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<?xml version="1.0" standalone="yes"?> <Paper uid="J90-1002"> <Title>AN INTERPRETATION OF NEGATION IN FEATURE STRUCTURE DESCRIPTIONS</Title> <Section position="3" start_page="0" end_page="0" type="metho"> <SectionTitle> 2 ROUNDS-KASPER LOGIC </SectionTitle> <Paragraph position="0"> 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:</Paragraph> <Paragraph position="2"> To define the semantics of this language, feature structures are defined as acyclic finite automata. These are formally defined as follows: Delinition 1. An acyelie finite automation is a 7-tuple A = (Q,~,F,&qo, F,?O, where: 1. Q is a nonempty finite set (of states), 2. z is a countable set (the alphabet), 3. F is a countable set (the output alphabet), 4. 6 : Q x ~ ---, Q is a finite partial function (the transition function), 5. qo ~ Q (the initial state), 6. F C Q (the set of final states), 7. X : F --~ r is a total function (the output function), 1 8. the directed graph (Q, E) is acyclic, where pEq iff for some l ~ ~, 6(p, l) = q, 9. for every q G Q, there exists a directed path from q0 to q in (Q, E), and 10. for every q C F, 6(q, l) is not defined for any 1. 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, ~&quot;A, I'A, 6A, qOA, FA, ~kA ) and B = ( Qs, ~,s, rs, 6s, qos, Fs, ),s), we say that A subsumes B (A E B) iff there is a homomorphism from A to B, i.e. there is a mapping h : QA ~ Qn such that: 1. h(6A( q, l) = tro(h(q), l), 2. tB(h(q)) = 1 A (q) for all q E FA, and 3. h(qoA) = qoB 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. 2 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, 6, q0, F, X). The satisfies relation (~) is defined as follows: Definition 3. An acyclic finite automation A = (Q, L, A, 6, q0, F, X) satisfies (~) a formula in the following cases: In the above, 6 is extended in the standard way to members of Z*, i.e. (5(q, E) = q and 5(q, wl) = ~(6(q, w),l) and A/I is the automaton obtained from A by making (5(q 0, l) 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 E_ B if and only if for every formula, 4), if A~ 4) thenB~ 4).</Paragraph> <Paragraph position="3"> Rounds and Kasper also showed that the satisfiability problem for their logic is NP-complete.</Paragraph> </Section> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> 3 PREVIOUS APPROACHES TO NEGATION </SectionTitle> <Paragraph position="0"> 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.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 3.1 CLASSICAL NEGATION </SectionTitle> <Paragraph position="0"> By classical negation, we mean an interpretation in which an automaton A satisfies a formula --7 4) if and only if it does not satisfy 4). 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:</Paragraph> <Paragraph position="2"> As can easily be seen, by the classical semantics, A ~ 4) and A E B, but B ~ 4).</Paragraph> <Paragraph position="3"> 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. 3 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.</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 3.2 INTUITIONISTIC LOGIC </SectionTitle> <Paragraph position="0"> Moshier and Rounds (1987) described an extension of the Rounds-Kasper logic, including an implication operator and hence, by extension, negation. The semantics is based on intuitionistic techniques. The notion of satisfying is replaced by one of forcing. Given a set of automata K, a formula 4), and A such that A ~ K, A forces in K ~4) (A F K ---~) if and only if for all B E K such that A E_ B, B does not force 4) in K. Thus, to show that a formula, 4), is satisfiable, we have to find a set K and an automaton A such that A forces in K 4).</Paragraph> <Paragraph position="1"> Moshier and Rounds also gave a complete proof system for their logic, and showed that the satisfiability problem, while decidable, was PSPACE-complete, thus making it even more intractable than the original Rounds-Kasper logic. Furthermore, Langholm (1989) has shown that not all formulae in the Moshier-Rounds logic can have hereditarily finite sets of minimal models. These computational problems, along with questions about the linguistic appropriateness of its semantics, render the linguistic value of the intuitionistic approach questionable.</Paragraph> </Section> <Section position="3" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 3.3 THREE-VALUED LOGIC </SectionTitle> <Paragraph position="0"> Here we take a look at how three-valued logic can be used to define the semantics of FDL. We also take a look at one particular interpretation of FDL that uses the automata of Section 2 as models. This interpretation is essentially the same one we presented earlier (Dawar and Vijay-Shanker 1989). This is an interpretation of negation that is intuitively appealing, formally simple, and computationally rto harder than the original Rounds-Kasper logic. The primary intention here (as in our earlier paper) is, however, to explore the use of three-valued logic in defining the semantics of FDL with negation. To this end, we will examine other interpretations also within the three-valued framework. Then, in the next section, we motivate a modified notion of automata models and redefine our interpretation with respect to it.</Paragraph> </Section> </Section> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> 3.3.1 THE THREE-VALUED FRAMEWORK </SectionTitle> <Paragraph position="0"> With each formula we associate the set (Tset) of automata that satisfy the formula, a set (Fset) of automata that contradict it, and a set (Uset) of automata that neither satisfy nor contradict it. 4 The Uset contains all automata that are not in either of the other two sets. Different interpretations of negation are obtained by varying definitions of what constitutes &quot;contradiction.&quot; The reason for Computational Linguistics Volume 16, Number 1, March 1990 113 Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions having some automata that neither satisfy nor contradict a formula is as follows: an automaton is to be viewed as a partial information structure. Given a description (formula), 4~, a feature structure A may not carry enough information to suggest that it satisfies or falsifies 4~. However, it may be possible to extend A to either satisfy or falsify ~b. For example, we will place \[person : third\] in the Uset of ~b = (number : singular/k person : third). Of course, this feature structure can be extended to falsify or satisfy 4~ as in: and pnumber : singular\] erson : third J number : plural\] erson : third J We will define the Tset and the Fset so that they are upward-closed with respect to subsumption for all formulae. Thus, we avoid the problem of nonmonotonicity associated with the classical interpretation of negation. In our logic, negation is defined so that an automaton A satisfies --~ if and only if it contradicts q~.</Paragraph> <Paragraph position="1"> Formally, the semantics is defined by a partial interpretation function, I. If WFF is the set of well-formed formulae of FDL, and A the set of acyclic finite automata, 5 the interpretation I is a partial function:</Paragraph> <Paragraph position="3"> dicts ~b 6 and is undefined otherwise. Thus, the following hold:</Paragraph> <Paragraph position="5"> We now look at one such interpretation function that uses the strong Kleene truth definition for conjunction and disjunction.</Paragraph> <Paragraph position="6"> Definition 4. The partial interpretation function I is defined as follows: 1. I(NIL, A) = True for all A; 2. I(TOP, A) = False for all A; 3. I(a, A) = True ifA is atomic and ~'(q0) = a I(a, A) = False if A is atomic and X(qo) = b for some b, b #: a (see Note 2.) l(a, A) is undefined otherwise; 4. I(l : ~p, A) = I(dp, All) if All is defined. (see Note 3.) I(l : ep, A) is undefined otherwise; 5. I(dp I A qb2, A) = True if I(q~ I, A) = True and I(qb 2, A) = True I(~b t A q~2, A) = False if I(~b l, A) = False or I(~b2, A) = False l(~b I A 4~2, A) is undefined otherwise; 6. /(~b I V ~b2, A) = True if I(~b x, A)= True orI(ep 2, A) = True /(~1 V ~2' A) = False if I(~bl, A) = False and I(~b2, A) = False /(q~l V ~b2, A) is undefined otherwise; 7. I(--CA A) = Trueif I(ep, A) = False I(--@, A) = False ifI(~b, A) = True I(-~, A) is undefined otherwise; 8. I(p I ~- p2, A) = True if 6(%, Pl) and 6(qo, P2) are defined and 6(%, Pl) = 6(%, P2) I(Pl ~ P2, ,4) = False if Alp I and Alp 2 are both defined and are not unifiable I(Pl ~ P2, A) is undefined otherwise (see Note 4.). where, q~, 4h, ~2 E WFF A = <Q,L,A, 6, qo, F,X>CA a,b@A l~L .Pl, P2 ~ L* NOTES formal language, since we find that defining implication in terms of negation and disjunction (i.e. 4~ =~ ~k-----~ v if) yields a semantics for implication that corresponds exactly to our intuitive understanding of implication. null 2. As one would expect, an atomic formula is satisfied by the corresponding atomic feature structure. On the other hand, only atomic feature structures are defined as contradicting an atomic formula. An interpretation of negation that defines a complex feature structure as contradiciting a (and hence satisfying ---a) is also possible. Our definition was motivated by the linguistic intention of the negation operator as given by Karttunen (1984), where, for instance, we require that an automaton satisfying the formula case : --ndative have an atomic value for the case feature. However, we now feel that this problem would best be dealt with in a multi-sorted logic and hence, in the interpretation we present in the next section, we have adopted the other alternative mentioned here.</Paragraph> <Paragraph position="7"> 3. In definition 4 above, we state that: I(1 : ep, A) = I(C/b, A~ l) ifA/l is defined. When A/l is defined, I(~, A/l) may still be True, False, or undefined. In any of these cases, I(1: cp, A) = l(dp, All). 7 I(l : dp, A) is not defined if A/l is not defined (as illustrated by the example given earlier where ~b ---- (person : third A number : singular). Not only is this condition required to preserve upward closure, it is also linguistically motivated. 14 Computational Linguistics Volume 16, Number 1, March 1990 Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions In the next section, we will make the distinction in feature structures between not being defined versus cannot be defined. In this section, we will say that I(l : 4~, A) is not defined if A is not defined for l and I(l : 4~, A) = false if A cannot be defined for I. 4. We have chosen to state that the set of automata that are incompatible with the formula Pl ~- P2 is not the set of automata for which 6(q0, P0 and 6(q0, P2) are defined and ~(qo, Pl) :~ 6(q0, P2), since such an automaton could subsume one in which 6(q0, Pl) = 6(qo, P2). Thus, we would lose the property of upward closure under subsumption. However, an automaton, A, in which/~(qo, Pl) and 6(qo, P2) are defined, and AlP l is not unifiable 8 with Alp 2 cannot subsume one in which tS(qo, Pl) = 6(q0, P2). The monotonicity property for the above interpretation can be stated as follows: Theorem 2. Tset(qb) is upward-closed under the subsumption relation for all formulae ~b.</Paragraph> <Paragraph position="8"> Proof. The proof is by induction on the structure of 4~ and can be found in Dawar 1988.\[7 We now take a look at some examples mentioned earlier and see how they are interpreted in the logic just defined. The first example expressed the agreement attribute of the verb sleep by the following formula: agreement : 7(person : third/k number : singular) (4) This formula is satisfied by any structure that has an agreement feature which, in turn, either has a person feature with a value other than third, or a number feature with a value other than singular. Thus, for instance, the first two structures satisfy the given formula, whereas the third structure is undefined with respect to the formula. cat : NP : second\]\] agreement : \[person \[agreement \[person: third \]\] : \[number : plural J\]</Paragraph> <Paragraph position="10"> On the other hand, for a structure to contradict formula (4), it must have an agreement feature defined for both person and number with values third and singular respectively. null Turning to another example mentioned earlier, the formula: null obj : type : reflexive V --l( subj : ref ~. obj : ref ) (5) is satisfied by the first two of the following structures, but is contradicted by the third (here co-index boxes are used to indicate co-reference of path-equivalence).</Paragraph> <Paragraph position="11"> We briefly examine here how the three-valued framework may be used to provide interpretations other than the one presented above.</Paragraph> <Paragraph position="12"> The classical interpretation of negation can, of course, be expressed by making I a total function such that wherever I(qL A) was previously undefined, it is now defined to be False.</Paragraph> <Paragraph position="13"> Moshier and Rounds consider a version in which forcing is always done with respect to the set of all automata, i.e. K*. This means that the set of feature structures that satisfy --~ is the largest upward-closed set of feature structures that do not satisfy ~b (i.e. the set of feature structures incompatible with q~). We can capture this in the three-valued framework described above by modifying the defini- null if Alp 1 and AlP 2 are both defined and are not unifiable or if A is atomic l(Pi ~&quot; Pc, A) is undefined otherwise.</Paragraph> <Paragraph position="14"> 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.</Paragraph> <Paragraph position="15"> 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 f : v if it is defined forfwith value v; it is incompatible with f: v if it is defined forfwith value x(x ~ v) and it is merely compatible with f: v if it is not defined forf. 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.</Paragraph> </Section> <Section position="6" start_page="0" end_page="0" type="metho"> <SectionTitle> 4 INTERPRETING FDL WITH AUGMENTED FEATURE STRUCTURES </SectionTitle> <Paragraph position="0"> We have seen examples (Section 1) of formulae that assert the existence of certain features. While 31 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 l.</Paragraph> <Paragraph position="1"> However, the formula -d:NIL is not satisfiable in the logic we have defined. This is because any automaton that does not have a feature labeled l subsumes one that does.</Paragraph> <Paragraph position="2"> We have, however, seen examples of formulae where 3 l occurs in the scope of a negation (for instance, Kasper \[1988b\] uses the formula 3Mood-type--, Rank : Clause).</Paragraph> <Paragraph position="3"> We certainly intend that such formulae be satisfiable.</Paragraph> <Paragraph position="4"> Since feature structures are partial information structures, if they are not defined for an attribute l, it could be due to lack of information about the value for the attribute l. On the other hand, here we wish to capture the fact that if a feature structure A satisfies the description --73 l, then not only is A not defined for l, but it is also the case that it cannot be defined for l. That is, it is erroneous to extend A to state a value for the attribute I.</Paragraph> <Paragraph position="5"> The problem stems from the fact that in the formula -7--1l, we are trying to capture the information that a feature structure not only does not have a value for the feature l, but cannot be extended to have a value for l; i.e.</Paragraph> <Paragraph position="6"> 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 l at any future time. This kind of &quot;negative&quot; information is not expressible in automata models as we have defined them. As they stand, they can only capture &quot;positive&quot; information. To include the negative information we need, we will define an augmented notion of feature structures and redefine our interpretation function accordingly.</Paragraph> <Paragraph position="7"> 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 &quot;error&quot; 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.</Paragraph> <Paragraph position="8"> In what follows, we capture the information of certain labels leading to &quot;error&quot; states without explicitly defining such states, 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.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 4.1 AUGMENTED FEATURE STRUCTURES </SectionTitle> <Paragraph position="0"> In this section, we give definitions relating to our augmented notion of f-structures. As we stated above, the augmentation consists of attaching to each nonfinal node in the f-structure graph a finite set of labels. These labels are exactly those for which we know that no outgoing arcs can be defined from that node. The set is finite since we require that our information structure at any point be finite. We formally define our extended notion of f-structure as follows: 9 Definition 5. An acyclic finite automaton is an 8-tuple ,4 == (Q,~,F, 6, qo, F,X,S), where: 1. Q is a nonempty finite set (of states), 2. 2; is a countable set (the alphabet), 3. F is a countable set (the output alphabet), 4. 6:Q x Z ~ Q is a finite partial function (the transition function), 5. qo E Q (the initial state), 16 Computational Linguistics Volume 16, Number 1, March 1990 Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions 6. F C Q (the set of final states), 7. X : F ~ r is a total function (the output function), 8. S:Q\F--* P'~&quot;~Y~) is a function from the nonfinal states to finite subsets of Z, 9. the directed graph (Q, E) is acyclic, where pEq iff for some I ~ Z, 6(p, l) = q, 10. for every q ~ Q, there exists a directed path from q0 toqin (Q,E), 11. for every q ~ F, 6(q, l) is not defined for any l, and 12. whenever I ~ S(q), 6(q, l) is not defined.</Paragraph> <Paragraph position="1"> We can now define the subsumption ordering on these structures as follows: Definition 6. Given two f-structures, A = ( Q.1, ~.1, rA, 61 , q0.1, FA, ~.1, S.1) and B = (Qn, ~n, F~, 6n, qon, Fn, ~n, Sn), we say that A subsumes B (A~B) iff there is a homomorphism from A to B, i.e. there is a mapping</Paragraph> <Paragraph position="3"> This definition of subsumption ensures that, for any automaton A, if I ~ S.1(6(q o, p)) then, for any automaton subsumed by A, the pathp is defined, but the path pl cannot be defined.</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 4.2 THE LANGUAGE </SectionTitle> <Paragraph position="0"> We now give the interpretation of FDL in terms of f-structure models as we have just defined them. The syntax of the language is the same as before.</Paragraph> <Paragraph position="1"> We first give the following auxiliary definitions: Definition 7. An f-structure A = (Q, 2;, I', 6, qo, F, ~, S) is: * atomic if and only if Q = F = {q0 }, * null if and only if Q = {q0} and F = ~ and * complex otherwise.</Paragraph> <Paragraph position="2"> We can now define the revised semantics: Definition 8. The (revised) partial interpretation function I is defined as follows: 1. I(NIL, A) = True for all A; 2. I( TOP, A) = False for all A; 3. I(a, A) = True ifA is atomic and k(q0) = a I(a, A ) = False if A is atomic with k (q0) :~ a or if A is complex l(a, A ) is undefined otherwise; 4. I(l:dp, A) = I(4~, A/l) if A/l is defined. I(l: 4~, A ) = False if l C S( qo ) or if A is atomic I( l : 4~, A ) is undefined otherwise; 5. l(ckl /~ 42, A) = True if 1(q~1, A ) = True and l(q~2, A ) = True</Paragraph> <Paragraph position="4"> We are now in a position to prove the following monotonicity property for our logic. We express it in terms of the knowledge (or information) ordering --<k on the truth values {_1_, True, False } defined by _1_ <k True, _1_ k False, True ~t k False and False ~t k True. In the following, I(~b, A) = _t_ is</Paragraph> <Paragraph position="6"> Proof. ~ Suppose for every formula, 4~, I(q~, A) --<k I(~b, B). Every pathp defined in A must also be defined in B, since I(p :NIL, A ) = True and hence I(p :NIL, B) = True. Since for every state qi in A, there is a path pf such that qi = 6A(qOA, Pi) we can define a map h such that h(qi) = 6B(qon, Pi ). To see that this map is indeed functional, note that, if there is a q E QA such that q = 6A(q0A, Pl) = 6A(q0.1, P2) for distinct Pl and P2, then I(Pl ~&quot; P2, A) = True. Thus l(p 1 ~ P2, B) = True and 6B(qos, Pl) and 6n(qo s, P2) do indeed describe the same state.</Paragraph> <Paragraph position="7"> One can immediately see that this map satisfies properties 1 and 2 of being a homomorphism given above in the definition of subsumption. To verify the other two conditions, note that if hA(qi) = a for some qi E A, then, I(pt:a, A) = True. Hence I(pi:a , B) = True and XB(6n(q0n, Pi)) = a. Thus condition 3 is satisfied. The argument for condition 4 is similar. We have, therefore, established that h is a homomorphism and hence that A v-- B. =~ The consequent is trivially true with I(4~, A) = _1_, so we will only consider the case when it is either True Computational Linguistics Volume 16, Number 1, March 1990 17 Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions or False. The proof is by induction on the structure of the formula.</Paragraph> <Paragraph position="8"> that q = 6 A (qoa, Pl ) = 6a(qoA, P2)&quot; Let h be a homomorphism witnessing A E_ B. Then, by the definition of a homomorphism, h(q) = 6n(qon, Pl ) = 6B(qon, P2) and therefore, I(p I ~- P2, B) = True.</Paragraph> <Paragraph position="9"> In the case in which l(p I ~ P2 , A ) = False, we have two possibilities. Either Alp I and Alp 2 are both defined and not unifiable, in which case, clearly by the definition of subsumption, the same will be true of B, or pl = wlx (choosing Px without loss of generality), for some label l and some paths (possibly empty) w and x such that I E Sa(6a(qo A, w)). But then, as we pointed out earlier, this would mean that the path wl and hence the path Pl cannot be defined in B either. Thus, in either case, l(pl ~- null But then, by induction hypothesis, I(ep, All) <-k I \[(4~, B/l) and therefore I(1 : c~, A) <-k I\] (l:q~, B).</Paragraph> <Paragraph position="10"> If All is not defined and I(l:C/k, A) = False, one of two possible cases applies: either A is atomic, in which case A = B or l E Sa(qoA), in which case I E Sn(qo B) by the definition of subsumption, and we are done.</Paragraph> <Paragraph position="11"> If I(4~/~ ~b, A ) = True, then I(4~, A ) = True and</Paragraph> <Paragraph position="13"> The argument is similar to the one in the previous case. --q, Since I(--~, A ) = True if and only if I(cb, A ) = False and vice versa, clearly I(--~, A ) __<k I(--~, B), since I(~b, A) _<k I(4~, B).~ The following simple corollary corresponds to the monotonicity result we established for our original three-valued semantics.</Paragraph> <Paragraph position="14"> Corollary. For all 4~, Tset(4~) is an upward-closed set. As we mentioned earlier in this section, Langholm (1989) describes negatively extended feature structures in a fashion very similar to what is described above. The interpretation he chooses for the description language is, however, intuRionistic in character. We believe that the modifications that we suggested to our interpretation (in Section 3.3.3) to capture the special case of intuitionistic logic in which forcing is always done with respect to K*, when applied to our new interpretation yield exactly the interpretation described by Langholm.</Paragraph> </Section> </Section> <Section position="7" start_page="0" end_page="0" type="metho"> <SectionTitle> 5 PROOF SYSTEM </SectionTitle> <Paragraph position="0"> 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.</Paragraph> <Paragraph position="1"> 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.</Paragraph> <Paragraph position="2"> We first introduce the notion of a labeled signed formula: Definition 9. A labeled signed formula is a triplet (w,X, q~), where w ~ L *, X ~ {True, False} and q~ WFF. (w, True, dp) will be written as wT4~, and ( w, False, 4a ) as wF4~.</Paragraph> <Paragraph position="3"> 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 ~ ~) in the following cases: A t = wTep if and only ifA/w is defined and I(4~,A/w) = Proof. Immediate from the definition of a closed set. \[\] Definition 12. A set of labeled signed formulae, c, is downward-saturated if and only if c is not closed, and 18 Computational Linguistics Volume 16, Number 1, March 1990 Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions</Paragraph> <Paragraph position="5"> wF(c~ V ~b) ~ c =~wF~ ~ c and wF~k ~ c Lemlna 2. If a finite set of labeled formulae, c, is downward-saturated, it is satisfiable.</Paragraph> <Paragraph position="6"> Proof. Consider the automaton, A ---- (Q, L, A,/~, q0, F, ~, S ), constructed from c as follows: 1. For every path w for which there is a formula ~b such that wT4~ ~ c or wF4~ ~ c, include a state qw in Q, with/~ defining a path from qo to qw labeled w. 2. For every pair of paths p~ = wx~ and PE ---- wx2 such that wT(x~ ~- x2) ~ c, let q,~ and q,2 be the same state.</Paragraph> <Paragraph position="7"> 3. For every formula wTa ~ c, include q~ in F and let h(qw) = a.</Paragraph> <Paragraph position="8"> 4. For every formula wFa ~ c, if there is no label I such that there is a state q~t ~ Q, then include qw in F and let h(q~) ---- b for any atomic value b such that b does not occur in any formula in c. ~1 5. For every formula wFl: NIL ~ c, include l in S(qw) 6. For every formula wF(p~ ~- P2) ~ c, if states q~p, and qwp~ are defined and neither of them is in F, then add new states q~ and q2 to Q and F, and for some label 1 that does not occur in any formula of c, define</Paragraph> <Paragraph position="10"> exactly one of the two states (say, q~p, ) is not in F, add just one new state q to Q and let 6(q~,, 1) = q for a label I that does not occur in c.</Paragraph> <Paragraph position="11"> If, however, one of the paths (say, p~) does not have the associated state (qwp,) defined, let p be the longest prefix ofp~ such that qw~ does exist, and let p~ = plx.</Paragraph> <Paragraph position="12"> Include I in S(q~).</Paragraph> <Paragraph position="13"> Claim 1 : The above construction of an automaton is well defined. null 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 ~ F and some label 1, 6(q, 1) is defined; or, for some state q and some label l, 1 ~ S(q) and ~(q, 1) 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.</Paragraph> <Paragraph position="14"> Claim 2: The automaton so constructed satisfies all the labeled signed formulae in c.</Paragraph> <Paragraph position="15"> 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, wFI:NIL, and wX(p~ ~ P2)) it follows immediately from the construction that they are satisfied by A. For the other cases (wX(q~ V ~b), wX(4J A ~b), and wX~ ~b), 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 (k) 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 I- d if and only if c 4= d and one of the following holds: 1. wT~4~ E candd = c U {wFdp) 2. wE-14~ ~ c and d ---- c U {wT~b} 3. wTl:dp E candd ---- c U {wlTdp, wTI:NIL} 4. wFl:c~ E candd = c U {wlFdp} 5. wFl:cb E candd = c U {wFI:NIL} 6. wlT(pl ~&quot; P2) E candd = c U {wT(lpl ~ lp2)} 7. wlF(p I = P2) E candd = c U {wF(lpl ~- lp2)} We denote by k* the reflexive and transitive closure of this entailment relation.</Paragraph> <Paragraph position="16"> Theorem 4. (Soundness) If c t-* d for sets of labeled signed formulae c and d, and d is downward-saturated, then c is satisfiable.</Paragraph> <Paragraph position="17"> Proof. This follows immediately from Lemma 2 and the fact that c I-* d implies c Cd.\[\] Lemma 3. For any set of labeled signed formulae c, there are only finitely many sets of labeled signed formulae d such that c I-* d.</Paragraph> <Paragraph position="18"> Computational Linguistics Volume 16, Number 1, March 1990 19 Anuj Dawar and K. Vijay-Shanker An Interpretation of Negation in Feature Structure Descriptions Proof. To prove this, we inductively define the notion of length of a formula, as follows:</Paragraph> <Paragraph position="20"> where length denotes string length.</Paragraph> <Paragraph position="21"> Also, define the length of a labeled signed formula wXC/ as ln(C/) + length(w). Let ~ be any labeled signed formula such that 4> ~ c but 4, C d for some d such that c I-* 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.E\] Lemma 4. For any set of labeled signed formulae c, if there is no set of labeled signed formulae d such that c Id, then c is either closed or downward-saturated.</Paragraph> <Paragraph position="22"> 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.E\] 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 l- 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 I- d. However, it is clear from the definition of entailment that if all such d are unsatisfiable, then so is c.I--\] 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 F-* d. Proof. By Lemma 3, there must be a d such that c I-* d and for no d' d i- 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 downwardsaturated. E\] Theorem 6. (NP-Completeness) The satisfiability problem for the logic we have defined is NP-Complete.</Paragraph> <Paragraph position="23"> Proof. It follows from the proof of Lemma 3 that the length of any derivation c I-* d is bounded by n 2, 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 NP-hard because the satisfiability problem for the Rounds-Kasper logic, which is a special case, is NP-hard. E\]</Paragraph> </Section> class="xml-element"></Paper>