical necessity for describing natural languages (Barton, Berwick, and Ristad 1987; Karttunen 1984), most practical feature structure systems will probably have NP-hard satisfiability problems. Given that we have to solve an NP-hard problem anyway, it seems reasonable to investigate the most expressive feature structure constraint language that has an NP-complete satisfiability problem. The Sch6nfinkel-Bernays' class, used in the manner described here, appears to be the most expressive language for feature structure constraints proposed in the literature so far whose satisfiability problem is no harder than NP.</Paragraph>
    <Section position="1" start_page="4" end_page="4" type="sub_section">
      <SectionTitle>
3.1 The SchSnfinkel-Bernays' Class
</SectionTitle>
      <Paragraph position="0"> The Sch6nfinkel-Bernays' class (hereafter SB) is the class of first-order closed prenex formulae without function symbols in which no existential quantifier occurs in the scope of any universal quantifier. That is, a formula is in SB iff it has no free variables and is of the form</Paragraph>
      <Paragraph position="2"> where a contains no quantifier symbols or function symbols. SB formulae are a proper subset of first-order formulae, and they are interpreted in exactly the same way as first-order formulae. The body a may contain boolean connectives (including negation), which can be used to express arbitrary boolean combinations of constraints.</Paragraph>
      <Paragraph position="3"> Unlike the satisfiability problem for full first-order logic, which is undecidable (co-recursively enumerable), the satisfiability problem for SB is decidable; in fact it is PSPACE-complete (Lewis and Papadimitriou 1981). Further, if SBn is the class of SB formulae with n or fewer universal quantifiers, then for any fixed n the satisfiability problem for SBn is NP-complete (Lewis 1980). In the applications described here, the number of universal quantifiers is fixed (i.e., it does not vary with the utterance or even with the grammar), so the corresponding satisfiability problems are all NP-complete.</Paragraph>
      <Paragraph position="4"> The class of SB formulae is interesting for other reasons besides its ability to express a wide range of linguistic constraints. As shown below, the class of SB formulae in clausal form constitute an extension of Datalog that allows disjunctive consequents.</Paragraph>
    </Section>
    <Section position="2" start_page="4" end_page="4" type="sub_section">
      <SectionTitle>
3.2 Formalizing Attribute-Value Structures Using SB
</SectionTitle>
      <Paragraph position="0"> SB is both simple and expressive enough that grammar designers might choose to state linguistic constraints directly in SB, rather than in terms of attributes and values. Nevertheless, it is important to understand how the properties of attribute-value structures can be stated in SB, since many of the techniques used to formalize them can be applied to other linguistically interesting structures as well.</Paragraph>
      <Paragraph position="1"> In fact there are several ways of formalizing attribute-value structures in SB, all of which seem to be linguistically equivalent. What follows is a formalization in SB that allows values to be used as attributes and allows attributes to be quantified over (this is handy for stating &amp;quot;sort constraints&amp;quot;), but no special claims are made for it over and above any other SB formalization.</Paragraph>
      <Paragraph position="2"> Following Johnson (1991b), attribute-value feature structures can be specified in SB in the following way. We can conceptualize of attribute-value arcs as instances of Computational Linguistics Volume 20, Number 1 a three-place relation arc, where arc(x, a~ y) means that there is an arc leaving node x labeled a pointing to node y.6 Of course, not all interpretations qualify as attribute-value structures; e.g., those which satisfy both arc(x~ a~ y) and arc(x~ a~ z) for some y ~ z violate the requirement that there is at most one arc with any given label leaving any node. We can express this requirement as an SB formula that is true in the intended interpretations (namely attribute-value feature structures).</Paragraph>
      <Paragraph position="3"> Vx Va Vy Vz arc(x~a~y) Aarc(x,a,z) ~ y = z. (1) Similarly, we can express the properties of the &amp;quot;attribute-value constants&amp;quot; with SB formulae. Let con be a property (i.e., a one-place relation) true of the &amp;quot;attribute-value constant&amp;quot; elements. These elements are required to have no arcs leaving them. The following formula expresses this requirement.</Paragraph>
      <Paragraph position="4"> VxVa Vy ~,, (con(x)Aarc(x~a~y)). (2) Note that the word &amp;quot;constant&amp;quot; in the name &amp;quot;attribute-value constant&amp;quot; is misleading here, since in this framework not all SB constant symbols will denote attribute-value &amp;quot;constants.&amp;quot; More precisely, being an 'attribute-value constant' is a property of an individual in an interpretation (i.e., an element of a feature structure), whereas being a constant is a property of a symbol in a formula. Constants can be used to denote complex attribute-value entities as well as attribute-value constants.</Paragraph>
      <Paragraph position="5"> Finally, we require that the names of attribute-value constants denote distinct attribute-value constants. We reserve a finite subset N of the constants of our language for use as the names of attribute-value constants, and require that they satisfy the following schemata. 7 For each c in N, con(c). (3) For each distinct pair Cl, c2 in N, cl ~ C2. (4) Schema (3) requires each symbol in N to denote an attribute-value constant, and schema (4) enforces distinctness in essentially the same manner as that used in the specification systems of algebraic data-type theory (Kapur and Musser 1987).</Paragraph>
      <Paragraph position="6"> Formulas (1) and (2) and the instances of schemata (3) and (4) can be regarded as defining attribute-value feature structures. These axioms are quite permissive: in 6 Johnson (1991a) and Smolka (1992) propose that an attribute-value arc labeled a from x to y be conceptualized as an instance of a two-place relation a(x, y). For most applications there is little substantive difference between these two approaches; the approach taken here allows attributes to be quantified over, e.g., to state sortal constraints, and permits values to be used as attributes, as in e.g., LFG (Kaplan and Bresnan 1982); for discussion and linguistic applications see also Johnson (1988).</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="4" end_page="11" type="metho">
    <SectionTitle>
7 As Patrick Blackburn (p.c.) points out, one consequence of this is that every model of these constraints
</SectionTitle>
    <Paragraph position="0"> will contain individuals corresponding to each attribute-value constant (since each constant symbol will be assigned a denotation). Whether this is desirable or problematic is debatable, but as he pointed out, it is easy to devise a conceptualization in which each attribute-value constant ci is conceptualized as a one-place predicate ci(.) that is true of at most one element. Under such a conceptualization (which can be formalized in SB as shown below) attribute-value constants would be the unique  members of one-element sorts.</Paragraph>
    <Paragraph position="1"> (i) For each c in N, Vx Vy c(x) A c(y) ~ x = y. (Uniqueness) (ii) For each c in N, Vx c(x) ~ con(x). (Constant property) (iii) For each distinct pair cl, c2 in N, Vx ~ (cl (x) A c2 (x)). (Disjointness) Mark Johnson Computing with Features as Formulae</Paragraph>
    <Paragraph position="3"> Three constraints expressed as formulae and also depicted graphically.</Paragraph>
    <Paragraph position="4"> addition to the usual finite acyclic feature structures, they allow infinite structures, cyclic structures, structures in which complex values serve as attributes, etc. While ruled out by fiat in standard treatments, admitting these additional structures causes no linguistic difficulties that I am aware of (in fact, some analyses crucially depend on their existence, as described in section 2.1.3 of Johnson \[1988\]), so in the interests of parsimony additional constraints that forbid them are not stipulated.</Paragraph>
    <Paragraph position="5"> In fact, because SB formulae possess the finite model property (i.e., if an SB formula has a model, then it has a finite model), restricting attention to finite models does not change the set of satisfiable SB formulae. Therefore it could have no effect on the set of well-formed utterances. Cyclic feature structures can be prohibited with a constraint formalizable in SB, as described in Johnson (1991b), and one can express a constraint in SB that requires that all attributes are &amp;quot;attribute-value constants&amp;quot; (even though there appears to be no linguistic motivation for such a constraint, and indeed, some analyses crucially depend on this not being the case, as pointed out in Johnson \[1988\]).</Paragraph>
    <Paragraph position="6"> To summarize, the simplest SB axioms defining attribute-value structures are quite permissive, allowing a wider range of structures to count as attribute-value structures than many other formalizations. However, all of the major restrictions on attribute-value structures discussed in the literature either have no effect whatsoever in this framework, or else can be directly stated as additional SB constraints.</Paragraph>
    <Section position="1" start_page="4" end_page="4" type="sub_section">
      <SectionTitle>
3.3 Expressing Feature Structure Constraints with SB
</SectionTitle>
      <Paragraph position="0"> In this approach, simple attribute-value constraints are represented by quantifier-free atomic formulae. For example, a constraint that the value of n's al arc is bl would be represented by the atom arc(n,al,bl), a constraint that the value of n's a2 arc is b2 is represented by arc(n, a2, b2), and a constraint that the value of n's al arc is the same as the value of its a2 arc is represented by the conjunction arc(n, all n') A arc(n, a2~ n') (n ~ is the single value of both arcs). These three constraints are depicted graphically in Figure 1. Note that the graphs in this figure are (depictions of) formulae, not attribute-value feature structures.</Paragraph>
      <Paragraph position="1"> Attribute-value &amp;quot;unification&amp;quot; is the conjunction and simplification of the formulae expressing the constraints to be unified. If all three constraints in the example of Figure 1 are conjoined together with axioms (1-3) above, then by (1) it follows that bl = n ~ = b2. Further, if bl and b2 are distinct constant symbols in N (thus they name attribute-value constants), then bl ~ b2 is an instance of (4), and the conjunction is therefore unsatisfiable. For further examples and a discussion of how the disjunction and negation of attribute-value constraints are transparently representable as SB formulae, see Johnson (1991a, 1991b).</Paragraph>
      <Paragraph position="2"> Computational Linguistics Volume 20, Number 1 A major motivation for using SB is that a wide variety of constraints, in addition to standard attribute-value constraints, can be expressed using it. This allows a grammar developer to introduce a wide variety of &amp;quot;designer features&amp;quot; with possibly idiosyncratic, customized properties, while guaranteeing that the composite system is decidable (usually in NP-time, as noted above).</Paragraph>
      <Paragraph position="3"> For example, suppose we want to impose sort restrictions of the following kind.</Paragraph>
      <Paragraph position="4"> To abbreviate the lexical entries of verbs we might introduce the one-place predicate 3rd-sg, where 3rd-sg(x) indicates that the value of x's person attribute is 3rd and x's number attribute is singular. This constraint can be expressed using the following SB formula.</Paragraph>
      <Paragraph position="5"> Vx 3rd-sg(x) ~ arc(x, person, 3rd) A arc(x, number, singular). (5) Similarly, constraints that restrict the possible values of certain attributes can be imposed. For example, one might want to require that the value of every arc labeled number is either singular or plural. This constraint can be expressed as the following SB formula.</Paragraph>
      <Paragraph position="6"> Vx Vy arc(x, number, y) ~ y = singular V y = plural. (6) These examples demonstrate only a small fraction of the variety of the feature structure constraints that can be expressed in SB. Even though all of these examples are based on attribute-value features, other sorts of features can be described in SB as well. For example, Johnson (1991a) shows how to formulate a variety of constraints on 'set-valued' features in SB.</Paragraph>
    </Section>
    <Section position="2" start_page="4" end_page="4" type="sub_section">
      <SectionTitle>
3.4 Expressing Tree Structure Constraints with SB Formulae
</SectionTitle>
      <Paragraph position="0"> Inspired by the work on description theory or 'D-theory' (Marcus, Hindle, and Fleck 1983; Vijay-Shanker 1992), this section shows how some elementary constraints on precedence and dominance in a tree can be expressed as SB formulae. It differs from that work in that different kinds of constraints are expressible (Vijay-Shanker was concerned with the formalization of a different kind of grammar), and that all of the constraints expressible in the system described below are decidable (this follows from the fact that they are defined and expressed using Sch6nfinkel-Bernays' formulae).</Paragraph>
      <Paragraph position="1"> These constraints are intended to appear as annotations on phrase structure rules (in the same way that attribute-value constraints do) and could be used to enforce a variety of &amp;quot;long-distance&amp;quot; relationships, such as the co- and contra-indexing constraints of binding theory (i.e., equality and inequality constraints on the values of index attributes). null The axiomatization begins by defining the primitive tree structure relations precedes and dominates. Once these primitive tree structure relations are defined, they can be used to approximate more complex relationships such as c-commands, as described below. All of these axioms are in the Sch6nfinkel-Bernays' class, so the satisfiability of arbitrary boolean combinations of such constraints is decidable.</Paragraph>
      <Paragraph position="2"> First, note that the standard definition of trees in terms of the binary relations &lt; (linear precedence) and D (domination) can be expressed directly as Sch6nfinkel-Bernays' formulae. The axioms presented below are just the definitions of trees given in Partee, ter Meulen, and Wall (1990) and Wall (1972) using the syntax of first-order logic. Axioms (7a-c) require that &lt; is a strict partial order, and axioms (8a-c) require Mark Johnson Computing with Features as Formulae that D is a weak partial order over the nodes in a tree. In what follows, N(x) is interpreted as meaning that x is a tree node.</Paragraph>
      <Paragraph position="4"> (10) requires that for each pair of nodes either one precedes the other or one dominates the other. Axiom (11) enforces the &amp;quot;no tangling&amp;quot; constraint.</Paragraph>
      <Paragraph position="6"> Finally, the following axioms (implicit in the standard treatments cited above) require the precedence and dominance relations to range over tree nodes.</Paragraph>
      <Paragraph position="8"> This concludes the specification of linear precedence and dominance relations over nodes. We now turn to the specification of other relations in terms of these. The proper dominance relation P can be defined in terms of dominance as follows.</Paragraph>
      <Paragraph position="9"> Vx Vy p(x, y) ~ x # y A D(x, y). (13) However, many interesting linguistic relations cannot be defined by Sch6nfinkel-Bernays' axioms. For example, the c-commands relation C is defined by the following formula (which says that x c-commands y iff x does not dominate y, and every node z that properly dominates x also properly dominates y).</Paragraph>
      <Paragraph position="10"> Vx vy C(x, y) ~ ~D(x, y) A Vz e(z, x) ~ P(z, y). (14) It is easy to see that this definition is not equivalent to a Sch6nfinkel-Bernays' formula by expanding the equivalence into two implications and moving the embedded quantifier out.</Paragraph>
      <Paragraph position="12"> of a universal quantifier. There are a number of ways to respond to this problem.</Paragraph>
      <Paragraph position="13"> First, we can abandon the attempt to work within the Sch6nfinkel-Bernays' class, and work with some other language. Rounds (1988) describes such a language called LFP, whose decidability follows from the fact that the domain of quantification is Computational Linguistics Volume 20, Number 1 restricted (just as in SB). However, it seems to be difficult to devise a decidable system capable of simultaneously expressing both tree structure and the variety of feature structure constraints that the SB approach described here can. Blackburn, Gardent, and Meyer-viol (1993) introduce a modal language L T for describing trees decorated with feature structures, whose satisfiability problem is undecidable. In the long run, such specialized &amp;quot;designer logics&amp;quot; may provide the most satisfying integration of tree structure and feature structure constraints.</Paragraph>
      <Paragraph position="14"> Second, the 'one-sided' approximation (14a) can be used in place of the correct axiom (14). The effect of using such one-sided approximations was investigated in Johnson (1991a). It was shown there that if ~ is a formula such as the one in (14) and ~/is the one-sided approximation (14a), then for any formula ~+(C) in which C only appears positively, ~ A ~+ (C) is satisfiable iff ~' A ~+ (C) is satisfiable. That is, if we are concerned only with positively occurring constraints, we can simplify (14) to (14a), i.e., ignore (14b), without affecting constraint satisfiability.</Paragraph>
      <Paragraph position="15"> Third, we can regard formulae such as (14) as the &amp;quot;macro&amp;quot; (15), used to expand constraints at the interface between the syntactic rules and the constraint solver. This &amp;quot;macro expansion&amp;quot; rewrites c-commands constraints into boolean combinations of constraints that the constraint solver can handle.</Paragraph>
      <Paragraph position="17"> The second and the third approaches differ in important ways. In the second approach, c-commands is a relation that is &amp;quot;understood&amp;quot; by the constraint solver (albeit only in its one-sided form), so it can be used to define other relations. In the third approach, c-commands constraints are not primitive constraints, so relations defined in terms of c-commands must also be expressible in terms of &amp;quot;macro expansion.&amp;quot; In the second approach, constraints are quantifier-free formulae (quantifiers appear only in the axioms), so the satisfiability problem is in NP. But in the third approach, macro expansion produces formulae that contain additional quantifiers, so the satisfiability problem may be PSPACE-complete.</Paragraph>
    </Section>
    <Section position="3" start_page="4" end_page="10" type="sub_section">
      <SectionTitle>
3.5 Limitations on Constraints Expressible with SB Formulae
</SectionTitle>
      <Paragraph position="0"> Of course, SB is not as expressive as full first-order logic. It is incapable of expressing functional relationships, since these require an existential quantifier inside the scope of a universal quantifier. This means, among other things, that it is impossible to state a constraint in SB requiring that a certain node must exist (as was noted in the discussion of c-command in the previous section) or that all nodes possess certain attributes. Thus for example, the following constraint, which requires that every tensed entity possess number and person attributes, is a first-order formula that is not in SB, since it requires a functional relationship between entities with tense attributes and the values of their number and person attributes.</Paragraph>
      <Paragraph position="1"> Vx Vy arc(x~ tensG y) ~ (3z arc(x~ number~ z)) A (3z arc(x~ person~ z)) (16) Similarly, a number of other extensions to the basic attribute-value framework discussed in the literature cannot be formalized in SB. Subsumption constraints, used in the treatment of (natural language) conjunction, are not expressible as SB formulae because the satisfiability problem for conjunctions of subsumption and attribute-value constraints is undecidable (D6rre and Rounds 1992). Positively occurring functional  Mark Johnson Computing with Features as Formulae (El) Vxx=x.</Paragraph>
      <Paragraph position="2"> (E2) VxVyx=y~y=x.</Paragraph>
      <Paragraph position="3"> (E3) Vx0... VXn xj = x 0 A P(Xl~... ~ xj~...~ xn) ~ P(Xl~...~ Xo~...~ Xn) for j = 1~...~ n, for every predicate symbol P appearing in ~.</Paragraph>
      <Paragraph position="4"> (E4) VXo...VXn xj =x0 --~ f(xl~...~Xj~...~Xn) =f(Xl~...~Xo~...~Xn)  for j = 1~...~ n, for every function symbol f appearing in ~.</Paragraph>
      <Paragraph position="5"> Figure 2 Equality axiom schemata for a first-order formula ~.</Paragraph>
      <Paragraph position="6"> uncertainty constraints, used in the LFG treatment of long-distance dependencies (Kaplan and Zaenen 1989) appear to have a decidable satisfiability problem (Kaplan and Maxwell 1988a), but the satisfiability problem for arbitrary boolean combinations of functional uncertainty constraints is undecidable (Keller 1991), so these cannot be expressed using SB formulae either (since the quantifier-free subclass of SB is closed under boolean operations).</Paragraph>
    </Section>
    <Section position="4" start_page="10" end_page="11" type="sub_section">
      <SectionTitle>
3.6 The Equality Relation
</SectionTitle>
      <Paragraph position="0"> In this paper the intended interpretation of the equality relation is identity; i.e., a = b if and only if a and b denote the same individual. However, for some purposes (e.g., in the least-fixed-point characterization of minimal models given below) this &amp;quot;special&amp;quot; interpretation of the equality complicates matters, and it is more convenient to treat the equality relation as a &amp;quot;normal&amp;quot; relation that is defined by a set of axioms E.</Paragraph>
      <Paragraph position="1"> The idea is that E has the property that a formula ~ is satisfiable under the identity interpretation of equality if and only if {9~} U E is satisfiable in an interpretation in which equality is not given any special treatment. In effect, the axioms E require that the equality relation denotes an equivalence relation, and permit the substitution of equals for equals. Together these imply that no predicate can distinguish equal individuals. This means that in terms of satisfiability and the consequence relation, exactly the same results are obtained irrespective of whether equality is treated as identity or defined by the axioms E.</Paragraph>
      <Paragraph position="2"> Such treatments of equality in first-order logic are well known and described in standard texts. For example, Chang and Lee (1973) give the axiom schemata in Figure 2, which generates syntactic equality axioms E for a first-order formula 9~, and prove that E has the properties just described, s What is important here is that for an SB formula ~ the instances of the axiom schemata are all SB formulae, and there are only finitely many instances of these schemata.</Paragraph>
      <Paragraph position="3"> This means that for an arbitrary SB formula ~ there is another SB formula ~ such that ~ is satisfiable with respect to an identity interpretation of equality if and only if A ~ is satisfiable with respect to an interpretation in which equality is treated like any other relation. Thus a method for determining the satisfiability of SB formulae without equality can be used to determine satisfiability of SB formulae in which equality is interpreted as identity.</Paragraph>
      <Paragraph position="4"> 8 Of course, (E4) has no instances if ~ is an SB formula, since SB formulae do not contain function symbols.</Paragraph>
      <Paragraph position="5">  Computational Linguistics Volume 20, Number 1 (17) Vx x = x.</Paragraph>
      <Paragraph position="6"> (18) Vx Vy x = y ~ y = x.</Paragraph>
      <Paragraph position="7"> (19) Vx Va Vy VXl x -~ Xl /X arc(x, a, y) ~ arc(x1, a, y).</Paragraph>
      <Paragraph position="8"> (20) Vx Va Vy Val a = al A arc(x, a, y) ~ arc(x, al, y).</Paragraph>
      <Paragraph position="9"> (21) Vx Va Vy Vyl y = yl Aarc(x,a,y) ~ arc(x,a,yl).</Paragraph>
      <Paragraph position="10"> (22) VCl Vc2 c = Cl A con(c) ~ con(c1).</Paragraph>
      <Paragraph position="11"> (23) Vx Vy x = y A 3rd-sg(x) ~ 3rd-sg(y).</Paragraph>
      <Paragraph position="12">  Figure 3 The equality axioms for arc, con, and 3rd-sg predicates.</Paragraph>
      <Paragraph position="13"> For example, consider the SB formulae in (1-4) and (5-16). These contain the three-place relation symbol arc and the one-place relation symbols con and 3rd-sg. The equality axioms obtained from schemata (El-E4) for any system of constraints that mention just these relations are given in Figure 3.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="11" end_page="16" type="metho">
    <SectionTitle>
4. Clausal Form and Disjunctive Datalog
</SectionTitle>
    <Paragraph position="0"> It is technically easier to work with a syntactically restricted class of SB formulae where the body of each formula has a particular syntactic form known as clausal form or Skolem standard form.</Paragraph>
    <Paragraph position="1"> Definition A clause is a formula of the form ,-~ o~1 V * * * V ~-, c~m V fll V * * * V ft,, where each oq and fly is an atomic formula (i.e., is of the form p(h,..., tn)), and m, n &gt; 0. A formula ~ is in clausal form iff it is a conjunction of clauses.</Paragraph>
    <Paragraph position="2"> The ,-~ ai are called negative literals and the fj are called positive literals. A clause for which m = 0 (i.e., one that consists solely of positive literals) is called a positive clause, and one for which n = 0 (i.e., one that consists solely of negative literals) is called a negative clause. A clause for which n = 1 is called a definite clause. A Horn clause is a clause for which n &lt; 1, i.e., either a negative clause or a definite clause. Abusing notation somewhat, a formula ~ in clausal form will sometimes also be treated as the set of the clauses that make up the conjunction ~. Similarly, because clauses are used as rewriting rules below, the clause ~-, o~1 V -.. V ,-~ o~ m V fl V ... V fin will sometimes be written as the equivalent implication oz I A ..-/x o~ m ~ fll V .-. V fin. A formula ~ in clausal form does not contain any quantifier symbols. As is standard, all variables in ~ are treated as implicitly universally quantified at the clausal level. Existentially quantified variables in SB formulae are inessential, in that they can always be directly replaced by Skolem constants.</Paragraph>
    <Paragraph position="3"> Restricting attention to SB formulae in clausal form imposes no real restriction on the class of constraints expressible. Standard procedures for transforming first-order formulae into clausal form, such as the ones described in Chang and Lee (1973), Duffy  Mark Johnson Computing with Features as Formulae (1991), or Genesereth and Nilsson (1987), transform SB formulae into SB formulae in clausal form.</Paragraph>
    <Paragraph position="4"> Interestingly, clausal form SB formulae correspond one-to-one with an extension to Datalog (Ullman 1988) that allows disjunctive &amp;quot;heads&amp;quot; or consequences. In notation borrowed from disjunctive logic programming (Kowalski 1979; Lobo, Minker, and Rajasekar 1992; Loveland 1987), the clauses above would be written as listed in the appendix. While the connection between logic programming and feature structures is well known (Ait-Kaci 1984; Ait-Kaci and Podelski 1993; Carpenter 1991, 1992; H6hfeld and Smolka 1988; Pereira 1987; Shieber 1992; Smolka 1992), this shows that the theory of feature structure constraints is also related to database theory as well.</Paragraph>
    <Paragraph position="5"> Negative clauses correspond to Datalog integrity constraints, and clauses with a single positive literal are definite clauses. Simple assertions, e.g., about the existence of arcs, consisting of exactly one positive literal are Datalog atomic clauses. Clauses with two or more positive literals cannot be expressed in Datalog itself, but require the disjunctive extension of Datalog. The appendix displays all of the SB formulae mentioned in this paper so far in clausal form in Datalog notation; (6'), (10 ~) and (7&amp;quot;) are expressed in the disjunctive extension to Datalog. In fact, the axioms defining attribute-value structures (1-4) and syntactic equality (El-E3) are all Horn Datalog clauses; i.e., the disjunctive extension is not needed for defining attribute-value feature structures.</Paragraph>
    <Paragraph position="6"> 5. Determining the Satisfiability of SB Formulae This section describes a forward-chaining algorithm for determining the satisfiability of SB formulae in clausal form. This algorithm is a nondeterministic variant of the semi-naive evaluation method for Datalog clauses in which the union-find algorithm is used to efficiently maintain equivalence classes of equal terms. It is also recognizable as a generalization of the standard unification algorithm for feature structures to arbitrary Horn SB constraints. 9 The treatment is informal because the goal of the section is to point out several important standard implementation techniques rather than to advance a totally new algorithm.</Paragraph>
    <Paragraph position="7"> The key intuition behind the algorithm is this. To demonstrate the satisfiability of a set S of clauses, it is sufficient to exhibit a set A of ground atoms drawn from the Herbrand base of S such that the following conditions hold (the next section proves this assertion).</Paragraph>
    <Paragraph position="8">  (a) (b) (c) ' ~ of any negative clause in S are For no ground instance ~ o~ 1 V .-. V ~ o~ m t in A. (If they were, then that clause would be falsified all of c~1,..., c% by A.) For each ground instance fl~ V * .. v fl~ of a positive clause in S, at least one of the fl\[ is in A.</Paragraph>
    <Paragraph position="9"> 'A. A ' ~fl~V..-Vfl~ofanimplicationin For each ground instance c h .. o~ m ! S, if all of the o~,..., o~ m are in A then so is at least one of the fl(,..., fl~. (In fact, the other two conditions are just special cases of this condition.) 9 It is a generalization of the algorithm described in Hegner (1991), which treats Horn combinations of attribute-value constraints.</Paragraph>
    <Section position="1" start_page="13" end_page="13" type="sub_section">
      <SectionTitle>
5.1 Naive Evaluation
</SectionTitle>
      <Paragraph position="0"> One could attempt to find such a set A in the following manner. First, one nondeterministically selects a fl~ from each of the ground instances of the positive clauses in S and adds these to A. Then one attempts to close A with respect to condition (c); if ' ' of some (ground instance of an) implication are in A, all of the antecedents oq,..., olm then one of the consequents fl~,..., fl~ is nondeterministically selected and added to A, unless at least one of them is already present. (Of course, all such nondeterministic paths might have to be investigated.) Periodically, condition (a) is checked; if it fails to hold, then this nondeterministic path on the search for A must be abandoned. Non-determinism arises solely from the presence of disjunction in consequents of clauses; if S is a set of Horn clauses then the fixed-point calculation proceeds deterministically.</Paragraph>
      <Paragraph position="1"> Ignoring the checking of condition (a), the method is essentially computing a fixed-point of the nonnegative clauses in S via a kind of iterative approximation known as naive evaluation. Naive evaluation is unnecessarily computationally inefficient. Once the set A is large enough to require an atom o~ to be added to A, naive evaluation &amp;quot;rediscovers&amp;quot; this requirement on all subsequent passes.</Paragraph>
    </Section>
    <Section position="2" start_page="13" end_page="13" type="sub_section">
      <SectionTitle>
5.2 Semi-Naive Evaluation
</SectionTitle>
      <Paragraph position="0"> Semi-naive evaluation avoids rediscovering the same fact in the same way by insisting that each time a clause is applied at least one of the antecedents was just discovered on the previous round (Ullman 1988, 1989). This is done by maintaining two sets of atoms, A and &amp;A, where A is the set of atoms discovered one or more iterations ago, and &amp;A is the set of atoms discovered at the last iteration. The nondeterministic semi-naive algorithm for computing a set A (if it exists) is sketched in Figure 4. In that algorithm choose is a &amp;quot;function&amp;quot; that nondeterministically picks one member from its set argument; it can be implemented using, e.g., backtracking. Ullman describes methods of matching clauses in S against the sets A and &amp;A that avoids calculating all of the ground instances of the clauses in S.</Paragraph>
      <Paragraph position="1"> The semi-naive algorithm can be used directly with the syntactic equality axioms given in Section 3.4 as a decision procedure for SB formulae, and hence for systems of feature structure constraints. However, the resulting system is inefficient because the equality axioms, specifically the instances of schemata (E2) and (E3), cause the &amp;quot;copying&amp;quot; of any atom containing an argument that appears in an equality atom to all members of the equivalence class containing that argument.</Paragraph>
      <Paragraph position="2"> For example, if p(a), q(b) and a = b are atoms in A, then instances of (E2) and (E3) ensure that p(b),q(a) and b = a will be added to &amp;A and thence to A. In general, if it is discovered that n constants al,.., an are equal, then A will ultimately contain the n 2 equalities ai = aj, 1 &lt; i &lt; n, 1 &lt; j &lt;_ n, as well as at least n &amp;quot;copies&amp;quot; of any predicate containing any ai.</Paragraph>
    </Section>
    <Section position="3" start_page="13" end_page="14" type="sub_section">
      <SectionTitle>
5.3 Union-Find and Equality
</SectionTitle>
      <Paragraph position="0"> As noted above, the equality axioms ensure that the relation that the equality symbol denotes is an equivalence relation and the substitutivity of equals for equals. In general, the union-find algorithm (Corman, Leiserson, and Rivest 1990; Gallier 1986; Nelson and Oppen 1980) maintains the equivalence classes of the equality relation far more efficiently than an approach that uses the syntactic equality axioms.</Paragraph>
      <Paragraph position="1"> The equivalence classes are encoded by associating each constant with a pointer that is either null or points to another constant, where a points to b only if a = b. These pointer correspond exactly to the &amp;quot;invisible pointers&amp;quot; used in standard implementations of the attribute-value unification algorithm.</Paragraph>
      <Paragraph position="2">  Mark Johnson Computing with Features as Formulae Input: A set of SB clauses S.</Paragraph>
      <Paragraph position="3"> Output: A set of ground clauses A iff S is satisfiable.</Paragraph>
      <Paragraph position="5"> clause in S and at least one of the o~; is in &amp;A, then fail,</Paragraph>
      <Paragraph position="7"> ,..., ~ is in &amp;A of an implication in S such that {oL~ Oq'n} C A, at least one of the o~ i and no flj~ is in A}, return A.</Paragraph>
      <Paragraph position="8"> Figure 4 The semi-naive algorithm for computing A. The find operation dereferences its argument, i.e., it follows these pointers until it reaches a constant with a null pointer, which is the equivalence classes' representative. Just as in the standard attribute-value unification algorithm, all arguments are always dereferenced before they are used.</Paragraph>
      <Paragraph position="9"> The union operation, called whenever an atom a = b is added to the set A, merges their equivalence classes by redirecting the pointer associated with the representative of one of them to point to the representative of the other. 1deg In this approach, only atoms that contain the redirected constant need to be added to &amp;A and thence to A. For example, if p(a) and q(b) are atoms in A and the equality a = b is discovered, causing a to be redirected to b, then only p(b) is added to &amp;A, and thence to A. Further, the &amp;quot;original&amp;quot; atom p(a) is no longer required; indeed, the new atom p(b) is exactly an argument-dereferenced variant of the old atom, so it is not necessary to copy the atom at all. In general, equalities between n items are represented by n - 1 nonnull pointers, and copying of atoms can be avoided by argument dereferencing.</Paragraph>
    </Section>
    <Section position="4" start_page="14" end_page="16" type="sub_section">
      <SectionTitle>
5.4 An Example
</SectionTitle>
      <Paragraph position="0"> This section presents a very simple example that demonstrates the semi-naive algorithm and the union-find techniques. The clauses used are the attribute-value axiom schemata (1-4) and the axioms defining the sort 3rd-sg (5), as well as the additional 10 The union-find algorithm achieves quasi-linear running time when it incorporates path compression and union by rank (Corman, Leiserson, and Rivest 1990).</Paragraph>
      <Paragraph position="2"> In the second iteration the antecedents of the first attribute-value axiom are satisfied, so &amp;A2 contains an equality atom.</Paragraph>
      <Paragraph position="3"> 3rd-sg(u), arc(u, number, v), con(sg), con(pl), con(3rd), I</Paragraph>
      <Paragraph position="5"> The equality atom causes v to be redirected to sg, and at this stage the inconsistency of the derived atom v = sg in &amp;A2 with the input constraint v ~ sg in S is detected. (It may be helpful to think of the constraint v ~ sg as the equivalent clause v = sg ~ false).</Paragraph>
      <Paragraph position="6"> The algorithm therefore returns with failure, indicating that the set S is unsatisfiable.</Paragraph>
      <Paragraph position="7"> At the point at which the inconsistency is detected, the set A contains the following atoms, where v =~ sg indicates that v is redirected to sg.</Paragraph>
      <Paragraph position="8"> { 3rd-sg(u)~ arc(u, number, v), con(sg), con(pl), con( 3rd), A 3 ~- arc(u, person, 3rd), arc(u, number, sg), v ~ sg The correspondence of this procedure to the standard attribute-value unification algorithm is quite strong. In this procedure, the attribute-value axiom (1) detects situations in which some node has two arcs with the same label pointing to, say, y and z. If such  Mark Johnson Computing with Features as Formulae a situation arises, the equality y = z is inferred, which results in y being redirected to z and causes all of the arcs leaving y to be added to &amp;A, where they will be compared with the arcs leaving z. The other attribute-value axiom schemata (2--4) detect constant-constant and constant-complex clashes, causing failure if one is found. Efficient processing demands that the atoms in A be indexed by their arguments to speed up the matching atoms with the antecedents of clauses. One way of doing this is to store on each constant a list of the atoms in which that constant appears. Such an index has the same structure as the standard graph encoding of feature structure constraints.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="16" end_page="19" type="metho">
    <SectionTitle>
6. A Fixed-Point Theorem
</SectionTitle>
    <Paragraph position="0"> We now turn to the theoretical justification of the bottom-up forward-chaining procedures sketched in the last section, and show that such methods will find a model for a set of SB formulae in clausal form if one exists. This section demonstrates that an SB formula in clausal form is satisfiable if and only if a bottom-up forward-chaining procedure finds a deductively closed set of atoms A. A similar theorem for the case in which all the clauses are Horn clauses is presented in Lloyd (1984); this section extends that work to arbitrary clauses.</Paragraph>
    <Paragraph position="1"> It presents a characterization of the models of an arbitrary first-order formula 9~ in clausal form in terms of the least-fixed points of a set {T~, X} of partial functions from Herbrand interpretations to Herbrand interpretations. These functions have the property that A is a Herbrand interpretation that satisfies ~ if and only if the least-fixed point of at least one of them is a submodel of A.</Paragraph>
    <Paragraph position="2"> For SB formulae this set of functions is finite and the least-fixed points are reached in a finitely bounded number of iterations. Since the procedures described in the last section calculate the least-fixed points of these functions, they can be used to determine the satisfiability of an arbitrary SB formula as well as all of its ground atomic consequences.</Paragraph>
    <Paragraph position="3"> The functions T~, x play a similiar r61e here to one that the transformation Tp plays in the least-fixed-point semantics of Horn clause programs. Informally, each function in the set {T~,x} corresponds to one whole sequence of nondeterministic choices of disjuncts in non-Horn clauses that could be made during an iterative approximation of the least-fixed point. This section is based on Sections 5 and 6 of Chapter I of Lloyd (1984), to which the reader should turn for further details.</Paragraph>
    <Paragraph position="4"> The fixed-point theorem holds for arbitrary first-order formulae in clausal form, but the set {T~,x} is finite if and only if ~ does not contain any function symbols, i.e., is an SB formula. Equality is not treated specially, so the formula 9~ must contain appropriate equality axioms, as mentioned above.</Paragraph>
    <Paragraph position="5"> Let U be the Herbrand universe with respect to ~ (i.e., the set of all terms that can be constructed using the constant and function symbols appearing in ~),11 and let B~ be the set of all ground atoms that can be formed using the predicate symbols of 9~ with elements of U as arguments. A Herbrand interpretation A is a subset of B~.</Paragraph>
    <Paragraph position="6"> Note that the set of Herbrand interpretations 2 B~ partially ordered by the subset relation forms a complete lattice. Further, U and hence B~ are finite iff ~ is an SB formula. (If ~ is in clausal form but is not an SB formula then it must contain a function symbol, so its Herbrand universe U is infinite.)  That is, a Herbrand interpretation trivially falsifies a set of clauses ~ just in case some negative clause in ~ is false in that interpretation. Clearly, any such interpretation cannot satisfy ~, but the converse does not hold: there are interpretations that do not trivially falsify 9~ but still do not satisfy ~ because they do not satisfy one or more of the nonnegative clauses in ~.</Paragraph>
    <Paragraph position="7"> We turn now to the nonnegative clauses in ~. The idea is that even if A does not IA- A t / I satisfy a ground instance a 1 &amp;quot;&amp;quot; am ~ fl V ..- V fin of some nonnegative clause in ~, we can extend A so that it does so by adding one of the f;. The chief technical difficulty here is caused by the nondeterminism involved in deciding which of the f\[ to add, and a device called a &amp;quot;choice function&amp;quot; is introduced to choose, for each ground instance of a clause, which of the atoms in its consequent will be added to A if its antecedent is contained in A. A choice function for a clause al A.., A am ~ fl V... V fin is therefore a function from all possible ways of grounding that clause to one of the fli. Definition A choice function for a clause c = (al A ... A am ~ fa V ... V ft,) is any function in (Vc -+ U) ~ \[1,..., n\], where Vc is the set of variables in the clause c.</Paragraph>
    <Paragraph position="8"> That is, a choice function for a clause is a function from variable assignments to an integer representing one of the clause's consequents. It is so named because for each variable assignment (i.e., each way of grounding the variables in ~) it &amp;quot;chooses&amp;quot; an atom from the consequent of the clause. Horn clauses have only one choice function, and negative clauses have no choice functions at all. Note that since U is finite for SB formulae, there are a finite number of variable assignment functions for any SB clause and hence only a finite number of choice functions for any SB clause.</Paragraph>
    <Paragraph position="9"> A choice function X for a set of clauses ~ is a function from 9~ to choice functions such that for each nonnegative clause c in ~, X(c) is a choice function for c (the value that X takes on negative clauses is ignored). Clearly, a choice function exists for every set of clauses.</Paragraph>
    <Paragraph position="10"> Given a set of clauses ~ and a choice function X for ~o, we define a function F~, x from Herbrand interpretations to Herbrand interpretations as follows.</Paragraph>
    <Paragraph position="11"> Definition F~, x is a function in 2 B* ~ 2 B* that is defined as follows.</Paragraph>
    <Paragraph position="13"> and for all variable assignments 0 : Vc ~ U such that 0(a,),...,0(a,) E A and i= X(c)(O)} Intuitively, F~, x corresponds to one nondeterministic step in the 'bottom-up' construction of a Herbrand model for ~ described in the previous section. If A makes the antecedent of some ground instance of some clause in 9~ true, then we use the choice function X to pick an atom in the consequent of that ground clause and add it to the interpretation. Different choice functions X represent different sequences of nondeterministic choices, and result in the construction of possibly different interpretations.  Mark Johnson Computing with Features as Formulae The following lemma, based directly on proposition 6.3 of Lloyd (1984), notes the continuity (and therefore the monotonicity) of F~, x.</Paragraph>
    <Paragraph position="14"> Lemma 1 The function F~, x is continuous. That is, if X is a directed subset of 2 B~ (i.e., every finite subset of X has an upper bound in X) then F~,x(lub(X)) = lub(F~,x(X)).</Paragraph>
    <Paragraph position="16"> The continuity of F~, x immediately implies the convergence of the sequence (F/~,x (O)); the value that it converges to is called the least-fixed point of F~, x, written Ifp(F~,x).</Paragraph>
    <Paragraph position="17"> Note that if ~ is in SB then there is an integer k such that lfp(F~,x) = Fk~,x(O); this follows directly from the monotonicity of F~, x and the finiteness of B~.</Paragraph>
    <Paragraph position="18"> The function F~, x and a condition requiring that the interpretation produced does not trivially falsify the set of clauses ~ together define the partial function T~, x.</Paragraph>
    <Paragraph position="19"> Definition T~, x is a partial function in 2 B~ --~ 2 B~ that is defined as follows.</Paragraph>
    <Paragraph position="20"> T~,x(A ) = F~,x(A ) if F~,x(A ) does not trivially falsify 9~, and is undefined otherwise. null Note that if the sequence (T/~,x(A)) is defined for all i then (T/~,x(A)) = i (F~, x (A)). T~, x enjoys the following kind of monotonicity.</Paragraph>
    <Paragraph position="21">  The following lemma shows that Herbrand models of 9~ contain fixed points of T~, x for some choice function X for 9~-Lemma 3 For all Herbrand interpretations A, A ~ 9~ iff there exists a choice function X for 9~ such that T/, x (A) _C A.</Paragraph>
    <Paragraph position="22">  We begin first with the left-to-right component of the proof. If A ~ 9~, then A does not trivially falsify ~, so T~, x (A) is defined. Now we show how to find for each satisfying interpretation A a choice function X such that T~x(A ) = A. Since A satisfies ~, for every nonnegative clause c = (o~1 A..-A O~m ~ fll V... V fin) in 9~ and for every variable assignment function 0 for the variables in c, if 0(o~1)~...~ 0(o~n) E A, then by the truth conditions for implication and disjunction, some O(fli) E A as well. Thus, for all 0 such that 0(oq)~... ~O(C~n) E A let X(c)(0) be any i such that O(fli) E A, and let X(c)(0) take any permissible value otherwise. Hence T~, x (A) = F~, x (A) = A.</Paragraph>
    <Paragraph position="23"> Now suppose T~,x(A ) C A. Since T~,x(A ) is defined, A does not trivially falsify any negative clause in ~. Let c = (o~1 A ... A O~m ~ fll V &amp;quot;.. V fin) be any nonnegative clause in ~, and let ~ E V~ ~ U be any variable assignment function for the variables in c. If 0(o~1)~... ~0(O~n) E A then O(fli) E T~,x(A ) C A as well, where i = X(c)(0), so A ~ c and hence A ~ 9~. \[\] The following theorem shows that a formula is satisfiable if and only if the least-fixed point of at least one of the T~, x exists. It justifies the decision procedures presented in the previous section, which operate by searching for such least-fixed points. The proof actually establishes something stronger, viz., that every Herbrand model of ~ is an extension of the least-fixed points of one or more of the T~ x. Thus an enumeration of all of the least-fixed points of the T~, x yields all of the &amp;quot;minimal models&amp;quot; of ~ (although it is not clear that these are in fact necessary for recognition or parsing, as discussed above).</Paragraph>
    <Paragraph position="24"> Theorem is satisfiable if and only if there exists a choice function X for ~ such that lfp(T~,x) exists.</Paragraph>
    <Paragraph position="25"> Proof If Ifp(T~,x) exists then by Lemma 3, lfp(T~,x) = ~. Now suppose A is a Herbrand interpretation that satisfies 9~- Lemma 3 asserts the existence of a choice function X such that T~,x(A ) exists and T~,x(A ) G A. By Lemma 2 and the fixed point property noted above lfp(T~,x) exists, since lfp(T~,x) = T~x(O ) C T~x(A ) C A. \[\] It is important to recognize that these &amp;quot;minimal models&amp;quot; are in general not upwardclosed: an extension A' of a model A can trivially falsify ~ even though A does not. This is essentially Moshier's (1988) and Pereira's (1987) observation that in the presence of negation the set of models is not upwardly closed.</Paragraph>
    <Paragraph position="26"> We conclude this section with the observation that the positive consequences of a formula 9~ can be &amp;quot;read off&amp;quot; its least-fixed points.</Paragraph>
    <Paragraph position="27"> Corollary is satisfiable iff for some choice function X for ~, lfp(T~,x) exists. Moreover, if fl = fll V ... V fin is any disjunction of ground atoms, 9~ ~ fl iff for all choice functions X for ~ such that lfp(T~,x) exists, at least one of the fli is in lfp(T~,x).</Paragraph>
  </Section>
  <Section position="8" start_page="19" end_page="20" type="metho">
    <SectionTitle>
7. Conclusion
</SectionTitle>
    <Paragraph position="0"> The main goal of this paper was to demonstrate from a computational perspective that Sch6nfinkel-Bernays' formulae are a natural generalization of (boolean combina- null Mark Johnson Computing with Features as Formulae tions of) attribute-value feature structure constraints. From a computational complexity perspective we noted that the satisfiability problem for SB formulae with a bounded number of quantifiers is NP-complete, so it is no harder than the satisfiability problem for disjunctive attribute-value constraints.</Paragraph>
    <Paragraph position="1"> From a more practical perspective, a semi-naive bottom-up evaluation strategy using union-find methods to handle equality generalizes the standard attribute-value &amp;quot;unification&amp;quot; algorithm to arbitrary SB constraints in clausal form. Because it treats standard attribute-value constraints in approximately the same way as the standard unification algorithm, and because it can incorporate the same kinds of indexing that the latter algorithm employs, the generalized algorithm should be able to determine the satisfiability of attribute-value constraints with approximately the same efficiency as the standard attribute-value unification algorithm.</Paragraph>
    <Paragraph position="2"> In generalizing attribute-value constraints to SB formulae, we noted that in clausal form the SB formulae constitute a disjunctive extension to Datalog, and that the standard attribute-value unification algorithm is closely related to a version of semi-naive evaluation algorithm used to evaluate Datalog clauses. This offers another perspective on feature structure constraints; they can be seen as kinds of databases containing information about the linguistic structures they describe.</Paragraph>
    <Paragraph position="3"> Perhaps the greatest weakness of this work is the lack of an efficient method for treating disjunctive constraints. The backtracking strategy suggested in the body of the paper can be extremely inefficient, even with 'toy' grammars. This problem is not unique to this approach; rather, it is endemic to most complex feature-based approaches to natural language processing, as evidenced by the volume of literature on the subject.</Paragraph>
    <Paragraph position="4"> As discussed in Section 3, the satisfiability problem for SB formula with a fixed number of universal quantifiers is NP-hard, so all known algorithms require exponential time in the worst case, and unless P=NP no tractable general-purpose algorithm for determining the satisfiability of SB formulae exists. With present technology, the best we can hope for is an algorithm that performs adequately on the types of problems that we actually encounter.</Paragraph>
    <Paragraph position="5"> Sometimes disjunctive constraints can be (automatically) transformed into nondisjunctive ones, thus avoiding the problem entirely. For example, Alshawi (1992) describes a technique attributed to Colmerauer for transforming disjunctions of finitedomain feature-value constraints into conjunctions. Kasper (1988) and Hegner (1991) point out that Horn clauses, although technically disjunctions, can be handled considerably more efficiently than general disjunctive constraints. The forward-chaining mechanisms that they propose for treating these constraints appear to be special cases of the semi-naive algorithm sketched in this paper.</Paragraph>
    <Paragraph position="6"> Unfortunately, I know of no general adequate method for handling the disjunctive constraints that arise in real grammars with acceptable efficiency. The techniques discussed by Maxwell and Kaplan (1991, 1992) seem most directly compatible with the approach described in this paper, and the methods described by Kasper (1987b), Eisele and D6rre (1988), and Emele (1991) might have important applications as well.</Paragraph>
  </Section>
class="xml-element"></Paper>