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<?xml version="1.0" standalone="yes"?> <Paper uid="P89-1003"> <Title>A THREE-VALUED INTERPRETATION OF NEGATION IN FEATURE STRUCTURE DESCRIPTIONS</Title> <Section position="1" start_page="0" end_page="0" type="metho"> <SectionTitle> A THREE-VALUED INTERPRETATION OF NEGATION IN FEATURE STRUCTURE DESCRIPTIONS </SectionTitle> <Paragraph position="0"/> </Section> <Section position="2" start_page="0" end_page="0" type="metho"> <SectionTitle> ABSTRACT </SectionTitle> <Paragraph position="0"> Feature structures are informational elements that have been used in several linguistic theories and in computational systems for natural-language processing. A logicaJ calculus has been developed and used as a description language for feature structures. In the present work, a framework in three-valued logic is suggested for defining the semantics of a feature structure description language, allowing for a more complete set of logical operators. In particular, the semantics of the negation and implication operators are examined. Various proposed interpretations of negation and implication are compared within the suggested framework. One particular interpretation of the description language with a negation operator is described and its computational aspects studied.</Paragraph> </Section> <Section position="3" start_page="0" end_page="22" type="metho"> <SectionTitle> 1 Introduction and Background </SectionTitle> <Paragraph position="0"> A number of linguistic theories and computational approaches to parsing natural language have employed the notion of associating informational dements, called feature structures, consisting of features and their values, with phrases. Rounds and Kasper \[KR86, RK86\] developed a logical calculus that serves as a description language for these structures.</Paragraph> <Paragraph position="1"> Several researchers have expressed a need for extending this logic to include the operators of negation and implication. Various interpretations have been suggested that define a semantics for these operators (see Section 1.2), but none has gained universal acceptance. In \[Per87\], Pereira set forth certain properties that any such interpretation should satisfy.</Paragraph> <Paragraph position="2"> In this paper we present an extended logical calculus, with a semantics in three-valued logic (based on Kleene's three-valued logic \[Kh52\]), that includes an interpretation of negation motivated by the approach given by Karttunen \[Kar84\]. We show that our logic meets the conditions stated by Pereira. We also show that the three-valued framework is powerful enough to express most of the proposed definitions of negation and implication. It therefore makes it possible to compare these different approaches.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 1.1 Rounds-Kasper Logic </SectionTitle> <Paragraph position="0"> In \[Kas87\] and \[RK86\], Rounds and Kasper introduced a logical formalism to describe feature structures with disjunctive specification. The language is a form of modal propositional logic (with modal operator &quot;:').</Paragraph> <Paragraph position="1"> In order to define the semantics of this language, feature structures are formally defined in terms of acyelic finite automata. These are finite-state automata whose transition graphs are acyclic. The formal definition may be found in \[RK86\].</Paragraph> <Paragraph position="2"> A fundamental property of the semantics is that the set of automata satisfying a given formula is upward-closed under the operation of subsumption. This is important, because we consider a formula to be only a partial description of a feature structure. The property is stated in the following theorem IRK86\]: Theorem 1.1 A C 8 if and only i/for every formula, ~, if A ~ ~ then B ~ cb.</Paragraph> </Section> <Section position="2" start_page="0" end_page="18" type="sub_section"> <SectionTitle> 1.2 The Problem of Adding Negation </SectionTitle> <Paragraph position="0"> Several researchers in the area have suggested that the logic described above should be extended to include negation and implication.</Paragraph> <Paragraph position="1"> Karttunen \[Kar84\] provides examples of feature structures where 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 tive possibilities. Kaxttunen also suggests an implementation technique to handle negative information.</Paragraph> <Paragraph position="2"> Johnson \[Joh87\], defined an Attribute Value Logic (AVL), similar to the Rounds-Kasper Logic, that included a classical form of negation. Kasper \[Kas88\] discusses an interpretation of negation and implication in an implementation of Functional Unification Grammar \[Kay79\] that includes conditionals. Kasper's semantics is classical, but his unification procedure uses notions similar to those of three-valued logic a .</Paragraph> <Paragraph position="3"> One aspect of the classical approach is that the prop-erty of upward-closure under subsumption is lost. Thus the evaluation of negation may not be freely interleaved with unification 2 In \[Kas88\], Kasper 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 by the following example from Pereira \[Per87\] which expresses the semantic constraint that the subject and object of a clause cannot be coreferential unless the object is a reflexive pronoun: oh3 : type : reflezive V -~(subj : ref ~ obj : re f) Moshier and Rounds \[MR87\] proposed an intuitionistic interpretation of negation that preserves upward-closure. They replace the notion of saris/action with one of modeltheoretic/arcing an described in Fitting \[Fit69\]. They also provide a complete proof system for their logic. The satisliability problem for this logic was shown to be PSPACEcomplete. null</Paragraph> </Section> <Section position="3" start_page="18" end_page="18" type="sub_section"> <SectionTitle> 1.3 Outline of this Paper </SectionTitle> <Paragraph position="0"> In the following section we will present our proposed solution in a three-valued framework, for defining the semantics of feature structure descriptions including negation 3.</Paragraph> <Paragraph position="1"> This solution is a formalization of the notion of negation in Karttunen \[Kar84\]. In Section 3 we will show that the framework of three-valued logic is flexible enough to express most of the different interpretations of negation mentioned above. In Section 4 we will show that the satisfiability problem for the logic we propose is NP-complete.</Paragraph> <Paragraph position="2"> We will now present our extended version of the Rounds-Kasper logic including negation. We do this by giving the semantics of the logic in a three-valued setting. This provides an interpretation of negation that is intuitively appealing, formally simple and computationally no harder than the original Rounds-Kasper logic.</Paragraph> <Paragraph position="3"> With each formula we associate the set (Tset) of automata that satis/y the formula, a set (Fset) of automata that contradict it and a set (Uset) of automata which neither satisfy nor contradict it 4. Different interpretations of negation are obtained by varying definitions of what constitutes &quot;contradiction.&quot; In the semantics we will define, we choose a definition in which contradiction is equivalent to essential incompatibility 5. 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 problems associated with the classical interpretation of negation. In our logic, negation is defined so that an automaton ,4 satisfies -,~b if and only if it contraPS1icts ~.</Paragraph> </Section> <Section position="4" start_page="18" end_page="18" type="sub_section"> <SectionTitle> 2.1 The Syntax </SectionTitle> <Paragraph position="0"> The symbols in the descriptive language, other than the connectives :, v, A,-, and ~ are taken from two primitive domains: Atoms (A}, and Labels (L).</Paragraph> <Paragraph position="1"> The set of well formed formulae (W), is given by: NIL;</Paragraph> <Paragraph position="3"/> </Section> <Section position="5" start_page="18" end_page="20" type="sub_section"> <SectionTitle> 2.2 The Semantics </SectionTitle> <Paragraph position="0"> Formally, the semantics is defined over the domain of partial functions from acycLic finite automata ~ to boolean values. null Definition 2.1 An acyclic finite automaton is a 7-tuple A =< Q, E, r, 6, qo, F, A >, where: 1. Q is a non-empty finite set (of states), ~. E is a countable set (the alphabet), 4A similar notion was used by Kasper \[Kas88\], who introduces the notion of compatibility. We shall comps.re this approach with ou~ in greeter detail in Section 3.4. Sln general, a feature structure is incompatible with a formula iPS the information it contains is inconsistent with that in the formula. We will distinguish two kinds of incompatibility. A feature structure is essentiall~/incompatible with a formula if the information in it contradicts the information in the formula. It is trivially incompatible with the formula if the inconsistency is due to an excess of mformtstion within the formula itself. Sin this paper we will not consider cyclic feature structures 3. r is a countable set (the output alphabet), 4. 6 : Q x E -&quot; Q is a finite partial/unction (the transition function), 5. qo ~ Q (the initial state), 6. F C Q (the set of final states), 7. A : F &quot;-* r is a total function (the output function), 8. the directed graph (Q, E) is acyclic, where pEq iff .for some 1 6 Z, 6(p, l) = q, 9..for every q ~ Q, there exists a directed path from qo to q in ( Q, E), and 10. for every q ~ F, 6(q, I) is not defined for any I. A formula ~ over the set of labels L and the set of atoms A is chaxacterized by a partial function: ~r, : {'41&quot;4 =< Q, L, A, 6, q0, F, A >} &quot;7&quot; {True, False} ~#,('4) is True iff &quot;4 satisfies ~b. It is False if'4 contradicts ~b r and is undefined otherwise. The formal definition is given below.</Paragraph> <Paragraph position="1"> Definition 2.2 For any formula C/~, the partial function .~'C/ over the set of acyclic finite automata, &quot;4 =< Q, L, A, 6, qo, F, A >, is defined as follows:</Paragraph> <Paragraph position="3"> if &quot;4 is atomic and A(qo) = b for some b, b # a (see Note ~.)</Paragraph> <Paragraph position="5"> if &quot;4/pa and &quot;4/P2 are both defined and are not unifiable is undefined otherwise (see Note 4.).</Paragraph> <Paragraph position="6"> Notes: I. We have not included an implication operator in the formal language, since we find that defining impllcation in terms of negation and disjunction (i.e ~b =~ ~b ~ -~@ V ~b) yields a semantics for implication that corresponds exactly to our intuitive understanding of implication.</Paragraph> <Paragraph position="7"> 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. Though a complex feature structure is clearly incompatible with an atomic formula we do not view it as being essentially incompatible with it. An interpretation of negation that defines a complex feature structure as contradicting a (and hence satisfying -,a) is also possible. However, our definition is motivated by the linguistic intention of the negation operator as given by Karttunen \[Kar84\]. Thus, for instance, we require that an automaton satisfying the formula case : &quot;.dative have an atomic value for the case feature.</Paragraph> <Paragraph position="8"> 3. In J. above, we state that: ~'~('4) = jr', ('4/1) if.Aft is defined. When &quot;4/l is defined, ~t ('4/I) may still 4.</Paragraph> <Paragraph position="9"> be True, False or undefined. In any of these cases, ~#(.A) -- ~I(.A/I) s. ~r~(.A) is not defined if .All is not defined. Not only is this condition required to preserve upward-closure, it is also linguistically motivated.</Paragraph> <Paragraph position="10"> Here again, we could have said that a formula of the form I : ~bz is contradicted by any atomic feature structure, but we have chosen not to do so for the reasons outlined in the previous note.</Paragraph> <Paragraph position="11"> We have chosen to state that the set of automata that are incompatible with the formula pz ~ p2 is not the set of automata for which 6(qo,pl) and 6(qo,p~) axe defined and 8(q0,pz) ~ 6(q0,p2), since such an automaton could subsume one in which 6(qo,px) = 6(q0,p~). Thus, we would lose the property of upward-closure under subsumption. However, an automaton, .4, in which 6(q0,pl) and 8(qo,p2) are defined and .A/p1 is not unifiable 9 with ~4/p2 cannot subsume one in which 6(q0,pa) = 6(q0,p2). As has been stated before, the set of automata that satisfy a given formula in the logic defined above is upward-closed under subsumption. This property is formally stated below. null Theorem 2.1 Given a formula ~b and two acyclie finite automata .4 and IJ, if ~(.A) is defined and .4 C B then y.(B) ~, defined and ;%(B) = 7.(~4).</Paragraph> <Paragraph position="12"> Proof: The proof is by induction on the structure of the formula. The details may be found in Dawar \[Daw88\].</Paragraph> </Section> <Section position="6" start_page="20" end_page="20" type="sub_section"> <SectionTitle> 2.3 Examples </SectionTitle> <Paragraph position="0"> We now take a look at the 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 : &quot;~(person : third A number : singular) (1) 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 following two structures satisfy the given formula: agreement: \[person: second\] SEquality here is strong equality (i.e. if .g,x(A\]l) is undefined other automata would have an undefined truth value for formula(1).</Paragraph> <Paragraph position="1"> Turning to the other example mentioned earlier, the formula: obj : type : reflexive x/&quot;~(subj : ref ~ obj : re f) (2) 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 or path-equivalence).</Paragraph> <Paragraph position="2"> \[obj. \[type-reflexive \]\] As we have stated before, the semantics for negation described in the previous section is motivated by the discussion of negation in Karttunen \[Kar84\], and that it is closely related to the interpretation of Kssper \[Kas88\]. In this section, we take a look at the interpretations of negation that have been suggested and how they may be related to interpretations in a three-valued framework.</Paragraph> </Section> <Section position="7" start_page="20" end_page="21" type="sub_section"> <SectionTitle> 3.1 Classical Negation </SectionTitle> <Paragraph position="0"> By classical negation, we mean an interpretation in which an automaton .4 satisfies a formula -~b if and only if it does not satisfy ~b. This is, of course, a two-valued logic. Such an interpretation is used by Johnson in his Attribute-Value Language \[Joh87\]. We can express it in our framework by making ~'~ a total function such that wherever 9re(A) was undefined, it is now defined to be False.</Paragraph> <Paragraph position="1"> Returning to our earlier example, we can observe that for formula(1) the structure \[ agreement: \[ person: third\] \] has a truth value of .false in the classical semantics but has an undefined truth value in the semantics we define.</Paragraph> <Paragraph position="2"> This illustrates the problem of non-monotonicity in the classical semantics since this structure does subsume one that satisfies formula (1).</Paragraph> </Section> <Section position="8" start_page="21" end_page="22" type="sub_section"> <SectionTitle> 3.2 Intultionistic Logic </SectionTitle> <Paragraph position="0"> In \[MR87\], Moshier and Rounds describe an extension of the Rounds-Kasper logic, including an implication operator and hence, by extension, negation. The semantics is based on intnitionistic techniques. The notion of satisfying is replaced by one of forcing. Given a set of automata/C, a formula ~b, and .A such that .4 ~ /C, .A forces in IC &quot;~b (,4 hn -~b) if and only if for all B ~/C such that A ~ B, B does not force ~b in/~. Thus, in order to find if a formula, ~b, is satisfiable, we have to find a set \]C and an automaton ~4 such that forces in IC ~.</Paragraph> <Paragraph position="1"> Moshier and Rounds consider a version in which forcing is always done with respect to the set of all automata, i.e. IC*. This means that the set of feature structures that satisfy --~b is the largest upward-closed set of feature structures that do not satisfy @ (i.e. the set of feature structures incompatible with ~b). We can capture this in the three-valued framework described above by modifying the definition of ~rC/ to make it False for all automata that are incompatible (trivially or essentially) with ~b (we call this new function ~r~). The definition of ~'~ differs from that of ~r+ in the following cases:</Paragraph> <Paragraph position="3"> if All is defined and</Paragraph> <Paragraph position="5"> if 8(qo, p,) and ~(qo, p2) are defined</Paragraph> <Paragraph position="7"> if A/p1 and .A/p2 are both defined and are not unifiable or if .4 is atomic ~'~(.4) is undefined otherwise .</Paragraph> <Paragraph position="8"> In the other cases, the definition of ~'~ parallels that of 7+.</Paragraph> <Paragraph position="9"> To illustrate the difference between ~'~ and 3r~, we define the following (somewhat contrived) formula: cb = (11 :avl2 : a) AI2 : b We also define the automaton ,4 = \[11 : b\] We can now observe that F~(A) is undefined but 3r~(A) = False. To see how this arises, note that in either system, the truth value of ,4 is undefined with respect to each of the conjuncts of C/i. This is so because ,4 can certainly be extended to satisfy either one of the conjuncts, just as it can be extended to contradict either one of them. But, for ~c'~#(.A) to be False, .4 must have a truth value of False for one of the conjuncts and therefore .~'C/(.4) is undefined. On the other hand, since .4 can never be extended to satisfy both conjuncts of ~ simultaneously, it can never be extended to satisfy ~b. Hence .4 is certainly incompatible with ~, but because this incompatibility is a result of the excess of information in the formula itself, we say that it is only trivially incompatible with ~.</Paragraph> <Paragraph position="10"> To see more clearly what is going on in the above example, consider the formula -~b and apply distributivity and DeMorgan's law (which is a valid equivalence in the logic described in the previous section, but not in the intuitionistic logic of this section) which gives us: -,~b = (-'la : a A &quot;./2 : a) V -~12 : b We can now see why we do not wish .4 to satisfy -~b, which would be the case if .~'~#(~4) were False.</Paragraph> <Paragraph position="11"> One justification given for the use of forcing sets other than /C* is the interpretation of formulae such as -~h : NIL. It is argued that since h : NIL denotes all feature structures that have a feature labeled h, -,h : NIL should denote those structures that do not have such a feature. However, the formula -~h : NIL is unsatisfiable both in the interpretation given in the last section as well as in the /C* version of intuitionistic logic. It is our opinion that the use of negation to assert the non-existence of features is an operation distinct from the use of negation to describe values mad should be described by a distinct operator. The present work attempts to deal only with the latter notion of negation. The authors expect to present in a forthcomi~ag paper a simple extension to the current semantics that will deal with issues of existence of features.</Paragraph> </Section> <Section position="9" start_page="22" end_page="22" type="sub_section"> <SectionTitle> 3.3 Karttunen's Implementation of Negation </SectionTitle> <Paragraph position="0"> As mentioned earlier, our approach was motivated by Karttunen's implementation as described in \[Kax84\]. 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 Ideg the formula then the constraints may be dropped. We feel that our definition of the Uset elegantly 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> </Section> <Section position="10" start_page="22" end_page="22" type="sub_section"> <SectionTitle> 3.4 Kasper's Interpretation of Negation and Conditionals </SectionTitle> <Paragraph position="0"> As mentioned earlier, Kasper ~Kas88\] used the operations of negation mad implication in extending Functional Unification Grammar. Though the semantics defined for these operators is a classical one, for the purposes of the algo.</Paragraph> <Paragraph position="1"> rithm Kasper identified three chases 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 Tact, Fset and User respectively. For instance, Kasper states that an automaton .A satisfies a formula f : v if it is defined for f with value v; it is incompatible with f : v if it is defined for f with value z (z ~ v) and it is merely compatible with f : v if it is not defined for f. 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> <Section position="4" start_page="22" end_page="22" type="metho"> <SectionTitle> 4 Computational Issues </SectionTitle> <Paragraph position="0"> In this section, we will discuss some computational aspects related to determining whether a formula is satisfiable or not. We will Show that the satisfiability problem is NP-complete, which is not surprising considering that the problem is NP-complete for the logic not involving negation (Rounds-Kasper logic).</Paragraph> <Paragraph position="1"> The NP-hardness of this problem is trivially shown if we observe that for any formula, ~b, without negation, Tset(C/) is exactly the set of automata that satisfy ~ according to the definition of satisfaction given by Rounds ldegIt is not clear whether falsification is equivalent to incomp~- ibility or only essential incompatibility, but from the examples involvin~ ease and agreement, we believe that only emJential incom- patibihty is intended. and Kasper \[KR86, RK86\] in their original logic. Since the satisfiabllity problem in that logic is NP-complete, the given problem is NP-haxd.</Paragraph> <Paragraph position="2"> In order to see that the given problem is in NP, we observe that a simple nondeterministic algorithm 11 can be given that is linear in the length of the input formula ~b and that returns a minimal automaton which satisfies ~b, provided it is satisfiable. To see this, note that the size (in terms of the number of states) of a minimal automaton satisfying ~b is linear in the length of C/ and verifying whether a given automaton satisfies ~b is a problem linear in the length of ~b and the size of the automaton. The details of the algorithm can be found in Dawar \[DawS8\].</Paragraph> </Section> class="xml-element"></Paper>