A COMPLETENESS THEOREM FOR THE LAMREK CALCULUS
O F SYNTACTIC CATEGORIES
by KOSTA D O ~ Ein
N Belgrade (Yugoslavia)
Inbroduction
In [5] LAMBEK
has formulated a calculus of syntactic categories, related to a calculus of residuals. which is meant t o underlie categorial grammars. I n [ I ] BUSZKOWSKI
ment,ions a nat,ural semigroup modelling of this calculus. without claiming completehas published in [ 2 ] a completeness theorem for t’he
ness. Recently. BUSZKOWSKI
product-free fragment of LAMBER’S
calculus (for various axiomatizations of this
it seems this is the first pufragment see [S] and 131). According to BUSZKOWSKI
calculus. This Completeness theorem of
blished result of this sort for LAMBEK’S
RURZKOWSKI
is given with respect to a new type of structures which he calls “category systems ”.
In t,his paper we shall prove a completeness theorem for the full Lambek calculus
with respect to partially ordered semigroup models using an interpretation somewhat
different from BUSZKOWSKI’S
interpretation of [11. More precisely! we shall consider
partially ordered semigroups (in the sense of [41, p. 153) and subsets in these semigroups which are closed from below (i.e.. if M is a subset, e , E M and el 5 e , , then
e , E X ) . Our proof will be entirely syntactical: i t will consist in an embe,dding of
LAMBEK’S
calculus in a first-order theory of partially ordered semigroups which has
unary predicates interpreted by subsets closed from below. These unary predicates
stand for syntactic categories. We shall suggest an intuitive interpretation of these
seniigroup models. After our main completeness result we shall consider completeness
calculus - in particular for the
results of the same type for fraginenh of LAMBEK’S
product-free fragment’.
1. The system L
The Lambek calculus is the following system which we shall call L.
The language of L, called 2’(L). consists of
(I) denumerably many primit.ive category expressions : P,, P,, . . .,
(11) three expressions for category operations: .>/: \,
(111)one expression for a category relat’,ion: +.
(‘ategory expressions in 9 ( L ) behave like terms. We shall use X I , X , . . . as schematic
letters for category expressions. If X , and X, are category expressions, then XI . X 2 ,
X , / X 2 and X , \ X , are category expressions, and X , + X , is a formula of 9 ( L ) .There
are no other formulae in 2’(L).
~
The a x i o m - s c h e m a t ) a and r u l e s of L are
X , .X ,
(c')
x3
-+
-
-+
x2\x3
5,+ X 2
(e)
--* x2\x3
(d')
XI
x , .'x,-+ x 3
X,+X3
-+ x3
2. The system S
The first-order t'heory of partially ordered semigroups we shall consider will lw
called S.
The language of S. called 2'(S), consists of
(i) denumerably many primitive term constants : p , , p 2 , . . ..
(ii) drnumerably many term variables: x, , x,. . .
(iii) one expression for a term operation :
+,
(iv) denuinerably many primitive category expressions: P I ,P, . . . . (identical with ( l ) ) ,
(v) two expressions for term relations: 5 , = .
(vi) the logical constant,s:
*.
&, v,
0,
7,
Vz,.
3si
We shall use a , , a,, . . . as schematic letters for terms. If a , and a , are terms, then
a , + a , is a term, and a , 2 a, and a , = a , are formulae. Category expressions in
Y ( S ) behave like unary predicates: so expressions of the form P f a , are formulae.
(Intuitively, Pp, means " a , is of the category Pi".)
Other formulae of Y ( S ) , involving logical constants from (vi), are built as usual.
The system S is axiomatized by the following
(1) logical p o s t u l a t e s :
the axioms and rules of the first-order predicate calculus involving the constants
of (vi).
"I
= a,,
(al = a 2 & A [ a , ] ) A[a,j, where the formula A[a,] is obtained from the formula A [ a , ] by replacing zero or more occurrences of a , by a,. with the usual
provisos for variables ;
(2) a x i o m - s c h e m a t a f o r p a r t i a l o r d e r :
5 a,
(2.2) ( a , 5 a , & a, 5 a 3 ) * a , 5 a 3 ;
(2.1) a ,
(2.3) (a,
I
5 Q,
& a2 5 a , )
a,
=
a,;
237
A (‘OAIPLETENESS THEOREM FOR THE LAMBEK CALCULUS OF SYNTACTIC CATEGORIES
(3) a x i o m - s c h e m a t a c o n n e c t i n g
(3.1) a ,
s
a2
(3.2) a ,
5
a,
2
( 3 . 3 ) (a,,
+
* a3 + a ,
*a,
a,
a3
& P,a,)
5 a,
5 a3
s with + and
category expressions:
+ a3.
+ a,,
* Pin,:
(4) s e m i g r o u p a x i o m - s c h e m a : ( a ,
+ a,) + a3 =
Q,
+ ( a 2 + a,3).
Next we shall introduce a system S’ closely related to S. Let 9 ( S ’ ) differ from 9 ( S )
only by having the expressions of (11) and (111) in addition to what we have in 9 ( S ) .
We have in 2’(S‘)complex unary predicates of the form X I . X , , X I/X,and X1\X2.
and also formulae of the form X , a , and XI -+ X,. It is clear that 2 ( L ) is a proper
su1)language of 9 ( S ’ ) .
The system S‘ differs from S by having in’ addition to (1) - (4) the following axiomschemata
( 5 ) ( X , . X,) a ,
0
3 2 , 3z,(X,r, & x,z, & a ,
5 x,
+ x,):
* Vz,(X,r, * X , ( a , + z2)L
aL 0 V ~ , ( x 2 ~*
2 ~yi(~
+ 2a i ) ) :
(6.1) ( X , / X , ) a ,
(6.2) (X,\X,)
(7) XI
4
x,0v21(x12,
* X,X,).
Tt is easily shown by induction on the complexity of XI that ( a , 5 a2 & X , a 2 )*
* X , a , is provable in S’ for every X , .
It is clear that (5)- (7) could be understood as specifying definitions which introduce
some abbreviations in 9 ( S ) , and we shall assume that ., /, \ and -+ have been introduced in 9 ( S ) by these definitions. Then it is trivial to show that the theorems of S
and S‘ coincide. We have introduced the system S’ only for convenience.
3. Embedding of L into S
We can prove the following lemma.
L e m m a 1 . If t,X,
-+
X , , then t , X ,
-+
X,.
This is done by a straightforward induction on the length of proof of X I
and we shall omit this proof.
-+
X , in L,
Sext we introduce the mapping t from 9 ( S ’ ) to the metalanguage of L satisfying
tk,) = p,.
t ( s , ) = the i-th schematic letter for category expressions,
+ a,) =
t ( X 1 ) = x,.
t(a1) . t(a2)>
t(a,
t(X,a,) = k,t(a,)
t ( X , ) (i.e. t,t(a,)
t ( a , a,) = t,t(a,) -+ t(%),
t(a, = a,) = t,t(a,) -b t(aJ and t,t(a,)
-+
-+
Xl),
s
t(X,
-+
X,)
=
-b
t L t ( X 1 )-+ t(X,f) (i.e. tLX,
t(a,),
-+
X,),
t ( A B ) = if t ( A ) , then t ( B ) ,
t(A 0 B ) = t ( A ) iff t ( B ) ,
f ( A& B ) = t ( A ) and t ( B ) ,
t ( A v B ) = t(A) or t ( B ) ,
t ( 1 A ) = not t ( A ) ,
t(Vx,A) = for eeery X I ,t ( A ) ,
t(3qA) = for some X i ,
f(A),
where A and B are formulae of Y ( S ’ ) .
R e m a r k . The translation of x, is the schematic letter “X,”, whereas the translation of the category expression X , is the category expression X I . However, since in
the proof of Lemma 2 below we shall be translating schemata for formulae of Y ( S ’ )
rather than these formulae, and since in these schemata the schematic letters
X , , 5,.. . might occur, in order to avoid confusion we shall translate the variables
X , . x 2 . . . by new schematic letters for category expressions: Y , , Y,, . . .
~
Then me can prove the following lemma.
L e m m a 2. If ks.A, theyt t ( A ) .
P r o o f . By induction on the length of proof of A in S’. It is trivial to show that
if A is an axiom from (1.1). then t ( A )is true. For the other axioms we have the following.
(1.2) This case follows immediately from (a).
(1.3) We show that if t,X, -+ X , and t,X, -+ X , and C [ X , ] , then C [ X , ] , where
C[X,] is obtained from the formula C [ X , ] of the metalanguage of L by replacing
zero or more occurrences of X , by X , . For that it is enough to show that if t,X, + X ,
and kLX2-+ X , , then k,X,[X,]
X,[X,] and kLX3[X2]-+ X , [ X , ] , where X , [ X , ]
--f
X,[X,]
by replaceing zero or more occurX,.This is done using the fact that 1, is closed under the mleh
is obtained from the category expression
rences of X , by
XL4XZ
(m)
X3’X4
x,. x, x,. x, ’
x,-,x, x , - + x ,
-+
(n)
+
X I+ x 2
(n‘)
&/X3
X,\X,
+
x,-+x,
x,\x,
(see [5], p. 164).
(What we have just shown, together with (a) and (e), guarantees that
-+
represents
a partial-ordering relation in L.)
(2) The cases (2.1) and (2.2)follow immediately from (a) and (e), and (2.3) is trivial.
(3) The cases (3.1) and (3.2) follow immediately from (m), and,(3.3) from (e).
(4)This case follows immediately from (b) and (b’).
(5) We show that
kLX3
-+
X I . X, iff for some Y , am! for some Y , . FLY, 3 X , and
t L Y z -+ X , and tLX3-+ Y , . Y,.
%I)
A C’OYPLETENESS THEOREM FOR THE LAMBEK CALCULUS OF SYNTACTIC CATEQORIES
From left to right we simply take 5, to be Y, and X, t o be Y,. For the other direction we have
x,
-+
Y , -+xl Y , - + X ,
(m)
Y , * Y , -+ x, . x,
Y, .Y,
x, x,. x,.
(e)
-+
(6.1) We show that
kLX3t X,/X, i f f for every Y , , if FLY2
-+
X,, then kLX3 Y,
-+
X, .
From left to right we have
For the other direction we have as an instance of the right-hand side that if
tLX, -+ X , , then tLX3. X, -+ X, . Using (a) we obtain kLX3. X, -+ X I , from which
tLX3-+ XJX, follows by (c).
(6.2) This case is analogous to case (6.1).
(7) We show that
kLX, -+ X, iff for every Y , , if FLY1 -+ X , , then k,Y,
-+
X,.
From left to right we use (e). For the other direction we have as an instance of the
right-hand side that if tLXl -+ X, , then kLX1-+ X , , from which !-,XI --* X , follows
using (a).
It remains only to remark that the rules of the predicate calculus hold in the metatheory of L. 0
As a corollary of this lemma we obtain that if ksrX, -+ X , , then kLXl t X,, which
together with Lemma 1 and the connection between S and S‘ gives the following
embedding theorem for L.
T h e o r e m 1 . tLXl -+ X , iff ksX,
-+
X,.
From this theorem we obt,ain our completeness theorem for L.
T h e o r e m 2. t,X,
-+ X
, iff X ,
-+
X, i s true in every model of S .
Let S, be the system which differs from S only by lacking (2.3);i.e., 5 in S, stands
for a preordering relation. Then by going over our proof it can easily be shown that
kLXl -+ X, iff k,,X, -+ X , , i.e., iff XI -+ X, is true in every model of S, .
4. An intuitive interpretation of S
The first-order theory of semigroups with unary predicates has a natural linguistic
stands for concatenation, and unary predinterpretation. Terms stand for words,
icates stand for syntactic categories. How can we interpret linguistically the ordering
relation =< of S (or S,)? There is probably more than one way to do that. One tentative suggestion is to read “ a , a,” as “ a L abbreviates a , or a , is equal to a,”.
(The relation ‘‘- abbreviates -” is neither reflexive nor symmetric.) It is easy to
+
s
240
KOSTA DOSEX
check that with this interpretation all the axioms of S are satisfied. The definition
given by (5) is also justified: it enables a3 tb be of the category X , . X , not only when
a 3 is equal t o the concatenation of the words a , of the category X , and a , of the catea,.
gory X , , hut also when a3 only abbreviates a word of the form a ,
+
5. Completeness of fragments of L
It can easily be seen t,hat our embedding theorem did not depend essentially on
the infinity of t8hesets of primitive category expressions and primitive term constants.
It’ could easily be adapted to cover finite cases.
It can also easily be seen that. our proof can be adapted to show that the system
obtained from L by omitt8ing(11) and (h’) (see [S]) can be embedded in S (or S,) without (4).
Let us now consider fragments of L obtained by making reductions in 2’(L). It can
easily be shown t h a t if from 2’(L) we omit / or \ (or both), and reject from L the rules
involving what we have omitted. the remaining calculus can still be embedded in S
(or Sl), without the definitions of (6.1) or (6.2), and is hence complete with respect
t o models of S (or Sl).
Next we shall consider the product-free fragment of L (i.e. the fragment without .).
Let, S, be t’he system obtained from S by omit’ting 5 from 9 ( S ) and rejecting all the
axioms involving 5 (i.e. the axioms of ( 2 ) and (3)). I n other words, S, is the firstorder theory of sernigroups with unary predicates. We show in the following lemma
t h a t S is a conswvative extension of S,.
L e m m a 3 . If A is a, fornairla of 9 ( S 2 ) ,then ks2A i f f t,A.
P r o o f . From left to right this is trivial. For the other direction it is enough t o check
t>hatthe axioms of ( 2 ) and (3) hold in S, when we replace 5 by =.
Then it, is possible to show t,he following theorem
T h e o r e m 3 . If . does nut occi~ri,n X I
-+
X , , then kLX, -+ X , iff k s 2 X 1+ X , .
P r o o f . From Theorem 1 we know t h a t kLX1-+ X , iff ksX, -+ X , . The expression 5 does not occur in XI -+ X , considered as a formula of 94s).Hence, by
Lemma 3 , k,X, -+ X , i f f ks,X, -+ X , . 0
So, the product-free fragment of L is complete with respect t o models of S,. This
should be connected with BuszKowsKI’s completeness theorem for this fragment
(see [ 2 ] ) .
6. Concluding remarks
A survey of our proof will show t h a t we have embedded L into the fragment of S
(or S , ) without
and 1.
It will also show t h a t we could have assumed the underlying first-order logic of S (or S,) to be intuitionistic rather than classical.
v
envisaged in [l] differs essentially from
The interpretation of L which BUSZKOWSKI
ours only in replacing a , 5 x f + x, in ( 5 ) by a , = xL x,. With ( 5 ) so modified it
can easily be shown t h a t if kLXI -+ X , , then t-s,X, + X , . We leave open the question
+
4 C O Y P I , E T E N E ~ STHEOREX FOR T H E L .I\IBhK CALCULI'S OF SYNTACTIC CATEGORIES
24 1
whether the converse also holds, i.e., a-liether L is not only sound but, also conipletc
wit 11 respect, to partially ordered semigroup models where the partial-ordering relation
is identity.
It is known that the first-order theory of semigroups is undecidable (see [?I, p. 86).
The system S ext'ends this theory only by adding the ordering relation and unary
predicates to the language. and t)he axioms of ( 2 ) and (3). It is easy to show tha't' S,
S1 and S, are conservative extensions of this theory (cf. the proof of Lemma 3). Hence,
a.11 these systenis are also undecidable. According t o our embedding theorems, the
system L, which was shown decidable in 151. gives decidable fragments of these syst'ems.
Referelires
[l] BWSZKOWSKI,
W.,Undecidability of some logical extensions of Ajdukiewicz-Lanibek calcul~is.
Stiidia Logica 37 (1978), 59-64.
[2] BUSZKOWSKI.
W., Compatibility of a categorial grammar wit)h a n associated category system.
This Zeitschrift 28 (1988), 239 - 838.
[3] COHEN,J. M.. The equivalence of two concepts of ca.tegorial grammar. Informat,ion and Control
10 (1967), 476-484.
[4] FWCHS,
L., Partially Ordered Algebraic Systems. Perganlon Press, Oxford 1963.
[5] LAXBEK,J., The mathematics of sentence structure. Amer. Math. Monthly 65 (1958), 154- 170.
[6] LAYBEK,
J., On the calculus of syntactic types. I n : Structure of Language and its Mathematical
Aspects (R. JACOBSOX,
ed.), Proc. 121h Symp. Appl. Math., American Math. SOC.,Providence,
R.1.1961, pp. 166-178, 864-265.
[7] TARSKI,
A, MOSTOWSKI,A., and K.M. ROBINSON,
Undecidable Theories. North-Holland Pnbl.
Comp., Amsterdam 1953.
[8] ZIELONKA,W., Axiomat,izability of ~~jdukiewicz-I,anibekcalculus by means of cancellation
schemes. This Zeitschrift 27 (1981). 315-Y24.
kiosta DoGen
JIatematiEki Instit ot
Knez Mihailova 35
1 1 OOO Belgrade
Yugoslavia
(Eingegangen am 17. Marz 1983)