Compositional Semantics for Linguistic Formalisms

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Compositional Semantics for Linguistic Formalisms

S h u l y W i n t n e r *

I n s t i t u t e for R e s e a r c h in C o g n i t i v e Science U n i v e r s i t y of P e n n s y l v a n i a

3401 W a l n u t St., S u i t e 4 0 0 A P h i l a d e l p h i a , P A 19018 s h u l y @ : t ± n c , c i s . u p e n n , e d u

A b s t r a c t

In what sense is a g r a m m a r the union of its rules? This p a p e r a d a p t s the notion of com- position, well developed in the context of programming languages, to the domain of linguistic formalisms. We s t u d y alternative definitions for the semantics of such formalisms, suggest- ing a d e n o t a t i o n a l semantics that we show to be compositional and fully-abstract. This facilitates a clear, m a t h e m a t i c a l l y sound way for defining g r a m m a r modularity.

1 I n t r o d u c t i o n

Developing large scale g r a m m a r s for natural languages is a complicated task, and the problems g r a m m a r engineers face when designing broad-coverage g r a m m a r s are reminiscent of those tackled by software engineering (Erbach and Uszkoreit, 1990). Viewing c o n t e m p o r a r y linguistic formalisms as very high level declara- tive p r o g r a m m i n g languages, a g r a m m a r for a natural language can be viewed as a program. It is therefore possible to a d a p t m e t h o d s and techniques of software engineering to the domain of n a t u r a l language formalisms. We believe t h a t any advances in g r a m m a r engineering must be preceded by a more theoretical work, concentrating on the semantics of grammars. This view reflects the situation in logic programming, where developments in alternative definitions for predicate logic semantics led to im- plementations of various program composition operators (Bugliesi et al., 1994).

This p a p e r suggests a denotational semantics tbr unification-based linguistic formalisms and shows t h a t it is compositional and fully- *I a m g r a t e f u l t o N i s s i m F r a n c e z for c o m m e n t i n g o n a n em'lier v e r s i o n of t h i s p a p e r . T h i s work was s u p p o r t e d by a n I R C S F e l l o w s h i p a n d N S F g r a n t S B R 8920230.

abstract. This facilitates a clear, m a t h e m a t i - cally sound way for defining g r a m m a r m o d u - larity. While most of the results we report on are p r o b a b l y not surprising, we believe t h a t it is i m p o r t a n t to derive t h e m directly for linguistic formalisms for two reasons. First, practi- tioners of linguistic formMisms usually do not view t h e m as instances of a general logic programming framework, b u t rather as first-class programming environments which deserve in- d e p e n d e n t study. Second, there are some cru- cial differences b e t w e e n c o n t e m p o r a r y linguistic formalisms and, say, Prolog: the basic elements - - t y p e d feature-structures - - are more general t h a n first-order terms, the notion of unification is different, and c o m p u t a t i o n s a m o u n t to parsing, rather t h a n SLD-resolution. The fact t h a t we can derive similar results in this new domain is encouraging, and should not b e considered trivial.

Analogously to logic p r o g r a m m i n g languages, the d e n o t a t i o n of g r a m m a r s can be defined using various techniques. We review alternative approaches, operational and denotational, to the semantics of linguistic formalisms in section 2 and show that they are "too crude" to s u p p o r t g r a m m a r composition. Section 3 presents an alternative semantics, shown to b e compositional (with respect to g r a m m a r union, a simple syntactic combination o p e r a t i o n on grammars). However, this definition is "too fine": in section 4 we present an adequate, compositional and fully-abstract semantics for linguistic formalisms. For lack of space, some proofs are omitted; an e x t e n d e d version is avail- able as a technical report (Wintner, 1999). 2 G r a m m a r s e m a n t i c s

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n a t u r a l to consider t h e n o t i o n of semantics of ratification-based formalisms. We review in this se(:tion t h e o p e r a t i o n a l definition of Shieber et a,1. (1995) a n d t h e d e n o t a t i o n a l definition of, e.g., P e r e i r a a n d Shieber (1984) or C a r p e n t e r (1992, pp. 204-206). We show t h a t these definitions are equivalent a n d t h a t n o n e of t h e m s u p p o r t s c o m p o s i t i o n a l i t y .

2.1 B a s i c n o t i o n s

W(, a s s u m e f a m i l i a r i t y w i t h theories of f e a t u r e s t r u c t u r e based u n i f i c a t i o n g r a m m a r s , as for- m u l a t e d by, e.g., C a r p e n t e r (1992) or Shieber (1992). G r a m m a r s are defined over typed fea- twre .structures (TFSs) which c a n be viewed as g e n e r a l i z a t i o n s of first-order t e r m s ( C a r p e n t e r , 1991). T F S s are p a r t i a l l y o r d e r e d by s u b s u m p - tion, w i t h ± t h e least (or m o s t general) T F S . A multi-rooted structure (MRS, see Sikkel (1997) ()r W i n t n e r a n d Francez (1999)) is a sequence of T F S s , w i t h possible r e e n t r a n c i e s a m o n g dif- fi;rent e l e m e n t s in t h e sequence. Meta-variables A , / 3 r a n g e over T F S s a n d a, p - over MRSs. M R S s are p a r t i a l l y o r d e r e d by s u b s u m p t i o n , de- n()ted '__', w i t h a least u p p e r b o u n d o p e r a t i o n ()f 'an'llfication, d e n o t e d 'U', a n d a g r e a t e s t lowest t)(mnd d e n o t e d 'W. We a s s u m e the existence of a. fixed, finite set WORDS of words. A lexicon associates w i t h every word a set of T F S s , its cat- egory. M e t a - v a r i a b l e a ranges o v e r W O R D S a n d .w -- over strings of words (elements of W O R D S * ) .

G r a m m a r s are defined over a signature of t y p e s a n d features, a s s u m e d to be fixed below.

D e f i n i t i o n 1. A r u l e is an M R S of length greater than or equal to 1 with a designated (fir'st) element, the h e a d o.f the rule. The rest of the elements .form the rule's body (which may be em, pty, in which case the rule is depicted a.s' a TFS). A l e x i c o n is a total .function .from WORDS to .finite, possibly empty sets o.f TFSs. A g r a m m a r G = (T¢,/:, A s} is a .finite set of ,rules TO, a lexicon £. and a s t a r t s y m b o l A s that is a TFS.

F i g u r e 1 depicts a n e x a m p l e g r a m m a r , 1 sup- pressing t h e u n d e r l y i n g t y p e hierarchy. 2

T h e d e f i n i t i o n of u n i f i c a t i o n is lifted to MRSs: let a , p be two M R S s of t h e s a m e length; t h e 'Grammars are displayed using a simple description language, where ':' denotes feature values.

2Assmne that in all the example grammars, the types s, n, v and vp are maximal and (pairwise) inconsistent.

A '~ = (~:at : .~)

{ ( c a t : s ) -+ (co, t : n ) ( c a t : v p ) ] 7~ = ( c a t : vp) ---> (c.at: v) ( c a t : n)

vp) + .,,)

Z2(John) = Z~(Mary) = { ( c a t : 'n)}

£(sleeps) = £(sleep) = £(lovcs) = {(co, t : v)}

F i g u r e 1: A n e x a m p l e g r a m m a r , G

unification of a a n d p, d e n o t e d c, U p, is t h e m o s t general M R S t h a t is s u b s m n e d by b o t h er a n d p, if it exists. O t h e r w i s e , t h e u n i f i c a t i o n .fails.

D e f i n i t i o n 2. A n M R S (AI,...,A~:) reduces to a T F S A with respect to a gram, m a r G (de- noted ( A t , . . . , A k ) ~(-~ A ) 'li~' th, ere exists a rule p E T~ such, that ( B , 1 3 1 , . . . , B ~ : ) = p ll

(_L, A1,...,

Ak) and B V- A. Wll, en G is under- stood from. the context it is om, itted. Reduction can be viewed as the bottom-up counterpart of derivation.

I f f , g, are f l m c t i o n s over t h e s a m e (set) do- m a i n , .f + g is )~I..f(I) U .q(I). L e t ITEMS = { [ w , i , A , j ] [ w E

WORDS*,

A is a T F S a n d i , j E { 0 , 1 , 2 , 3 , . . . } } . L e t Z = 2 ITEMS. M e t a - variables x, y r a n g e over i t e m s a n d I - over sets of items. W h e n 27 is o r d e r e d b y set inclusion it forms a c o m p l e t e lattice w i t h set u n i o n as a least u p p e r b o u n d (lub) o p e r a t i o n . A f l m c t i o n T : 27 -+ 27 is m o n o t o n e if w h e n e v e r 11 C_/2, also T(I1) C_ T(I2). It is c o n t i n u o u s i f t b r every chain I1 C_ /2 C_ . . . , T ( U j < ~/.i) = U j < ~ T ( I j ) . I f a f u n c t i o n T is m o n o t o n e it has a least fixpoint ( T a r s k i - K n a s t e r t h e o r e m ) ; if T is also continu- ous, t h e fixpoint c a n be o b t a i n e d by i t e r a t i v e a p p l i c a t i o n of T to t h e e m p t y set (Kleene the- orem): lfp(T) = T S w , where T I " 0 = 0 a n d T t n = T ( T t ( n - 1)) w h e n 'n is a succes- sor o r d i n a l a n d (_Jk<n(T i" n) w h e n n is a limit ordinal. [image:2.612.308.522.89.184.2]

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be correct (observables-preserving): if G1 - G 2 , then Ob(G1) = Ob(G2).

Let 'U' be a composition operation on grammars and '•' a combination o p e r a t o r on deno- rations. A (correct) semantics ' H ' is compo- .s'itional (Gaifinan and Shapiro, 1989) if when-

ever ~ 1 ~ : ~G2~ a n d ~G3] -- ~G4], also ~G, U G3~ = [G2 U G4]. A semantics is com- m u t a t i v e (Brogi et al., 1992) if ~G1 U G 2 ] = ~G,~ • [G2~. This is a stronger notion t h a n (:ompositionality: if a semantics is c o m m u t a t i v e with respect to some o p e r a t o r then it is compositional.

2.2 A n o p e r a t i o n a l s e m a n t i c s

As Van E m d e n and Kowalski (1976) note, "to define an operational semantics for a program- ruing language is to define an implementational i n d e p e n d e n t interpreter for it. For predicate logic the p r o o f p r o c e d u r e behaves as such an interpreter." Shieber et al. (1995) view parsing as a. deductive process t h a t proves claims a b o u t the g r a m m a t i c a l s t a t u s of strings from assumptions derived from the grammar. We follow their in- sight and n o t a t i o n and list a deductive s y s t e m for parsing unification-based grammars.

D e f i n i t i o n 3. The deductive parsing s y s t e m associated with a g r a m m a r G = (7~,F.,AS} is defined over ITEMS and is characterized by:

A x i o m s : [a, i, A, i + 1] i.f B E Z.(a) and B K A ; [e, i, A , i] if B is an e-rule in T~ and B K_ A

Goals: [w, 0, A, [w]] where A ~ A s

I n f e r e n c e rules:

[wx , i l , A1, i l l , . . . , [Wk, ik, Ak , Jk ] [Wl " " " Wk, i, A , j]

if .'h = i1,+1 .for 1 <_ l < k and i = il and J = Jk and ( A 1 , . . . , A k ) =>a A

W h e n an item [ w , i , A , j ] can be deduced, applying k times the inference rules associ- z~ted with a g r a m m a r G, we write F-~[w, i, A, j]. W h e n the n u m b e r of inference steps is irrele- vant it is omitted. Notice that the domain of items is infinite, and in particular that the num- ber of axioms is infinite. Also, notice t h a t the goal is to deduce a T F S which is s u b s u m e d by the start symbol, and when T F S s can be cyclic, there can be infinitely many such T F S s (and, hence, goals) - see W i n t n e r and Francez (1999).

D e f i n i t i o n 4. The operational d e n o t a t i o n o.f a g r a m m a r G is EG~o,, = { x IF-v; :,:}. G1 - o p

G2 iy ]C1 o, = G2Bo ,

We use the operational semantics to define the language generated by a g r a m m a r G: L ( G ) = { ( w , A } [ [w,O,A,l',,[] E [ G ] o , } . Notice that a language is not merely a set of strings; rather, each string is associated with a T F S through the d e d u c t i o n procedure. Note also t h a t the start s y m b o l A ' does not play a role in this definition; this is equivalent to assuming that the start s y m b o l is always the most general TFS, _k.

The most natural observable for a g r a m m a r would be its language, either as a set of strings or a u g m e n t e d by TFSs. T h u s we take Ob(G) to be L ( G ) and by definition, the operational semantics '~.] op' preserves observables.

2.3 D e n o t a t i o n a l s e m a n t i c s

In this section we consider d e n o t a t i o n a l semantics through a fixpoint of a t r a n s f o r m a t i o n a l operator associated with grammars. -This is es- sentially similar to the definition of Pereira and Shieber (1984) and C a r p e n t e r (1992, pp. 204- 206). We then show t h a t the d e n o t a t i o n a l semantics is equivalent to the operational one.

Associate with a g r a m m a r G an o p e r a t o r 7 ~ that, analogously to the i m m e d i a t e conse- quence o p e r a t o r of logic programming, can be thought of as a "parsing step" o p e r a t o r in the context of g r a m m a t i c a l formalisms. For the following discussion fix a particular g r a m m a r G = ( n , E , A ~ ) .

D e f i n i t i o n 5. Let T c : Z -+ Z be a trans- f o r m a t i o n on sets o.f items, where .for every I C_ ITEMS, [ w , i , A , j ] E T(~(I) iff either

• there exist Y l , . . . , y k E I such that Yl = [w1,,iz,Al,jt] .for" 1 < 1 <_ k and il+l = jz f o r 1 < l < k and il = 1 and j k = J and

( A 1 , . . . , A k ) ~ A and w = "w~ .. • wk; or • i = j a n d B is an e-rule in G a n d B K A

and w = e; or

• i + l = j and [w[ = 1 a n d B G 12(w) and B K A .

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of logic p r o g r a m m i n g languages, define a fixpoint semantics for unification-based grammars by taking the least fixpoint of the parsing step o p e r a t o r as the d e n o t a t i o n of a grammar. D e f i n i t i o n 6. The f i x p o i n t d e n o t a t i o n of a g r a m m a r G is ~G[.fp = l.fp(Ta).

G1

=--.fp G2 iff ~ti,( T<; ~ ) = l fp(Ta~).

The denotational definition is equivalent to the operational one:

T h e o r e m 1. For x E ITEMS,

X E lfp(TG) iff

~-(? x.

The p r o o f is t h a t [ w , i , A , j ] E T a $ n iff F-7;,[w, i, A, j], by induction on n.

C o r o l l a r y 2. The relation '=fp' is correct: whenever G1 =.fp G2, also Ob(G1) = Ob(a2).

2 . 4 C o m p o s i t i o n a l i t y

While the operational and the denotational semantics defined above are s t a n d a r d for com- plete grammars, they are too coarse to serve as a model when the composition of grammars is concerned. W h e n the denotation of a grammar is taken to be ~G]op, i m p o r t a n t character- istics of the internal s t r u c t u r e of the g r a m m a r are lost. To d e m o n s t r a t e the problem, we intro- duce a natural composition o p e r a t o r on grammars, namely union of the sets of rules (and the lexicons) in the c o m p o s e d grammars.

D e f i n i t i o n 7. / f GI = <T¢1,

~1,

A~) and

G2

=

(7-~2,E'2,A~) are two grammars over the same signature, then the u n i o n of the two gram- mars, denoted G1 U G2, is a new grammar G = (T~, £, AS> such that T~ = 7~ 1 (.J 7"~2, ft. = ff~l + ff~2 and A s = A~ rq A~.

Figure 2 exemplifies g r a m m a r union. Observe that for every G, G', G O G' = G' O G.

• P r o p o s i t i o n 3. The equivalence relation '=op' is not compositional with respect to Ob, {U}.

Proof. Consider the grammars in figure 2.

~a:~o,, = l a d o . = {["loves",/, ( c a t : v ) , i + 1]l i > 0} b u t tbr I = {["John loves John", i, ( c a t : s ) , i + 3 I i >_ 0}, I C_

[G1UG4]op

whereas

I ~ [G1UGa~op. Thus Ga =-op G4 b u t

(Gl (2 Go) ~op (G1 tO G4), hence

'~--Op'

is not compositional with respect to Ob, {tO}. []

G1 : A s = (cat :.s)

(co, t : s) -+

(c.,t:

,,,,) (co, t : vp)

C(John) = {((:.t : n)}

a 2 : A s = (_1_)

(co, t : vp) -+

(co, t: v)

(cat : vp) -+ ( c a t : v ) ( c a t : n ) /:(sleeps) = / : ( l o v e s ) = { ( c a t : v)} G o : A s = (&)

/:(loves) = { ( c a t : v)} G 4 : A s = (_1_)

( c a t : v p )

-+ (co, t: v )

( c a t : n )

C(loves) = {(cat: v)}

G1 U G2 : A s = (cat : s)

(co, t : ~) -+

(~:o,t: ,,,,)

(~.at : vp) (cat : vp) -~ (co, t : v)

(cat: vp) --+ (cat: v) ( c a t : n) /:(John) = { ( c a t : n)}

£(sleeps) = £(loves) = { ( c a t : v)} G 1 U G a : A s = (cat : s)

(cat: s) --+ ( c a t : n) (cat: vp)

C(John) = {(cat: ',,,)}

£(loves) = { ( c a t : v)} GI U G4 : A s = (cat : s)

(co, t : ~) +

(co.t: ,,,.) (cat: vp)

(co, t : vp) -~

( c a t : , , )

(co, t : ~) /:(John) = { ( c a t : n)}

/:(loves) = { ( c a t : v)}

Figure 2: G r a m m a r union

The implication of the above proposition is that while g r a m m a r union might be a natural, well defined syntactic operation on grammars, the s t a n d a r d semantics of grannnars is too coarse to s u p p o r t it. Intuitively, this is because when a g r a m m a r G1 includes a particular rule p that is inapplicable for reduction, this rule contributes nothing to the d e n o t a t i o n of the grammar. B u t when G1 is combined with some other grammar, G2, p might be used for reduction in G1 U G2, where it can interact with the rules of G2. We suggest an alternative, fixpoint based semantics for unification based g r a m m a r s that naturally s u p p o r t s compositionality.

3 A c o m p o s i t i o n a l s e m a n t i c s

[image:4.612.309.531.86.428.2]

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tion of a g r a m m a r .

D e f i n i t i o n 8. The a l g e b r a i c d e n o t a t i o n o.f G is ffGffa I = T a . G1 - a t G2 iff Tal = TG2.

Not only is the algebraic semantics composi- tionM, it is also c o m m u t a t i v e with respect to g r a m m a r union. To show t h a t , a composition o p e r a t i o n on d e n o t a t i o n s has to be defined, and we tbllow Mancarella a n d Pedreschi (1988) in its definition:

Tc;~ • To;., =

),LTc,

(~) u Ta2 ( 5

T h e o r e m 4. The semantics '==-at ' is commuta- tive with respect to grammar union and '•': for e, vcry two grammars G1, G2, [ a l f f a t " ~G2ffal =

: G I [-J G 2 ff (tl .

Proof. It has to be shown that, for every set of items L Tca~a., (I) = T a , ( I ) u Ta.,(I).

• if x E TG1 (I) U TG~, (I) t h e n either x G Tch (I) or x E Ta.,(I). From the definition of g r a m m a r union, x E T G 1 u G 2 ( I ) in any case.

• if z E Ta~ua.,(I) t h e n x can be added by either of the three clauses in the definition of Ta.

- if x is a d d e d by the first clause t h e n

there is a rule p G 7~1 U T~2 t h a t licenses the derivation t h r o u g h which z is added. T h e n either p E 7~1 or p G T~2, b u t in any case p would have licensed the same derivation, so either

~ Ta~ (I) or • ~ Ta~ (I).

- if x is a d d e d by the second clause t h e n

there is an e-rule in G1 U G2 due to which x is added, a n d by the same rationale either x C TG~(I) or x E TG~(I).

- if x is a d d e d by the t h i r d clause t h e n

there exists a lexical category in £1 U £2 due to which x is added, hence this category exists in either £1 or £2, and therefore x C TG~ (I) U TG2 (I).

[]

Since '==-at' is c o m m u t a t i v e , it is also compositional w i t h respect to g r a m m a r union. In- tuitively, since TG captures only one step of

the c o m p u t a t i o n , it cannot capture interactions among different rules in the (unioned) g r a m m a r , and hence taking To: to be the d e n o t a t i o n of G yields a compositional semantics.

T h e Ta operator reflects the s t r u c t u r e of the g r a m m a r b e t t e r t h a n its fixpoint. In other words, the equivalence relation induced by TG is finer t h a n the relation induced by lfp(Tc). T h e question is, how fine is the ' - a l ' relation? To make sure t h a t a semantics is not too fine, one usually checks the reverse direction.

D e f i n i t i o n 9. A f u l l y - a b s t r a c t equivalence relation ' - ' is such that G1 =- G'2 'i,.[.-f .for all G, Ob(G1 U G) = Ob(G.e U G).

P r o p o s i t i o n 5. Th, e semantic equivalence re- lation '--at' is not fully abshuct.

Proof. Let G1 be the g r a m m a r A~ = ± ,

£1 = 0,

~ = { ( c a t : ~) -~ (~:.,t : ,,,,p) (c.,t : vp), ( c a t : up) -~ (,:..t : ',,.p)}

and G2 be the gramm~:r A~ = 2 ,

Z:2 = O,

n ~ = {(~at : .~) -~ (~,.,t : .,p) ( . a t : ~p)}

• G1 ~ a t G2: because tbr I = {["John loves M a r y " , 6 , ( c a t : np),9]}, T(;I(I ) = I b u t To., (I) = O

• for all G, Ob(G U G~) = Ob(G [3 G2). T h e only difference between GUG1 a n d GUG2 is the presence of the rule (cat : up) -+ (cat : up) in the former. This rule can c o n t r i b u t e n o t h i n g to a d e d u c t i o n procedure, since any item it licenses m u s t already be deducible. Therefore, any item deducible w i t h G U G1 is also deducible with G U G2 a n d hence Ob(G U G1) ---- Ob(G U G,2).

[]

A b e t t e r a t t e m p t would have been to consider, instead of TG, the fbllowing o p e r a t o r as the d e n o t a t i o n of G: [G]i d = A I . T a ( I ) U I. In other words, the semantics is T a + Id, where I d is the identity operator. Unfortunately, this does not solve the problem, a s '~']id' is still not

fully-abstract.

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4 A f u l l y a b s t r a c t s e m a n t i c s

We have shown so far t h a t ' H f p ' is not compositional, and t h a t ' H i d ' is compositional b u t not fully abstract. T h e "right" semantics, therefore, lies somewhere in between: since the choice of semantics induces a natural equivalence on grammars, we seek an equivalence that is cruder thzm ' H i d ' b u t finer t h a n 'H.fp'. In this section we a d a p t results from Lassez and Maher (1984) a.nd Maher (1988) to the domain of unification- b~Lsed linguistic formalisms.

Consider the following semantics for logic programs: rather t h a n taking the operator asso- d a t e d with the entire program, look only at the rules (excluding the facts), and take the mean- ing of a program to be the function that is obtained by an infinite applications of the operator associated with the rules. In our framework, this would a m o u n t to associating the following operator with a grammar:

D e f i n i t i o n 10. L e t RG : Z -~ Z be a trans- f o r m a t i o n on sets o.f items, where .for every [ C ITEMS, [ w , i , A , j ] E R G ( I ) iff there exist Y l , . . . , Y k E I such that yl = [ w z , i t , A l , j d .for 1 _ < l _ < k and il+t = jl .for 1 < l < k and i, = 1 a n d . j k = J and ( A 1 , . . . , A k ) ~ A and

"~1) ~ 'tl) 1 • • • ? U k .

Th, e f u n c t i o n a l denotation of a g r a m m a r G is /[G~.f,,, = ( R e + I d ) ~ = End-0 (RG + I d ) n. Notice that R w is not R G "[ w: the f o r m e r is a f u n c t i o n " d f r o m sets of i t e m s to set of items; the latter is

a .set of items.

Observe that R c is defined similarly to Ta (definition 5), ignoring the items added (by Ta) due to e-rules and lexical items. If we define the set of items I ' n i t c to be those items that are a.dded by TG i n d e p e n d e n t l y of the argument it operates on, then for every g r a m m a r G and every set of items I, T a ( I ) = R a ( I ) U I n i t a . Re- lating the functional semantics to the fixpoint one, we tbllow Lassez and Maher (1984) in prov- ing that the fixpoint of the g r a m m a r transfor- mation o p e r a t o r can be c o m p u t e d by applying the fimctional semantics to the set I n i t G . D e f i n i t i o n 11. For G = ( h g , £ , A ~ ) , I n i t c = {[e,i,A,i] [ B is an e~-rule in G and B E_A} U { [ a , i , A , i + 1J I B E £ ( a ) .for B E A }

T h e o r e m 6. For every g r a m m a r G,

(R.c + fd.) (z',,.itcd = tb(TG)

Proof. We show that tbr every 'n., ( T ~ + I d ) n = (E~.-~ ( R e + Id) ~:) (I'nit(;) by induction on

Tt.

For n = 1, ( T c + I d ) ~[ 1 = (Tc~ + I d ) ( ( T a + I d ) ~ O) = (Tc, + Id)(O). Clearly, the only items added by TG are due to the second and third clauses of definition 5, which are exactly I n i t a . Also, ( E ~ = o ( R a + Id)~:)(Initc;) = ( R a + I d ) ° ( I n i t c ) = I'nitc;.

Assume that the proposition holds tbr n - 1, that is, ( T o + Id) "[ (',, - 1) = t~E'"-2t~'a:=0 txta + I d ) k ) U n i t e ) . T h e n

( T a + I d ) $ n = definition of i"

(TG + I d ) ( ( T a + I d ) ~[ (v, - 1)) = by the induction hypothesis

~n--2

( T a + I d ) ( (

k=0(RG +

I d ) k ) ( I n i t a ) ) = since T a ( I ) = R a ( I ) U I n i t a

E n - 2

( R a + I d ) ( ( k=Q(Rc; + I d ) ~ ' ) ( I n i t a ) ) U I n i t a =

(Ra

+ (Ra + Id) k) (1',,,its,)) = ( y ] n - 1 / R _{k=0 ~,} ,

Id)h:)(Init(:)

, G - I -

Hence (RG + I d ) ~ ( I n i t ( ; = (27(; + I d ) ~ w =

lfp( TG ) . []

The choice of 'Hfl~' as the semantics calls for a different notion of' observables. The denotation of a g r a m m a r is now a flmction which reflects an infinite n u m b e r of' applications of the g r a m m a r ' s rules, b u t completely ignores the e- rules and the lexical entries. If we took the observables of a g r a m m a r G to be L ( G ) we could in general have ~G1].f,. = ~G2]fl~. b u t Ob(G1) 7 ~ Ob(G2) (due to different lexicons), that is, the semantics would not be correct. However, when the lexical entries in a g r a m m a r (including the e- rules, which can be viewed as e m p t y categories, or the lexical entries of traces) are taken as in- put, a natural notion of observables preservation is obtained. To guarantee correctness, we define the observables of a g r a m m a r G with respect to a given input.

D e f i n i t i o n 12. Th, e o b s e r v a b l e s of a gram- m a r G = ( ~ , / : , A s} with respect to an in- put set of items I are Ot, (C) = {(',,,,A) I

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C o r o l l a r y 7.

The semantics '~.~.f ' is correct:

'llf G1 =fn G2 then .for every I, Obl(G1) =

Ol, ( a,e ).

The above definition corresponds to the pre- vious one in a n a t u r a l way: when the input is taken to be

Inita,

the observables of a g r a m m a r are its language.

T h e o r e m 8.

For all G, L(G) = Obinita(G).

P'moJ:

L(G) =

definition of

L(G)

{ (',,,, A) I [w, O, A,

I 1] e I[C]lo,,}

=

definition 4

{(w,

A) [ F-c [w, O, A,

=

by t h e o r e m 1

{<w, A> I [,w, 0, A,

Iwl]

e

l.fp(Ta)} =

by t h e o r e m 6

{(,w,

A)

I

[w, O, A, [wl]

e [ G ] f n ( I n i t G ) } = by definition 12

Obt,,,~tc; (G)

[]

.To show t h a t the semantics

'Hfn'

is compositional we must define an o p e r a t o r for combining denotations. Unfortunately, the simplest operator, ' + ' , would not do. However, a different o p e r a t o r does the job. Define

~Gl~.f~ • [G2~f~

to 1)e ([[G1]l.fn +

[G2~f~) °'.

T h e n

'H.f~'

is c o m m u t a - tive (and hence compositional) with respect to ~•' a n d 'U'.

T h e o r e m 9. fiG1 U

G2~fn = ~Gl]fn " ~G2~.fn.

The p r o o f is basically similar to the case of logic p r o g r a m m i n g (Lassez and Maher, 1984) and is detailed in W i n t n e r (1999).

T h e o r e m 10.

The semantics '~'[fn' is fully

abstract: ,for every two grammars G1 and G2,

'llf .for" every grammar G and set of items I,

Obr(G1 U G) = ObI(G2 U G), then G1 =fn G2.

T h e p r o o f is constructive: assuming that G t ~f;~ G2, we show a g r a m m a r G (which de- t)ends on G1 and G2) such t h a t

Obt(G1 U G) ¢

Obr(G2 U G).

For the details, see W i n t n e r (1999).

5 C o n c l u s i o n s

This p a p e r discusses alternative definitions for the semantics of unification-based linguistic formalisms, culminating in one that is b o t h compositional and fully-abstract (with respect to g r a m m a r union, a simple syntactic c o m b i n a t i o n operations on grammars). This is mostly an a d a p t a t i o n of well-known results from h)gic programming to the ti'amework of unification-based linguistic tbrmalisms, and it is encouraging to see that the same choice of semantics which is compositional and fiflly-abstra(:t for Prolog t u r n e d out to have the same desirable proper- ties in our domain.

The functional semantics '~.].f,' defined here assigns to a g r a m m a r a fimction which reflects the (possibly infinite) successive application of g r a m m a r rules, viewing the lexicon as input to the parsing process. We, believe that this is a key to m o d u l a r i t y in g r a m m a r design. A grammar m o d u l e has to define a set of items t h a t it "exports", and a set of items t h a t can be "imported", in a similar way to the declaration of interfaces in p r o g r a m m i n g languages. We are currently working out the details of such a definition. An i m m e d i a t e application will facilitate the i m p l e m e n t a t i o n of g r a m m a r development systems t h a t s u p p o r t m o d u l a r i t y in a clear, m a t h e m a t i c a l l y sound way.

The results r e p o r t e d here can b e e x t e n d e d in various directions. First, we are only concerned in this work with one c o m p o s i t i o n operator, g r a m m a r union. B u t alternative o p e r a t o r s are possible, too. In particular, it would b e in- teresting to define an o p e r a t o r which combines the information encoded in two g r a m m a r rules, for example by unifying the rules. Such an operator would facilitate a separate development of g r a m m a r s along a different axis: one m o d u l e can define the syntactic c o m p o n e n t of a grammar while another m o d u l e would account for the semantics. T h e composition o p e r a t o r will unify each rule of one m o d u l e with an associated rule in the other. It remains to be seen w h e t h e r the g r a m m a r semantics we define here is compositional and fully a b s t r a c t with respect to such an operator.

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over a given signature, it is more realistic to assume separate, interacting signatures. We hope to be able to explore these directions in the fu- ture.

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