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<?xml version="1.0" standalone="yes"?> <Paper uid="J91-2001"> <Title>Features and Formulae</Title> <Section position="3" start_page="134" end_page="136" type="metho"> <SectionTitle> 3 Reference markers in DRT correspond approximately to the referential indices associated with NPs in GB theory. </SectionTitle> <Paragraph position="0"> Computational Linguistics Volume 17, Number 2</Paragraph> <Paragraph position="2"> The result of unifying the value of the refs-out attribute of Figure 7 with s in Figure 6 this > def= Tnum= sg u agrv L pers = 3rd these .def = + num = pl ~agr-- LPers= 3rd U' V' salmon > IP red= salmdegn3rd\] 1 e, Lagr =f\[Pe rs= swim rpre i pers nUm sgll agr= 3rd subj =e g ~ tense = pres Feature structures demonstrating interaction of negative values and unification to negative feature values; specifically, some apparently plausible extensions lose the associativity property of unification. Example 6 Consider the feature structures in Figure 9, which might be assigned to the singular determiner this, the plural determiner these, the noun salmon, and the verb swim (the latter two structures are the same as those depicted in Figures 2 and 5). These can be used to analyze utterances such as these salmon swim and (the ill-formed utterance) this salmon swim, which involve the unification of u, e', and e or u', e', and e, respectively. Suppose a negative value is interpreted as a constraint that a feature structure either satisfies or does not satisfy, and suppose further that in Figure 9 the negative feature constraint f is satisfied by the value f'. Then e' and e in Figure 9 unify, and moreover further unification of e ~ with either u ~ or u succeeds, undesirably in the latter case. (Reinterpreting the negative constraint f so that f' fails to satisfy it does not help, since the unification of e, e', and u ~ should succeed). On the other hand, we obtain the results we desire if e' is unified with u or u ~ before being unified with e. If e' is first unified with u, then f~ is unified with v, and further unification of e' with e fails, since v does not satisfy f. If e' is first unified with u' then f' is unified with v' and further unification of e' with e succeeds, since v' does satisfy f. Thus under this interpretation Johnson Features and Formulae of negation and unification, the success or failure of a sequence of unifications depends on the order in which they are performed. 4</Paragraph> </Section> <Section position="4" start_page="136" end_page="140" type="metho"> <SectionTitle> 2. Feature Structures and Function-free Formulae </SectionTitle> <Paragraph position="0"> These problems have generated a considerable body of work on the mathematical properties of feature structures and the constraints and operations that apply to them.</Paragraph> <Paragraph position="1"> Following Kaplan and Bresnan (1982), Pereira and Shieber (1984), Kasper and Rounds (1986, 1990), and Johnson (1988, 1990a) the constraints that determine the feature structures are regarded as formulae from a language for describing feature structures, rather than as feature structures themselves.</Paragraph> <Paragraph position="2"> Disjunction and negation appear only in expressions from the description language, rather than as components of the feature structures that these expressions describe. Thus the lexical entries in the examples above will be interpreted as formulae that constrain the feature structures that can be associated with these lexical items in a syntactic tree, rather than the feature structures themselves. For example, the feature matrices depicted in Figures 2, 4-6, and 9 will be interpreted as graphical depictions of formulae expressing constraints on linguistic objects, rather than the linguistic objects that satisfy these constraints. This avoids any need to refer to &quot;negative&quot; or &quot;disjunctive&quot; objects as entities appearing in a feature structure.</Paragraph> <Paragraph position="3"> As explained below, the familiar attribute-value &quot;unification algorithm&quot; can be interpreted as computing the atomic consequences of a purely conjunctive formula (where the graphs it operates on are data structures efficiently representing such formulae), and unification failure corresponds to the unsatisfiability of that conjunction (Kasper and Rounds 1990; Johnson 1988, 1990a; Pereira 1987).</Paragraph> <Paragraph position="4"> The most widely known model of feature structures and constraint language is the one developed to explain disjunctive feature values by Kasper and Rounds (1986, 1990) and Kasper (1986, 1987). The Kasper-Rounds treatment resolves the difficulties in interpreting disjunctive values by developing a specialized language for expressing these constraints. Various proposals to extend the Kasper-Rounds approach to deal with negative feature values are described by Moshier and Rounds (1987), Moshier (1988), Kasper (1988), Dawar and Vijayashanker (1989, 1990), Langholm (1989); other extensions to this framework are discussed by D6rre and Rounds (1989), Smolka (1988, 1989), and Nebel and Smolka (1989); and Shieber (1989) discusses the integration of such feature systems into a variety of parsing algorithms.</Paragraph> <Paragraph position="5"> One difficulty with this approach is that the constraint language is &quot;custom built,&quot; so important properties, such as compactness and decidability, must be investigated from scratch. Moreover, it is often unclear if the treatment can be extended to handle other types of feature structures as well. Rounds (1988) proposes a model for set-valued features, but he does not provide a language for expressing constraints on such set-valued entities, or investigate the computational complexity of systems of such constraints.</Paragraph> <Paragraph position="6"> This paper follows an alternative strategy suggested in Johnson (1990a): axiomatize the relevant properties of feature structures in some well-understood language (here first-order logic) and translate constraints on these structures into the same language.</Paragraph> <Paragraph position="7"> 4 It is possible to avoid these problems by augmenting feature structures with &quot;inequality arcs,&quot; as was first proposed (to my knowledge) by Karttunen (1984) and discussed in Johnson (1990a), Johnson (in press) and pages 67-72 of Johnson (1988). However, it is hard to justify the existence of such arcs if feature structures are supposed to be linguistic objects (rather than data structures that represent formulae manipulated during the parsing process).</Paragraph> <Paragraph position="8"> Computational Linguistics Volume 17, Number 2 Thus the satisfiability problem for a set of constraints on feature structures is reduced to the satisfiability problem for the axioms conjoined with the translation of these constraints in the target language. Importantly, techniques used to determine satisfiability in the target language can be used to determine the satisfiability of the feature constraints as well. In this paper the properties of attribute-value structures and constraints on them are expressed in a decidable class of first-order formulae: this means that the satisfiability problem for such formulae, and hence the feature constraints that they express, is always decidable.</Paragraph> <Paragraph position="9"> Of course, some linguistic analyses make use of feature structure constraint systems that can encode undecidable problems. For example, subsumption constraints, which are useful in the description of agreement phenomena in coordination constructions (Shieber 1989) can be used to encode undecidable problems, as D6rre and Rounds (1989) have shown. Clearly such constraints cannot be expressed in a decidable class, but often they can be axiomatized in other standard logics. Johnson (1991) shows how (positively occurring) subsumption constraints can be axiomatized in first-order logic, and sketches treatments of sort constraints and nonmonotonic devices such as ANY values (Kay 1985) and 'constraint equations' (Kaplan and Bresnan 1982) can be formalized in second-order logic using circumscription.</Paragraph> <Section position="1" start_page="137" end_page="138" type="sub_section"> <SectionTitle> 2.1 Axiomatizing Feature Structures with Function-Free Formulae </SectionTitle> <Paragraph position="0"> This section shows how the important properties of feature structures can be axiomatized using formulae from the Sch6nfinkel-Bernays class, which is the class of first-order formulae of the form 3X1. . . Xn~yl . . . yng~ where ~ contains no function symbols or quantifiers. (Thus no existential quantifier can appear in the scope of a universal quantifier.) This class of formulae was chosen because it is both decidable (see e.g. Lewis and Papadimitriou 1981) and can express the quantification needed to describe the particular set operations proposed here, as well as a variety of other interesting types of feature structures and constraints. (For a general discussion of decidable classes see Gurevich 1976 and Dreben and Goldfarb 1979.) The next section shows how the various kinds of constraints on feature structures described above can be translated into this class of formulae, so any system of such feature constraints is decidable as well.</Paragraph> <Paragraph position="1"> The elements of a feature structure, both complex and constant, constitute the domain of individuals in the intended interpretation. The attributes are binary relations over this domain, s We proceed by axiomatizing the conditions under which an interpretation corresponds to a well-formed feature structure, formulating them in essentially the same way as Smolka (1988, 1989) does.</Paragraph> <Paragraph position="2"> The axiomatization begins by describing the properties of the constant elements of attribute-value structures. The attribute-value constants are the denotation of certain constant symbols of the language of first-order logic, but not all constants (of the first-order language) will denote attribute-value constants since it is convenient to have constants that denote other entities as well. The following axiom schemata express the requirement that attribute-value constants have no attributes and that all attribute5 This differs from earlier work (Johnson 1988) in which values and attributes were both conceptualized as individuals. In fact, research in progress indicates that it is advantageous to conceptualize of attributes as individuals and attribute relations in terms of a 3-place relation arc, where arc(x, a, y) is true iff the value of x's attribute a is y. This permits the quantification over attributes needed to define both simple and parameterized sorts to be expressed.</Paragraph> <Paragraph position="3"> Johnson Features and Formulae value constants are distinct; i.e., that distinct attribute-value constants denote different entities.</Paragraph> <Paragraph position="4"> 1. For all attribute-value constants c and attributes a, V x -~a(G x).</Paragraph> <Paragraph position="5"> 2. For all distinct pairs of attribute-value constants Cl, c2, Cl # c2.</Paragraph> <Paragraph position="6"> The next axiom schema requires attributes to be single-valued.</Paragraph> <Paragraph position="7"> 3. For all attributes a, V xyz a(x,y) Aa(x,z) ~ y = z.</Paragraph> <Paragraph position="8"> This completes the axiomatization of attribute-value feature structures. 6 The claim is that any interpretation that satisfies these axioms is an attribute-value structure, i.e. 1-3 constitute a definition of attribute-value structures. Such interpretations can be viewed as (possibly infinite and disconnected) directed graphs, where the individuals constitute the graph's nodes and the attribute relations the arcs between those nodes. Thus these axioms admit a much wider class of models than do most other treatments of feature structures (e.g., Kasper and Rounds (1990) require feature structures to be a certain type of finite automata). In fact it is easy to add axioms requiring attribute-value structures to have additional properties such as acyclicity. But since the axioms that define attribute-value structures are in effect assumptions that stipulate the nature of linguistic entities, we obtain a more general theory the weaker these axioms are. Thus 1-3 are intended to stipulate only the properties of attribute-value structures that are required by linguistic analyses.</Paragraph> <Paragraph position="9"> Note that the partiality of attributes is of crucial importance: if attributes were required to be total rather than partial functions, we could not axiomatize them with formulae from the Sch6nfinkel-Bernays class. (An axiom schema requiring attributes to be total functions would have instances of the form Vx 3y a(x,y), which do not belong to the Sch6nfinkel-Bernays class).</Paragraph> </Section> <Section position="2" start_page="138" end_page="140" type="sub_section"> <SectionTitle> Example 7 </SectionTitle> <Paragraph position="0"> The interpretation corresponding to the attribute-value structure depicted in Figure 1 has as its domain the set D = {seem, like, john, sg, 3rd, mary, pres, none} U {e0,..., es}.</Paragraph> <Paragraph position="1"> The attributes denote relations on D x D. For example, pred denotes the relation {leo~seem), (el, like), le2,johnl~ (e3~ maryl}. It is straightforward to check that all of the axioms hold in this interpretation.</Paragraph> <Paragraph position="2"> Instead of providing entities in the interpretation that serve as the denotation for &quot;disjunctive&quot; or &quot;negative&quot; features, we follow Kasper and Rounds (1986, 1990), Moshier and Rounds (1987), and Johnson (1988, 1990) in permitting disjunction and negation only in the constraint language. Since the classical semantics of disjunction and negation for first-order languages is consistent and monotonic, a consistent, monotonic semantics for negative and disjunctive feature constraints follows directly. (An example is presented below; see Johnson (1990) and especially Section 2.10 of Johnson (1988) for further discussion).</Paragraph> <Paragraph position="3"> We turn now to the set-valued features. The most straightforward way of introducing set-valued features would be to combine some standard axiomatization of set theory with the axiomatization of attribute-value structures just presented. Unfortunately, 6 This axiomatization is finite iff the sets of attribute symbols and constant symbols are finite. In the intended computational and linguistic applications this is always the case.</Paragraph> <Paragraph position="4"> Computational Linguistics Volume 17, Number 2 all of the formulations of set-theory I am aware of, such as Zermelo-Fraenkel settheory, are expressed in languages whose satisfiability problem is undecidable. While this does not imply that the satisfiability problem for set-valued feature-structure constraints is also undecidable (since the feature constraint language may have restricted expressiveness), it does mean that its decidability cannot be shown by noting that a translation into a decidable class of formulae exists.</Paragraph> <Paragraph position="5"> Also, as an anonymous reviewer points out, since the intended linguistic applications only require finite sets and operations such as union and intersection, standard theories of sets (such as Zermelo-Fraenkel set-theory) are much more powerful than needed.</Paragraph> <Paragraph position="6"> Instead, we axiomatize just those properties of set-valued features that our feature constraints require using formulae from the Sch6nfinkel-Bernays class. We interpret the two-place relation in as the membership relation; in(x, y) is true in a model iff x is a member of y. We place no restrictions on this relation, but in other formulations axioms of foundation and extensionality could be used to ensure that the in relation can be interpreted as the set-membership relation of Zermelo-Fraenkel set theory. Thus this axiomatization presented here will admit models in which the values of set-valued features do not have these properties. 7 These additional properties of the set-membership relation don't seem to be needed in linguistic analyses, so such stipulations are not made here.</Paragraph> <Paragraph position="7"> The axiom of foundation requires that all sets are well founded; i.e., that the transitive closure of the set-membership relation is irreflexive, or more informally, that no set directly or indirectly contains itself as a member. Versions of set-theory that relax this restriction have been proposed by, e.g., Aczel (1988), and Rounds (1988) argues that non-well founded sets may be appropriate models of set values in feature structures. The paradoxes associated with non-well founded set theories are avoided here because the axiom of comprehension that asserts the existence of paradoxical sets is not included in this axiomatization; i.e., the only way of defining a set is either by explicitly listing its members or by means of union and intersection operations.</Paragraph> <Paragraph position="8"> The axiom of extensionality requires that if sets $1 and $2 contain exactly the same members then $1 = $2; without extensionality it is possible for two different sets to contain exactly the same members. Admittedly the primary reason for omitting an extensionality axiom is that it does not appear to be axiomatizable using Sch6nfinkel-Bernays' formulae, but three other reasons motivate this decision.</Paragraph> <Paragraph position="9"> First, as noted in Shieber (1986) and in Example 1 above, feature structures in general are not extensional (e.g., two distinct attribute-value elements can have exactly the same attributes and values), and it seems reasonable to treat set-values in a nonextensional fashion as well.</Paragraph> <Paragraph position="10"> Second, extensionality could produce undesirable interactions with the attribute-value component of feature structures. Since set-valued features can also have attributes (for example, in LFG (Kaplan and Bresnan 1982)) a conjunction is associated with a set-value that also has attributes), extensionality would prohibit there being 7 In fact there are Sch6nfinkel-Bernays axioms that require the in relation to be acyclic. Define a new relation, say in + , by the axioms</Paragraph> <Paragraph position="12"> Then in any model in + denotes a superset of the transitive closure of the in relation. The following axiom requires that this transitive closure is irreflexive, i.e. that no set is contained in itself.</Paragraph> <Paragraph position="13"> v s ~in + (s, s).</Paragraph> <Paragraph position="14"> Johnson Features and Formulae two set-valued features that contain exactly the same elements but that differ on the value of some attribute, something a linguistic analysis might reasonably require. Third, as far as I am aware, no linguistic analysis requires sets to be extensional. Appealing to the same general considerations that were used to justify the attribute-value axioms, since the assumption that sets are extensional is not required, the stipulation is not made here.</Paragraph> <Paragraph position="15"> It is necessary to define some predicates that describe set-values. We begin by presenting a general first-order axiomatization of these predicates, and then approximate these with formulae from the Sch6nfinkel-Bernays class.</Paragraph> <Paragraph position="16"> Most of the definitions are straightforward, and are given without explanation, s The unary predicate null is true of an element iff that element has no members.</Paragraph> </Section> </Section> <Section position="5" start_page="140" end_page="140" type="metho"> <SectionTitle> 4. Vx null(x) ~ -~y in(y,x) </SectionTitle> <Paragraph position="0"> The ternary relation union(x, y, z) is true only if every element in z is in x or y.</Paragraph> </Section> <Section position="6" start_page="140" end_page="140" type="metho"> <SectionTitle> 5. Vxyz union(x,y,z) ~ Vu in(u,z) ~ in(u,x) V in(u,y) </SectionTitle> <Paragraph position="0"> The ternary relation intersection(x, y, z) is true only if every element in z is in x and in y.</Paragraph> </Section> <Section position="7" start_page="140" end_page="147" type="metho"> <SectionTitle> 6. Vxyz intersection(x,y,z) ~ Vu in(u,z) *--* in(u,x) A in(u,y) </SectionTitle> <Paragraph position="0"> The binary relation singleton(u, x) is true if and only if u is the only member of x.</Paragraph> <Paragraph position="1"> 7. Vux singleton(u, x) ~ in(u, x) A Vv in(v, x) ~ u = v Unfortunately the axioms 4-7 do not belong to the Sch6nfinkel-Bernays class, so we cannot guarantee the decidability of systems of constraints defined using them simply by noting a translation into this class exists. However, in all of the linguistic applications I am aware of these predicates always appear positively, and in this case these axioms can be replaced by the corresponding &quot;one-sided&quot; axioms given below. (The predicate null is an exception, since some HPSG analyses (Pollard and Sag 1987) require the set of unsaturated arguments of some phrases to be non-null. However, it is possible to require that a set s is nonempty by introducing a new constant u and require that in(u, s).) 4'. Vxy null(x) --* -,in(y, x) 5'. Vuxyz union(x,y,z) --. (in(u,z) ~ in(u,x) V in(u,y)) 6'. Vuxyz intersection(x,y,z) --. (in(u,z) ~ in(u,x) A in(u,y)) 7'. Vuxv singleton(u, x) --. (in(u, x) A in(v, x) --* u = v) These one-sided definitions are incorrect when these predicates appear negatively (i.e., in the scope of an odd number of negation symbols after all other proposition connectives have been expressed in terms of A, V, and -,). For example, an interpretation 8 In the following axioms all of the connectives are to be interpreted as right-associative. Computational Linguistics Volume 17, Number 2 with an empty in relation can satisfy =null(x). As Johan van Benthem and the anonymous reviewer independently pointed out to me, it is possible to prove that so long as the relations null, union, intersection, and singleton appear only positively in linguistic constraints, any model satisfying 4'-7' differs from a :model satisfying 4-7 at most in the denotation of these relations; other relations, in particular the attribute relations or even the in relation, are not affected by the one-sided approximation. The following proposition expresses this.</Paragraph> <Paragraph position="2"> Let x be a tuple of variables, A be any relation symbol, ~(A) be any formula in which A appears only positively, and ~(x) be a formula in which A does not appear. Then (i) .M ~ @(A) A Vx A(x) *-+ ~(x) if and only if there is a model M' differing from A4 only on the denotation it assigns to A such that (ii) A4' ~ @(A) A Vx A(x) -+ ~(x).</Paragraph> <Paragraph position="3"> The left to right direction is obvious. The proposition follows from right to left as follows. Let A,/' be any model that satisfies (ii). A model A4 that satisfies (i) can be constructed as follows. Let A4 be the model that agrees with A4' except possibly on A, where IAIM = IAI~, U ~Ax~(x)~,. Now we check that M satisfies (i). Since ~A~ 2 ~A~, and A appears only positively in @(A), A4 ~ ~(A). Further dr4 ~ VxA(x) *-- ~(x) by construction. Since A does not appear in ~(x), ~Ax~(x)~ = ~Ax~(x)~,, and since ~A~M D \[A~,, M ~ VxA(x) --+ ~(x) as well. Thus M satisfies (i) as required. In fact we have shown something stronger; the denotation of A in A4' is a subset of the denotation of A in d~4. *</Paragraph> <Section position="1" start_page="141" end_page="143" type="sub_section"> <SectionTitle> 2.2 Expressing Constraints </SectionTitle> <Paragraph position="0"> A feature structure is specified implicitly by means of the constraints that it must satisfy. This section shows how such constraints can be translated into quantifies-free and function-free prenex formulae. There is a plethora of different notations for expressing these constraints: the constraint requiring that the value of attribute a of (the entity denoted by) x is (the entity denoted by) y is written as Ix a / -- y in PATR-II (Shieber 1986), as (x a) = y in GFG (Kaplan and Bresnan 1982), and as x(a) ~ y in Johnson (1988), for example. Here we express attribute-value constraints using the attribute relations a, so this constraint would be expressed as a(x, y). Set-valued constraints are expressed using the relations in, null, union, and singleton defined in the previous section. The propositional connectives are used to express negative and disjunctive feature constraints. This section shows how constraints on feature bundles can be specified using equality, the attribute relations, and the set predicates axiomatized in the last section. (In fact as far as the theoretical results of this paper are concerned all that is important is that the constraints are taken to mean the same thing as these formulae, irrespective of the notation in which they are expressed.) e ~, e', f~, f&quot; and g&quot; are constants of the first-order language that are not attribute-value constants. 9 8a. pred(e'~ salmon) A agr(e', f') A pers(f', 3rd) 8b. pred(g&quot;, swim)A tense(g&quot;, pres)A subj(g&quot;, e&quot;)A agr(e',f&quot;)A num(f&quot;, sg)A pers(f&quot;, 3rd).</Paragraph> <Paragraph position="1"> Example 3 (continued) The lexical entry for the determiner die of Figure 4 is the following formula, where x is a (non-attribute-value) constant that denotes the feature structure of the determiner, and y and z are constants that are not attribute-value constants.</Paragraph> <Paragraph position="2"> 9. cat(x, determiner) A agr( x, y ) A (case(x, nom ) V case(x, acc ) ) A (number(y, plural) V (number(y, singular)A gender(y, feminine) ) ) Example 4 (continued) The lexical entry for the verb swim of Figure 5 is the following formula, where g is a constant that denotes the feature structure of the verb, and e and f are constants that are not attribute-value constants. 1deg 10. pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e,f)A -~(num(f, sg)A pers(f , 3rd) ) The lexical entries for the determiners this and these of Figure 9 are the following formulae, where u, v, u ~ and v ~ are constants that are not attribute-value constants, and u denotes the feature structure of this and u ~ denotes the feature structure of these. 11. def(u, +) A agr(u, v) A num(v, sg) A pers(v, 3rd) 12. def(u', +) A agr(u', v') A num(v',pl) A pers(v',3rd) 9 Instead of naming all of the nonroot attribute-value elements with constants as is done here, it is possible to merely assert their existence using an existential quantification. For example, the lexical entry for salmon could be the formula 3f'pred(e', salmon) A agr(e', f I) A pers(f , 3rd) where fr is an existentially quantified variable. This formulation has the advantage that no 'renaming' is needed when determining subsumption of systems of attribute-value constraints. (The subsumption relation between systems of constraints is used in certain types of 'unification based' parsers (Shieber 1989).) That is, a system of constraints represented by a formula ~ subsumes another system of constraints represented by 8 iff A ~ 0 ~ ~, where A is the conjunction of the axioms defining the relevant types of feature structures.</Paragraph> <Paragraph position="3"> 10 The formulation (10) of the negative constraint depicted in Figure 5 does not imply that f has either a num or pers attribute. Conceivably, one might want to interpret such a negative constraint as requiring f to have both num and pers attributes with values differing from either sg or 3rd, respectively. The formula below expresses this interpretation.</Paragraph> <Paragraph position="4"> pred(g, swim) A tense(g, pres) A subj(g, e) A agr(G f) A hum(f+ u) A pers(f , v) A -~(u = sg A v = 3rd) Computational Linguistics Volume 17, Number 2 Example 5 (continued) The lexical entries for she and woman of Figure 6 are the formulae (13) and (14), where u denotes the feature structure of the pronoun, v denotes the feature structure of the noun, and w, s, s', s', i, and i' are constants that are not attribute-value constants. 13. cat(u, np) A refs-in(u, s) A refs-out(u, s) A index(u, i) A in(i, s) 14. cat(v, n)A index(v, i')A refs-in(v, s')A refs-out(v, s&quot;)A singleton(i', w)A union(s', w, s&quot;) In general then, a system of feature structure constraints can be viewed as a function-free and quantifier-free formula. These constraints are satisfiable if and only if there is an interpretation that simultaneously satisfies the corresponding formula and the axioms presented in the previous section, or equivalently, the conjunction of this formula and the relevant axioms from the axiomatization. This conjunction is itself a formula from the Sch6nfinkel-Bernays class, and so the satisfiability problem for systems of feature structure constraints is decidable.</Paragraph> <Paragraph position="5"> Further, we can apply results on the computational complexity of the satisfiability problem for the Sch6nfinkel-Bernays class to determine the computational complexity of the satisfiability problem for systems of such feature constraints. Since (universal) quantifiers appear only in the axiomatization of feature structures and not in the feature constraints themselves, the number of quantifiers appearing in the conjunction of the feature constraints and the axiomatization is a constant, and does not vary with the size of the system of feature constraints. By Proposition 3.2 of Lewis (1980), the satisfiability problem for a formula F with u universal quantifiers in the Sch6nfinkel-Bernays class requires nondeterministic time polynomial in IFI u, so the problem is in NP. The reductions presented in Kasper and Rounds (1986) and Johnson (1988) can be used to show that the problem is NP-hard, so the satisfiability problem for feature constraints with set-values (as defined above) is NP-complete.</Paragraph> </Section> <Section position="2" start_page="143" end_page="147" type="sub_section"> <SectionTitle> 2.3 Unification and Satisfaction </SectionTitle> <Paragraph position="0"> This section discusses the relationship between unification and the axiomatization presented above.</Paragraph> <Paragraph position="1"> Unification identifies or merges exactly the elements that the axiomatization implies are equal. The unification of two complex elements e and e' causes the unification of the values of all attributes a that are defined on both e and e'. Similarly, the conjunction of the formulae e = e', a(6 f), a(e', f') and the axioms given above implies that f = f, since axiom schema (3) requires that attributes are single valued.</Paragraph> <Paragraph position="2"> Similarly, the unification of two attribute-value structures fails either when two distinct constant elements are unified (a constant-constant clash) or when a constant and a complex element are unified. The formula x = x ~ is unsatisfiable under exactly the same circumstances in the theory axiomatized above. The formula x = x' conjoined with x = c and x' = c' for distinct attribute-value constants c, c' is unsatisfiable, since c C/ c' by axiom schema (2). Also, x = x' is unsatisfiable when conjoined with a(x,y) for any y and x' = c, since ~a(c,y) by axiom schema (1).</Paragraph> <Paragraph position="3"> If attention is restricted to purely conjunctive attribute-value systems, the corresponding formulae can be represented as a directed graph, where nodes represent (first-order) constants, and an arc labeled a from x to y encodes the atom a(x,y).</Paragraph> <Paragraph position="4"> Then the standard attribute-value 'unification algorithm' can be used as a specialized inference procedure that takes as input such a graph encoding of a conjunction of Johnson Features and Formulae attribute-value relations and returns (the graph encoding of) the conjunction of all of their atomic consequences.</Paragraph> <Paragraph position="5"> As Kasper (1986, 1987) noted in a different setting, the steps of the attribute-value unification algorithm are just applications of the axioms 1-3. It 'forward chains' using axiom schema (3) (for which the graph representation provides efficient indexing), and checks at each step that 1 and 2 are not falsified; if they are falsified the unification algorithm halts and reports a unification failure. Atomic equalities x = y are represented by a 'forwarding pointer' from x to y (as in the UNION-HND algorithm (Gallier 1986; Nelson and Oppen 1980; Johnson in press)).</Paragraph> <Paragraph position="6"> Example 2 (continued) The unification of e ~ and e&quot; in Figures 2 and 3 corresponds to conjoining the formula e ~ = e&quot; to the conjunction of 8a and 8b, resulting in the formula 15a.</Paragraph> <Paragraph position="7"> 15a. e'= e&quot;A pred(e', salmon) A agr(e', f') A pers(f', 3rd)A pred(g&quot;, swim)A tense(g&quot;, pres)A subj(g&quot;, e&quot;)A agr(e&quot;, f&quot;)A num~f&quot;, sg )A pers(~&quot;, 3rd). This formula can be simplified by substituting e r for e&quot; to yield 15b (this substitution corresponds exactly to the first step of the unification algorithm, viz. redirecting e&quot; to e0. The affected subformulae are in boldface below.</Paragraph> <Paragraph position="8"> 15b. e' = e&quot;A pred(e'~ salmon)A agr(e',f')A pers~f', 3rd)A pred(g&quot;~ swim)A tense(g&quot;, pres)A subj(g&quot;~ e')A agr(e'~f&quot;)A num(f&quot;, sg)A pers(f&quot;, 3rd). Since 15b contains the conjunction of agr(e',f') and agr(e',f&quot;), axiom schema (3) requires that f' = f&quot;, so 15b can be further simplified by substituting f' for f&quot; to yield 15c.</Paragraph> <Paragraph position="9"> 15c. e' = e&quot;A f' = f&quot;A pred(e', salmon)A agr(e',f')A pers(f', 3rd)A pred(g&quot;, swim)A tense(g&quot;~ pres)A subj(g&quot;, e')A agr(e'~ f')A num~f'~ sg)A pers(f'~ 3rd).</Paragraph> <Paragraph position="10"> The duplicate occurrences of agr(d,f') and pers(fr,3rd) can be deleted, yielding 15d (these last two steps correspond exactly to the unification of f~ and f&quot; in Figure 3). 15d. e' = e&quot;A f'= f&quot;A pred(e', salmon)A agr(e',f')A pers(f',3rd)A pred(g&quot;, swim )A tense(g&quot;, pres )A subj(g&quot;, e')A num(f', sg).</Paragraph> <Paragraph position="11"> No further simplifications are possible, and 15d is satisfiable. In fact 15d describes the structure depicted in Figure 3, as expected.</Paragraph> <Paragraph position="12"> The standard unification algorithm is unable to handle negative constraints correctly, as noted above. However, because negation is interpreted declaratively (in fact, classically) in the first-order language used to express constraints here, its treatment is straightforward and unproblematic, and suggests ways of extending the unification algorithm to cover these cases (Johnson 1990b, to appear).</Paragraph> <Paragraph position="13"> The unification of e ~ and e (i.e. the lexical entries for salmon and swim) of Figure 9 corresponds to the conjunction of the formula e = e I to the conjunction of 8a and 10, which is the formula 16a.</Paragraph> <Paragraph position="14"> 16a. e = e'A pred(e', salmon)A agr(e',f)A pers(f', 3rd)A pred(g, swim)A tense(g, pres)A subj(g, e)A agr( G f)A -~(num(f , sg)A pers(f , 3rd) ) This can be simplified by straightforward applications of axiom schema (3), equality substitution, and propositional equivalences to obtain 16b.</Paragraph> <Paragraph position="15"> 16b. e = e'A f = f'A pred(G salmon)A pers(f, 3rd)A pred(g, swim)A tense(g, pres)A subj(g, e)A agr(G f)A -~num(f , sg).</Paragraph> <Paragraph position="16"> This formula could be depicted as in Figure 10, where such matrices are now to be understood as graphical depictions of formulae. The further unification of e' with u, the lexical entry for this, corresponds to the conjunction of e ~ = u to the conjunction of the formulae 16b and 11, which is the formula 16c.</Paragraph> <Paragraph position="17"> 16c. e = e'A f = f'A e' = uA pred(G salmon)A pers(f, 3rd)A pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e, f)A -~num(f , sg)A def(u, +)A agr(u, v)A num(v, sg)A pers(v, 3rd).</Paragraph> <Paragraph position="18"> By substituting e for both e I and u in 16c, we obtain 16d.</Paragraph> <Paragraph position="19"> 16d. e = e'A f = f'A e = uA pred(e, salmon)A pers(f, 3rd)A pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e,f)A -~num(f, sg)A def(e~ +)A agr(e~ v)A num(v, sg)A pers(v, 3rd).</Paragraph> <Paragraph position="20"> Again, since 16d contains the conjunction of agr(e,f) and agr(G v), axiom schema (3) requires that f = v, so 16d can be further simplified by substituting f for v, yielding 16e. 16e. e = e'A f = f'A e = uA f = vA pred(e, salmon)A pers(f, 3rd)A pred(g, swim)A tense(g, pres)A subj(g,</Paragraph> <Paragraph position="22"> A graphical depiction of the formula 16f' The formula 16e is unsatisfiable, since it contains conjunction of both num(f, sg) and its negation -~num(f, sg). This is the desired result, since the utterance this salmon swim is ill formed.</Paragraph> <Paragraph position="23"> On the other hand, the unification e' in 16b (c.f. Figure 10) is with u', the lexical entry for these, corresponds to the conjunction of e' = u', 16b and 12, which is the formula 16c'.</Paragraph> <Paragraph position="24"> 16c'. e = e'A f = f'A e' = u'A pred(e, salmon)A pers(f, 3rd)A pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e, d)A -~numff , sg)A def(u', +)A agr(u', v')A num(v', pl)A pers(v', 3rd).</Paragraph> <Paragraph position="25"> By following the same steps as were used to simplify 16c to 16e, 16c' can be simplified to 16e'.</Paragraph> <Paragraph position="26"> 16e'. e = e'A f =f'A e = u'A f = v'A pred(e, salmon)A pers(f, ard)A pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e, f)A -~nurn(f , sg)A def(e, +)A agr(e,f)A num(f , pl)A pers(f , 3rd).</Paragraph> <Paragraph position="27"> One of duplicate conjuncts agr(e,f) can be deleted, and since num(f, pl) implies -~num(f, sg) (by instances sg pl of (2) and Vxyz num(x,y) A num(x,z) --+ y = z of (3)), 16e' can be further simplified to 16f', where -~num(f, sg) has also been deleted. 16f'. e = e'A f = f'A e = u'A f = v'A pred(e, salmon)A pers(f, 3rd)A pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e, f)A def(e, +)A num(f , pl)A pers(f, 3rd).</Paragraph> <Paragraph position="28"> This formula is satisfiable, as desired, since the utterance these salmon swim is well formed. This formula could be depicted as in Figure 11, where again the matrix is to be understood as a graphical depiction of the formula 16f'.</Paragraph> <Paragraph position="29"> The set-valued examples are somewhat more complicated because they involve quantification. null union(d, w, s&quot;)A null(s').</Paragraph> <Paragraph position="30"> Now singleton(i', w) A union(d, w, s&quot;) implies by axioms (5) and (7) that Vu in(u~ s') *-* u = i' V in(u, s'). Further, since null(s') implies by axiom (4) that Vu-~in(u, s'), it follows that 17a is equivalent to 17b.</Paragraph> <Paragraph position="31"> 17b. cat(v, n)A index(v, i')A refs-in(v, s')A refs-out(v, s')A singleton(i', w)A union(s',w,s&quot;)A null(s')A Vu (in(u,s&quot;) ~ u = i').</Paragraph> <Paragraph position="32"> Unifying the value of the refs-out attribute of Figure 7 with the value of the refs-in attribute of u in Figure 6 corresponds to conjoining s = s&quot; with the conjunction of 17b and 13, yielding 17c.</Paragraph> <Paragraph position="33"> 17c. s = s&quot;A cat(u, np)A refs-in(u,s)A refs-out(u,s)A index(u, i)A in(i,s)A cat(v, n)A index(v, i')A refs-in(v, s')A refs-out(v, s')A singleton( i', w)A union(s',w,s&quot;)A null(s')A Vu (in(u,s') ~ u = i').</Paragraph> <Paragraph position="34"> This can be simplified by substituting s for s&quot; and noting that Vu (in(u, s) ~ u = i') and in(i, s) implies that i = i', as required.</Paragraph> <Paragraph position="35"> 17d. s = s'A cat(u, np)A refs-in(u, s)A refs-out(u, s)A index(u, i)A in(i, s)A cat(v, n)A index(v, i)A refs-in(v, s')A refs-out(v, s)A singleton(i, w)A union(d, w, s)A null(s')A Vu (in(u, s) ~ u = i).</Paragraph> </Section> </Section> class="xml-element"></Paper>