In nite Set Uni cation with Application to Categorial ... - Semantic Scholar
In nite Set Uni cation with Application to Categorial Grammar JACEK MARCINIEC
Adam Mickiewicz University Pozna«, Poland [email protected]
Abstract In this paper the notion of uni er is extended to the in nite set case. The proof of existence of the most general uni er of any in nite, uni able set of types (terms) is presented. Learning procedure, based on in nite set uni cation, is described. Keywords: categorial grammar, uni cation, discovery procedure.
1 Introduction and preliminaries A discovery procedure for determining categorial grammar from ( nite) lin guistic data has been studied in Buszkowski [1, 2] as well as van Ben them [9]. Further modi cations of the initial algorithms were presented in Buszkowski & Penn [3] and Marciniec [8]. All the procedures men tioned make use of uni cation of nite sets. The learning procedure given by Kanazawa [4, 5] admits as an input in nite sequences of structures but only those which constitute the whole categorial language. An attempt to admitting arbitrary in nite sets of postulates leads to the need of considering uni cation with respect to in nite sets. Fundamental concepts related to uni cation maintain their sense when referred to in nite sets of terms (or types). However, not all their properties
from: Studia Logica 58: 339355, 1997.
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2 can be generalised, and those which admit generalisation require dierent proofs, since the standard ones essentially rely on niteness (see [7]). In this paper we prove the existence of the most general uni er of any in nite, uni able set of types by reduction of this problem to the nite case. A uni able set of types always contains a sort of nite representation a gist the most general uni er of which can be naturally extended to that of the whole set. The above fact, employed in the theory of categorial grammar, makes it possible to de ne a notion of compact consequence operator on the universe of functor-argument structures. Tarski's conditions, in particular compactness, of this operator can be shown using methods of Kanazawa [4, 5], based on an analysis of categorial grammar. However, in the scope of consistent data, it is possible to prove these facts in a dierent (more fundamental) way, by reducing them to general properties of uni cation of in nite sets. The paper is organised as follows. At rst we establish the notation and recapitulate some essential notions concerning uni cation and classical cate gorial grammar. In Section 2 we construct the rudiments of the uni cation of in nite sets. Finally, in Section 3, we give the account of learning procedure based on in nite set uni cation. The author acknowledges the supervision of this research by Wojciech Buszkowski. We de ne basic notions used in this paper. The set of all functor-argument structures on the set of atoms V , FS (V ), is de ned as the smallest set satis fying the following conditions: V FS (V ); if A ; : : : ; An 2 FS (V ) then (A ; : : : ; An)i 2 FS (V ); for all n > 2; 1 6 i 6 n: If A = (A ; : : : ; An)i 2 FS (V ) then Ai is the functor whereas each Aj , for j 6= i, is an argument in structure A. For any S FS (V ), SUB (S ) denotes the smallest set such that: S SUB (S ); if (A ; : : : ; An)i 2 SUB (S ) then Aj 2 SUB (S ); for all j = 1; : : : ; n: SUB (S ) consists of all substructures of structures from S . For technical reasons we designate: 1
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3 SUB a(S ) = fAj 2 SUB (S ) : (9A 2 SUB (S ))(A = (A1; : : : ; An)i ^ i 6= j )g SUB f (S ) = fAi 2 SUB (S ) : (9A 2 SUB (S ))(A = (A1 ; : : : ; An)i )g SUB a(S ) (SUB f (S )) contains all the structures playing the role of arguments (functors) in substructures from SUB (S ). The length l(A) of a functor-argument structure A 2 FS (V ) is de ned as follows: 8 if A 2 V > < 1X n l(A) = > l(Aj ) if A = (A1; : : : ; An)i : j =1
The height h(A) of functor-argument structure A 2 FS (V ) we de ne: A2V h(A) = 0max fh(A ) : j = 1; : : : ; ng + 1 ifif A = (A ; : : : ; An)i j We will write, that a set T FS (V ) is bounded with respect to length (height) if (9N 2 N)(8t 2 T )(l(t) 6 N ) ((9N 2 N)(8t 2 T )(h(t) 6 N ) ). l(T ) (h(T )) will denote max fl(t) : t 2 T g (max fh(t) : t 2 T g) respectively. We admit the case l(T ) = @ (h(T ) = @ ). In what follows we assume that each mapping : P ?! FS (Q) expands to a mapping from FS (P ) to FS (Q): (1) ((a ; : : : ; an)i) = ((a ); : : : ; (an ))i: For any set S of arguments of some mapping , [S ] will denote the image of S . A set Pr = Const [ Var , where Const \ Var = ;, is called a set of primitive types. Members of Tp = FS (Pr ) will be called types. For any T Tp we set: Var (T ) = SUB (T ) \ Var , Const (T ) = SUB (T ) \ Const , Pr (T ) = SUB (T ) \ Pr . A substitution is any mapping : Pr ?! Tp ful lling the condition (c) = c, for any c 2 Const . By we will denote the set of all substitutions. We say that a substitution uni es a set of types T if [T ] is a singleton. A set T Tp is said to be uni able if there exists a uni er of T . A uni er of T is called a most general uni er (mgu) of T if, for any uni er of T , there is a substitution such that = . For any uni able and nite set of types T , there exists an mgu of T , and it can be eectively found (cf. [7]). Two types u and v are said to be variants, if there are substitutions and such that v = (u) and u = (v). If and are mgu-s of some T Tp then [T ] and [T ] are variants. 1
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4 Let T = fT ; : : : ; Tng be a nite family of (possibly in nite) nonempty sets of types. A substitution is a uni er of T if it is a uni er of each Ti, i = 1; : : : ; n. A uni er of T is called an mgu of T if, for any uni er of T , there is a substitution such that = . The problem of uni ability of T can easily be reduced to that of set of types. De nition 1. Let T = fT ; : : : ; Tng be a family of countable sets of types. Assume Ti = fti ; ti ; : : : g, for all i = 1; : : : ; n. In case Ti is nite we set tji = ti ,Nfor all j > card (Ti). De ne tj = (tj ; : : : ; tjn) , for all j 2 N. We denote T = ftj gj2N. Fact 1. Let T = fT ; : : : ; Tng be a family of countable sets of types. Then: is a uni er of T i is a uni er of N T is the mgu of T i is the mgu of N T By a classical categorial grammar we mean a triple G = (VG ; IG; sG ) where a nite set of atoms VG is called the lexicon of G, a mapping IG : VG ?! 2Tp is called the initial type-assignment of G, sG 2 Pr c is called the principal type of G. The terminal type-assignment TG : FS (VG ) ?! 2Tp is de ned as follows: TG(v) = IG(v); for v 2 VG ; TG((A ; : : : ; An)i) = fai 2 Tp : (9(a ; : : : ; an)i 2 TG(Ai))(8j 6= i)(aj 2 TG(Aj ))g: Throughout this paper we will use the expression grammar in the meaning of classical categorial grammar. For any grammar G and t 2 Tp , we de ne a category of type t: CAT G (t) = fA 2 FS (VG ) : t 2 TG(A)g: FL(G) = CAT G (sG) will be called the F-language generated by G. A grammar G is said to be nite (rigid) i card (IG(v)) < @ (card (IG(v)) = 1), for all v 2 VG . A set L is rigid (rigidly describable) if L = FL(G) (L FL(G)) for some rigid grammar G. A grammar G is compatible with L FS (V ) if L FL(G). For any grammars G and H we write G H i VG = VH ; sG = sH and IG(v) IH (v), for all v 2 VG . 1
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2 In nite set uni cation In this section we will prove the existence of the most general uni er of any in nite, uni able set of types. We will also show that an mgu of any set is
5 always established on the basis of its nite subset. All the results presented below are still valid if we consider arbitrary terms instead of types. Fact 2. If T is uni able then Const (T ) is nite. From now on we assume that the set Const is nite. Fact 3. If T is bounded with respect to length and Pr (T ) is nite then T is nite. De nition 2. Let T Tp . We say that a substitution respects an equiv alence relation on SUB (T ) if (8a; b 2 Pr (T ))(a b ) (a) = (b)). Fact 4. If there exists a substitution respecting relation then: (8a; b 2 Const )(a b ) a = b) By [t] we denote the equivalence class of type t with respect to the equivalence relation . De nition 3. Let T Tp . For any equivalence relation SUB (T ) SUB (T ) we de ne a mapping [ ] : Pr (T ) ?! 2Pr T : ([p]) if p 2 Var (T ) [ p] = Var fpg if p 2 Const (T ) Denote [ Pr ] = f[ p] : p 2 Pr (T )g. According to (1) [ ] expands to a mapping from FS (Pr (T )) to FS ([[Pr ] ). Fact 5. If respects then, for all u; v 2 Tp, [ u] = [ v] implies (u) = (v). De nition 4. Let S FS (P ); T FS (Q). By structural induction we de ne relation D(S; T ) SUB (S ) SUB (T ): S T D(S; T ); if h(A ; : : : ; An)i; (B ; : : : ; Bn)ii 2 D(S; T ) then hAj ; Bj i 2 D(S; T ); for all j = 1; : : : ; n Fact 6. For all T Tp the relation D(T; T ) is re exive and symmetric. By T we denote the transitive closure of D(T; T ). Notice that T SUB (T ) SUB (T ): ( )
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Fact 7. For any sets of types U and T : If U T then U T .
We skip the proof of the above fact since the next one uses similar tech nique. Fact 8. Let T be a set of types. Then: (8u; v 2 SUB (T )) [u T v ) (9S T )(card (S ) < @ ^ u S v)] Proof. Let u; v 2 SUB (T ). Denote the set whose existence is to be proved by S (u; v). We consider hu; vi 2 D(T; T ) at rst. If u; v 2 T then of course S (u; v) = fu; vg. Assume now that u = (u ; : : : ; un)i, v = (v ; : : : ; vn)i and our thesis holds for u and v. Then we put S (uj ; vj ) = S (u; v), for all j = 1; : : : ; n. Let nally u T v. There exist u ; : : : ; uk such that u = u, uk = v and hSuj ; uj i 2 D(T; T ) for j = 1; : : : ; k ? 1. By Fact 7 we can set S (u; v) = k? j S (uj ; uj ). Fact 9. Each uni er of T respects T . Proof. Let be a uni er of T . By depth induction we will show that, for all a; b 2 SUB (T ), if ha; bi 2 R(T; T ) then (a) = (b). For a; b 2 T , the above holds by the de nition of uni er. Suppose a = (a ; : : : ; an)i; b = (b ; : : : ; bn)i and ha; bi 2 R(T; T ). Then if our thesis holds for a and b, that is (a) = (b), then it also holds for each aj and bj , j = 1; : : : ; n, by the de nition of substitution. If u T v then u = u = : : : = uk = v and huj ; uj i 2 R(T; T ), for j = 1; : : : ; k ? 1. From the rst part of the proof we get immediately (u) = (u ) = : : : = (uk) = (v). Below we will write [ ] T instead of [ ] . Fact 10. For all u; v 2 T if [ u] T = [ v] T then: (2) (8a 2 SUB (u))(9b 2 SUB (v))([[a] T = [ b] T ^ a fu;vg b): Proof. Assume [ u] T = [ v] T . We prove (2) by depth induction on a. If a = u then (2) holds by assumption. Let a = (a ; : : : ; an)i and there ex ists b 2 SUB (v) such that [ a] T = [ b] T and a fu;vg b. Since [ a] T = ([[a ] T ; : : : ; [ an] T )i there must exist b ; : : : ; bn such that b = (b ; : : : ; bn)i. Then of course [ aj ] T = [ bj] T and aj fu;vg bj for j = 1; : : : ; n, which nishes the proof. 0
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Corollary 1. For all u; v 2 T such that [ u] T = [ v] T we have: (8p 2 Pr (u))(9q 2 Pr (v))(p fu;vg q): Proof. Our thesis follows immediately from Fact 10 since [ a] T = [ b] T and a 2 Pr implies b 2 Pr .
Lemma 1. Let T be a set of types bounded with respect to length. Then ind (T ) is nite. Proof. Let M = l(T ). We assume that Var = fx ; x ; : : : g. Denote Var M = fx ; : : : ; xM g 1
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SM = fa 2 FS (Var M ) : l(a) = card(Var (a))g: Clearly, for any a 2 SM , l(a) 6 M , each variable occurs in a at most once and, consequently, SM is nite. It is also easy to see, that SUB (SM ) = SM . One easily proves the following: (3) (8u 2 Tp )(l(u) 6 M ) (9a 2 SM )(9 2 )(l(u) = l(a) ^ ( (a) = u)) Assume SM = fa ; : : : ; amg. For i = 1; : : : ; m, we de ne substitutions:
i(xj ) =
1
x i? xj (
M +j
1)
if j 6 M otherwise
We denote SM = f i(ai) : i = 1; : : : ; mg. Each element of SM is a variant of some element of SM , and each is build using dierent variables. Clearly, (8t ; t 2 SM )(SUB (t ) \ SUB (t ) = ;). Recall De nition 4. To simplify notation we will write u ; t instead of hu; ti 2 D(T; SM ). We de ne a function: 1
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f : SUB (T )= ?! 2SUB S (
M)
T
by setting:
f (K) = ft 2 SUB (SM ) : (9u 2 K)(u ; t)g:
8 Since card (Rng (f )) < @ , it su ces to show that f is one-to-one. First, by depth induction, we prove: (4) if u ; t and v ; t then hu; vi 2 D(T; T ): For t 2 SM the above is obvious since t 62 SUB (SM ) n ftg. Assume now that (4) holds, for arbitrary t = (t ; : : : ; tn)i. We will show that it also holds, for each tj , 1 6 j 6 n. Let uj ; tj and vj ; tj . Since t is the only structure in SUB (SM ) containing tj , there must exist u = (u ; : : : ; un)i and v = (v ; : : : ; vn)i such that u ; t and v ; t. Then, by induction, hu; vi 2 D(T; T ) and consequently huj ; vj i 2 D(T; T ). It follows from (4) that f (K ) \ f (K ) 6= ; implies K = K , for all K ; K 2 SUB (T )= . To prove that f is on-to-one we have to show that ; 62 Rng (f ). Again, by depth induction on u we derive: (5) (8u 2 SUB (T ))(9t 2 SUB (SM ))(u ; t ^ l(u) = l(t) ^ (9 2 )(u = (t)): For u 2 T (5) follows immediately from (3). Suppose u = (u ; : : : ; un)i ; t, l(u) = l(t) and u = (t) for some 2 . Then there must exist t ; : : : ; tn such that t = (t ; : : : ; tn)i. Clearly uj ; tj , uj = (tj ) and l(uj ) = l(tj ), which nishes the proof of (5). Consequently, f is one-to-one and therefore SUB (T )= T is nite. Corollary 2. If T is bounded with respect to length then [ Pr] T is nite. De nition 5. Let T Tp. A set of types U T is said to be a gist of T if: (6) [ T ] T = [ U ] T (7) (8a; b 2 Pr (U ))(a T b , a U b): Since U = T ful ls (6) and (7), a gist of T always exists. We aim to show that, for any set of types bounded with respect to length, there is a nite gist of this set. First, we need a number of auxiliary facts. Fact 11. Let U be a gist of T . Then: (8A 2 [ Pr ] T )(A \ Pr (U ) 6= ;): Proof. Let A 2 [ Pr ] T and p 2 A. There is some t 2 T such that p 2 Pr (t). By (6) [ t] T = [ u] T for some u 2 U . By Fact 10 there exists q 2 Pr (u) such that [ p] T = [ q] T which nishes the proof since q 2 A = [ p] T . 0
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9 The above fact justi es the following de nition. De nition 6. Let U be a gist of T . A function U : [ Pr ] T ?! Pr is said to be a choice function for U if U (A) 2 A\ Pr (U ), for all A 2 [ Pr ] T . Clearly U is one-to-one. We x U , for the given U . Let U be a substitution such that: U (p) = U ([[p] T ):
(8)
The above de nition is correct, since for any c 2 Const we have: U (c) = U ([[c] T ) = U (fcg) = c:
For any substitution respecting U we de ne the extension "U of (with respect to U ): (9) "U =
U
Observe that the above de nition does not depend on the choice of U . Lemma 2. Let T be a set of types. If T is bounded with respect to length then [ T ] T is nite. Proof. Let U be an arbitrary gist of T and U be a choice function for U . Then U is one-to-one correspondence between [ T ] T and U [T ]. Since l( U (t)) = l(t) for any t 2 T , U [T ] is also bounded with respect to length. By Fact 3 U [T ] is nite, because Pr ( U [T ]) = U ([[Pr ] T ) is nite, by Lemma 1. Theorem 1. Let T be a set of types bounded with respect to length. There exists a nite gist of T . Proof. First we de ne a set of types U : for each A 2 [ T ] T we choose one t 2 T such that [ t] T = A and put it in U . By Lemma 2, U is nite. Also, for any W such that U W T , we have [ U ] T = [ W ] T = [ T ] T . We prove: 0
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10 By Corollary 1 there exists b 2 Pr (u) such that a fu;wg b. By Fact 7, since fu; wg W , we have a W b. According to Fact 8 for any a; b 2 SUB (T ) such that a T b there exists a nite set S (a; b) T such that a S a;b b. Let D = ffa; bg Pr (U ) : (a T b) ^ (a 6U0 b)g. We set: (
(11) U = U [ 0
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fa;bg2D
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Clearly U is nite. We will show that U ful ls (7). Let a; b 2 Pr (U ) and a T b. By (10) there exist a0; b0 2 Pr (U ) such that a U a0; b U b0. Since U is transitive, it su ces to show that a0 U b0. If a0 U0 b0 then also a0 U b0. If a0 6U0 b0 then, by (11), S (a0; b0) U . Consequently a0 U b0 which nishes the proof. Fact 12. Let U be a gist of T and be a substitution respecting U . Then: (12) "U (a) = (a) for all a 2 Pr (U ) (13) "U respects T Proof. To prove (12) assume x 2 Var (U ). Then U (x) T x and, by (7), U (x) U x, which implies ( U (x)) = (x), since respects U . To prove that "U respects T choose p ; p 2 Pr (T ) ful lling p T p . Then there exist t ; t 2 T such that p 2 Pr (t ) and p 2 Pr (t ). By (6) there are u ; u 2 U such that [ u ] T = [ t ] T and [ u ] T = [ t ] T . By Fact 10 there exist q ; q 2 Pr (U ) such that [ p ] T = [ q ] T and [ p ] T = [ q ] T . Then q T p T p T q . By (7) also q U q , and since respects U , we get (q ) = (q ). Finally "U (p ) = ( U (p )) = ( U (q )) = (q ) = (q ) = ( U (q )) = ( U (p )) = "U (p ). Fact 13. Let T be a set of types. If respects T then, for any gist U of T , we have: "U = . Proof. Suppose U be a gist of T . Since U (x) T x for all x 2 Pr and respects T we have: (x) = ( U (x)) = "U (x). Lemma 3. Let U be a gist of T and be a substitution respecting U . Then, "U uni es T i uni es U . Proof. Assume uni es U . For each t 2 T , there exists u 2 U such that [ t] T = [ u] T . Then "U (t) = "U (u) = (u). 0
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Corollary 3. A set of types is uni able if and only if its gist is uni able. Lemma 4. Let T be a uni able set of types, and let U be a gist of T . For
any mgu of U "U is an mgu of T . Proof. Let be an mgu of U . Since any uni er of T is also a uni er of U , there exists such that = . Then: ("U (x)) = (( U (x))) = ( U (x)) = "U (x) = (x). The last equality holds by Fact 9 and Fact 13.
We have an obvious statement: Fact 14. If T is uni able then T is bounded with respect to length. As a consequence we get: Theorem 2. Let T be an in nite, uni able set of types. Then there exist a nite set U T , mgus of T and of U , such that [T ] = [U ]. Proof. Since T is uni able, by Theorem 1, there exists a nite gist U of T . For any mgu of U , = "U is an mgu of T and [T ] = "U [T ] = "U [U ] = [U ]. Corollary 4. Let T be an in nite, uni able set of types. Then, there exists an mgu of T . Theorem 3. Let T = fT ; : : : ; Tng be a uni able family of nonempty, pos sibly in nite sets of types. There exist a family U = fU ; : : : ; Ung , mgus of T and of U such that, for all i = 1; : : : ; n, Ui is nite, Ui Ti and [Ti] = [Ui]. Proof. Let T = fT ; : : : ; Tng beNsuch that at least one its member is in nite. Recall De ni on 1. By Fact 1 T = ftj gj2N is aNuni able set of types. By Theorem 2 there N exist a nite set N N, mgu ofj T and mgu of ftj gj2N such that [ T ] = [ftj gj2N ]. We put Ui = fti gj2N , for i = 1; : : : ; n. The following example shows that an in nite set of types need not be uni able, even if each its nite subset is uni able: 1
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Theorem 4. Let T be in nite, not uni able set of types. If T is bounded with respect to height then there exists a nite, not uni able set of types U T .
Proof. We prove the theorem by induction on h(T ). Observe that it su ces to consider the case l(T ) = @0. Indeed, if l(T ) < @0 then by Theorem 1 there exists a nite gist of T , which, by Corollary 3, is not uni able. Suppose h(T ) = 0. Then T Pr . Since T is not uni able there must exist c1; c2 2 Const (T ) such that c1 6= c2. Put U = fc1; c2g. Assume our thesis holds for any T 0 such that 0 < h(T 0) < h(T ). Suppose there exist u = (u1; : : : ; un)i 2 T and v = (v1; : : : ; vm)j 2 T such that m 6= n or i 6= j . Then U = fu; vg is not uni able. Suppose there exist n; i such that each element of T n Pr has a form (t1; : : : ; tn)i . De ne j (T ) as the set of all tj such that (t1; : : : ; tn)i 2 T , for some tk , 1 6 k 6 n, k 6= j . There exists k 2 f1; : : : ; ng such that l(k (T )) = @0. Since h(k (T )) < h(T ), by induction, there exists a nite, not uni able set Uk k (T ). By the de nition of k there also exists a nite set U T such that Uk k(U ). Clearly U is not uni able. From the above Theorem and Fact 1 one easily obtains: Theorem 5. Let T = fT1; : : : ; Tng be a not uni able family of nonempty, possibly in nite sets of types such that, for all i = 1; : : : ; n, Ti is bounded with respect to height. There exists a not uni able family U = fU1 ; : : : ; Ung such that, for all i = 1; : : : ; n, Ui Ti and Ui is nite.
3 Categorial consequence In this section we prove that each rigid language is determined by a nite set of functor-argument structures. At rst we generalize Buszkowski's discovery procedure (cf. [1, 2]) to the in nite sample case. De nition 7. Designate a type s 2 Const . Let A 2 FS (V ). Let V A be A any one-to-one function from SUB a(A) to Var . We de ne a relation 7! SUB (A) Tp : A A 7! s A B 7! V A(B ) for B 2 SUB a(A) A A if (B ; : : : ; Bn )i 7! ti and (8j 6= i)(V A (Bj ) = tj ) then Bi 7! (t ; : : : ; tn)i 1
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De nition 8. Let L FS (V ) be countable. We assume that L = fS ; S ; : : : g. Also, for each i > 1 we x a one-to-one function V S : SUB a(Si) ?! Var , 1
such that:
2
i
(14) Rng (V S +1 ) \ n
n [ i=1 L
Rng (V S ) = ;: i
We de ne a relation 7! SUB (L) Tp : L
(15) 7!=
1 [
i=1
7S ! i
De nition 9. Let L FS (V ) be countable. Denote VL = V \ SUB (L). By GF (L) we will denote the grammar de ned as follows:
VGF L = VL ; sGF L = s; L IGF L (v) = ft 2 Tp : v 7! tg Lemma 5. For all A 2 SUB (L) we have: L TGF L (A) = ft 2 Tp : A 7! tg Proof. We prove the lemma by structural induction. For A 2 V our thesis holds by the de nition of GF (L). Suppose ti 2 TGF L ((A ; : : : ; An)i). By the de nition of terminal type assignment there exist tj ; j 6= i such that tj 2 TGF L (Aj ) and (t ; : : : ; tn)i 2 L L TGF L (Ai). By induction Aj 7! tj and Ai 7! (t ; : : : ; tn)i. There exists l 2 N S such that Ai 7! (t ; : : : ; tn)i. It follows from the de nition of 7! that there are S B ; : : : ; Bn such that Bi = Ai, V S (Bj ) = tj for j 6= i and (B ; : : : ; Bn )i 7! ti . Also, for any A 2 FS (V ) and B 2 SUB (A): L if V A (B ) = t then (8C 2 SUB (L))(C 7! t ) C = B ): Consequently, Bj = Aj , for j = 1; : : : n S L To nish the proof suppose that (A ; : : : ; An)i 7! ti. Then (A ; : : : ; An)i 7! ti for some Sl 2 L. Since Aj 2 SUB a(Sl) for all j 6= i there exist tj ; j 6= i such S that V S (Aj ) = tj for j 6= i. By the de nition of 7! we get Ai 7! (t ; : : : ; tn)i and nally, by induction, tj 2 TGF L (Aj ); j 6= i and (t ; : : : ; tn)i 2 TGF L (Ai) which entails ti 2 TGF L ((A ; : : : ; An)i). ( )
( )
( )
( )
( )
1
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( )
l
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Corollary 5. FL(GF (L)) = L De nition 10. For any grammar G and a substitution we de ne the gram
mar [G]: V G = VG ; s G = sG; (8v 2 VG )(I G (v) = (IG (v)): Fact 15. For any substitution , grammar G and A 2 FS (VG ): (TG(A)) T G (A): Corollary 6. For any grammar G and substitution , FL(G) FL( [G]) In what follows we will write that a substitution uni es a grammar G if it uni es the family fIG(v) : v 2 V g. Theorem 6. Let L FS (V ) be nonempty (possibly in nite). Then, for any grammar G, the following conditions are equivalent: (i) G is rigid and compatible with L, (ii) G = [GF (L)] for some uni er of GF (L). Proof. (ii))(i) follows from Corollary 6. We prove (i))(ii). Recall de nitions 7 and 8. We de ne a substitution: (9l 2 N)(V S (A) = x) (x) = xTG(A) ifotherwise. Since G is rigid and IG (v) 6= ; for all v 2 SUB (L), it is enough to show that (IGF L (v)) IG(v) for all v 2 V . By structural induction on t we prove: [ ]
[ ]
[ ]
[ ]
l
( )
L (16) (8t 2 Tp )(8A 2 SUB (L))(A 7! t ) (t) = TG(A)) L If t 2 Var then A 7! t implies t = V S (A) for some Sl 2 L. If t = s then L A 7! t implies A 2 L. Suppose now that (16) is true for all tj , j = 1; : : : ; n. L Let Ai 7! (t ; : : : ; tn)i for some Ai 2 SUB (L). There exists l 2 N such that S L Ai 7! (t ; : : : ; tn)i. Denote Aj = V ?S (tj ), for j 6= i. Then (A ; : : : ; An)i 7! L ti and Aj 7! tj for j 6= i. By induction (ti) = TG ((A ; : : : ; An)i) and (tj ) = TG (Aj ) for j 6= i which implies, by the de nition of terminal type assignment, ((t ; : : : ; tn)i) = TG(Ai). From (16) immediately follows (8v 2 V )(8t 2 Tp )(t 2 IGF L (v) ) (t) = IG(v)). l
1
l
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15
Corollary 7. For any nonempty L FS (V ), L is rigidly describable if and only if GF (L) is uni able.
De nition 11. Let L be rigidly describable. By Theorem 3 there exists an mgu of GF (L). The grammar RG (L) = [GF (L)], where is the mgu of GF (L), will be called the rigid grammar determined by L.
Lemma 6. Let G and G be two uni able grammars such that G G . 1
2
1
2
Let 1 and 2 be mgus of G1 , G2 respectively. Then there exists substitution such that 2[G2] = 1[G1].
Proof. Since is also a uni er of G , there must exist a substitution such that = . Then [G ] = [G ] = [ [G ]]. 2
2
1
1
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1
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1
Corollary 8. Let L FS (V ) be rigidly describable and L L. Then there
exists a substitution such that RG (L) = [RG (L1 )]:
1
Proof. Recall the construction of GF (L). For each Si 2 L1 we choose the L1 L same function V S to de ne both 7! and 7! . Then GF (L1) GF (L), so our thesis follows from Lemma 6. i
Corollary 9. Let L be rigidly describable. Then, for any grammar G the
following conditions are equivalent:
(i) G is rigid and compatible with L, (ii) G = [RG (L)], for some substitution . Proof. Let RG (L) = [GF (L)]. Since is an mgu of GF (L), any uni er of GF (L) has a form , so our thesis follows immediately from Theorem 6.
Theorem 7. Let L be rigidly describable and nonempty. Then: (17) L FL(RG (L)) (18) L L ) FL(RG (L )) FL(RG (L)) 1
1
(19) FL(RG (FL(RG (L)))) = FL(RG (L)) (20) (9L0 L)(card (L0) < @ ^ FL(RG (L0)) = FL(RG (L))): 0
16 Proof. (17) is an immediate consequence of Corollaries 5, 6 and 9. (18) follows from Corollaries 6 and 8. To justify (19) it su ces to prove that FL(RG (FL(RG (L)))) FL(RG (L)). According to Corollary 9 there exists such that RG (L) = [RG (FL(RG (L)))] so our thesis holds by Corollarry 6. Finally we will prove (20). By Theorem 3, there exists a nite grammar G GF (L) such that RG (L) = [G] for some mgu of G. By the de nition L of GF (L), t 2 IG(v) implies v 7! t. For all v 2 V and t 2 IG(v) we choose S Sv;t 2 L such that v 7! t. Denote L0 = fSv;tgv2V;t2I (v). Clearly L0 is nite and G GF (L0) GF (L). By Lemma 6 there are substitutions and such that RG (L) = [RG (L0)] and RG (L0) = [[G]] = [RG (L)]. Our thesis holds by Corollary 6. v;t
G
De nition 12. Let L FS (V ). A categorial consequence of L Cn (L) is de ned as follows: (RG (L)) if L is rigidly describable Cn (L) = FL otherwise FS (VL )
In order to prove that Cn satis es Tarski's conditions we need some additional characteristics of rigid grammars. De nition 13. A rigid grammar G is said to be adequate if the following hold: A 62 SUB (FL(G)) implies TG(A) = ;, if CAT G ((t ; : : : ; tn)i) 6= ; then CAT G(tj ) 6= ;, for all 1 6 j 6 n, j 6= i. 1
For any grammar G, denote Var (G) =
[
v2V
rate the following lemma due to Kanazawa1 :
Var (IG (v)): We will incorpo
G
Lemma 7. For any L FS (V ), RG (L), if exists, is adequate, if G is adequate then card (Var (G)) 6 card(VG ),
Instead of adequate Kanazawa [5] uses equivalent in the scope of rigid grammars expression: grammar with no useless type. 1
17
if G; G0 are adequate grammars such that G0 = [G] for some but
there is no such that G = [G0] then card (Var (G0)) < card (Var (G)).
Fact 16. Cn satis es Tarski's conditions: (21) (8L FS (V ))(L Cn (L)) (22) (8L ; L FS (V ))(L L =) Cn (L ) Cn (L )) (23) (8L FS (V ))(Cn (Cn (L)) = Cn (L)) (24) (8L FS (V ))(9L0 L)(card (L0) < @ ^ Cn (L0) = Cn (L)) 1
2
1
2
1
2
0
Proof. Conditions (21), (22) and (23) easily follow from Theorem 7, since FS (V ) is not rigidly describable. We will prove (24) assuming L is not rigidly describable. Let L FS (V ) be an in nite, not rigidly describable set of structures. We consider two cases. Case 1: for each v 2 V , IGF (L)(v) is bounded with respect to height. Then our thesis easily follows from Theorem 5. Case 2: there exists v0 2 V such that IGF (L)(v0) is not bounded with respect to height. Suppose each nite subset of L is rigidly describable. Then there exists an in nite sequence L0 L1 : : : of nite subsets of L such that, for all i 2 N,
(25) h(IRG L (v )) < h(IRG L +1 (v )): (
i)
0
(
i
)
0
By Corollary 8, there exist substitutions i such that RG (Li ) = i[RG (Li)], for all i 2 N. Observe that the existence of i such that RG (Li ) = i[RG (Li )] would contradict (25). Finally, since each RG (Li ) is adequate, we have card (Var (RG (Li )) < card (Var (RG (Li ))), for all i 2 N. This however is impossible since any adequate and rigid grammar G admits at most card (VG ) variables. +1
+1
+1
18
References [1] W. Buszkowski, Solvable Problems for Classical Categorial Gram mars, Bull. Pol. Acad. Scie. Math. 35 (1987), pp. 373382. [2] W. Buszkowski, Discovery Procedures for Categorial Grammars, in [6]. [3] W. Buszkowski and G. Penn, Categorial Grammars Determined from Linguistic Data by Uni cation, Studia Logica XLIX, 4 (1990) pp. 431454. [4] M. Kanazawa, Identi cation in the Limit of Categorial Grammars, Journal of Logic, Language and Information, to appear. [5] M. Kanazawa, Learnable Classes of Categorial Grammars, Disserta tion, Stanford University, 1994. [6] E. Klein and J. van Benthem (eds), Categories, Polymorphism an Uni cation, Universiteit van Amsterdam, Amsterdam, 1987. [7] J. W. Lloyd, Foundations of Logic Programming, Sprringer-Verlag, Berlin, 1987. [8] J. Marciniec, Learning Categorial Grammars by Unication with Neg ative Constraints, Journal of Applied Non-Classical Logics, 4 (1994), pp. 181-200. [9] J. van Benthem, Categorial Equations, in [6]. Institute of Linguistics Adam Mickiewicz University ul. Mi¦dzychodzka 5 60371 Pozna« Poland address for correspondence:
ul. w. Rocha 4A/3 61-142 Pozna« Poland
Faculty of Mathematics and Computer Science Adam Mickiewicz University ul. Matejki 48/49 60769 Pozna« Poland
Adam Mickiewicz University Pozna«, Poland [email protected]
Abstract In this paper the notion of uni er is extended to the in nite set case. The proof of existence of the most general uni er of any in nite, uni able set of types (terms) is presented. Learning procedure, based on in nite set uni cation, is described. Keywords: categorial grammar, uni cation, discovery procedure.
1 Introduction and preliminaries A discovery procedure for determining categorial grammar from ( nite) lin guistic data has been studied in Buszkowski [1, 2] as well as van Ben them [9]. Further modi cations of the initial algorithms were presented in Buszkowski & Penn [3] and Marciniec [8]. All the procedures men tioned make use of uni cation of nite sets. The learning procedure given by Kanazawa [4, 5] admits as an input in nite sequences of structures but only those which constitute the whole categorial language. An attempt to admitting arbitrary in nite sets of postulates leads to the need of considering uni cation with respect to in nite sets. Fundamental concepts related to uni cation maintain their sense when referred to in nite sets of terms (or types). However, not all their properties
from: Studia Logica 58: 339355, 1997.
1
2 can be generalised, and those which admit generalisation require dierent proofs, since the standard ones essentially rely on niteness (see [7]). In this paper we prove the existence of the most general uni er of any in nite, uni able set of types by reduction of this problem to the nite case. A uni able set of types always contains a sort of nite representation a gist the most general uni er of which can be naturally extended to that of the whole set. The above fact, employed in the theory of categorial grammar, makes it possible to de ne a notion of compact consequence operator on the universe of functor-argument structures. Tarski's conditions, in particular compactness, of this operator can be shown using methods of Kanazawa [4, 5], based on an analysis of categorial grammar. However, in the scope of consistent data, it is possible to prove these facts in a dierent (more fundamental) way, by reducing them to general properties of uni cation of in nite sets. The paper is organised as follows. At rst we establish the notation and recapitulate some essential notions concerning uni cation and classical cate gorial grammar. In Section 2 we construct the rudiments of the uni cation of in nite sets. Finally, in Section 3, we give the account of learning procedure based on in nite set uni cation. The author acknowledges the supervision of this research by Wojciech Buszkowski. We de ne basic notions used in this paper. The set of all functor-argument structures on the set of atoms V , FS (V ), is de ned as the smallest set satis fying the following conditions: V FS (V ); if A ; : : : ; An 2 FS (V ) then (A ; : : : ; An)i 2 FS (V ); for all n > 2; 1 6 i 6 n: If A = (A ; : : : ; An)i 2 FS (V ) then Ai is the functor whereas each Aj , for j 6= i, is an argument in structure A. For any S FS (V ), SUB (S ) denotes the smallest set such that: S SUB (S ); if (A ; : : : ; An)i 2 SUB (S ) then Aj 2 SUB (S ); for all j = 1; : : : ; n: SUB (S ) consists of all substructures of structures from S . For technical reasons we designate: 1
1
1
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3 SUB a(S ) = fAj 2 SUB (S ) : (9A 2 SUB (S ))(A = (A1; : : : ; An)i ^ i 6= j )g SUB f (S ) = fAi 2 SUB (S ) : (9A 2 SUB (S ))(A = (A1 ; : : : ; An)i )g SUB a(S ) (SUB f (S )) contains all the structures playing the role of arguments (functors) in substructures from SUB (S ). The length l(A) of a functor-argument structure A 2 FS (V ) is de ned as follows: 8 if A 2 V > < 1X n l(A) = > l(Aj ) if A = (A1; : : : ; An)i : j =1
The height h(A) of functor-argument structure A 2 FS (V ) we de ne: A2V h(A) = 0max fh(A ) : j = 1; : : : ; ng + 1 ifif A = (A ; : : : ; An)i j We will write, that a set T FS (V ) is bounded with respect to length (height) if (9N 2 N)(8t 2 T )(l(t) 6 N ) ((9N 2 N)(8t 2 T )(h(t) 6 N ) ). l(T ) (h(T )) will denote max fl(t) : t 2 T g (max fh(t) : t 2 T g) respectively. We admit the case l(T ) = @ (h(T ) = @ ). In what follows we assume that each mapping : P ?! FS (Q) expands to a mapping from FS (P ) to FS (Q): (1) ((a ; : : : ; an)i) = ((a ); : : : ; (an ))i: For any set S of arguments of some mapping , [S ] will denote the image of S . A set Pr = Const [ Var , where Const \ Var = ;, is called a set of primitive types. Members of Tp = FS (Pr ) will be called types. For any T Tp we set: Var (T ) = SUB (T ) \ Var , Const (T ) = SUB (T ) \ Const , Pr (T ) = SUB (T ) \ Pr . A substitution is any mapping : Pr ?! Tp ful lling the condition (c) = c, for any c 2 Const . By we will denote the set of all substitutions. We say that a substitution uni es a set of types T if [T ] is a singleton. A set T Tp is said to be uni able if there exists a uni er of T . A uni er of T is called a most general uni er (mgu) of T if, for any uni er of T , there is a substitution such that = . For any uni able and nite set of types T , there exists an mgu of T , and it can be eectively found (cf. [7]). Two types u and v are said to be variants, if there are substitutions and such that v = (u) and u = (v). If and are mgu-s of some T Tp then [T ] and [T ] are variants. 1
0
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4 Let T = fT ; : : : ; Tng be a nite family of (possibly in nite) nonempty sets of types. A substitution is a uni er of T if it is a uni er of each Ti, i = 1; : : : ; n. A uni er of T is called an mgu of T if, for any uni er of T , there is a substitution such that = . The problem of uni ability of T can easily be reduced to that of set of types. De nition 1. Let T = fT ; : : : ; Tng be a family of countable sets of types. Assume Ti = fti ; ti ; : : : g, for all i = 1; : : : ; n. In case Ti is nite we set tji = ti ,Nfor all j > card (Ti). De ne tj = (tj ; : : : ; tjn) , for all j 2 N. We denote T = ftj gj2N. Fact 1. Let T = fT ; : : : ; Tng be a family of countable sets of types. Then: is a uni er of T i is a uni er of N T is the mgu of T i is the mgu of N T By a classical categorial grammar we mean a triple G = (VG ; IG; sG ) where a nite set of atoms VG is called the lexicon of G, a mapping IG : VG ?! 2Tp is called the initial type-assignment of G, sG 2 Pr c is called the principal type of G. The terminal type-assignment TG : FS (VG ) ?! 2Tp is de ned as follows: TG(v) = IG(v); for v 2 VG ; TG((A ; : : : ; An)i) = fai 2 Tp : (9(a ; : : : ; an)i 2 TG(Ai))(8j 6= i)(aj 2 TG(Aj ))g: Throughout this paper we will use the expression grammar in the meaning of classical categorial grammar. For any grammar G and t 2 Tp , we de ne a category of type t: CAT G (t) = fA 2 FS (VG ) : t 2 TG(A)g: FL(G) = CAT G (sG) will be called the F-language generated by G. A grammar G is said to be nite (rigid) i card (IG(v)) < @ (card (IG(v)) = 1), for all v 2 VG . A set L is rigid (rigidly describable) if L = FL(G) (L FL(G)) for some rigid grammar G. A grammar G is compatible with L FS (V ) if L FL(G). For any grammars G and H we write G H i VG = VH ; sG = sH and IG(v) IH (v), for all v 2 VG . 1
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2 In nite set uni cation In this section we will prove the existence of the most general uni er of any in nite, uni able set of types. We will also show that an mgu of any set is
5 always established on the basis of its nite subset. All the results presented below are still valid if we consider arbitrary terms instead of types. Fact 2. If T is uni able then Const (T ) is nite. From now on we assume that the set Const is nite. Fact 3. If T is bounded with respect to length and Pr (T ) is nite then T is nite. De nition 2. Let T Tp . We say that a substitution respects an equiv alence relation on SUB (T ) if (8a; b 2 Pr (T ))(a b ) (a) = (b)). Fact 4. If there exists a substitution respecting relation then: (8a; b 2 Const )(a b ) a = b) By [t] we denote the equivalence class of type t with respect to the equivalence relation . De nition 3. Let T Tp . For any equivalence relation SUB (T ) SUB (T ) we de ne a mapping [ ] : Pr (T ) ?! 2Pr T : ([p]) if p 2 Var (T ) [ p] = Var fpg if p 2 Const (T ) Denote [ Pr ] = f[ p] : p 2 Pr (T )g. According to (1) [ ] expands to a mapping from FS (Pr (T )) to FS ([[Pr ] ). Fact 5. If respects then, for all u; v 2 Tp, [ u] = [ v] implies (u) = (v). De nition 4. Let S FS (P ); T FS (Q). By structural induction we de ne relation D(S; T ) SUB (S ) SUB (T ): S T D(S; T ); if h(A ; : : : ; An)i; (B ; : : : ; Bn)ii 2 D(S; T ) then hAj ; Bj i 2 D(S; T ); for all j = 1; : : : ; n Fact 6. For all T Tp the relation D(T; T ) is re exive and symmetric. By T we denote the transitive closure of D(T; T ). Notice that T SUB (T ) SUB (T ): ( )
1
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6
Fact 7. For any sets of types U and T : If U T then U T .
We skip the proof of the above fact since the next one uses similar tech nique. Fact 8. Let T be a set of types. Then: (8u; v 2 SUB (T )) [u T v ) (9S T )(card (S ) < @ ^ u S v)] Proof. Let u; v 2 SUB (T ). Denote the set whose existence is to be proved by S (u; v). We consider hu; vi 2 D(T; T ) at rst. If u; v 2 T then of course S (u; v) = fu; vg. Assume now that u = (u ; : : : ; un)i, v = (v ; : : : ; vn)i and our thesis holds for u and v. Then we put S (uj ; vj ) = S (u; v), for all j = 1; : : : ; n. Let nally u T v. There exist u ; : : : ; uk such that u = u, uk = v and hSuj ; uj i 2 D(T; T ) for j = 1; : : : ; k ? 1. By Fact 7 we can set S (u; v) = k? j S (uj ; uj ). Fact 9. Each uni er of T respects T . Proof. Let be a uni er of T . By depth induction we will show that, for all a; b 2 SUB (T ), if ha; bi 2 R(T; T ) then (a) = (b). For a; b 2 T , the above holds by the de nition of uni er. Suppose a = (a ; : : : ; an)i; b = (b ; : : : ; bn)i and ha; bi 2 R(T; T ). Then if our thesis holds for a and b, that is (a) = (b), then it also holds for each aj and bj , j = 1; : : : ; n, by the de nition of substitution. If u T v then u = u = : : : = uk = v and huj ; uj i 2 R(T; T ), for j = 1; : : : ; k ? 1. From the rst part of the proof we get immediately (u) = (u ) = : : : = (uk) = (v). Below we will write [ ] T instead of [ ] . Fact 10. For all u; v 2 T if [ u] T = [ v] T then: (2) (8a 2 SUB (u))(9b 2 SUB (v))([[a] T = [ b] T ^ a fu;vg b): Proof. Assume [ u] T = [ v] T . We prove (2) by depth induction on a. If a = u then (2) holds by assumption. Let a = (a ; : : : ; an)i and there ex ists b 2 SUB (v) such that [ a] T = [ b] T and a fu;vg b. Since [ a] T = ([[a ] T ; : : : ; [ an] T )i there must exist b ; : : : ; bn such that b = (b ; : : : ; bn)i. Then of course [ aj ] T = [ bj] T and aj fu;vg bj for j = 1; : : : ; n, which nishes the proof. 0
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Corollary 1. For all u; v 2 T such that [ u] T = [ v] T we have: (8p 2 Pr (u))(9q 2 Pr (v))(p fu;vg q): Proof. Our thesis follows immediately from Fact 10 since [ a] T = [ b] T and a 2 Pr implies b 2 Pr .
Lemma 1. Let T be a set of types bounded with respect to length. Then ind (T ) is nite. Proof. Let M = l(T ). We assume that Var = fx ; x ; : : : g. Denote Var M = fx ; : : : ; xM g 1
2
1
and
SM = fa 2 FS (Var M ) : l(a) = card(Var (a))g: Clearly, for any a 2 SM , l(a) 6 M , each variable occurs in a at most once and, consequently, SM is nite. It is also easy to see, that SUB (SM ) = SM . One easily proves the following: (3) (8u 2 Tp )(l(u) 6 M ) (9a 2 SM )(9 2 )(l(u) = l(a) ^ ( (a) = u)) Assume SM = fa ; : : : ; amg. For i = 1; : : : ; m, we de ne substitutions:
i(xj ) =
1
x i? xj (
M +j
1)
if j 6 M otherwise
We denote SM = f i(ai) : i = 1; : : : ; mg. Each element of SM is a variant of some element of SM , and each is build using dierent variables. Clearly, (8t ; t 2 SM )(SUB (t ) \ SUB (t ) = ;). Recall De nition 4. To simplify notation we will write u ; t instead of hu; ti 2 D(T; SM ). We de ne a function: 1
2
1
2
f : SUB (T )= ?! 2SUB S (
M)
T
by setting:
f (K) = ft 2 SUB (SM ) : (9u 2 K)(u ; t)g:
8 Since card (Rng (f )) < @ , it su ces to show that f is one-to-one. First, by depth induction, we prove: (4) if u ; t and v ; t then hu; vi 2 D(T; T ): For t 2 SM the above is obvious since t 62 SUB (SM ) n ftg. Assume now that (4) holds, for arbitrary t = (t ; : : : ; tn)i. We will show that it also holds, for each tj , 1 6 j 6 n. Let uj ; tj and vj ; tj . Since t is the only structure in SUB (SM ) containing tj , there must exist u = (u ; : : : ; un)i and v = (v ; : : : ; vn)i such that u ; t and v ; t. Then, by induction, hu; vi 2 D(T; T ) and consequently huj ; vj i 2 D(T; T ). It follows from (4) that f (K ) \ f (K ) 6= ; implies K = K , for all K ; K 2 SUB (T )= . To prove that f is on-to-one we have to show that ; 62 Rng (f ). Again, by depth induction on u we derive: (5) (8u 2 SUB (T ))(9t 2 SUB (SM ))(u ; t ^ l(u) = l(t) ^ (9 2 )(u = (t)): For u 2 T (5) follows immediately from (3). Suppose u = (u ; : : : ; un)i ; t, l(u) = l(t) and u = (t) for some 2 . Then there must exist t ; : : : ; tn such that t = (t ; : : : ; tn)i. Clearly uj ; tj , uj = (tj ) and l(uj ) = l(tj ), which nishes the proof of (5). Consequently, f is one-to-one and therefore SUB (T )= T is nite. Corollary 2. If T is bounded with respect to length then [ Pr] T is nite. De nition 5. Let T Tp. A set of types U T is said to be a gist of T if: (6) [ T ] T = [ U ] T (7) (8a; b 2 Pr (U ))(a T b , a U b): Since U = T ful ls (6) and (7), a gist of T always exists. We aim to show that, for any set of types bounded with respect to length, there is a nite gist of this set. First, we need a number of auxiliary facts. Fact 11. Let U be a gist of T . Then: (8A 2 [ Pr ] T )(A \ Pr (U ) 6= ;): Proof. Let A 2 [ Pr ] T and p 2 A. There is some t 2 T such that p 2 Pr (t). By (6) [ t] T = [ u] T for some u 2 U . By Fact 10 there exists q 2 Pr (u) such that [ p] T = [ q] T which nishes the proof since q 2 A = [ p] T . 0
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9 The above fact justi es the following de nition. De nition 6. Let U be a gist of T . A function U : [ Pr ] T ?! Pr is said to be a choice function for U if U (A) 2 A\ Pr (U ), for all A 2 [ Pr ] T . Clearly U is one-to-one. We x U , for the given U . Let U be a substitution such that: U (p) = U ([[p] T ):
(8)
The above de nition is correct, since for any c 2 Const we have: U (c) = U ([[c] T ) = U (fcg) = c:
For any substitution respecting U we de ne the extension "U of (with respect to U ): (9) "U =
U
Observe that the above de nition does not depend on the choice of U . Lemma 2. Let T be a set of types. If T is bounded with respect to length then [ T ] T is nite. Proof. Let U be an arbitrary gist of T and U be a choice function for U . Then U is one-to-one correspondence between [ T ] T and U [T ]. Since l( U (t)) = l(t) for any t 2 T , U [T ] is also bounded with respect to length. By Fact 3 U [T ] is nite, because Pr ( U [T ]) = U ([[Pr ] T ) is nite, by Lemma 1. Theorem 1. Let T be a set of types bounded with respect to length. There exists a nite gist of T . Proof. First we de ne a set of types U : for each A 2 [ T ] T we choose one t 2 T such that [ t] T = A and put it in U . By Lemma 2, U is nite. Also, for any W such that U W T , we have [ U ] T = [ W ] T = [ T ] T . We prove: 0
0
0
0
0
(10) (8W T )(U W ) (8a 2 Pr (W ))(9b 2 Pr (U ))(a W b)) 0
0
Let U W T and a 2 Pr (W ). Then there exists w 2 W such that a 2 Pr (w). By the de nition of U , there is some u 2 U such that [ w] T = [ u] T . 0
0
0
10 By Corollary 1 there exists b 2 Pr (u) such that a fu;wg b. By Fact 7, since fu; wg W , we have a W b. According to Fact 8 for any a; b 2 SUB (T ) such that a T b there exists a nite set S (a; b) T such that a S a;b b. Let D = ffa; bg Pr (U ) : (a T b) ^ (a 6U0 b)g. We set: (
(11) U = U [ 0
[
fa;bg2D
)
0
S (a; b)
Clearly U is nite. We will show that U ful ls (7). Let a; b 2 Pr (U ) and a T b. By (10) there exist a0; b0 2 Pr (U ) such that a U a0; b U b0. Since U is transitive, it su ces to show that a0 U b0. If a0 U0 b0 then also a0 U b0. If a0 6U0 b0 then, by (11), S (a0; b0) U . Consequently a0 U b0 which nishes the proof. Fact 12. Let U be a gist of T and be a substitution respecting U . Then: (12) "U (a) = (a) for all a 2 Pr (U ) (13) "U respects T Proof. To prove (12) assume x 2 Var (U ). Then U (x) T x and, by (7), U (x) U x, which implies ( U (x)) = (x), since respects U . To prove that "U respects T choose p ; p 2 Pr (T ) ful lling p T p . Then there exist t ; t 2 T such that p 2 Pr (t ) and p 2 Pr (t ). By (6) there are u ; u 2 U such that [ u ] T = [ t ] T and [ u ] T = [ t ] T . By Fact 10 there exist q ; q 2 Pr (U ) such that [ p ] T = [ q ] T and [ p ] T = [ q ] T . Then q T p T p T q . By (7) also q U q , and since respects U , we get (q ) = (q ). Finally "U (p ) = ( U (p )) = ( U (q )) = (q ) = (q ) = ( U (q )) = ( U (p )) = "U (p ). Fact 13. Let T be a set of types. If respects T then, for any gist U of T , we have: "U = . Proof. Suppose U be a gist of T . Since U (x) T x for all x 2 Pr and respects T we have: (x) = ( U (x)) = "U (x). Lemma 3. Let U be a gist of T and be a substitution respecting U . Then, "U uni es T i uni es U . Proof. Assume uni es U . For each t 2 T , there exists u 2 U such that [ t] T = [ u] T . Then "U (t) = "U (u) = (u). 0
1
1
1
1
1
1
2
1
2
2
2
2
2
2
2
2
1
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2
2
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2
2
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2
11
Corollary 3. A set of types is uni able if and only if its gist is uni able. Lemma 4. Let T be a uni able set of types, and let U be a gist of T . For
any mgu of U "U is an mgu of T . Proof. Let be an mgu of U . Since any uni er of T is also a uni er of U , there exists such that = . Then: ("U (x)) = (( U (x))) = ( U (x)) = "U (x) = (x). The last equality holds by Fact 9 and Fact 13.
We have an obvious statement: Fact 14. If T is uni able then T is bounded with respect to length. As a consequence we get: Theorem 2. Let T be an in nite, uni able set of types. Then there exist a nite set U T , mgus of T and of U , such that [T ] = [U ]. Proof. Since T is uni able, by Theorem 1, there exists a nite gist U of T . For any mgu of U , = "U is an mgu of T and [T ] = "U [T ] = "U [U ] = [U ]. Corollary 4. Let T be an in nite, uni able set of types. Then, there exists an mgu of T . Theorem 3. Let T = fT ; : : : ; Tng be a uni able family of nonempty, pos sibly in nite sets of types. There exist a family U = fU ; : : : ; Ung , mgus of T and of U such that, for all i = 1; : : : ; n, Ui is nite, Ui Ti and [Ti] = [Ui]. Proof. Let T = fT ; : : : ; Tng beNsuch that at least one its member is in nite. Recall De ni on 1. By Fact 1 T = ftj gj2N is aNuni able set of types. By Theorem 2 there N exist a nite set N N, mgu ofj T and mgu of ftj gj2N such that [ T ] = [ftj gj2N ]. We put Ui = fti gj2N , for i = 1; : : : ; n. The following example shows that an in nite set of types need not be uni able, even if each its nite subset is uni able: 1
1
1
x (x ; x) ((x ; x) ; x) (((x ; x) ; x) ; x) ... 1
2
1
3
4
1
1
1
1
1
12
Theorem 4. Let T be in nite, not uni able set of types. If T is bounded with respect to height then there exists a nite, not uni able set of types U T .
Proof. We prove the theorem by induction on h(T ). Observe that it su ces to consider the case l(T ) = @0. Indeed, if l(T ) < @0 then by Theorem 1 there exists a nite gist of T , which, by Corollary 3, is not uni able. Suppose h(T ) = 0. Then T Pr . Since T is not uni able there must exist c1; c2 2 Const (T ) such that c1 6= c2. Put U = fc1; c2g. Assume our thesis holds for any T 0 such that 0 < h(T 0) < h(T ). Suppose there exist u = (u1; : : : ; un)i 2 T and v = (v1; : : : ; vm)j 2 T such that m 6= n or i 6= j . Then U = fu; vg is not uni able. Suppose there exist n; i such that each element of T n Pr has a form (t1; : : : ; tn)i . De ne j (T ) as the set of all tj such that (t1; : : : ; tn)i 2 T , for some tk , 1 6 k 6 n, k 6= j . There exists k 2 f1; : : : ; ng such that l(k (T )) = @0. Since h(k (T )) < h(T ), by induction, there exists a nite, not uni able set Uk k (T ). By the de nition of k there also exists a nite set U T such that Uk k(U ). Clearly U is not uni able. From the above Theorem and Fact 1 one easily obtains: Theorem 5. Let T = fT1; : : : ; Tng be a not uni able family of nonempty, possibly in nite sets of types such that, for all i = 1; : : : ; n, Ti is bounded with respect to height. There exists a not uni able family U = fU1 ; : : : ; Ung such that, for all i = 1; : : : ; n, Ui Ti and Ui is nite.
3 Categorial consequence In this section we prove that each rigid language is determined by a nite set of functor-argument structures. At rst we generalize Buszkowski's discovery procedure (cf. [1, 2]) to the in nite sample case. De nition 7. Designate a type s 2 Const . Let A 2 FS (V ). Let V A be A any one-to-one function from SUB a(A) to Var . We de ne a relation 7! SUB (A) Tp : A A 7! s A B 7! V A(B ) for B 2 SUB a(A) A A if (B ; : : : ; Bn )i 7! ti and (8j 6= i)(V A (Bj ) = tj ) then Bi 7! (t ; : : : ; tn)i 1
1
13
De nition 8. Let L FS (V ) be countable. We assume that L = fS ; S ; : : : g. Also, for each i > 1 we x a one-to-one function V S : SUB a(Si) ?! Var , 1
such that:
2
i
(14) Rng (V S +1 ) \ n
n [ i=1 L
Rng (V S ) = ;: i
We de ne a relation 7! SUB (L) Tp : L
(15) 7!=
1 [
i=1
7S ! i
De nition 9. Let L FS (V ) be countable. Denote VL = V \ SUB (L). By GF (L) we will denote the grammar de ned as follows:
VGF L = VL ; sGF L = s; L IGF L (v) = ft 2 Tp : v 7! tg Lemma 5. For all A 2 SUB (L) we have: L TGF L (A) = ft 2 Tp : A 7! tg Proof. We prove the lemma by structural induction. For A 2 V our thesis holds by the de nition of GF (L). Suppose ti 2 TGF L ((A ; : : : ; An)i). By the de nition of terminal type assignment there exist tj ; j 6= i such that tj 2 TGF L (Aj ) and (t ; : : : ; tn)i 2 L L TGF L (Ai). By induction Aj 7! tj and Ai 7! (t ; : : : ; tn)i. There exists l 2 N S such that Ai 7! (t ; : : : ; tn)i. It follows from the de nition of 7! that there are S B ; : : : ; Bn such that Bi = Ai, V S (Bj ) = tj for j 6= i and (B ; : : : ; Bn )i 7! ti . Also, for any A 2 FS (V ) and B 2 SUB (A): L if V A (B ) = t then (8C 2 SUB (L))(C 7! t ) C = B ): Consequently, Bj = Aj , for j = 1; : : : n S L To nish the proof suppose that (A ; : : : ; An)i 7! ti. Then (A ; : : : ; An)i 7! ti for some Sl 2 L. Since Aj 2 SUB a(Sl) for all j 6= i there exist tj ; j 6= i such S that V S (Aj ) = tj for j 6= i. By the de nition of 7! we get Ai 7! (t ; : : : ; tn)i and nally, by induction, tj 2 TGF L (Aj ); j 6= i and (t ; : : : ; tn)i 2 TGF L (Ai) which entails ti 2 TGF L ((A ; : : : ; An)i). ( )
( )
( )
( )
( )
1
( )
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( )
l
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14
Corollary 5. FL(GF (L)) = L De nition 10. For any grammar G and a substitution we de ne the gram
mar [G]: V G = VG ; s G = sG; (8v 2 VG )(I G (v) = (IG (v)): Fact 15. For any substitution , grammar G and A 2 FS (VG ): (TG(A)) T G (A): Corollary 6. For any grammar G and substitution , FL(G) FL( [G]) In what follows we will write that a substitution uni es a grammar G if it uni es the family fIG(v) : v 2 V g. Theorem 6. Let L FS (V ) be nonempty (possibly in nite). Then, for any grammar G, the following conditions are equivalent: (i) G is rigid and compatible with L, (ii) G = [GF (L)] for some uni er of GF (L). Proof. (ii))(i) follows from Corollary 6. We prove (i))(ii). Recall de nitions 7 and 8. We de ne a substitution: (9l 2 N)(V S (A) = x) (x) = xTG(A) ifotherwise. Since G is rigid and IG (v) 6= ; for all v 2 SUB (L), it is enough to show that (IGF L (v)) IG(v) for all v 2 V . By structural induction on t we prove: [ ]
[ ]
[ ]
[ ]
l
( )
L (16) (8t 2 Tp )(8A 2 SUB (L))(A 7! t ) (t) = TG(A)) L If t 2 Var then A 7! t implies t = V S (A) for some Sl 2 L. If t = s then L A 7! t implies A 2 L. Suppose now that (16) is true for all tj , j = 1; : : : ; n. L Let Ai 7! (t ; : : : ; tn)i for some Ai 2 SUB (L). There exists l 2 N such that S L Ai 7! (t ; : : : ; tn)i. Denote Aj = V ?S (tj ), for j 6= i. Then (A ; : : : ; An)i 7! L ti and Aj 7! tj for j 6= i. By induction (ti) = TG ((A ; : : : ; An)i) and (tj ) = TG (Aj ) for j 6= i which implies, by the de nition of terminal type assignment, ((t ; : : : ; tn)i) = TG(Ai). From (16) immediately follows (8v 2 V )(8t 2 Tp )(t 2 IGF L (v) ) (t) = IG(v)). l
1
l
1
1
1
l
1
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( )
15
Corollary 7. For any nonempty L FS (V ), L is rigidly describable if and only if GF (L) is uni able.
De nition 11. Let L be rigidly describable. By Theorem 3 there exists an mgu of GF (L). The grammar RG (L) = [GF (L)], where is the mgu of GF (L), will be called the rigid grammar determined by L.
Lemma 6. Let G and G be two uni able grammars such that G G . 1
2
1
2
Let 1 and 2 be mgus of G1 , G2 respectively. Then there exists substitution such that 2[G2] = 1[G1].
Proof. Since is also a uni er of G , there must exist a substitution such that = . Then [G ] = [G ] = [ [G ]]. 2
2
1
1
2
2
2
1
1
1
Corollary 8. Let L FS (V ) be rigidly describable and L L. Then there
exists a substitution such that RG (L) = [RG (L1 )]:
1
Proof. Recall the construction of GF (L). For each Si 2 L1 we choose the L1 L same function V S to de ne both 7! and 7! . Then GF (L1) GF (L), so our thesis follows from Lemma 6. i
Corollary 9. Let L be rigidly describable. Then, for any grammar G the
following conditions are equivalent:
(i) G is rigid and compatible with L, (ii) G = [RG (L)], for some substitution . Proof. Let RG (L) = [GF (L)]. Since is an mgu of GF (L), any uni er of GF (L) has a form , so our thesis follows immediately from Theorem 6.
Theorem 7. Let L be rigidly describable and nonempty. Then: (17) L FL(RG (L)) (18) L L ) FL(RG (L )) FL(RG (L)) 1
1
(19) FL(RG (FL(RG (L)))) = FL(RG (L)) (20) (9L0 L)(card (L0) < @ ^ FL(RG (L0)) = FL(RG (L))): 0
16 Proof. (17) is an immediate consequence of Corollaries 5, 6 and 9. (18) follows from Corollaries 6 and 8. To justify (19) it su ces to prove that FL(RG (FL(RG (L)))) FL(RG (L)). According to Corollary 9 there exists such that RG (L) = [RG (FL(RG (L)))] so our thesis holds by Corollarry 6. Finally we will prove (20). By Theorem 3, there exists a nite grammar G GF (L) such that RG (L) = [G] for some mgu of G. By the de nition L of GF (L), t 2 IG(v) implies v 7! t. For all v 2 V and t 2 IG(v) we choose S Sv;t 2 L such that v 7! t. Denote L0 = fSv;tgv2V;t2I (v). Clearly L0 is nite and G GF (L0) GF (L). By Lemma 6 there are substitutions and such that RG (L) = [RG (L0)] and RG (L0) = [[G]] = [RG (L)]. Our thesis holds by Corollary 6. v;t
G
De nition 12. Let L FS (V ). A categorial consequence of L Cn (L) is de ned as follows: (RG (L)) if L is rigidly describable Cn (L) = FL otherwise FS (VL )
In order to prove that Cn satis es Tarski's conditions we need some additional characteristics of rigid grammars. De nition 13. A rigid grammar G is said to be adequate if the following hold: A 62 SUB (FL(G)) implies TG(A) = ;, if CAT G ((t ; : : : ; tn)i) 6= ; then CAT G(tj ) 6= ;, for all 1 6 j 6 n, j 6= i. 1
For any grammar G, denote Var (G) =
[
v2V
rate the following lemma due to Kanazawa1 :
Var (IG (v)): We will incorpo
G
Lemma 7. For any L FS (V ), RG (L), if exists, is adequate, if G is adequate then card (Var (G)) 6 card(VG ),
Instead of adequate Kanazawa [5] uses equivalent in the scope of rigid grammars expression: grammar with no useless type. 1
17
if G; G0 are adequate grammars such that G0 = [G] for some but
there is no such that G = [G0] then card (Var (G0)) < card (Var (G)).
Fact 16. Cn satis es Tarski's conditions: (21) (8L FS (V ))(L Cn (L)) (22) (8L ; L FS (V ))(L L =) Cn (L ) Cn (L )) (23) (8L FS (V ))(Cn (Cn (L)) = Cn (L)) (24) (8L FS (V ))(9L0 L)(card (L0) < @ ^ Cn (L0) = Cn (L)) 1
2
1
2
1
2
0
Proof. Conditions (21), (22) and (23) easily follow from Theorem 7, since FS (V ) is not rigidly describable. We will prove (24) assuming L is not rigidly describable. Let L FS (V ) be an in nite, not rigidly describable set of structures. We consider two cases. Case 1: for each v 2 V , IGF (L)(v) is bounded with respect to height. Then our thesis easily follows from Theorem 5. Case 2: there exists v0 2 V such that IGF (L)(v0) is not bounded with respect to height. Suppose each nite subset of L is rigidly describable. Then there exists an in nite sequence L0 L1 : : : of nite subsets of L such that, for all i 2 N,
(25) h(IRG L (v )) < h(IRG L +1 (v )): (
i)
0
(
i
)
0
By Corollary 8, there exist substitutions i such that RG (Li ) = i[RG (Li)], for all i 2 N. Observe that the existence of i such that RG (Li ) = i[RG (Li )] would contradict (25). Finally, since each RG (Li ) is adequate, we have card (Var (RG (Li )) < card (Var (RG (Li ))), for all i 2 N. This however is impossible since any adequate and rigid grammar G admits at most card (VG ) variables. +1
+1
+1
18
References [1] W. Buszkowski, Solvable Problems for Classical Categorial Gram mars, Bull. Pol. Acad. Scie. Math. 35 (1987), pp. 373382. [2] W. Buszkowski, Discovery Procedures for Categorial Grammars, in [6]. [3] W. Buszkowski and G. Penn, Categorial Grammars Determined from Linguistic Data by Uni cation, Studia Logica XLIX, 4 (1990) pp. 431454. [4] M. Kanazawa, Identi cation in the Limit of Categorial Grammars, Journal of Logic, Language and Information, to appear. [5] M. Kanazawa, Learnable Classes of Categorial Grammars, Disserta tion, Stanford University, 1994. [6] E. Klein and J. van Benthem (eds), Categories, Polymorphism an Uni cation, Universiteit van Amsterdam, Amsterdam, 1987. [7] J. W. Lloyd, Foundations of Logic Programming, Sprringer-Verlag, Berlin, 1987. [8] J. Marciniec, Learning Categorial Grammars by Unication with Neg ative Constraints, Journal of Applied Non-Classical Logics, 4 (1994), pp. 181-200. [9] J. van Benthem, Categorial Equations, in [6]. Institute of Linguistics Adam Mickiewicz University ul. Mi¦dzychodzka 5 60371 Pozna« Poland address for correspondence:
ul. w. Rocha 4A/3 61-142 Pozna« Poland
Faculty of Mathematics and Computer Science Adam Mickiewicz University ul. Matejki 48/49 60769 Pozna« Poland
