A Logical Semantics for Feature Structures

A LOGICAL SEMANTICS FOR FEATURE STRUCTURES Robert T. Kasper and William C. Rounds ElectricalEngineering and Computer Science Department University of Michigan Ann Arbor, Michigan 48109 Abstract temic G r a m m a r [3]. They also include grammar formalisms which have been developed as computational tools, such as Functional Unification G r a m m a r (FUG) [7], and PATR-II [14]. In these computational formalisms, unificat/on is the primary operation for matching and combining feature structures. Feature structures are called by several different names, including f-structures in LFG, and functional descriptiona in FUG. Although they differ in details, each approach uses structures containing sets of attributes. Each attribute is composed of a label/value pair. A value may be an atomic symbol, hut it may also be a nested feature structure. The intuitive interpretation of feature structures may be clear to linguists who use them, even in the absence of a precise definition. Often, a precise definition of a useful notation becomes possible only after it has been applied to the description of a variety of phenomena. Then, greater precision may become necessary for clarification when the notation is used by many different investigators. Our model has been developed in the context of providing a precise interpretation for the feature structures which are used in FUG and PATR-II. Some elements of this logical interpretation have been partially described in Kay's work [8]. Our contribution is to give a more complete algebraic account of the logical properties of feature structures, which can be used explicitly for computational manipulation and mathematical analysis. Proofs of the mathematical soundness and completeness of this logical treatment, along with its relation to similar logics, can be found in [12]. Unification-based grammar formalisms use structures containing sets of features to describe linguistic objects. Although computational algorithms for unification of feature structures have been worked out in experimental research, these algcwithms become quite complicated, and a more precise description of feature structures is desirable. We have developed a model in which descriptions of feature structures can be regarded as logical formulas, and interpreted by sets of directed graphs which satisfy them. These graphs are, in fact, transition graphs for a special type of deterministic finite automaton. This semantics for feature structures extends the ideas of Pereira and Shieber [11], by providing an interpretation for values which are specified by disjunctions and path values embedded within disjunctions. Our interpretati6n differs from that of Pereira and Shieber by using a logical model in place of a denotational semantics. This logical model yields a calculus of equivalences, which can be used to simplify formulas. Unification is attractive, because of its generality, but it is often computations/]), inefficient. Our mode] allows a careful examination of the computational complexity of unification. We have shown that the consistency problem for formulas with disjunctive values is NP-complete. To deal with this complexity, we describe how disjunctive values can be specified in a way which delays expansion to disjunctive normal form. 1 Background: U n i f i c a t i o n in G r a m m a r Several different approaches to natural language grammar have developed the notion of feature structures to describe linguistic objects. These approaches include linguistic theories, such as Generalized Phrase Structure Grammar (GPSG) [2], Lexical Functional Grammar (LFG) [4], and Sys- 257 2 Disjunction and Non-Local Values Karttunen [5] has shown that disjunction and negation are desirable extensions to PATR-II which are motivated by a wide range of linguistic local path values in Section 5.4. die : ,,¢reement : 3 number : 8ia¢ aumber : pl ] Logical Formulas for Feature Structures The feature structure of Figure 1 can also be represented by a type of logical formula: Figure 1: A Feature Structure containing Value Disjunction. die = case : (hOrn V acc) A phenomena. He discusses specifying attributes by disjunctive values, as shown in Figure 1. A ~alue disjuactioa specifies alternative values of a single attribute. These alternative values may be either atomic or complex. Disjunction of a more general kind is an essential element of FUG. Geaera/ disjunction is used to specify alternative groups of multiple attributes, as shown in Figure 2. Karttunen describes a method by which the basic unification procedure can be extended to handle negative and disjunctive values, and explains some of the complications that result from introducing value disjunction. When two values, A and B, are to be unified, and A is a disjunction, we cannot actually unify B with both alternatives of A, because one of the alternatives may become incompatible with B through later unifications. Instead we need to remember .a constraint that at least one of the alternatives of A must remain compatible with B. An additional complication arises when one of the alternatives of a disjunction contains a value which is specified by a non-local path, a situation which occurs frequently in Functional Unification Grammar. In Figure 2 the obj attribute in the description of the adjunct attribute is given the value < actor >, which means that the obj attribute is to be unified with the value found at the end of the path labeled by < actor > in the outermost enclosing structure. This unification with a non-local value can be performed only when the alternative which Contains it is the only alternative remaining in the disjunction. Otherwise, the case = objective attribute might be added to the value of < actor > prematurely, when the alternative containing adjunct is not used. Thus, the constraints on alternatives of a disjunction must also apply to any non-local values contained within those alternatives. These complications, and the resulting proliferation of constraints, provide a practical motivation for the logical treatment given in this paper. We suggest a solution to the problem of representing non- a~'eement : ( (gender : fern A number : sing) V n u m b e r : pl) This type of formula differs from standard propositional logic in that a theoretically unlimited set of atomic values is used in place of boolean values. The labels of attributes bear a superficial resemblance to modal operators. Note that no information is added or subtracted by rewriting the feature matrix of Figure 1 as a logical formula. These two forms may be regarded as notational variants for expressing the same facts. While feature matrices seem to be a more appealing and natural notation for displaying linguistic descriptions, logical formulas provide a precise interpretation which can be useful for computational and mathematical purposes. Given this intuitive introduction we proceed to a more complete definition of this logic. 4 A Logical Semantics As Pereira and Shieber [11] have pointed out, a grammatical formalism can be regarded in a way similar to other representation languages. Often it is useful to use a representation language which is disctinct from the objects it represents. Thus, it can be useful to make a distinction between the domain of feature structures and the domain of their descriptions. As we shall see, this distinction allows a variety of notational devices to be used in descriptions, and interpreted in a consistent way with a uniform kind of structure. 258 4.1 Domain of Feature Structures The graphs feature feature quired: PATR-II system uses directed acyclic (dags) as an underlying representation for structures. In order to build complex structures, two primitive domains are re- cat ~ subj S = [ case = nominative ] actor = < sub.7' > voice = passive goal = < subj > cat = pp adjunct = prep = by obj = < actor > = [ case = objective ] mood = declarative ] mood interrogative ] f Figure 2: Disjunctive specificationcontaining non-local values, using the notation of F U G . (b) unification is equivalent to a statemerge operation; 1. atoms (A) 2. labels (L) 4. the techniques of automata theory become available for use with feature structures. The elements of both domains are symbols, usually denoted by character strings. Attribute I ~ belt (e.g., acase~) are used to mark edges in a dag, and atoms (e.g., "gen z) are used as primitive values at vertices which have no outgoing edges. A dag may also be regarded as a transition graph for a partially specified deterministic finite automaton (DFA). This automaton recognises strings of labels, and has final states which are atoms, as well as final states which encode no information. An automaton is formally described by a tuple .~ = ( Q , L , 5,qo, F) A consequence of item 3 above is that the dis- ," tinction between type identity and token identity it clearly revealed by an automaton; two objects are necessarily the same token, if and only if they are represented by the same state. One construct of automata theory, the Nerode relation, is useful to describe equivalent paths. If #q is an automaton, we let P(A) be the set of all paths of ~4, namely the set {z E L* : 5(q0, z) is defined }. The Nerode relation N ( A ) is the equivalence relation defined on paths of P(~) by letting where L is the set of labels above, 6 is a partial function from Q × L to Q, and where certain elements of F m a y be atoms from the set A. W e require that ~ be connected, acyclic,and have no transitions from any finalstates. D F A s have several desirable properties as a domain for feature structures: 4.2 Domain Logical of Descriptions: Formulas We now define the domain FML of logical formulas which describe feature structures. Figure 3 defines the syntax of well formed formulas. In the following sections symbols from the Greek alphabet axe used to stand for arbitrary formulas in FML. The formulas N I L and T O P axe intended to convey gno information z and ~inconsistent information s respectively. Thus, N I L corresponds to a unification variable, and T O P corresponds to unification failure. A formula l : ~bwould indicate that a value has attribute l, which is itself a value satisfying the condition ~b. 1. the value of any defined path can be denoted by a state of the automaton; 2. finding the value of a path is interpreted by running the automaton on the path string; 3. the automaton captures the crucial properties of shared structure: (a) two paths which axe unified have the same state as a value, 259 2. 11 ~ T O P never; NIL TOP aEA ~< 191 > , . . . , < 19, >] where each 19~ E L* l:~bwherelELand~bEFML 3. /l ~ a ¢ = ~ /I is the one-state automaton a with no transitions; 4. A ~ E ¢ = ~ E is a subset of an equivalence class of N(~); 5. A ~ l : cb ¢=~ A / l is defined and A / I ~ ~; ¢v¢ Figure 3: The domain, FML, of logical formulas. where ~ / I is defined by a subgraph of the aut o m a t o n A with start state 5(qo, l), that is Conjunction and disjunction will have their ordinary logical meaning as operators in formulas. An interesting result is t h a t conjunction can be used to describe unification. Unifying two structures requires finding a structure which has all features of both structures; the conjunction of two formulas describes the structures which satisfy all conditions of b o t h formulas. One difference between feature structures and their descriptions should be noted. In a feature structure it is required that a particular attribute have a unique value, while in descriptions it is pouible to specify, using conjunction, several values for the same attribute, as in the formula s bj : (19e.so. : 3) ^ s bj: ira = (Q,L, 6, qo, F), then .~/l = (Q', L, 6, 6(qo, l), f ' ) ; where Qi and F ' are formed from Q and F by removing any states which are unreachable from 6(q0, : 4.4 A feature structure satisfying such a description will contain a unique value for the attribute, which can be found by unifying all of the values that are specified in the description. Formulas may also contain sets of paths, denoting equivalence classes. Each element of the set represents an existing path starting from the initial state of an automaton, and all paths in the set are required to have a c o m m o n endpoint. If E = I < z >, < y >~, we will sometimes write E as < z > = < y >. This is the notation of PATRII for pairs of equivalent paths. In subsequent sections we use E (sometimes with subscripts) to stand for a set of paths that belong to the same equivalence class. 4.3 0. Any formula can be regarded as a specification for the set of a u t o m a t a which satisfy it. In the case of conjunctive formulas (containing no occurences of disjunction) the set of a u t o m a t a satisfying the formula has a unique minimal element, with respect to subsumption.* For disjunctive formulas there m a y be several minimal elements, but always a finite number. Calculus of Formulas It is possible to write m a n y formulas which have an identical interpretation. For example, the formulas given in the equation below are satisfied by the same set of a u t o m a t a . case : (gen V ace V dat) A case : ace = case : ace In this simple example it is clear that the right side of the formula is equivalent to the left side, and t h a t it is simpler. In more complex examples it is not always obvious when two formulas are equivalent. Thus, we are led to state the laws of equivalence shown in Figure 4. Note that equivalence (26) is added only to make descriptions of cyclic structures unsatisfiable. I n t e r p r e t a t i o n of Formulas We can now state inductively the exact conditions under which an a u t o m a t o n Jl satisfies a formula: 1A subsumption order can be defined for the domain of automata, just as it is defined for dags by Shieber [15]. A formal definition of subsurnption for this domain appears in [12]. 1. A ~ N I L always; 260 Failure: = TOP (1) ¢ A TOP = TOP CANIL = ~b = T O P , Va, b 6 A a n d a # b = TOP (2) (3) (4) (s} (6} l : TOP Conjunction (unification}: aAb aAl:¢ /:¢AZ:,#, = t:(¢A¢) Disjunction: ¢ v NIL = ¢vTOP = z:¢v~:¢ = (7) is) (9) NIL t:(¢v¢) Commutative: ¢A¢ = ¢ ^ ¢ ¢v¢ = ¢v¢ (1o) (11) Associative: (¢^¢)^x = ¢^(¢^x) (n) (¢v¢)vx = ¢,v(¢vx) (13) ¢A~ = ~b 4v4 @ (14) (15) Idempotent: = Distributive: (~v¢)^x = (~^x) v ( ¢ ^ x ) (~,A¢)Vx = (~VX)^(¢VX) (16) (17) = ~, (18) = 4, (19) Absorption: (¢A¢)V~ (¢v¢)A¢ Path Equivalence: E1 A E 2 E, ^ E2 ----- E2 whenever E1 _C E2 (20) = (21) E1 ^ (E2 u { z y I ~ e El}) for any y such that 3 z : z ~ E l and zy E E2 EAz:c -- EA(A y:c) wherexeE (22) glEE E = E A {z} if" z is a prefix of a string in E (23) (24) = NIL (2s) = T O P for any E such that there are strings (26) l:E {,) E z, z y E E and y # e Figure 4: Laws of Equivalence for Formulas. 261 5 Complexity of Disjunctive Descriptions exponent depends on the number of disjunctions in the formula (in the worst case). To date, the primary benefit of using logical formulas to describe feature structures has been the clarification of several problems that arise with disjunctive descriptions. 5.1 5.3 Most of the systems which are currently used to implement unification-based grammars depend on an expansion to disjunctive normal form in order to compute with disjunctive descriptions. 2 Such systems are exemplified by Definite Clause G r a m m a r [10], which eliminates disjunctive terms by multiplying rules which contain them into alternative clauses. Kay's parsing procedure for Functional Unification G r a m m a r [8] also requires expanding functional descriptions to DNF before they are used by the parser. This expansion may not create much of a problem for grammars containlng a small number of disjunctions, but if the grammar contains 100 disjunctions, the expansion is clearly not feasible, due to the exponential sise of the DNF. Ait-Kaci [1] has pointed out that the expansion to DNF is not always necessary, in work with type structures which are very similar to the feature structures that we have described here. Although the NP-completeness result cited above indicates that any unification algorithm for disjunctive formulas will have exponential complexity in the worst case, it is possible to develop algorithms which have an average complexity that is less prohibitive. Since the exponent of the complexity function depends on the number of disjunctions in a formula, one obvious way to improve the unification algorithm is to reduce the number of disjunctions in the formula be/ors ezpan.sion to DNF. Fortunately the unification of two descriptions frequently results in a reduction of the number of alternatives that remain consistent. Although the fully expanded formula may be required as a final result, it is expedient to delay the expansion whenever possible, until after any desired unifications are performed. The algebraic laws given in Figure 4 provide a sound basis for simplifying formulas containing disjunctive values without expanding to DNF. Our calculus differs from the calculus of AitKaci by providing a uniform set of equivalences for formulas, including those that contain disjunction. These equivalences make it possible to ~ NP-completeness of consistency problem f o r f o r m u l a s One consequence of describing feature structures by logical formulas is that it is now relatively easy to analyse the computational complexity of various problems involving feature structures. It turns out that the satisfiability problem for CNF formulas of propositional logic can be reduced to the consistency (or satisfiability) problem for formulas in FML. Thus, the consistency problem for formulas in FML is NPcomplete. It follows that any unification algorithm for FML formulas will have non-polynomial worst-case complexity (provided P ~ NP!), since a correct unification algorithm must check for consistency. Note that disjunction is the source of this complexity. If disjunction is eliminated from the domain of formulas, then the consistency problem is in P. Thus systems, such as the original PATR-II, which do not use disjunction in their descriptions of feature structures, do not have to contend with this source of NP-completeness. 5.2 Disjunctive Normal Avoiding expansion to DNF Form A formula is in disjt, neti,~s normal form (DNF) if and only if it has the form ~1 V ... v ~bn, where each ~i is either 1. s E A 2. ~bx A . . . A ~bm, where each ~bl is either (a) lx : . . . : lk : a, where a E A, and no path occurs more than once (b) [< pl > , . . . , < p~ >], where each p~ E L*, and each set denotes an equivalence class of paths, and all such sets disjoint. The formal equivalences given in Figure 4 allow us to transform any satisfiable formula into its disjunctive normal form, or to T O P if it is not satisfiable. The algorithm for finding a normal form requires exponential time, where the 2One exception is K a r t t u n e n ' s i m p l e m e n t a t i o n , which was described in Section 2, b u t it h a n d l e s only value disjunctions, a n d does n o t h a n d l e non-local p a t h values e m b e d d e d w i t h i n disjunctions. 262 eliminate inconsistent terms before expanding to DNF. Each term thus eliminated may reduce, by as much as half, the sise of the expanded formula. 5.4 R e p r e s e n t i n g Non-local P a t h s The logic contains no direct representation for non-local paths of the type described in Section 2. The reason is that these cannot be interpreted without reference to the global context of the formula in which they occur. Recall that in Functional Unification G r a m m a r a nonlocal path denotes the value found by extracting each of the attributes labeled by the path in successively embedded feature structures, beginning with the entire structure currently under consideration. Stated formally, the desired interpretstion of I :< p > is A~l:<p> voice = p a s s i v e goal = < s u b j > cat = pp prep = by adjenct = obj = < actor > = [ case----objective ] Figure 5: Functional Description containing non-local values. in the context o f ~ 3B ~ and 3 w E L * : voice : p a s s i v e ^ goal :< s u b j > ^ a d j u n c t : (eat : pp ^ prep : by ^ obj :< actor > ^ obj : ease : objective) E / t o = A a n d 5 ( q o , , l) = 5(qo, ,p). This interpretation does not allow a direct comparison of the non-local path value with other values in the formula. It remains an unknown quantity unless the environment is known. Instead of representing non-local paths directly in the logic, we propose that they can be used within a formula as a shorthand, but that all paths in the formula must be expanded before any other processing of the formula. This p a t h e x p a n s i o n is carried out according to the equiva~ lences 9 and 6. After path expansion all strings of labels in a formula denote transitions from a common origin, so the expressions containing non-local paths can be converted to the equivalence class notation, using the schema 11 : . . . : I n : < p > path expansion voice : p a s s i v e ^ goal :< sub3" > ^ a d j u n c t : eat : pp ^ a d j u n c t : prep : by ^ a d j u n c t : obj :< actor > ^ a d j u n c t : obj : ease : objective path equivalence ==~ voice : p a s s i v e ^ [< goat >, < s u b j >] ^ a d j u n c t : cat : pp /~ a d j u n c t : prep : by ^ [< a d j u n c t obj >, < actor >] ^ a d j u n c t : obj : case : objective = [<11 . . . . ,In > , < p >]. Consider the passive voice alternative of the description of Figure 2, shown here in Figure 5. This description is also represented by the first formula of Figure 6. The formulas to the right in Figure 6 are formed by Figure 6: Conversion of non-local values to equivalence classes of paths. 1. applying path expansion, 2. converting the attributes containing nonlocal path values to formulas representing equivalence classes of paths. By following this procedure, the entire functional description of Figure 2 can be represented by the logical formula given in Figure 7. 263 number of disjunctions in a formula without expanding to DNF. cat : s A s u b j : case : n o m i n a t i v e A Note that path expansion does not require an expansion to full DNF, since disjunctions are not multiplied. While the DNF expansion of a formula may be exponentially larger than the original, the path expansion is at most quadratically larger. The size of the formula with paths expanded is at most n x p, where n is the length of the original formula, and p is the length of the longest path. Since p is generally much less than n the size of the path expansion is usually not a very large quadratic. ((vdce : ac~ve ^ [< acto,. >, < subj >i) V (voice : pas~ve ^ |< goal >, < subj >] A a d j u n c t : cat : pp A adjunct : prep : by A [< a d j u n c t obj >, < actor >] ^ a d j u n c t : obj : case : objective)} ^ ( m o o d : declarative V mood : interrogative) 5.5 and General Disjunction The path expansion procedure illustrated in Figure 6 can also be used to transform formulas containing value disjucntion into formulas containing general disjunction. For the reasons given above, value disjunctions which contain non-local path expressions must be converted into general disjunctions for further simplification. While it is possible to convert value disjunctions into general disjunctions, it is not always possible to convert general disjunctions into value disjunctions. For example, the first disjunction in the formula of Figure 7 cannot be converted into a value disjunction. The left side of equivalence (9) requires both disjuncts to begin with a common label prefix. The terms of these two disjuncts contain several different prefixes (voice, actor, s u b j , goat, and a d j u n c t ) , so they cannot be combined into a common value. Before the equivalences of section 4 were formulated, the first author attempted to implement a facility to represent disjunctive feature structures with non-local paths using only value disjunction. It seemed that the unification algorithm would be simpler if it had to deal with disjuncti+ns only in the context of attribute values, rather than in more general contexts. While it w ~ possible to write down grammatical definitions using only value disjunction, it was very difficult to achieve a correct unification algorithm, because each non-local path was much like an unknown variable. The logical calculus presented here clearly demonstrates that a representation of general disjunction provides a more direct method to determine the values for non-local paths. Figure 7: Logical formula representing the description of Figure 2. It is now possible to unify the description of Figure 7 (call this X in the following discussion) with another description, making use of the equivalence classes to simplify the result. Consider unifTing X with the description Y Value Disjunction = actor : case : n o m i n a t i v e . The commutative law (10) makes it possible to unify Y with any of the conjuncts of X. If we unify Y with the disjunction which contains the vo/ce attributes, we can use the distributive law (16) to unify Y with both disjuncts. When Y is unified with the term containing [< a d j u n c t obj >, < actor >], the equivalence (22) specifies that we can add the term a d j u n c t : obj : case : n o m i n a t i v e . This term is incompatible with the term a d j u n c t : obj : case : objective, and by applying the equivalences (6, 4, 1, and 2) we can transform the entire disjunct to T O P . Equivalence (8) specifies that this disjunction can be eliminated. Thus, we are able to use the path equivalences during unification to reduce the 264 6 Implementation tures with other grammatical formalisms based on logic, such as DCG [10] and LFP [13]. The calculus described here is currently being implemented as a program which selectivelyapplies the equivalences of Figure 4 to simplify formulas. A strategy (or algorithm) for simplifying formulas corresponds to choosing a particular order in which to apply the equivalences whenever more than one equivalence matches the form of the formula. The program will make it possible to test and evaluate differentstrategies,with the correctness of any such strategy following directly from the correctness of the calculus. While this program is primarily of theoreticalinterest,it might yield useful improvements to current methods for processing feature structures. The original motivation for developing this treatment of feature structures came from work on an experimental parser based on Nigel [9], a large systemic grammar of English. The parser is being developed at the USC/Information Sciences Instituteby extending the PATR-II system of SRI International. The systemic grammar has been translated into the notation of Functional Unification Grammar, as described in [6]. Because References [1] Ait-Kaci, H. A New Model of Computation Based on a Calculus of Type Subsumption. P h D thesis, University of Pennsylvania, 1984. [2] Gazdar, G., E. Klein, G.K. Pullum, and I.A. Sag. Generalized Phrase Structure Grammar. BlackweU Publishing, Oxford, England, and Harvard University Press, Cambridge, Massachusetts, 1985. [3] G.R. Kress, editor. Halliday: System and Function in Language. Oxford University Press, London, England, 1976. [4] Kaplan, R. and J. Bresnan. Lexical Functional Grammar: A Formal System for Grammatical Representation. In J. Bresnan, editor, The Mental Representation of Grammatical Relations. MIT Press, Cambridge, Massachusetts, 1983. [5] Karttunen, L. Features and Values. In Proceedings of the Tenth International Conference on Computational Linguistics, Stanford University, Stanford, California, July 2-7, 1984. [6] Kasper, R. Systemic Grammar and Functional Unification Grammar. In J. Benson and W. Greaves, editors, Proceedings of the I ~ h International Systemics Workshop, Norwood, New Jersey: Ablex (forthcoming). [7] Kay, M. Functional Grammar. In Proceedings of the Fifth Annual Meeting of the Berkeley Linguistics Society, Berkeley Linguistics Society, Berkeley, California, February 17-19, 1979. [8] Kay, M. Parsing in Functional Unification Grammar. In D. Dowty, L. Kartunnen, and A. Zwicky, editors, Natural Language Parsing. Cambridge University Press, Cambridge, England, 1985. [9] Mann, W.C. and C. Matthiessen. Nigel: A Systemic Grammar for Text Generation. USC / Information Sciences Institute, RR83-105. Also appears in R. Benson and J. Greaves, editors, Systemic Perspectives on Discourse: Selected Papers Papers from the Ninth International Systemics Workshop, Ablex, London, England, 1985. this grammar contains a large number (several hundred) of disjunctions, it has been necessary to extend the unification procedure so that it handles disjunctive values containing non-local paths without expansion to DNF. We now think that this implementation of a relatively large grammar can be made more tractable by applying some of the transformations to feature descriptions which have been suggested by the logical calculus. 7 Conclusion W e have given a precise logical interpretation for feature structures and their descriptions which are used in unification-basedgrammar formalisms. This logic can be used to guide and improve implementations of these grammmm, and the processors which use them. It has allowed a closer examination of several sources of complexity that are present in these grammars, particularly when they make use of disjunctive descriptions. W e have found a set logical equivalences helpful in suggesting ways of coping with this complexity. It should be possible to augment this logic to include characterizationsof negation and implication, which we are now developing. It m a y also be worthwhile to integrate the logic of feature struc- 265 [10] Pereira, F. C. N. and D. H. D. Warren. 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M. An Introduction to Uai~ation-bo~ed Approaches to Grammar. Chicago: University of Chicago Press, CSLI Lecture Notes Series (forthcoming). 266