A Generalized Greibach Normal Form for Definite Clause Grammars
Marc Dymetman CCRIT, Communications Canada 1575 boul. Chomedey Laval (Qutbec) H7V 2X2, Canada [email protected]
Abstract In other words, there does not exist, in general, an al- gorithm for deciding whether a definil~ clause grammar An arbitrary definite clause grammar can be transforaled DCGI accepts a string String, i.e. whether there exists into a so-called Generalized Greibach Normal Form some linguistic structure S such that DCG1 "analyses (GGNF), a generalization of the classical Greibach Nor- String into S'. On the other hand, ander the condi- mat Form (GNF) for context-free grammars. tion that DCCx is offline-parsable, that is, under the The normalized definite clause grammar is declara- condition that the context-free skeleton of DCG1 is not tively equivalent to the original definite clause grammar, infinitely ambiguous, then the parsing problem is indeed that is, it assigns the same analyses to the same strings. decidable [7]. Offline-parsability of the original grammar is reflected in The fact that the parsing problem with DCGt is de- an elementary textual property of the transformed gram- cidable in theory should be carefully distinguished from mar. When this property holds, a direct (top-down) Pro- the fact that a given algorithm for parsing is able to ex- log implementation of the normalized grammar solves ploit this potentiality. A parsing algorithm for which this the parsing problem: all solutions are enumerated on is the case, that is, an algorithm which is able, for any backtracking and execution terminates. offline-parsable DCGx, to decide whether a given string When specialized to the simpler case of context-free is accepted or not by DCG1 is said to be strongly stable grammars, the GGNF provides a variant to file GNF, [7]. Most parsing algorithms are not strongly stables, a where the transformed context-free grammar not only notable exception being the "Earley deduction" algorithm generates the same strings as the original grammar, but [7], once modified according to the proposals in [10].4 also preserves their degrees of ambiguity (this last prop- Top-down implementations--and in particular the erty does not hold for the GNF). standard Prolog implementation of a DCG---are espe- The GGNF seems to be the first normal form result cially fragile in this respect, for they fail to terminate as for DCGs. It provides an explicit factorization of the soon as the grammar contains a left-recursive rule such potential sources of uudeeidability for the parsing prob- as: lem, and offers valuable insights on the computational structure of unification grammars in general. vp(VP2) ~ vp(VPt),pp(PP), { combine(V P1, P P, V P2)}. 1 Introduction In [2], automatic local transformations were performed on a DCG in order to eliminate some limited, but frequent From the point of view of decidability, the parsing prob- in practice, types of left-recarsion in such a way that lem with a definite clause grammar is much more com- the resulting grammar could be directly implemented in plex than the corresponding problem with a context-free Prolog. This initial work led us naturally to the more grammar. The parsing problem with a context-live gram- general question: mar is decidable, i.e. there exists an algorithm which de- Question: Is it possible to automatically transform an termines whethex a given string is accepted or not by the grammar;t by contrast, the parsing problem with a deft- assumption that auxiliary predicates are unifications takes care of this interfering problem. nite clause grammar is undecidable, even if the auxiliary aAn important cl~s of offline-parsable grammars---which Prolog predicates are restricted to be unifications.2 we will call here the class of explicitly offline-parsable grammars--is the class of DCGs whose context free skeleton t For instance the standard top-down parsing algorithm using is a proper context-free grammar, that is, a grammar without a GNF of the grammar. rules of the type A ~ [ ] (empty productions) or of the type 2This assumption, or a similar one, is always made when A ~ B (chain rules) [1]. This subclass is much less problem- the decidability proparfies of logical (or unification) grammars atic to parse than the full class of offline-parsuble DCGs (for are studied. The reason is simple: if the auxiliary predicates instance a left-comer parsing algorithm will work). However, are defined through an unrestricted Prolog program, there is it is an easy consequence of the GGNF result that, for any no means of guaranteeing that calling some auxill~y predicate offline-parsable DCG, there exists an explicitly offline-parsable will not result in nontexmination, for reasons quite independent DCG equivalent to it. from the structure of the grammar under consideration. The 4Stuart Shieber, personal communication.
AcrEs DE COLIN'G-92, NANT~.s,23-28 hOt]T 1992 3 6 6 PROC. OF COLING-92, NANTES, AUG. 23-28, 1992 arbitrary oflliue-parsable DUG1 into an equivalent 2. Auxiliary clauses, constituting an autonomous def- DCG2 in such a way that the standard top-down ex- inite program, defining rite auxiliary predicates ap~ ecution of DCG2 terminates (in success or failure) pcaring in the right-hand sides of nonterminal rules. for any given input string? These clauses am written: To answer this question (in the positive), we have taken p(7~ ..... 7)) :- f~ , an indirect route, and, adapting to the case of definite clause grammars some of the standard techniques used where fl is some sequence of predicate goals. to transform context-free grammars into GNF, s have established the following results: A definite grammar scheme DGS is syntactically iden- tical to a definite clause grammar, except for the fact that: 1. Generalized Greibach Normal Form for definite clause grammars It is possible to translorm any 1. Tile '/} arguments appearing in the nonterminal and definite clause grammar DCG1 (oflline-parsable or auxiliary predicate goals ,are restricted to beiug vari- not) into a definite clause grammar DUG2 equiva- ables: no constants or complex terms are allowed; lent to DCGI~at is, assigning the same analyses to the same strings--and which is in a certain form, 2. Only nonterminal clauses appear in the definite called the Generalized Greibach Normal Form, or grannnar scheme, but no auxiliary clause; the auxil- GGNF, of DCG1. iary predicates which may appear in the right-hand sides of clauses do not receive a definition. 2. Explicit representation of offline-parsability in the GGNF The offline-parsability of DCGI is equiva- A definite grammar scheme can be seen as an un- lent to an elementary textual property of DCG2: the completely specified definite clause grammar, that is, a so-called "unit subgrammar" contains no cycle, i.e. definite clause granmtar "lacking" a definition for the no nonterminal calling itself directly or indirectly. ~ auxiliary predicates {p(Xt ..... X,)} that it "uses". The auxiliary predicates are "free": the interpretation of p is 3. Termination condition for top-down parsing with the not fixed a priori, but can be an arbitrarily chosen n-ary GGNF If DCG1 is offline-parsable, and such that relation on a certain Herbrand universe of terms. 7 its auxiliary predicates it are unifications, then, for any input String, the standard top-down (Prulog) Example 1 The following clauses define a definite execution of DCG2 enumerates all solutions to the grammar scheme DGSI :s parsing problem for String and terminates. a l(X)--* a3(E), a l(A), a2(B), {q(E,A, B, X)} 2 The Generalized Greibach Normal al(X) ~ [oh], {pl(X)} .2(X) ~ [dull, {p2(X)} Form for Definite Clause Gram- a3(X) ~ [ ], {p3(X)} mars In this definite grammar scheme, only variables appear 2.1 Definite Grammar Schemes as argutnents; the auxiliary predicates pl, p2, p3 and q do not receive a definition. As usually defined (see [6]), a definite clause grammar consists in two separate sets of clauses: If a definite program definiug these auxiliary predicates 1. Nonterminal clauses, written as: is added to the definite grammar scheme, one obtains a full-fledged definite clause grammar, which can be inter- a(Tl ..... 7h) --, a , preted in the usual manner. where n is a nonterminal, T~ ..... T. am terms (vari- Conversely, every definite clause grammar can be seen able.s, constants, or complex terms), and a is a se- as the conjunction of a definite grammar scheme and of quence of "terminal goals" [term], of "nontenni- a definite clause program. In order to do so, a minor nal goals" b(T{,..., 7;'), and of"auxiliary predicate transforntation must be performed: each complex term goals" {p(Ti', .... T~')}. T appearing as an argument in the head or body of a nonterminal clause must be replaced by a variable X, SSee the Appendix for some indications on the methods used. this variable being implicitly constrained to unify with T SThus. a side-effect of the GGNF is to provide a deci- through the addition of an ad-hoc unification goal in the sion procedure for the problem of knowing whether a DCG is body of the clause (see [3]). offline-parsable or not. This is equivalent to deciding whether a eomext-free grammar is infinitely ambiguous or not, a problem rThe domain of interpretation can in fact be any set. Taking the decidability of which seems to be "quasi.folk-knowledge", it to be the Ilerbrand universe over a certain vocabulary of although I was innocent of this until the fact was brought to my functional symbols permits to "simulate" a DCG, by fixing the attention by Yves Schabes, among others: the proof is more or interpretation of the free auxiliary predicates in this domain. less implicit in the usual technique to make a CFG "cycle-free", Another linguistically relevant domain of interpretation is the [1, p. 150] . See also [4] for a special case of this problem. set of directed acyclic graphs built over a certain vocabulary (Caveat. The notion of "cycle" in "cycle-free" is technically of labels and atomic symbols, which pemtits tile simulation of different from the notion used here, which simply means: cycle unification grammars of the PATR-II type. in the graph associated with the relation "callable from". See SThe usual symbol for the initial nonterminal is s; we prefer note 10.) to use al for reasons of notational coherence.
ACRES DE COLING-92, NAhq~S, 23-28 AO~r 1992 3 6 7 PROC, O1: COLING-92, NAh'rt:,s, AUG. 23-28, 1992 Example 2 Consider the following definite clause gram- 2.2 GGNF for definite clause grammars mar: Structure of the GGNF The definite grammar scheme DGS, on the terminal vocabulary V, having Q as its al(cons(B, A)) ---, a3(E), al(A), a2(B). set of auxiliary predicates, is said to be in Generalized Greibach Normal Form if: al(X) -. [oh], {pl(x)}. a2(f) ~ toni]. 1. The nonterminals of DGS are partitioned in three a3(X) ---* [ ], {p3(X)}. distinct subsets: A = {al}, called the initial set; U, called the set of unit nonterminals; N, called the set of co-unit nonterminals. pl(nil). p3(r). 2. The rules of DGS are partitioned into three groups of rules, called the factorization group (defining the Let us define two new predicates p2 and q by the fol- elements of A), the unit group (defining the elements lowing clauses: of U), and the co-unit group (defining the elements of N), graphically presented in the following man- p2(/). net: q(E, A, B, cons(B, A)). factorization rules: definition of the elements of A then the definite clause grammar above can be rewritten as: unit rules: definition of the elements of U al(X) ---, a3(E), al(A), a2(B), {q(E, A, B, X)}. al(X) -. [oh], {pl(x)}. co-unit rules: definition of the elements of N a2(X) ---, [oui], {p2(X)}. 3. Factofization rules are taken among the two follow- a3(X) --* [ 1, {p3(x)}. ing rules:
pl(nil). al(Xl ..... X.) --, nl(X1 ..... X,~) , p2(f). and/or: pS(r). al(Xl,...,Xn) "~ ul(Xl ..... Xn) , q(E, A, B, cons(B, A)). where nl E N and ul E U, and where n E N is that is, in the form of a definite grammar scheme to the arity of the initial nonterminal al. which has been added a set of auxiliary clauses defining 4. Unit rules are of the form: its auxiliary predicates. This definite grammar scheme is in fact identical with DGSI (see previous example). u(Xl ..... X,,) --* tl , In the sequel of this paper, we will be interested, not where u E U is a unit nonterminal of arity m, m E directly in transformations of definite clause grammars, N, and where H is a finite sequence of nonterminal but in transformations of definite grammar schemes. The unit goals of U, of auxiliary predicates of Q, or is transformation of a definite grammar scheme DGS into the empty string []. The group of unit rules forms a DGS ~ will respect the following conditions: subscheme of the GGNF definite grammar scheme • The auxiliary predicates of DGS and of DGS' are (see below). the same; 5. Co-unit rules are of the form: • For any definite clause program P which defines the auxiliary predicates in DGS (and therefore also n( Xt , . . . , X~ ) --+ [term] Af , those in DGSI), the definite clause grammar DCG obtained through the adjunction of P to DGS where n E N is a co-unit nonterminal of arity k, has the same denotational semantics as the definite k E N. where [term] E V and where .hi is a finite clause grammar DCG' obtained through the adjunc- sequence of terminal goals of V, of nonterminal unit tion of P to DGS j. goals of U, of auxiliary predicates of Q, or of non- terminal co-unit goals of N. Under the preceding conditions, DGS and DGS' are stud to be equivalent definite grammar schemes. 6. The context-free skeleton of DGS, considered as a The grammar Vansformatious thus defined are, in a context-free grammar, is reduced. 9 certain sense, universal transformations:, they are valid °A context-free grammar is said to be reduced iff it all its independently from the interpretation given to the auxil- nonterminals are accessible from al and are productive (see [5, iary predicates. pp. 73-78])
ACrEs DE COLING-92, Nx~rI~s. 23-28 AO~T 1992 3 6 8 P~toc. Or COL1NG-92, NANTES, AU(~. 23-28, 1992 The nonterminals of A, U, and N am defined in func- I. If tile unit subscheme does not contain a cycle, then, tion of one another, as well as in function of [ ], of the for arty input string Siring, the standard top-down terminals of V, and of the auxiliary predicates of Q, ac- parsing algorithm terminates, after enumerating all cording to the definitional hierarchy illastrated below: the analyses for Siring;
2. If the unit subscheme contains a cycle, the top-down parsing algorithm can terminate or not, depending on the definition given to the auxiliary predicates.
We give below three exmnples of definite grammar G i schemes, and of the equivalent definite grammar schemes in GGNF. v :o:: ti: 2.5 Examples
2.3 Structure of the unit subscheme and Example 3 Consider the definite grammar scheme DGS1 given in Example 1, repeated below: offline-parsability One can remark that: al(X)-~ a3(E), al(A), a2(B), {q(E,A, B,X)} ul(X) ~ [oh], {pl(X)} • The group of unit rules is closed: file definition of a2(X) --, [dull, {p2(X)} unit nontenninals involves only unit nonterminals (but no co-unit nonterminal). For this reason, the ~3(x) -, [ j, {p3(x)) group of unit roles is called the unit subscheme (or, loosely, the unit subgrammar) of the GGNF definite The following definite grammar scheme DGS;u is in grammar scheme. GGNF and is equivalent to DGSI:
• The unit subscheme can only generate the empty al(X) --~ nl(X) string [ ]. ¢ The unit subscheme of the GGNF is said to contain a cycle iff there exists a unit nonterminal u(X1, .... X,,~) which "calls itself recursively", directly or indirectly, in- nl(X) --* [oh], {pl(X)} side this group./° One can show that this property is nl(X) --* [dill, {pl(Y)}, h(Y,X) equivalent to the fact that the context-free skeleton of h(Y,X) ~ [dull, {p2(B)}, {p3(E)}, DGS is infinitely ambiguous, or, in other words, to the fact that DGS is not offline-parsable [3]. {q(E, Y, I3, X)} h(Y,X)--, [oui], {p2(B)}, {p3(E)}, 2.4 'lop-down parsing with the GGNF {q(E, 1I, B, Z)}, h(Z, X)
Let DGS be a definite grammar scheme in GGNF, having Suppose P is any auxiliary definite program which Q for its set of auxiliary predicates. Assume that every defines the auxiliary predicates p l,p2,p3, q. Then the element p of Q, of arity n, is defined tlLrough a head definite clause grammars DCGt and DCG2, obtained clauset ~, of the form: by adjunction of this program to DGSt and DGS2, re- spectively, are equivalent. p(TI ..... T,,). The unit subscheme of DGS~ does not contain a cy- cle (it is empty12). One can conclude that DCGI, as where 7'1, ... ,Tn can be any terms; In other words, the well as DCG~, are offline-parsable. If, moreover, it is auxiliary predicates are constrained to be simply unifica- assumed that P defines the auxiliary predicates as be- tions. Let DCG be the definite clause grammar obtained ing unifications, then it can be concluded that top-down through adjunction of these clauses to DGS. The gram- parsing with DCG~ will enumerate all possible analyses mar DCG has the following properties: for a given string and terminate. t°For example, the scheme: For instance, assume that the auxiliary program con- sists in the following four clauses (see Example 2): t~t(X) ~ u2(g), u3(Z).... ,,2(x) ~ udY). m(Z) .... pl(nil). p2(f). contains a cycle in ul • p3(r). n We use the terminology 'head clause' for a clause without body. A more standard terminology would be 'unit clause', q(E, A, fl, cons(B, A)). but this would conflict with our technical notion of "unit' (a nonterminai generating the empty string [ ]). 12See note 14.
ACRESDE COLING-92, NAhn'as, 23-28 AOl)r 1992 3 6 9 PROC. OF COLING-92, NANTES, AUG. 23-28, 1992 Then, DCGt becomes: The GGNF of DGSs is DGS4 below:
al(X)-* a3(E), at(A), a2(B), {q(E,A,B,X)}. al(X) --, nt(X) al(X)--~ [oh], {pl(X)}. al(X) ---* ut(X) a2(X) ---, [oui], {p2(X)}. a3(X)--, [1, {p3(X)}. ul(X) --* u2(Y), {r(X,Y)} u2(x) ~ [], {q(X)] pt(nil). p2(f). nt(X) --* [ouil, nl(Y), {v(x, Y)} p3(r). nl(X) --, [oui], ulW), {p(X, Y)} q(E, A, B, cons(B, A)). From an inspection of DGS4 it can be concluded that: and DCG2: • The unit subscheme does not contain a cycle} 4 at(X) ---* nl(X) Therefore DGS4, and consequently DGSa, is offline-parsable. nt(X) ---* [ohl, {pl(X)} nl(X)---.* [oh], {pt(Y)}, h(Y,X) • If DCG3 (resp. DCG4) is the definite clause gram- h(Y, X)~ [null, {p2(B)}, {p3(E)], mar obtained through the adjunction to SDG3 (resp. {q(E, Y, B, X)} SDG4) of clauses defining the auxiliary predicates p, q, r, then DCG3 and DCG4 are equivalent; Fur- h(Y, X)--~ [null, {p2(B)}, {p3(E)}, thermore, ff these definitions make p, q, r unification {q(E, V, B, Z)}, h(Z, X) predicates, then top-down parsing with DCG4 ter- minates, after enumerating all solutions. pl(nit). Example 5 Consider the following definite grammar p2(f). scheme DGS~: p3(r). q(E, A, B, cons(B, A)). al(X) --* [oh], {p(X)} at(X) ---, al(Y), {q(X,Y)} These two definite clause grammars are declaratively equivalent. They both accept strings of the form: The GGNF of DGS~ is DGS6 given below:
oh oui ... oui al(X) ---, nt(X)
where oui is repeated k times, k E N, and assign to each of these strings the (single) analysis represented by the u(X, X) ~ [] term: u(X,Y) --* {q(X,Z)}, u(Z,Y)
ifk=0 : nil, nl(X) ~ [oh], {PW)}, u(X, Y) ifk > 0 : cons(f .... eons(f, nil)...) (cons repeated k times.) From an inspection of DGS6 it can be concluded that:
On the other hand, from the operational point of view, • The unit subscheme contains acycle. Therefore nei- if a top-down parsing algorithm is used, DCGt loops on ther DGS6 nor DGS~ are oflline-parsable. any input string,ts while DCG2 enumerates all solutions on backtracking---here, zero or one solution, depending • If DCG5 (resp. DCG6) is the definite clause gram- on whether the string is in the language generated by the mar obtained through the adjunction to DGS5 (resp. grammar--and terminates. DGS~) of clauses defining the auxiliary predicates p, q, then DCG~ and DCG6 are equivalent; Example 4 Consider the following definite grammar scheme DGSz: • Even if p,q are defined as unifications, top-down parsing with DCG6 may not terminate. al(X)---, [null, at(Y), {p(X,Y)} Regarding the last point, let us show that different ~l(X) --. ~2(Y), {~(X,Y)} definitions for p and q result in different computational a2(X) .-* [], {q(X)} behaviors: taRemark that DCG1 is left recursive in a "vicious" (covert) a4 It can easily be shown that, iff this is the ease, then the way: nontexminal al calls itself, net immediately, but arc* unit nonterminals can be completely eliminated, as in the case calling a3, which does not consume anything in the input string. of example 3 above.
ACn'ESDE COLING-92, NANTES,23-28 AOUT 1992 3 7 0 PRoc. OF COLING-92, NANTES, Auo. 23-28. 1992 Situation 1 Assume that p,q are defined Appendix: Some indications on the through the following clauses: transformation method
p(nil). We can only give here some brief indications in tile hope q(f(X), X). that rite interested reader will be motivated to look into the full description given ill [3]. We start with some In such a situation, top-down parsing with comments on the GGNF in the CFG case and then move DCG6 of the input string oh does not on to the case of definite grammar schemes. terminate: an infinite number of solutions CFGs, Algebraic Systems, and the GGNF. The most (X = nil, X = f(nil), . . .) are enumerated powerful transformation methods existing for context- on backtracking and the program loops, t5 free grammars are "algebraic ("matrix based" [8]) ones Situation 2 Assume that p,q are defined by relying on the concepts of formal power series and al- the following clauses: gebraic systems (see [5, 91). Using such concepts, a context4ree grammar such as: p(X) :- fail. al ~ ala2 I a2 I [] q(nil, nil). a~ --, [v]
The first clause defines p as being the is refommlated into the algebraic system: 'false' (omitting giving a clause for p would have the same result). In such a al = ala2+a2+ l situation, top-down parsing with DCG6 a s = [ v ] terminates. which represents a fixpoint equation in the variables (or "nontermitmls") al,a2 on a certain algebraic structure 3 Conclusions: (a non-commutative semiring) of formal power series Noo<
The GNF for this grantmar is fire grammar: nl ~ [ohl u al ~ [oh] and it can be verified that it preserves degrees of ambiguity. This difference, which may be considered minor in tile case of The original grammar assigns an infinite degree of ambiguity CFGS, plays an important role in the transformation of DCGs. to [oh], while its GNF does not (in fact it is easy to show that 1o Furthermore, in the case at hand, they represent file unique a GNF can never be infinitely ambiguous). On the other hand, solution to this system.
Adds DE COLING-92. NANTES.23-28 AO~r 1992 3 7 1 PI~oc. OF COLING-92. NANTES, AUG. 23-28. 1992 where "equals can be replaced by equals", and (3) they Acknowledgments possess a rich algebraic structure (addition, multiplica- tion) which endows them with mathematical perspicuity Thanks to C. ]3oitet, M. Boyer, F. Guenthner, F. Perrault, and power. Y. Schabes, M. Simard, and G. van Noord for comments There are some substantial differences between the and suggestions. transformation steps used to obtain the GGNF of an al- gebraic system and the standard ones used to obtain its GNF, the principal one lying in the necessity to preserve References degrees of ambiguity at each step. In the GNF case, the [1] A.V. Aho and J.D. Ullman. The Theory of Pars- initial step consists in first transforming the initial sys- ing, Translation and Compiling, volume 1: Parsing. tem into a proper system (a notion analoguous to that of Prentice-Hall, Englewood Cliffs, NJ, 1972. proper CFG)--an operation which does not preserve de- grees of ambiguity--and then performing the main trans- [2] M. Dymetman and P. lsabelle. Reversible logic formation. For this reason, the transformation steps in the grammars for machine translation. In Proceedings GGNF case must be formulated in a more global way, of the Second International Conference on Theoret- which, among other complications, involves the use of ical and Methodological Issues in Machine Trans- certain identities on regular languages) 9 However, there lation of Natural Languages, Pittsburgh, PA, June are also important similarities between the GNF and the 1988. Carnegie Mellon University. GGNF transformations, among them the observation that the elementary algebraic system in the variable a on the [3] M. Dymetman. Transformations de gram- vocabulary V = {Iv], It]}: maires logiques. Applications au probl~me de la rdversibilit~ en Traduction Automatique. ThSse a = a Iv] + [t] d'Etat. Universit6 de Grenoble, Grenoble, France, has the unique solution a = [t] Iv]*, an observation which July 1992. can be much generalized, and which plays a central role in both cases. [4] A. Haas. A generalization of the offline-parsable DCGs, Mixed Systems, and the GGNF. In ordcr grammars. In Proceedings of the 27th Annual Meet- to define the GGNF in the case of Definite Grammar ing of the ACL, 237-42, Vancouver, BC, Canada, Schemes (or, equivalently. DCGs), we have introduced June 1989. so-eallcd mixed systems, a generalization of algebraic [5] Harrison, M.A. 1978. Introduction to Formal Lan- systems capable of representing association of structures guage Theory. Reading, MA: Addison-Wesley. to strings. Without going into details, let's consider the following definite grammar scheme: [6] Pereim, EC.N. and D.H.D. Warren. 1980. Definite Clause Grammars for Language Analysis. Artificial ai(x ) -+ al(y)ax(z){p(z,y,z)} I lntelligence:13, 231-78. a~(y) {q(z,y)} I [ ] {r(x)} a2(~:) -+ [vl{s(z)} [7] Pereira, F.C.N. and D.H.D. Warren. 1983. Parsing This scheme is reformulated as the mixed system: as Deduction. In Proceedings of the 21th Annual Meeting of the ACL, 137-44. Cambridge, MA, June. alx = aly a2z IY~z + a2y qYz + r,: a2~ = [v] s~ [8] Rosencmntz D. J. 1967. Matrix equations and nor- real forms for context-free grammars. JACM 14:3, In this system, the variables (or nonterminals) at, a~ are 501-07. seen as functions: /g -~ B<
AcrEs DE COLING-92, NANTES, 23-28 AO'O'r 1992 3 7 2 PROC. OF COLING-92, NANTES, AUG. 23-28, 1992