Gosse Bouma* Rijksuniversiteit Groningen Unification-based grammar formalisms use feature structures to represent linguistic knowledge. The only operation defined on feature structures, unification, is information-combining and monotonic. Several authors have proposed nonmonotonic extensions of this formalism, as for a linguistically adequate description of certain natural language phenomena some kind of default reasoning seems essential. We argue that the effect of these proposals can be captured by means of one general, nonmonotonic, operation on feature structures, called default unification. We provide a formal semantics of the operation and demonstrate how some of the phenomena used to motivate nonmonotonic extensions of unification-based formalisms can be handled. 1. Introduction While monotonicity isoften desirable from a formal and computational perspective, it is at odds with a considerable body of linguistic work. Default principles, default rules, and default feature-values can be found in many linguistic formalisms and are used prominently in work on phonology, morphology, and syntax. In spite of their great expressive power and flexibility, unification-based grammar formalisms (see Shieber 1986a, for an introduction) are in general not very successful in modeling such devices. Unification is an information-combining, monotonic, operation on feature structures, whereas the implementation ofdefault devices typically requires some form of nonmonotonicity. In this paper, we present a nonmonotonic operation on feature structures, which enables us to implement the effects of a number of default devices used in linguistics. As the operation is defined in terms of feature structures only, an important characteristic of unification-based formalisms, namely that linguistic knowledge is encoded in the form of feature structures, is preserved.
In the next section, we present an overview of linguistic phenomena that are best described using defaults. We also argue that previous proposals for handling these phenomena in a unification-based setting are unsatisfactory. Section 3 provides the formal background for the central part of the paper, Section 4, in which a definition of default unification is presented. Section 5 briefly presents ome applications of this operation and the final section draws some conclusions concerning the role of nonmonotonicity in unification-based formalisms. 2. Previous Work There are a number of phenomena that suggest hat unification-based grammar formalisms might profit from the addition of some form of nonmonotonicity, and several authors have in fact suggested such extensions. In this section, we argue that these proposals uffer from a number of shortcomings. Most importantly, previous proposals have either been highly restricted in scope or have been presented in very informal * Computational Linguistics Department, Postbus 716, 9700 AS Groningen, The Netherlands (~) 1992 Association for Computational Linguistics Computational Linguistics Volume 18, Number 2 terms, thus leaving a number of questions concerning the exact behavior of the proposed extensions unanswered.
An overview of the issues that call for the addition of non-monotonic reasoning and of some of the proposals in that direction is presented below. • • • Exceptional Rules. Consider a language in which the vast majority of verbs cannot precede its subject, whereas a small number of exceptional verbs can. The rule accounting for inverted structures would probably require that verbs occurring in it be marked as +INV (i.e. (INV) : +). As a consequence, all regular verbs must be marked explicitly as --INV (to prevent them from occurring in the inversion rule). Note that, in a unification-based grammar, there is no need to mark the exceptional verbs as +INV, which leads to the rather counterintuitive situation that regular verbs need to be marked extra, whereas the exceptional ones can remain underspecified. Amore natural solution would be to assign all verbs the specification qNV by default (either by means of template inheritance or by means of lexical feature specification defaults as used in Generalized Phrase Structure Grammar \[GPSG; Gazdar et al. 1985\]) and to overwrite or block this specification in the exceptional cases. The possibility of incorporating an overwrite operation in a unification-based formalism is mentioned in Shieber (1986a, p. 60). Feature Percolation Principles. Both GPSG and Head-driven Phrase Structure Grammar (HPSG; Pollard and Sag 1987) adopt the so-called Head Feature Convention (HFC). In GPSG, the HFC is a default principle: head features will normally have identical values on mother and head, but specific rules may assign incompatible values to specific head features. In unification-based formalisms, it is impossible to express this principle directly. Adding the constraint (Xo head) = (Xi head) to every rule of the form Xo --~ X1... Xn (with Xi (1 K i < n) the head of the rule and assuming all head features to be collected under head) will not do, as it rules out the possibility of exceptions altogether. Shieber (1986b) therefore proposes to add this constraint conservatively, which means that, if the rule already contains conflicting information for some head feature f, the constraint is replaced by a set of constraints (Xo head f') = (Xi head f'), for all head features f' # f. Structuring the Lexicon. Flickinger, Pollard, and Wasow (1985), Flickinger (1987), De Smedt (1990), Daelemans (1988), and others, have argued that the encoding and maintenance of the detailed lexical descriptions typical for lexicalist grammar formalisms benefits greatly from the use of (nonmonotonic) inheritance. In Flickinger, Pollard, and Wasow (1985), for instance, lexical information is organized in the form of frames, which are comparable to the templates (i.e., feature structures that may be used as part of the definition of other feature structures) of PATR-II (Shieber 1986a). A frame or specific lexical entry may inherit from more general frames. Frames can be used to encode information economically and, perhaps more importantly, as a means to express linguistic generalizations. For instance, all properties typical of verbs are defined in the VERB-frame, and properties typical of auxiliaries are defined in the AUX-frame. The AUX-frame may inherit from the VERB-frame, thus capturing the fact that an auxiliary is a kind of verb.
Gosse Bourna Feature Structures and Nonmonotonicity In this approach, amechanism that allows inheritance of information by default (i.e., a mechanism in which local information may exclude the inheritance of more general information) is of great importance. Without such a mechanism, a frame may contain only properties that hold without exception for all items that inherit from this frame. In practice, however, one often wants to define the properties that are typical for a given class in the form of a frame, without ruling out the possibility that exceptions might exist. In unification-based formalisms, templates can play the role of frames, but as unification is used to implement inheritance, nonmonotonic inheritance is impossible. • Inflectional Morphology. In PATR-II the lexicon is a list of inflected
word forms associated with feature structures. The only tools available
for capturing lexical generalizations are templates (see above) and lexical
rules. Lexical rules may transform the feature structure of a lexical entry.
An example is the rule for agentless passive (Shieber 1986a, p. 62), which
transforms the feature structure for transitive past participles into a feature
structure for participles occurring in agentless passive constructions.
Lexical rules can only change the feature structure of a lexical entry, not
its word form, and thus, the scope of these rules is rather restricted.
While the examples in Flickinger, Pollard, and Wasow (1985) and Evans
and Gazdar (1989a,b) suggest hat the latter restriction can be easily
removed, it is not so obvious how a unification-based grammar
formalism can cope with the combination of rules and exceptions typical
for (inflectional) morphology. For instance, it is possible to formulate a
rule that describes past tense formation in English, but it is not so easy
to exclude the application of this rule to irregular verbs and to describe
(nonredundantly) past tense formation of these irregular verbs. Evans
and Gazdar (1989a,b) present he DATR-formalism, which, among other
things, contains a nonmonotonic inference system that enables an elegant
account of the blocking-phenomenon just described. The examples used
throughout their presentation are all drawn from inflectional
morphology and illustrate once more the importance of default
reasoning in this area of linguistics. • Gapping. In Kaplan (1987) it is observed that gapping constructions and
other forms of nonconstituent conjunction can be analyzed in Lexical
Functional Grammar (Bresnan and Kaplan, 1982) as the conjunction of
two functional-structures (f-structures), one of which may be incomplete.
The missing information in the incomplete f-structure can be filled in if it
is merged with the complete f-structure, using an operation called
priority union. Priority union of two f-structures Aand B is defined as an
operation that extends A with information from B that is not included
(or filled in) in A. As not all information in B is present in the priority
union of A and B, this operation introduces nonmonotonicity.
The proposals for incorporating the kind of default reasoning that is required for each of the phenomena bove are both rather diverse and idiosyncratic and, furthermore, suffer from a number of shortcomings.
The Head Feature Convention and Feature Specification Defaults of GPSG, for instance, appear to be motivated with a very particular set of linguistic phenomena in mind and also are rather intimately connected to peculiarities of the GPSG-formalism. What Computational Linguistics Volume 18, Number 2 is particularly striking is the fact that two different conceptions of default appear to play a role: a head feature is exempt from the HFC only if this would otherwise lead to an inconsistency, whereas a feature is exempt from having the value specified in some feature specification default (among others) if this feature covaries with another feature.
Overwrite and add conservatively are also highly restricted operations. From the examples given in Shieber (1986a) it seems as if overwriting can only be used to add or substitute (nonmonotonically) one atomic feature value in a given (possibly complex) feature structure (which acts as default). Add conservatively, on the other hand, is only used to add one reentrancy (as far as possible) to a given feature structure (which acts as nondefault). An additional restriction is that add conservatively iswell behaved only for the kind of feature structures used in GPSG (that is, feature structures in which limited use is made of covariation or reentrancy). Consider for instance the example in (1). 1 Adding the constraint (Xo head) = (X1 head) to (1) conservatively could result in either la or lb. Example 1 Go head X1 head Xo head a. X1 head X0 head b. X1 head
As add conservatively and overwriting are, in a sense, mirror images of each other, it is tempting to generalize the definitions of these operations and to think of them as operations on arbitrary feature structures, whose effect is equivalent o that of priority union. Thus, given two feature structures FSD (the default) and FSND (the nondefault), adding FSp to FSND conservatively would be equivalent o overwriting FSD with FSND, and to the priority union of FSND and FSD (i.e. FSND/FSD in the notation of Kaplan \[1987\]). However, in light of the example above, it should be clear that such a generalization is highly problematic. Other examples worth considering are 2 and 3. 1 Whether this kind of situation can occur in GPSG probably depends on whether one is willing to
conclude from examples uch as: S\[COMPo~\] ---* {\[SUBCATa\]}, H\[COMP NIL\] (Gazdar et al. 1985, p. 248) that covariation of arbitrary categories i in principle not excluded in this formalism. Gosse Bouma Feature Structures and Nonmonotonicity Example 2 Example 3 g:r~a FSND= \[ g:b \] Again, if we try to combine the two feature structures along the lines of any one of the operations mentioned above, there are at least two possible results (note that in Example 3, we could either preserve the information that features f and g are reentrant, or preserve the information that f : a), and there is no telling which one is correct.
Two conclusions can be drawn at this point. First of all, on the basis of the examples just given, it can be concluded that a nonmonotonic operation on feature structures that relies (only) on the fact that the result should be consistent must be very restricted indeed, as more generic versions will always run into the problem that there can be several mutually exclusive solutions to solving a given unification conflict. Second, claims that the operations add conservatively, overwriting, and priority union are equivalent are unwarranted, as no definitions of these operations are available that are sufficiently explicit to determine what their result would be in moderately complex examples uch as 1-3.
The approach exemplified by Flickinger (1987) and others is to use a generalpurpose knowledge representation formalism to represent linguistic information and model default inheritance. Feature structures are defined as classes of some sort, which may inherit from other, more generic, classes. The inheritance strategy used says that information in the generic class is to be included in the specific class as well, as long as the specific class does not contain local information that is in conflict with the information to be inherited. Such an inheritance strategy will run into problems, however, if reentrancies are generally allowed. For instance, think of the examples presented above as involving a generic class FSD from which a specific class FSND inherits. The inheritance procedure in, for instance, Flickinger (1987, p. 59ff) does not say anything about which one of the possible results will be chosen.
The work of Evans and Gazdar (1989a,b), finally, is not easily incorporated in a unification-based formalism, as they use semantic nets instead of feature structures to represent linguistic information. That is, although the syntax of DATR is suggestively similar to that of, for instance, PATR-II, DATR descriptions do in fact denote graphs that differ rather substantially from the graphs used to represent feature structures (see Evans and Gazdar 1989b). The nonmonotonic reasoning facilities of DATR therefore are not directly applicable in a unification-based formalism either.
We conclude that a formally explicit definition of a nonmonotonic operation on feature structures is still missing. In particular, the interaction of reentrancy and nonmonotonicity isa subtle issue, which has not been given the attention itdeserves. That there is a need for nonmonotonic devices is obvious from the fact that several authors have found it necessary to introduce partial solutions for dealing with nonmonotonicity in a unification-based setting. The intuitions underlying these proposals appear to be compatible, if not identical, and thus it seems attractive to consider an operation that subsumes the effects of the proposals o far. Default Unification, as defined below, is an attempt o provide such an operation. Computational Linguistics Volume 18, Number 2 3. Feature Structures and Unification Feature structures are often depicted as matrices of attribute-value pairs where values are either atoms or feature structures themselves and, furthermore, values may be shared by different attributes in the feature structure. Feature structures can be defined using a description language, such as the one found in PATR-II (Shieber 1986a) or in Kasper and Rounds (1986; 1990). For instance, 4a is a description of 4b. Example 4 a. ( (f) (g f) (g f) f'a = = = a a Igg) Following the approach of Kasper and Rounds (1986; 1990), and others, we represent feature structures formally as finite (acyclic) automata (the definition below is taken from Dawar and Vijay-Shanker 1990): Definition A finite acyclic automaton A is a 7-tuple
(Q, ~, P, 6, q0, F, "~/where: 1. Q is a nonempty finite set of states, 2. 3. 4. G is a countable set (the alphabet), 1 ~ is a countable set (the output alphabet), ~ : Q x G --* Q is a finite partial function (the transition function), 5.
7. q0EQ, FC_Q, )~ : F --* P is a total function (the output function), 8. the directed graph (Q, E) is acyclic, where pEq iff for some l E Y~,6(p,I) = q, 9. 10. for every q E F, 6(q, I) is not defined for any l. for every q E Q, there exists a directed path from q0 to q in (Q, E), and We will frequently write QA, GA, etc. for the set of states of automaton A, the alphabet
of A, etc.
The relationship between the matrix notation and the automaton concept should be obvious. The following automaton M is, for instance, equivalent o the matrix in 4b. Gosse Bourna Feature Structures and Nonmonotonicity Example 5 QM ~M FM 6M(qo,f) = ~M(qo~g) "~" 92 = = = Note that ~M(qo,gf) = ~M(qo,gg) = 93, 2 which represents the fact that the two paths (gf) and (gg) are reentrant. Unification is defined in terms of subsumption, arelation that imposes a partial ordering on automata: {q0,ql,q2,q3} {f,g} {a} ql ~SM(q2,g) = ~M(q2,f) FM /~M(ql) AM(93 ) = = = = q3 q3 {ql,q3} a a Definition An automaton A subsumes an automaton B (A _ B) iff there is a homomorphism h
from A to B such that:
Intuitively, A u B if B extends the information in A. A = B if A _ B and B U A. Unification of two automata A and B (A U B) is the least upper bound of these automata under subsumption. If no upper bound exists, unification fails.
The semantics of descriptions (sets of formulae of the description language) is given in terms of satisfaction: Definition An automaton A = (Q, G, F, 6, q0, F, ;~) satisfies a description D (A ~ D) or a formula
(A ~ q~) in the following cases: A~D iff for all q~ E D :A ~ q~, A~a iff Q = F = {q0} and &(q0) = a, A ~ (p) -- D iff 6(qo, p) is defined and qo/P ~ D, A ~ (pl) = (p2) iff 6(qo, pl) = ~(q0, p2).
qo/P is the automaton obtained from A by making ~(q0, P) the initial state and removing all inaccessible states. There is always a unique minimal element in the subsumption hierarchy that satisfies a description D. This element is the denotation of D. 3 2 rS(q, pl) is defined for pl ff ~* as 6(6(q, p), 1). Computational Linguistics Volume 18, Number 2 4. Default Unification Default reasoning with feature structures requires the ability to modify feature structures nonmonotonically. Unification does not have this ability, as it can only replace a feature structure by more specific instances of that structure. Below, we define default unification as an operation that merges parts of one feature structure (the default argument) with another feature structure (the nondefault argument). We write AU!B for the default unification of the default feature structure A and the nondefault feature structure B. The operation has the following characteristics: 1. It has a declarative semantics and is procedurally neutral. That is, if A -- A' and B = B', then (AU!B) = (A'U!B'). 2. It is monotonic only with respect to the nondefault argument. That is, B U (AU!B) is always true, but in general A U (AU!B) will not hold. 3. It never fails. If A is fully incompatible with B, (AU!B) = B. 4. It gives a unique result. 5. Reentrancies in the nondefault argument may be replaced by a weaker set of reentrancies if necessary (this is the add conservatively operation of Shieber (1986b)).
Intuitions about default unification appear to be more clear in those cases where feature structures do not contain any reentrancies. Therefore, we will first define default unification for this case, moving to the general case in Section 4.2. Section 4.3. deals with the incorporation of add conservatively. 4.1 Default Unification without Reentrancies Subsumption suggests astraightforward efinition of an operation that has properties 1-4 above. Definition Default Unification (first version) AU!B = A ~ U B, where A ~ is the maximal (i.e. most specific) element in the subsumption ordering such that A' r- A and A ~ U B is defined.
From this definition of U!, it follows immediately that properties 1-3 hold. The fact that default unification has a unique result follows from the fact that A' is unique (up to isomorphism). 4 Note furthermore that from tile requirement that A ~ must be the maximal it follows that no information contained in A is left out in AU!B unnecessarily. Gosse Bouma Feature Structures and Nonmonotonicity
An example of default unification is presented below (where nil is used to represent the empty feature structure): Example 6 A = f:a g: g: ii! a f:a \] B = g. :b\] \] f;a a ! g: \[ g : nii na f:a A'UB= f: f:a g: g: g:b
The definition of default unification above relies crucially on the fact that there is a unique maximal element A' unifiable with B. In Section 2, we argued that such an approach is only feasible for a limited domain. In particular, once reentrancies are introduced, A' is no longer guaranteed to be unique, and the definition above is therefore not easily generalized. Fortunately, it is also possible to define AU!B without requiring unifiability of some element A t with B explicitly. This definition, which will be extended below, defines AU!B in terms of the difference of the two arguments A and B. Definition Difference (first version) The difference of two automata A and B is the maximal element A - B that meets the following conditions: 1. A-BU_A, 2. if 8a-B(q0,p) is defined, then there is no prefix p' of p such that 6B(qO,P') E FB, 3. if ~a-B(q0,p) E FA-B then ~8(q0,P) is undefined. Definition Default Unification (second version) Atd!B = (A - B) U B.
It should be obvious that characteristics 1-3 continue to hold. Uniqueness follows in this case from the fact that the difference operation will give a unique result. (A - B can be constructed from A by checking for each state in A whether it must be removed Computational Linguistics Volume 18, Number 2 or not and ensuring that the resulting automaton is connected.) For instance, assuming A and B to be defined as in Example 6, we find that A - B is: Example 7 Note that in A - B all parts that are identical in A and 13 are removed, whereas this was not the case for A', as defined in Definition 3.1. The outcome of default unification, however, is identical in both cases. The reason for this restriction on A - B will become apparent below.
While default unification monotonically extends the nondefault argument (i.e. B E AU!B) and nonmonotonically extends the default argument, the operation itself is monotonic in its default argument and nonmonotonic in its nondefault argument. The theorem below proves monotonicity for the default argument; that is, a more specific default argument will lead to a more specific outcome of default unification: Theorem 1 For all feature structures A, B, and C, not containing reentrancies, if A F- B then (ALl!C) _ (BU!C). Proof It suffices to show that (A - C) G (B - C), or in other words:
2. if 6A-C(qo,P) is defined then 6B-c(q0,p) is defined. If these two conditions are met, there is a homomorphism from A - C to B - C as required by the definition of subsumption. (Remember that there are no reentrancies.)
Case (1): If .X(6A-C(qO, p)) = a, then (i))~(6B(qo, p)) = a (since A - C _G_ A U B) and from the definition of A - C it follows that (ii) there is no prefix p' of p such that 6c(qo, p') c Fc nor is ~c(q0,p) defined. From (i) and (ii) it follows that ~B-c(q0,p) is defined and that ~(6~-c(q0, p)) = a.
Case (2): If ~A-C(q0, p) is defined and 6A-C(qO, p) (~ FA-C (otherwise this case reduces to case (1)), it follows that (i) 6B(q0, p) is defined, and (ii) there is no prefix p~ of p such that ~c(q0, p') E Fc. From (i) and (ii) it follows that ~B-C(q0, p) is defined. •
Note, however, that addition of nondefault information does not necessarily lead to a more specific result. That is, the dual of Theorem 1. does not hold: Example 8 if B E_ C then (ALl!B) G (ALl!C) The reason is that addition of nondefault information may lead to a larger amount of default information being removed, and thus, the resulting feature-structures AU!B and AU!C can be incompatible. An example that falsifies 8 is presented below. Gosse Bouma Feature Structures and Nonmonotonicity Example 9 A= ~:a\] B= L~:b\] Au!B=
: b AU!C = b b
Finally, for feature structures without reentrancies, the following distribution law holds: Theorem 2 For all feature structures A, B, and C, not containing any reentrancies and such that A U B is defined, (A U B)U!C -- (AU!C) td (BU!C) Proof Since (AUB)U!C = ((AUB)-C)WC and (AU!C)U(BU!C)= ((A-C)UC)W((B-C)UC)= (A - C) U (B - C) U C, it suffices to prove that (A U B) - C = (A - C) U (B - C). Let D=(AOB)-CandE=(A-C)u(B-C).Itmustbeshownthat(1)DEEand(2) EGD.
Case (1): If A(6o(qo,p)) = a, then (i) A(~AuB(qo,p)) = a and thus A(6A(qo,p)) = a or A(~B(q0,p)) = a (since there are no reentrancies) and (ii) there is no prefix p' of p such that 6c(qo,p') c Fc, nor is 8c(qo,p) defined. From (i) and (ii) it follows that A(6A-c(qo,p)) = a or A(SB-C(qo,P)) = a, and thus that A(6E(qo, p)) = a. Similarly, if 8D(qo, p) is defined (but not an end state), it follows that 6,~(qo, p) or 8B(qO, p) is defined and that there is no prefix p' of p such that 6c(qo, p) EFc. Therefore, either 8A-C(qO, p) or 6B-C(qo,p) is defined, and thus 8E(qO,P) is defined. It follows that D _u E.
Case (2): If A(SE(qo,p)) = a, then )~(SA-c(qo,p)) = a or A(SB-c(qo,p)) = a (since there are no reentrancies). Therefore, A(SAuB(qo,P)) = a and also, there is no prefix p' of p such that 6c(qo,p') E Fc, nor is ~c(qo,P) defined. It follows that A(6D(qo,p)) = a. Similarly if ~E(qO, p) is defined but not an end state, 6D(qO, p) is defined. It follows that EUD. •
As long as Theorem 2. holds, it is possible to define default unification by decomposing the default argument into simpler feature structures and adding these (nonmonotonically) to the nondefault argument. This approach appears to underlie some of the previous proposals, but is inadequate once reentrancies enter the picture. 4.2 Default Unification with Reentrancies Taking reentrancies into account requires an extension of the difference operation. If we allow either default or nondefault information to refer to an extension of a nondefault or default reentrancy, respectively, there is in general no unique maximal element subsuming A and unifiable with B. A slight modification of Examples 2 and 3 will illustrate this. Example 10 Computational Linguistics Volume 18, Number 2 Example 11 A= \[ f : : \[T\]~ \[~ : : a\] \ \] B \[ g : \[f : b\] \]
In Example 10, there is default information that refers to an extension of a nondefault reentrancy. A - B could be constructed from .4 by removing either the fact that (if) : a or (gf) : b. In 11, nondefault information refers to an extension of a default reentrancy. In this case, we could either remove the reentrancy (and the fact that (gf) : a) or remove the fact that (if) : a and (gf) : a and preserve the reentrancy. Neither solution subsumes the other. To avoid such problems, it is best to avoid interaction between reentrancies and other information altogether and to treat reentrant nodes in a similar fashion as atomic nodes. That is, we remove default reentrancies if they refer to defined parts of the nondefault automaton, and default information in general is removed if it refers to extensions of nondefault reentrancies. Thus, the difference operation can be extended as follows: Definition Difference (final version) The difference of A and B is the maximal element A - B in the subsumption ordering that meets the following conditions:
A-BU_A, 2. if 6A-B(qO,P) is defined, then there is no prefix pt of p such that 6B(qo,p') E FB or 6B(qo,P') = 6B(qo,P")(p' p"), 3. if 6A-~(qO,P) C FA-B then 6B(qo,p) is undefined,
undefined. The definition of default unification remains as before: Definition Default Unification (= second version) AU!B --- (A - B) t3 B.
Again, characteristics 1-4 of default unification mentioned in the introduction of this section hold. Uniqueness of the result follows from the fact that A - B is unique. (A - B can be constructed in this case as follows: for all paths p, if 6A(q0, p) = 6A(q0, P'), and p is defined in B, introduce a new value for ~A(q0, p) such that the automata that have ~A(q0, p) and 6A(qo, p') as initial state are isomorphic. Next, check for all states in the modified automaton whether they must be removed and ensure that the resulting automaton is connected.)
The monotonicity properties of default unification also remain as before. The theorem below is the relevant generalization of Theorem 1. Theorem 3 For all feature structures A, B, and C, if A G B then (AU!C) U (BU!C) Proof It suffices to show that A - C ___ B - C, or in other words:
Gosse Bouma Feature Structures and Nonmonotonicity
3. if ~x-c(q0, p) is defined, then ~B-C(q0~p) is defined. If these three conditions are met, there is a homomorphism from A - C to B - C as required by the definition of subsumption.
Case (1): If /~(6A-c(qo,p)) = a, then (i) ~(~B(qO,P)) = a ( since A - C E A _G B) and (ii) from the definition of A - C, it follows that there is no prefix p' of p such that 6c(q0, p') EFc or 6c(qo, p') = 6c(q0~ p"), nor is ~c(q0, p) defined. From (i) and (ii) it follows that ~B-C(q0~ p) is defined and that A(~B-c(q0, p)) = a.
Case (2): Similarly, if 6A-C(qO,p) = ~A-C(q0,p'), then (i) 6B(qo,p) = ~B(qo~P'), and (ii) there is no prefix p' of p such that ~c(q0,p') E Fc or 6c(qo,p') = 6c(qo,p') nor is 6c(q0, p) defined. From (i) and (ii) it follows that 6B-c(q0, p) = 6B-c(q0, p').
Case (3): If 6A-c(qo,p) is defined and ~A-C(q0,p) ~ FA-C (otherwise this case reduces to case (1)) and 6A-C(qO,P) not reentrant (otherwise this case reduces to case (2)), it follows that (i) ~B(qo,P) is defined, and (ii) there is no prefix p' of p such that 6c(qo,p') E Fc or 6c(qo,p') = ~c(q0,p"). From (i) and (ii) it follows that ~B-C(qO, P) is defined. • The distribution law, however, continues to hold only in one direction: Theorem 4 For all feature structures A, B, and C, such that A U B is defined, (AU!C) U (BU!C) £(A u B)u!c Proof As in the previous section, it suffices to prove that (A - C) U (B - C) __U (A U B) - C. From the fact that X E X' and Y G Y' implies (X u Y) G (X' u Y'), it follows that ((A - C) U (B - C)) £- (A U B). Now, as in the previous proof, if some path p is atomic, reentrant, or merely defined in (A- C) U (B - C), it follows that (i) p is atomic, reentrant, or defined in A U B and (ii) there is no atomic or reentrant path p' in C that is a prefix of p, nor is p defined in C if p is atomic or reentrant in (A - C) u (B - C). It follows that p is atomic, reentrant, or defined in (A U B) - C. •
An illustration of this result is given below. Note that 12 also illustrates that the converse of Theorem 4. no longer holds. Example 12 A = B= \[g:a\] c= \[g:b\] Computational Linguistics Volume 18, Number 2 4.3 Add Conservatively Defining default unification as (A - B) U B will fail to capture the idea of Shieber's (1986b) add conservatively, as the difference operation completely removes a default reentrancy ifone of the paths leading to it is also defined in the nondefault argument. However, linguistic applications, such as an encoding of the Head Feature Convention, indicate that a more subtle approach should be taken. In particular, if a default structure contains the information that (P/= (P'), whereas in the nondefault structure (pl) is defined for some feature l, we want to treat only I as an exception to the general rule that (P/= (P'/, and preserve the information that Ipl') = (p'l' I (for l' # I).
We implement his idea using the following operation: Definition Let A and B be automata. The extension of A relative to B (Ext(A, B)) is the minimal
(i.e. most general) element Ext(A, B) such that
2. A G Ext(A~B), if ~A(qO,P) = ~A(qO,p') and ~B(qo,pql) is defined (for some pql E ~*), then ~Ext(A,B) (qo, pql') = ~Ext(A,B) (qo, p'ql') (wherever possible) for all 1 /E G.
The automaton A is extended, sometimes omewhat redundantly, with reentrant paths that are extensions of paths already reentrant in A. Ext(A, B) is nevertheless usually more informative than A itself, as the addition of a path pl blocks unification with feature structures in which p receives an atomic value. Note furthermore that path extensions are not always possible; that is, if 6A(qO,p) E FA and 6B(qo, pl) is defined, there is no extension of A in which pl is defined. (This explains the wherever possible). In order to get all relevant path-extensions, G will in general be the set of all features defined in the grammar, although in particular cases G can be restricted to a smaller set (the set of head-features, for instance).
We are now ready to give a definition of default unification that incorporates the effects of add conservatively. To avoid confusion, we use the operator t3ac! for this extended version of default unification. Definition Default Unification (final version) AOac!B = (Ext(A, B) - B) 0 B.
An example of default unification involving reentrancies is presented below. We assume that the set of features G = {f,g}. Example 13 A = B = Ii\[r:a \]\] Ext(A, B) = f:F1 I f'\[\] X:\[ f:\[ X:\[ Gosse Bouma Feature Structures and Nonmonotonicity Ext(A,B) - B = g \[ fg : n~ \] \] (Ext(A, B) - B) U B = The example shows that default unification is slightly more restrictive than add conservatively, since the original reentrancy isremoved even though A and B would have been unifiable. The reason is of course that this will guarantee uniqueness of the result of default unification, whereas this is not the case for add conservatively.
In this section, we sketch how default unification can be incorporated in a grammar formalism and argue briefly that this can be an alternative for some of the extensions mentioned in Section 2. 5.1 Nonmonotonic Template Inheritance In grammar formalisms such as PATR-II, feature structures are defined as sets of equations and templates. Each equation or template denotes a feature structure (i.e. the minimal feature structure that satisfies the equation or the equations that make up the template definition), and the denotation of a set of such elements is simply the unification of all their denotations. Incorporation of default unification requires that a distinction is made between default and nondefault information. In the notation used here, nondefault information is prefixed by a "!'. The feature structure denoted by a definition that contains both default and nondefault information is arrived at by first unifying all default information and unifying all nondefault information. Next, the two feature structures are combined by means of default unification (tAac!).
If templates are incorporated as default information, the feature structure denoted by the template is inherited nonmonontonically. (Monotonic inheritance is possible as well of course: this is achieved by prefixing a template with "!'.) As an illustration, consider the following fragment, in which an attempt is made to encode some of the peculiarities of the English auxiliary system in a lexicalist grammar: Example 14 NP VERB VP AUX : ( : ( :( : ( (cat) = n (nform) = norm (cat) = v (aux) -- (inv) = VERB (subcat first) = NP (subcat rest) = empty VERB !(aux) = + !(inv) = + (subcat first) = VP (subcat rest first) = NP (subcat rest rest I -- empty !(subcat first subcat first nform I =
(subcat rest first nform I ). ). ). ). Computational Linguistics Volume 18, Number 2 Adding the equations !(aux} : + and !(inv} : +5 to the definition of AUX has an effect comparable to that of the overwrite-operation of (C;hieber 1986a, p. 60). The AUX template inherits from VERB by default, but the equations just mentioned block inheritance of the values for (inv} and (aux}. However, default unification allows us to do more. An auxiliary does not subcategorize for an ordinary NP subject, nor does it subcategorize for a complement VP that subcategorizes for an ordinary NP subject. Rather, the restrictions to be placed on the nform of the subject are inherited from the embedded VP: Example 15 a. it will annoy Kim that she lost b. *Sue will annoy Kim that she lost This dependency between elements of the subcat list is encoded in the final equation, which also suppresses (or overwrites) the default value for (nform}. The denotation of AUX is thus: Example 16 cat : v aux : + inv : + first : subcat : rest :
The nonmonotonic inheritance regime is flexible enough to allow for exceptions to exceptions. Gazdar et al. (1985, p. 65) observe that at least in some dialects of English, the auxiliary might cannot occur in inverted structures. This is expressed in the following lexical entry, in which might inherits nonmonotonically from AUX, which itself inherits nonmonotonically from VERB: Example 17 might :( AUX
!(inv}=- ).
There is an important difference between the approach to nonmonotonic inheritance sketched here and the majority of inheritance-based formalisms used for Knowledge Representation, which has to do with the way in which templates are evaluated. If a template is used as part of the definition of another feature structure, all we need to know to determine the denotation of this feature structure is the denotation of this cat : v aRx : -inv : subcat : \[ \ ] f \[cat:np i nform r s : t \[\] : 1 rest : empty
. \[ cat : np flrst:\[nform:~\] \] rest : empty Gosse Bouma Feature Structures and Nonmonotonicity template (which is a feature structure). How this template was defined (as a set of equations or as a combination of (more general) templates, as a combination of default and nondefault information or not) is completely irrelevant o its meaning. Thus, the denotation of AUX would remain as before, if we defined it as: Example 18 AUX : ( (cat) = v (aux) = + (inv) = + . Consequently, the denotation of might is not affected by this change in definition either.
The role of classes (or frames) in inheritance-based systems, however, as described in, for instance, Touretzky (1986), is rather different. To determine the denotation of a class might that inherits from a class AUX, we not only need to know the contents of AUX, but also the classes from which AUX inherits. The latter is important for resolving multiple-inheritance onflicts. If the class might inherits from both AUX and VERB, for instance, and AUX in its turn inherits from VERB as well, information inherited from AUX must take precedence over information from VERB, as the former is more specific than the latter. In our nonmonotonic inheritance mechanism for templates, such reasoning is impossible. Adding the template VERB as default information to the definition of the template (or lexical entry) might would lead to a unification failure of the default information, and thus the definition as a whole would be considered as illegal. 6 This is as it should be, we believe, given the fact that the inheritance hierarchy as such should not play a role in determining the meaning of templates. The denotation of the template AUX is the feature structure in 16 (i.e., whether it is defined as in 14 or as in 18 is irrelevant), and from that it is impossible to conclude that AUX inherits from VERB, and thus the kind of reasoning used to justify the resolution of feature conflicts used in Touretzky (1986) is not applicable in our case. 5.2 Lexical Defaults The definition of auxiliaries above is still unsatisfactory in that it predicts that auxiliaries subcategorize for verbal complements that are specified as (aux) = -. Clearly, this requirement is too strong (although it is correct for the auxiliary do). One way to solve this problem is to redefine the AUX-template as: Example 19 AUX : . ° . (subcat first cat) = v (subcat first subcat first) = NP (subcat first subcat rest) = empty 6 Of course, it is possible to combine incompatible default information ifwe impose the correct ordering
explicitly. This can be done by using definitions (i.e. a set of equations in brackets) in definitions: might : ( (VERB !AUX)
!(inv) -= -). This is equivalent to the definition of might given in 17, albeit more complex and possibly misleading. Computational Linguistics Volume 18, Number 2 This solution seems inelegant, however, as it reconstructs part of the VP-template in order to express the correct subcategorization requirements. Thus, the obvious generalization that an auxiliary subcategorizes for a VP is missed by this redefinition. The source of this inelegance is the fact that VP inherits from VERB, and that VERB contains default information about properties typical for verbs. However, while these properties hold for the vast majority of verbs, it is not the case that if an element subcategorizes for a verbal complement, he default properties need to hold for the complement as well. What is needed here is a distinction between properties that hold by default for all members of a class and default properties that can be assumed to hold if a lexical item subcategorizes for members of this class. While the latter can be expressed safely by means of templates, the former are more adequately expressed in the form of lexical defaults.
The extension of unification-based formalisms witlh lexical defaults can be implemented using default unification. The effect of lexical defaults is comparable to that of lexical Feature Specification Defaults in GPSG. A lexical default is a statement of the form Name: Ant ~ Cons, where Ant and Cons are feature structure descriptions. The interpretation of lexical defaults is that the feature structure of each lexical entry that is subsumed by the antecedent of a lexical default is extended, by means of default unification, with the contents of the consequent. Lexical entries are thus compiled in two stages: first, the denotation of the feature structure description is computed and next, the lexical defaults are applied to this feature structure.
Consider for example the following lexical defaults: Example 20 FSD1 : FSD2 : VERB ) ~ ((aux)=-). (aux)=- ) ~ ( (inv)=- ). The fragment in 14 and 17 is assumed to be redefined as follows: Example 21 VERB: VP: AUX : might: Each verbal lexical item will be extended with the information (aux) = -, unless it is an auxiliary of course, since in that case, the lexical entry is already specified as (aux) = +. Only nonauxiliary verbs are extended with the information (inv) = (FSD2). 7 Auxiliaries remain unspecified for this feature, thus capturing the fact that (cat) = v VERB (subcat first) = NP (subcat rest) = empty ). VERB (aux) = + (subcat first) = VP • .. AUX (inv) = ). ). ). Gosse Bouma Feature Structures and Nonmonotonicity auxiliaries can, but not necessarily do, occur in inverted structures. 8 The exceptional character of might is expressed in this case by adding explicitly the information that it cannot invert.
The problem sketched at the beginning of this section is now resolved. An auxiliary subcategorizes for a VP, which in its turn inherits from the template VERB. However, since the latter template no longer contains default information that should hold for lexical entries only, an auxiliary no longer subcategorizes for verbal complements that are (aux) : -. Auxiliaries that subcategorize for a restricted set of verbal complements, such as do, which requires a (aux) : - complement, can be encoded by adding the relevant constraint to their lexical entries. 5.3 Specialization of Reentrancies Another important property of default unification is that it enables us to define exceptions to a reentrancy. Consider for instance the following GPSG rule (where H indicates the head of the rule): Example 22 s x 2 \[-subj\] The symbol S can be analyzed as the feature structure in 23. Applying the Head Feature Convention to the rule in 22 amounts to adding to H all head features compatible with head features in S. Using default unification, this is implemented in 24 as a default reentrancy that equates the head features of S and H. Example 23 S :( (head n) = -
(head v) = +
(head bar) = 2
(head subj) = + . Example 24 S-rule X0 ~ X1 X2; (Xo) = s (Xl head bar) = 2 !(X2 head subj) = (Xo head) = (X2 head) The final equation in 24 both implements the HFC and defines X2 as the head daughter. An exception to the reentrancy is the fact that IX2 head subj) = -, which is therefore represented as nondefault information. In this approach, the HFC is part of the rules itself and thus, the effect of Shieber's (1986b) special-purpose compilations tep, which adds the HFC conservatively, is achieved directly.
implementing the effect of FSD2 above as a feature coocurrence restriction. 8 Note that, as in GPSG, the feature INV plays a double role by indicating both an item's potential to
occur in inverted structures as well as indicating whether a given structure is inverted or not. Computational Linguistics Volume 18, Number 2 6. Conclusions We have shown in the preceding sections that it is possible to incorporate nonmonotonicity in a unification-based formalism, while at the same time preserving the idea that linguistic knowledge is represented in the form of feature structures.
In spite of their great flexibility, unification-based formalisms are in general not very well equipped to deal with linguistic rules or generalizations that have a default character and for which exceptions exist. In Sections 4 and 5 we hope to have demonstrated that a nonmonotonic operation on feature structures combined with straightforward extensions of the description languages used in unification-based formalisms enables a satisfactory account of the phenomena mentioned in the introduction. The applications illustrate that default unification can be used to give linguistically appealing implementations of certain natural anguage phenomena, not that it would be impossible to account for these facts using unification only. Thus, default unification serves to extend the expressive power of unification-based formalisms, but leaves the representation method of unification-based formalisms, in which linguistic objects are represented as feature structures, unchanged. Comparing default unification to earlier proposals, we believe that an advantage of our approach is that it is general, in the sense that one operation is used to achieve the effects of overwriting, add conservatively, nonmonotonic template inheritance, and priority union. Also, whereas previous proposals do not seem to be well behaved for feature structures containing reentrancies, default unification is defined for feature structures of arbitrary complexity.
D6rre et al. (1990) suggest hat the use of nonmonotonic devices in unificationbased formalisms will, for the time being, be limited to off-line extensions of these formalisms; that is, extensions whose effect can be computed at compile time and result in ordinary feature structures. They also note that while there may be linguistic arguments in favor of more dynamic notions of default reasoning, from a computational point of view the off-line approach is clearly preferred. Default unification, as used in the previous section, is an example of an off-line extension, as the effects of nonmonotonic template inheritance, lexical defaults, and the meaning of rule definitions in which default and non-default information is combined, can be computed at compile time. Again, this emphasizes the point that incorporation of default unification in principle only extends the expressive power of unification-based formalisms. Acknowledgments A syntactic approach to default unification is presented in Bouma (1990). The reactions on that paper made it clear to me that default unification should be defined not only for feature structure descriptions, but also for feature structures themselves. For helpful questions, suggestions, and comments on the material presented here, I would like to thank Bob Carpenter, John Nerbonne, audiences in Tilburg, Groningen, Tiibingen, and Dhsseldorf, and three anonymous CL reviewers. References Bouma, Gosse (1990). "Defaults in
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