A number of linguistic theories and computational ap­proaches to parsing natural language have employed the notion of associating informational elements, consisting of features and their values, with phrases. Such elements, called feature structures, have been used in linguistic theories, such as Generalized Phrase Structure Grammar (GPSG; Gadzar et al. 1985) and Lexical Functional Gram­mar (Kaplan and Bresnan 1983), and in computational formalisms, such as Functional Unification Grammar (Kay 1979) and PATR-II (Shieber 1984).

Rounds and Kasper introduced a logical formalism to describe feature structures with disjunctive specification (Kasper 1987; Kasper and Rounds 1986; Rounds and Kasper 1986). The language is a form of modal preposi­tional logic. To define the semantics of this language, feature structures are formally defined as acyclic finite automata. The detailed definition is given in Section 2. A fundamental property of the semantics is that it is mono-tonic in the sense that the set of automata satisfying a given formula is upward-closed under the operation of subsump-tion. This is important, because we consider a formula to be only a partial description of a feature structure. This property is precisely formulated in Section 2.

Several researchers have expressed a need for extending this logic to include the operators of negation and implica­tion. These two are related in that, in most logical systems, it is possible to use one to define the other (in the presence of a disjunction operator). In this paper, we shall concen­trate on the problem of extending the logic to include negation, while also showing that it yields a satisfactory interpretation of implication.

Karttunen (1984), for instance, provides examples of feature structures in which a negation operator might be useful. For instance, the most natural way to represent the number and person attributes of a verb such as sleep would be to say that it is not third person singular, rather than expressing it as a disjunction of the other possibilities. We express this agreement constraint by the following formula:

Pereira (1987) provides the following example formula that expresses the semantic constraint that the subject and object of a clause cannot be coreferential unless the object is a reflexive pronoun:

In this section, we take a look at the calculus developed by Rounds and Kasper to describe feature structures. The symbols in the language are taken from two primitive domains:

2. Labels (L).

The set of well-formed formulae ( W), is given by:

a where a G A I : <j> where / G L and 4> G W

To define the semantics of this language, feature struc­tures are defined as acyclic finite automata. These are forma lly defined as follows:

Definition 1. An acyclic finite automation is a 7-tuple

4. ô : Q x 2 —«• g is a finite partial function (the transition function), 6. FC6 (the set of final states),

8. the directed graph (Q, E) is acyclic, where pEq iff for some / G 2, ô(p, I) = q,

9. for every q G Q, there exists a directed path from q0to q in (Q, E), and 10. for every q G F, 8(q, I) is not defined for any /.

We can define a partial ordering of information on acyclic finite automata. This partial ordering is given by. the subsumption relation, defined as follows:

Definition 2. Given two acyclic finite automata, A = < Qa, 2U, rA, hA, q0A, FA, XA ) and B = ( QB, 2B, TB, 8B, q0B, Fß. \b)> we sav tnat A subsumes B {A ç B) iff there is a homomorphism from A to B, i.e. there is a mapping h : Qa —" Qb such that:

Unification, which is the primary information-combining operation on feature structures, can now be simply defined as the operation of finding a least upper-bound (if any upper-bound exists) under the above ordering.

We can now give the semantics of a formula over the set of labels L and the set of atoms A. The domain over which this is done is the set of acyclic finite automata A = Q, L, A, 5, q0, F,X). The satisfies relation (h) is defined as follows:

Definition 3. An acyclic finite automation A = (Q, L,A, ô, q0, F, X) satisfies (h= ) a formula in the following cases:

r and K. Vijay-Sbanker

A 1= NIL always

In the above, ô is extended in the standard way to members of 2*, i.e. ô(q, e) = q and ô(q, wl) = ô(ô(q, w),l) and A/1 is the automaton obtained from /I by making 5(q0, /) the initial state and eliminating all unreachable states.

A fundamental property of the semantics given above is that the set of automata satisfying a given formula is upward-closed under the operation of subsumption. The property is stated in the following theorem (Rounds and Kasper 1986):

Theorem 1. A ç B if and only if for every formula, (p, if A 1= (p then B 1= (p.

Rounds and Kasper also showed that the satisfiability problem for their logic is NP-complete.

In this section we examine the problem of adding a nega­tion operator to the language described in the previous section. We do this by presenting various approaches to defining the semantics of the extended language. We look at these approaches in terms of both their linguistic appro­priateness and their computational properties. We will also show that the framework of three-valued logic that we present can be used as a basis for comparison of the different approaches.

By classical negation, we mean an interpretation in which an automaton A satisfies a formula —i <p if and only if it does not satisfy <p. Johnson (1987) defined an Attribute Value Logic (AVL), similar to the Rounds-Kasper Logic, that included a classical form of negation. Smolka (1988) pre­sented a classical semantics for negation in a Rounds-Kasper-like framework. While such approaches are appro­priate under one view of feature structures, they are not satisfactory from the viewpoint of feature structures seen as partial descriptions. This is because the crucial property of monotonicity is lost, as can be seen from the following example:

Example 1.

person : second] person : second

number : singular <p = ~~l(person : second A number : singular)

As can easily be seen, by the classical semantics, A h= and AIZB, but B \f<p.

Kasper (1988a) discusses an interpretation of negation and implication in an implementation of Functional Unifi­cation Grammar that is extended to include conditionals. Kasper's semantics is classical, but his unification proce­dure uses notions similar to those of three-valued logic.Kasper also localized the effects of negation by disallowing path expressions within the scope of a negation. This restriction may not be linguistically warranted as can be seen from Pereira's formula example in Section 1.

if A/px and A/p2 are both defined

and are not unifiable or if A is atomic I(Pi = P2< A) is undefined otherwise.

As mentioned earlier, our approach was motivated by Karttunen's implementation as described in Karttunen 1984. In the unification algorithm given, negative con­straints are attached to feature structures or automata (which themselves do not have any negative values). When the feature structure is extended to have enough informa­tion to determine whether it satisfies or falsifies the for­mula, then the constraints may be dropped. We feel that our definition of the Uset captures the notion of associating constraints with automata that do not have sufficient infor­mation to determine whether they satisfy or contradict a given formula.

As discussed in Section 3.1, Kasper (1988a) used the operations of negation and implication in extending Func­tional Unification Grammar. Though the semantics defined for these operators is a classical one, for the purposes of the algorithm Kasper identified three classes of automata asso­ciated with any formula: those that satisfy it, those that are incompatible with it, and those that are merely compatible with it. We can observe that these are closely related to our Tset, Fset, and Uset respectively. For instance, Kasper states that an automaton A satisfies a formula / : v if it is defined for/ with value v; it is incompatible with/: v if it is defined for/with value x{x ¥= v) and it is merely compati­ble with / : v if it is not defined for /. In three-valued logic, we incorporate these notions into the formal semantics, thus providing a formal basis for the unification procedure given by Kasper. Our logic also gives a more uniform treatment to the negation operator, since we have removed the restriction that disallowed path equivalences in the scope of a negation.

We have seen examples (Section 1) of formulae that assert the existence of certain features. While 3 / is not a formula in the Rounds-Kasper syntax, we can regard it as syntactic sugar for the formula I: NIL, which is indeed satisfied exactly by those automata that have a feature /.

However, the formula —\l:NIL is not satisfiable in the logic we have defined. This is because any automaton that does not have a feature labeled / subsumes one that does. We have, however, seen examples of formulae where 3/ occurs in the scope of a negation (for instance, Kasper [1988b] uses the formula 3Mood-type — Rank : Clause). We certainly intend that such formulae be satisfiable.

Since feature structures are partial information struc­tures, if they are not defined for an attribute /, it could be due to lack of information about the value for the attribute /. On the other hand, here we wish to capture the fact that if a feature structure A satisfies the description -i 3 /, then not only is A not defined for /, but it is also the case that it cannot be defined for /. That is, it is erroneous to extend A to state a value for the attribute /.

The problem stems from the fact that in the formula -13/, we are trying to capture the information that a feature structure not only does not have a value for the feature /, but cannot be extended to have a value for /; i.e. we have the information that, in the current context, the information structure that we are building is not going to acquire a value for the feature / at any future time. This kind of "negative" information is not expressible in auto­mata models as we have defined them. As they stand, they can only capture "positive" information. To include the negative information we need, we will define an augmented notion of feature structures and redefine our interpretation function accordingly.

To use the analogy with finite state automata, note that in a deterministic fsa we often consider states that do not have outgoing arcs defined on certain labels as having those arcs leading to an "error" state. Since we view fsas as complete structures, this distinction between arcs that are not defined and those that cannot be defined is unimpor­tant. However, when we view our automata models as partial information structures, we must distinguish be­tween the case in which a feature is simply not defined (leaving open the possibility that it may be defined in some extension) and the case in which we know that a certain feature cannot be defined.

In what follows, we capture the information of certain labels leading to "error" states without explicitly defining such s tates, but by attaching to each state in the structure a finite set of labels. This set contains those labels that cannot be defined from that state. We already have an elementary form of this notion in our restriction on final states, when we specify that they cannot have any outgoing arcs. We are effectively saying that no label can be defined from these states. We formalize all these notions below.

4. b:Q x 2 Ô is a finite partial function (the transition function),

r and K. Vijay-Shanker

8. S:Q\F — pfi"™ is a function from the nonfinal states to finite subsets of 2,

9. the directed graph (Q, E) is acyclic, where pEq iff for some / G 2, ô(p, I) = q,

10. for every q G Q, there exists a directed path from q0to q in (Q,E), 12. whenever / G S(q), ô(q, I) is not defined.

We can now define the subsumption ordering on these structures as follows:

This definition of subsumption ensures that, for any automaton A, if / G SA(ô(q0, p)) then, for any automaton subsumed by A, the path p is defined, but the path pi cannot be defined.

In this section, we give a proof system for the logic de­scribed above that is essentially an adaptation of the tab­leau proof system described by Moshier and Rounds (1987) for their intuitionistic interpretation of the feature logic.

The proof system works, not with individual formulae, but with sets of labeled signed formulae. The Moshier-Rounds tableau proof method worked with sets of sets of labeled signed formulae. However, this extra level of com­plexity is not needed here.

We first introduce the notion of a labeled signed formula:

Definition 9. A labeled signed formula is a triplet (w, X, 0>, where wGI'JG {True, False] and <p G WFF. (w, True, 0> will be written as wTcp, and (w, False, 0) as wF<p.

We can now define the notion of an f-structure satisfy­ing a labeled signed formulae:

Definition 10. An f-structure, A, satisfies a labeled signed formula $ (written A \= <J>) in the following cases:

A \= wT<p if and only if A/w is defined and I(<p,A/w) = True

A \= wF(p if and only if Ajw is defined and I(<p,A/w) = False Definition 11. A set c of labeled signed formulae is closed if and only if at least one of the following holds:

• wTa,wTl:NIL G c,

• wFNIL G c, or

• wTTOPG c

for some / G L,w,p G L*, x G L+,a,b G A and0 E WFF.

Lemma 1. Any closed set of labeled signed formulae is unsatisfiable.

Proof. Immediate from the definition of a closed set.D Definition 12. A set of labeled signed formulae, c, is downward-saturated if and only if c is not closed, and

r and K. Vijay-Shanker

wTl.NIL G c =>wlF<t> Ecor wFl.NIL G c =*wT(lpx **lp2) Ec =>wF(lpx ~lp2)Ec =>wTpx:NIL G c and Lemma 2. If a finite set of labeled formulae, c, is down­ward-saturated, it is satisfiable.

Proof. Consider the automaton, A = (Q, L,A, ô, q0, F, X,S), constructed from c as follows:

1. For every path w for which there is a formula 0 such that wT4> E c or wF0 G c, include a state qw in Q, with ô defining a path from q0 to qw labeled w.

2. For every pair of paths px = wxx and p2 = wx2 such that wrOc, « x2) E c, let çpl and qp2 be the same state.

3. For every formula wTa £ c, include q,w in F and let X(?J = a.

4. For every formula wFa G c, if there is no label / such that there is a state qwl E Q, then include qw in F and let \(qw) — b for any atomic value b such that b does not occur in any formula inc. 5. For every formula wFl : NIL E c, include / in S(qw)

6. For every formula wF (px « />2) G c, if states «7,^ and qwpi are defined and neither of them is in F, then add new states qx and q2 to g and F, and for some label / that does not occur in any formula of c, define KqwPl, I) = Qi and 5(4^, /) = qx with Xfa, ) = a and M?2) = ^ where a and Z> are distinct atomic values. If exactly one of the two states (say, qWfi ) is not in F, add just one new state q to Q and let &(qKPl, I) = q for a label / that does not occur in c.

If, however, one of the paths (say, px ) does not have the associated state (qwpi ) defined, let p be the longest prefix of px such that qwp does exist, and let px = plx. Include / in S(qp).

The above construction of an automaton is well de­fined.

We need to verify that the above definition yields an automaton that meets our definition of an acyclic finite automaton without any conflicts. The possible conflicts that could arise would be that: X does not define a function; the graph of the automaton had a cycle; for some q E F and some label /, ô(q, I) is defined; or, for some state q and some label /, / E S(q) and &(q, I) is defined. However, in each of these cases, it is easy to see that were it to arise in the construction given above, the original set c would in fact be closed, contradicting the hypothesis that it is downward-saturated. Claim 2:

The automaton so constructed satisfies all the labeled signed formulae in c.

We establish this claim by induction over the structure of the formulae in c. For the base cases (namely labeled signed formulae of the forms: wXa, wXNIL, wFl : NIL, and wX(px « p2)) it follows immediately from the con­struction that they are satisfied by A. For the other cases (wX{cj> V }j/), wX(4)A\p), and wX-i<t>), their sub-formulae are also in c since it is downward-saturated. But by the induction hypothesis, these sub-formulae are satisfied by A. That completes the result.□

The entailment relation (h) on sets of labeled signed formulae is defined as follows:

Definition 13. Let c and d be two sets of labeled signed formulae. Then c \- d if and only if c 4= d and one of the following holds:

We denote by K* the reflexive and transitive closure of this entailment relation.

Theorem 4. (Soundness) If c (-* </ for sets of labeled signed formulae c and <7, and d is downward-saturated, then c is satisfiable.

Proof. This follows immediately from Lemma 2 and the fact that cV* d implies c Ed.U Lemma 3. For any set of labeled signed formulae c, there are only finitely many sets of labeled signed formulae d such that c h* d.

r and K. Vijay-Shanker

ln(<t> V^) = ln{4> A\p) = ln{<t>) + Inty) + 1 where length denotes string length.

Also, define the length of a labeled signed formula wX<j> as ln(4>) + lengthiyv). Let $> be any labeled signed formula such that $ c but «I» G d for some d such that c V* d. Observe that the length of $ is bounded by the length of the longest formula in c and that $ does not contain any symbols that do not occur in c. The result follows. □ Lemma 4. For any set of labeled signed formulae c, if there is no set of labeled signed formulae d such that c h d, then c is either closed or downward-saturated.

Proof. Clearly, if c is closed under all the entailment rules listed above, then it satisfies all the implications listed in the definition of downward saturation. Hence, if it is not downward-saturated, it must be closed. □ Lemma 5. For any satisfiable set of labeled signed formu­lae c that is not downward-saturated, there is a satisfi­able set of labeled signed formulae d such that c h d.

Proof. Since c is satisfiable, it is not closed. Since it is not downward-saturated, by hypothesis, there must be a d such that c Y d. However, it is clear from the definition of entailment that if all such d are unsatisfiable, then so is Theorem 5. (Completeness) For any satisfiable set of labeled signed formulae, c, there is a downward-satu­rated set of labeled signed formulae d such that c\-* d.

Proof. By Lemma 3, there must be a d such that cY* d and for no d'd V d'. All such d are either closed or downward-saturated by Lemma 4. However, not all of them can be closed since then by Lemma 5, c would be unsatisfiable. Hence, at least one of them is downward-saturated. □ Theorem 6. (NP-Completeness) The satisfiability prob­lem for the logic we have defined is NP-Complete.

Proof. It follows from the proof of Lemma 3 that the length of any derivation cV* dis bounded by n, where n is the sum of the lengths of the formulae in c. Since this bound is polynomial, the problem is in NP. It is 7V.P-hard because the satisfiability problem for the Rounds-Kasper logic, which is a special case, is NP-hard. □

A logical formalism with a complete set of logical operators has come to be accepted as a means of describing feature structures. While the intended semantics of most of these operators is well understood, the negation and implication operators have raised some problems, leading to a variety of approaches in their interpretation.

In Dawar and Vijay-Shanker 1989 and the present work, we introduced the framework of three-valued logic as a means of defining the semantics of a feature structure description language with negation. This framework per­mits us to say that a formula such as —i/ : 0 does not have a truth value defined in a feature structure that does not have a feature /. This enables us to define an interpretation that, unlike the classical approach to negation, is monotonie, as a logic describing partial structures should be.

We presented one particular interpretation of FDL within this three-valued framework and compared it with other approaches to defining the semantics of negation. We showed that several different such approaches could be cast in the three-valued framework. In particular, we showed that the special case of the Moshier-Rounds intuitionistic approach, in which forcing is always considered with re­spect to K* could be captured in our framework.

One motivation cited by Moshier and Rounds for consid­ering forcing sets other than K* was so that formulae of the form —il:NIL could be considered satisfiable. The same reason led us to examine an augmented notion of feature structure models for FDL that yields an interpretation that is conceptually simple, motivated by the preservation of monotonicity, and is computationally no harder than the original Rounds-Kasper logic. We also showed that our interpretation meets the conditions set out by Pereira (1987) for a satisfactory interpretation of negation.