PROOF FIGURES AND STRUCTURAL OPERATORS 
FOR CATEGORIAL GRAMMAR" 
Guy Barry, Mark Hepple t, Nell Leslie and Glyn Morrill I 
Centre for Cognitive Science, University of Edinburgh 
2 Buccleuch Place, Edinburgh EBB 9LW, Scotland 
guy@cogsci, ed. ac.uk, arh@cl, cam. ac. uk, 
neil@cogs c i. ed. ac. uk, Glyn. Norrill@let. ruu. nl 
ABSTRACT 
Use of Lambek's (1958) categorial grammar for lin- 
guistic work has generally been rather limited. There 
appear to be two main reasons for this: the nota- 
tions most commonly used can sometimes obscure the 
structure of proofs and fail to clearly convey linguistic 
structure, and the cMculus as it stands is apparently 
not powerful enough to describe many phenomena en- 
countered in natural language. 
In this paper we suggest ways of dealing with both 
these deficiencies. Firstly, we reformulate Lambek's 
system using proof figures based on the 'natural de- 
duction' notation commonly used for derivations in 
logic, and discuss some of the related proof-theory. 
Natural deduction is generally regarded as the most 
economical and comprehensible system for working 
on proofs by hand, and we suggest that the same 
advantages hold for a similar presentation of cate- 
gorial derivations. Secondly, we introduce devices 
called structural modalities, based on the structural 
rules found in logic, for the characterization of com- 
mutation, iteration and optionality. This permits the 
description of linguistic phenomena which Lambek's 
system does not capture with the desired sensitivity 
and gencrallty. 
LAMBEK CATEGORIAL 
GRAMMAR 
PRELIMINARIES 
Categorial grammar is an approach to language de- 
scription in which the combination of expressions is 
governed not by specific linguistic rules but by general 
logical inference mechanisms. The point of departure 
can be seen as Frege's position that there are certain 
'complete expressions' which are the primary bear- 
ers of meaning, and that the meanings of 'incomplete 
expressions' (including words) are derivative, being 
* We would like to thank Robin Cooper, Martin Picker- 
ing and Pete Whitelock for comments and discussion relat- 
ing to this work. The authors were respectively supported 
by SERC Research Studentship 883069"/1; ESRO Re- 
search Studentshlp C00428722003; ESPRIT Project 393 
and Cognitive Science/HCI Research Initiative 89/CS01 
and 89/CS25; SERC Postdoctoral Fellowship B/ITF/206. 
I Now at University of Cambridge Computer Labora- 
tory, New Musctuns Sitc, Pembroke Street, Cambridge 
(31}2 3Q(;, Engl~u.l. 
1 Now at OTS, '\]'rans 1O, 3512 JK Utrecht, Netherlands. 
their contribution to the meanings of the expressions 
in which they occur. We s.uppese that linguistic ob- 
jects have (at least) two components, form (syntactic) 
and meaning (semantic). We refer to sets of such ob- 
jects as categories, which axe indexed by types, and 
stipulate that all complete expressions belong to cat- 
egories indexed by primitive types. We then recur- 
sively classify incomplete expressions according to the 
meems by which they combine (syntactically and se- 
mantically) with other expressions. 
In the 'syntactic calculus' of Lambek (1958) (var- 
iously known as Lambek categoriai grammar, Lambek 
calculus, or L), expressions are classified by means of 
a set of bidirectional types as defined in (1). 
(1) a. If X is a primitive type then X is a type. 
b. If X and Y are types then X/Y and Y\X 
are types. 
X/Y (resp. Y\X) is the type of incomplete expres- 
sions that syntactically combine with a following (resp. 
preceding) expression of type Y to form an expression 
of type X, and semantically are functions from mean- 
ings of type Y to meanings of type X. 
Let us assume complete expressions to be sen- 
tences (indexed by the primitive type S), noun phrases 
(NP), common nouns (N), and non-finite verb phrases 
(VP). By the above definitions, we may assign types 
to words as follows: 
(2) John, Mary, Suzy := NP 
man, paper := N 
the := NP/N 
likes, read := (NP\S)/NP 
quickly :--- (NP\S)\(NP\S) 
without := ((NP\S)\(NP\S))/VP 
understanding := VP/NP 
We represent the form of a word by printing it in 
italics, and its meaning by the same word in boldface. 
For instance, the form of the word "man ~ will be 
represented as man and its meaning as man. 
PROOF FIGURES 
We shall present the rules of L by means of proo~ 
fi~res, based on Prawitz' (1965) systems of 'nat- 
ural deduction'. Natural deduction was developed 
by Gentzen (1936) to reflect the natural process of 
mathematical reasoning in which one uses a number 
of in/erence tulsa to justify a single proposition, the 
conclusion, on the basis of having justifications of a 
number of propositions, called assumptions. During 
- 198 - 
a proof one may temporarily make a new assumption 
if one of the rules licenses the subsequent withdrawal 
of this assumption. The rule is said to discharge the 
assumption. The conclusion is said to depend on the 
undischarged assumptions, which are called the by. 
potheses of the proof. 
A proof is usually represented as a tree: with the 
assumptions as leaves and the conclusion at:the root. 
Finding a proof is then seen as the task of filling this 
tree in, and the inference rules as operations on the 
partially completed tree. One can write the infer- 
ence rules out as such operations, but as these are 
rather unwieldy it is more usual to present the rules 
in a more compact form as operations from a set of 
subproofs (the premises) to a conclusion, as follows 
(where m >_ I and n >_ 0): 
(3) . : \[Y.l\] d , \[y.n\] d 
.R' Z 
This states that a proof of Z can be obtained from 
proofs of X1 .... , Xm by discharging appropriate oc- 
currences of assumptions Y, ..... I/,. The use of 
square brackets around an assumption indicates its 
discharge. R is the name of the rule, and the index i 
is included to disambiguate proofs, since there may be 
an uncertainty as to which rule has discharged which 
assumption. 
As propositions are represented by formulas in 
logic, so linguistic categories are represented by type 
formulas in L. The left-to-right order of types indi- 
cates tim order in which the forms of subexpressions 
are to be concatenated to give a composite expres- 
sion derived by the proof. Thus we must take note 
of the order and place of occurrence of the premises 
of the rules in the proof figures for L. There is also 
a problem with the presentation of the rules in the 
compact notation as some of the rules will be written 
us if they had a number of conclusions, as follows: 
(4) : 
Xl -.. X,~ ,,, /r~ 
~q ... Y. 
This rule should be seen as a shorthand for: 
(,~) : : 
Xl ... Xm tt 
Y, ... Y.. 
Z 
If the rules are viewed in this way it will be seen that 
they do not violate the single conclusion nature of the 
figures. 
As with standard natural deduction, for each con- 
nective there is an elimination rule which gates how 
a type containing that connective may be consumed, 
and an introduction rule which states how a type con- 
raining that connective may be derived. The elimi- 
nation rule for / states that a proof of type X/Y 
followed by a proof of type Y yields a proof of type 
X. Similarly the elimination rule for \ states that 
a proof of type Y\X preceded by a proof of type Y 
yields a proof of: type X. Using the notation above, 
we may write these rules as follows: 
: : b. i:. . (6) a. x)r., i'le v\x\~ 
X X 
We shall give a semantics for this calculus in the same 
style as the traditional functional semantics for intu- 
itionistic logic (Troelstra 1969; Howard 1980). In the 
two rules above, the meaning of the composite expres- 
sion (of type X) is given by the functional application 
of the meaning Of the functor expression (i.e. the one 
of type X/Y or Y\X) to the meaning of the argument 
expression (i.e. the one of type Y). We represent func- 
tion application :by juxtaposition, so that likes John 
means likes applied to John. 
Using the rules \[E and \E, we may derive "Mary 
likes John" as a sentence as follows: 
(7) Mary likes John 
(NP\S)/NP NP /P. 
,i 
NP NP\S 
S 
The meaning of the sentence is read off the proof by 
interpreting the/E and \E inferences as function ap- 
plication, giving the following: 
(8) (likes John) Mary 
The introduction rule for / states that where the 
rightmost a~sumption in a proof of the type X is of 
type Y, that assumption may be discharged to give 
a proof of the type X/Y. Similarly, the introduction 
rule for \ states that where the leftmost assumption 
in a proof of the type X is of type Y, that assumption 
may be discharged to give a proof of the type Y\X. 
Using the notation above, we may write these rules 
as follows: 
(9) a. ~\]' b. \[v.\]' 
.~1I, v\x \X ~ 
Note however that this notation does not embody the 
conditions that ihave been stated, namely that in/I 
Y is the rightmost undischarged assumption in the 
proof of X, and:in \I Y is the leftmost undischarged 
assumption in the proof of X. In addition, L carries 
the condition that in both/I and \I the sole assump- 
tion in a proof cannot be withdrawn, so that no types 
are assigned to the empty string. 
In the introduction rules, the meaning of the re- 
sult is given by lambd&-abstraction over the meaning 
of the discharged assumption, which can be repre- 
sented by a variable of the appropriate type. The re- 
lationship between lambda-abstraction and function 
application is given by the law of t-equality in (10), 
- 199 - 
where c~\[/~lV \] means '~ with//substituted for #'. (See 
llindley and Seldin 1986 for a full exposition of the 
typed lambda-calculus.) 
(10) (xv\[o,\])//= o,\[//Iv\] 
Since exactly one assumption must be withdrawn, the 
resulting lambda-terms have the property that each 
binder binds exactly one variable occurrence; we refer 
to this as the 'single-bind' property (van Benthem 
1983). The rules in (9) are analogous to the usual 
natural deduction rule of conditionalization, except 
that the latter allows withdrawal of any number of 
assumptions, in any position. 
The \]I and \l rules are commonly used in con- 
structions that are assumed in other theories to in- 
volve 'empty categories', such as (11): 
(11) (John is the man) who Mary likes. 
We assume that the relative clause modifies the noun 
"man" and hence should receive the type N\N. The 
string "Mary likes" can be derived as of type S/NP, 
and so assignment of the type (N\N)/(S/NP) to the 
object relative pronoun "who" allows the analysis in 
(12) (cf. Ades and Steedma n 1982): 
(12) who Mary likes 
(NP\S)/NP \[NP\]aIE 
NP ' NP\S\E 
S 
(N\N)I(S/NP) SINP III ./F. 
N\N 
The meaning of the string can be read off the proof by 
interpreting /I and \I as lambda-abstraction, giving 
the term in (13): 
(is) who (Ax\[(likes ~) Mary\]) 
Note that this mechanism is only powerful enough 
to allow constructions where the extraction site is 
clause-peripheral; for non-peripheyad extraction (and 
multiple extraction) we appear to need an extended 
logic, as described later. 
DERIVATIONAL EQUIVALENCE AND 
NORMAL FORMS 
In the above system it is possible to give more than 
one proof for a single reading of a string. For exam- 
pie, corot)are the derivation of "Mary likes John" in 
(7), and the corresponding lambda-term in (8), with 
the derivation in (14) and the iambda-term in (15): 
(14) Mary likes Jolm 
(NP\S)/NP \[NPp 
NP NP\S\E /E 
S 
S/NP/I1 NP 
.i~, S 
(15) (Az\[(likcs x) Mary\]) John 
By the definition in (10), the terms in (8) and (15) are 
//-equal, and thus have the same meaning; the proofs 
in (7) and (14) are said to exhibit derivationai equiva- 
lence. The relation of derivational equivalence clearly 
divides the set of proofs into equivalence classes. We 
shall define a notion of normal form for proofs (and 
their corresponding terms) in such a way that each 
equivalence class of proofs contains a unique normal 
form (cf. Hepple mad Morrill 1989). 
We first define the notions of contraction and re.due. 
tion. A contraction schema R D C consists of a par- 
ticular pattern R within proofs or terms (the redez) 
and an equal and simpler pattern C (the contractum). 
A reduction consists of a series of contractions, each 
replacing an occurrence of a redex by its contractum. 
A normal form is then a proof or term on which no 
contractions are possible. 
We define the following contraction schemas: weak 
contraction in (16) for proofs, and t-contraction in 
(17) for the corresponding lambda-terms. 
(16) ~. V.\]' 
" I> Y 
v ,/v, ~" 
X 
b. \[r.), 
: X i l> ~, v\x \I 
XV. k X 
(17) (~y\[,~\]),o ~ ~,Laly\] 
From (10) we see that t-contraction preserves mean- 
ing according to the standard functional interpreta- 
tion of typed lambda.calculus. Therefore the cor- 
responding weak contraction preserves the semantic 
functional interpretation of the proof; in addition it 
preserves the syntactic string interpretation since the 
redex and contractum contain the same leaves in the 
same order. For example, the proof in (14) weakly 
contracts to theproof in (7), and correspondingly the 
term in (15) //-contracts to the term in (8). The 
results of these contractions cannot be further con- 
tracted and so ~re the respective results of reduction 
to weak normal form and//.normal form. 
Weak contraction in L strictly decreases the size 
of proofs (e.g. the number of symbols in a contractum 
is always less than that in a redex), and//-contraction 
in. the single-bind lambda-calculus strictly decreases 
the size of terms. Thus there is strong normalization 
with respect to these reductions: every proof (term) 
reduces to a weak normal form (//-normal form) in 
a finite number of steps. This has as a corollary 
(normalization) that every proof (term) has a nor- 
mad form, so that normal forms are fully represen- 
tative: every proof (term) is equal to one in normal 
form. Since reductions preserve interpretations, an 
interpretation of a normal form will always be the 
- 200 - 
same as that of the original proof (term). Thus re- 
stricting the search to just such proofs addresses the 
problem of derivational equivalence, while preserving 
generality in that all interpretations are found• 
Proofs in L and singie-bind lambda-terms (like 
the more general cases of intuitionistic proofs and full 
lambda-terms) exhibit a property called the Church- 
Itosser property) from which it follows that normal 
forms are unique. 2 
For formulations of L that are oriented to pars- 
ing, defining normal forms for proofs provides a basis 
for handling the so-called 'spurious ambiguity' prob- 
lem, by providing for parsing methods which return 
all aml only normal form proofs. See KSnig (1989) 
~t,,d lh:pl,lc (1990). 
STRUCTURAL MODALITIES 
From a logical perspective, L can be seen as the weak- 
est of a hierarchy of implicational sequent logics which 
differ in the amount of freedom allowed in' the use 
of assumptions. The higl,est of these is (the impli- 
cational fragment of) the logistic calculus LJ intro- 
duced in Gentzen (1936). Gentzen formulated this 
calculus ia terms of sequences of propositions, and 
then provided explicit structural rules to show the 
permitted ways to manipulate these sequences. The 
structural rules are permutation, which allows the or- 
der of the assumptions to be changed; contraction, 
which allows an assumption to be used more than 
once; and toeakening, which allows an assumption to 
be ignored. For a discussion of the logics generated 
by dropping some or all of these structural rules see 
e.g. van Benthem (1987). 
Although freely applying structural rules 'are clear- 
ly not appropriate in categorial grammars for linguis- 
tic description, commutable, iterable and optional el- 
eme,ts do occur in natural language. This suggests 
that we should have a way to indicate that structural 
operatiops are permissible on specific types, while still 
forbidding tl,eir general application. To achieve this 
we propose to follow the precedent of the e~ponen- 
lial operators of Girard's (1987) linear sequent logic, 
which lacks the rules of contraction and weakening, 
by s.ggesting a similar system of operators called 
structnral sodalities, tiers we shall describe a sys- 
tem of universal sodalities, which allow us to deal 
with the logic of commutable, iterable and: optional 
extractions, a 
For each universal sodality we shall present an 
elimination rule, and one or more 'operational rules', 
whicl, are essentially controlled versions of structural 
1 This is the property that if a proof (term) M reduces 
to two proofs (terms) NI, N2, then there is a proof (term) 
to wlfich both NI and N2 reduce. 
2The above remarks also extend to a second form of re- 
duction, strong reduction/11-reduction, which we have not 
space to describe here. See Morrill et aL (1990). 
aThe name is dmseJa because the elimination and in- 
troduct;on rules appropriate to each operator turn out to 
be those for the unlvcrsal ,nodality in the \]nodal logic $4. 
See Dosen (1990), 
rules. (Introduction rules can also be defined, but we 
omit these here for brevity and because they axe not 
required for the linguistic applications we discuss.) 
Note that these operators are strictly formal devices 
and not geared towards specific linguistic phenom- 
ena. Their use fat the applications described, which 
are suggested purely for illustration, may lead to over- 
generation in some cases. 4 
COMMUTATION 
The type AX is assigned to an item of type X which 
may be freely permuted. A hu the following infer- 
ence rules: 
: : : • .. 
(18) ,~'x ~x ~" ~" ~x 
' ~E ~-~Prm Prn~ 
X Y Z~X AX Y 
From these rules we see that an occurrence of an item 
of type X in any position may be derived from an item 
of type AX. 
We may use this operator in a treatment of rein. 
tivization that will allow not only peripheral extrac- 
tion as in (198), but also non-peripheral extraction as 
in (19b): 
(19) a. (Here is the paper) which Snzy read. 
b. (Here is the paper) which Suzy read 
quickly. 
We shall generate these examples by assuming that 
"which z licenses extraction from any position in the 
body of the relative clause. We may accomplish this 
by giving Uwhich~ the type (N\N)/(S/ANP) (cf. the 
extraction operator T of Moortgat (1988)). This al- 
lows the derivations in (20a-b) (see Figure 1), which 
correspond to the lambda-terms in (21a-b) respec- 
tively: 
(21) a. which (Az\[(read z) Suzy\]) 
b. which (Az\[(qulckly (read z)) Suzy\]) 
ITERATION 
The type X ~ is assigned to an item of type X which 
may be freely permuted and iterated, t has the fol- 
lowing inference rules: 
• . . 
• --JE xtTPrm -------Prm I X Y X t Y 
f Con 
X t X t 
'In Morrill et 4/. (1990) we give a system of wodali- 
ties that differs from the present proposal in several re- 
spects. There aretwo unidirectional commutation modal. 
itiea rathe¢ than the single bidirectional sodality given 
here, and a single operational rule is associated with each 
of the universal modalities. We ahto suggest a (more ten- 
tative) system of swlstenfial modalltles for dealinl$ with 
elements that are themselves commutable, iterable or 
optional. 
- 201 - 
(2o) which Suzy 
(N\N)/(S/ANP) 
NP 
N\N 
l). wldch Suzy read 
NP 
(N\N)/(S/~NP) 
N\N " 
re~d 
(NP\S)/NP NP 
• /E NP\Sw. 
S , ,/F 
S/&NP ./E 
(NP\S)/NP 
NP\S 
quickly 
(NP\S)\(NP\S) \[~NP\]I PrmLX 
~NP (NP\S)\(NP\S) &E 
NP "/E 
..... NP\S\E 
S 
S/ANP/I' -/E 
\E 
(24) wlfich Suzy read 
(N\N)/(S/NP)') 
NP 
N\N 
(NP\S)/NP 
without 
((NP\S)\(NP\S))/VP 
NP\S 
(NP\S)\(NP\S) 
Np t 
'E NP 
/~. 
S 
S/NP1111 
.IE 
understanding 
VP/NP 
VP 
NP\S.\ E 
\[NP~p ! Con 
Np I Np t 
'E 
NP ,/E 
./E 
Prm t (NP\S)\(NP\S) 
\E 
(2s) too long 
PredP/(ForP/NPII) 
for Suzy to concentrate 
(ForP/VP)/NP NP ./F, 
ForP/VP VP ,/E 
ForP 
PredP 
ForP 
, ,/P 
ForP/NP u ...... 
/E 
\[NP II\] I Wknll 
Figure 1. Derivations illustrating use of structural modalities 
- 202- 
One or more occurrences of items of type X in any 
position may be derived from an item of type X ~. 
We may use this modality in t treatment of mul- 
tiple ¢xtraction. Consider tits parasitic gap construc- 
tion in (23): 
(23) (Here is the paper) which Susy read without 
understanding. 
In order to generate both this example and the ones 
iu (19), we shall now assume that ~which" licenses 
extraction not just from any position in the body of 
a relative clause, but from any number of positions 
greater than or squad to one. We may do this by al- 
tering the type of awhich ~ to (N\N)/(S/NPt). Since 
h~s 'all the inference rules of A, tl~e derivations in 
(20) v~iil still go througl, with the new type. In addi- 
tion~ the icon inference ~ule allows the derivation of 
(23) given in (24) (see Figure 1), and the correspond- 
i,g term in (25/: 
(25.) which (Az\[((without (understanding z)) 
(read z)) Suzy\]) 
OPTIONALITY 
The type X II is assigned to an item of typeX which 
may be freely permuted, iterated and omitted. I has 
the following inference rules: 
Prm X ~" '~ 'X n 
xll Y Y 
Zero or more occurrences of items of type X in any 
position may be derived from an item of type X ~\[. 
We n|ay use this modality in a treatment of op 
tional extraction, ors illustrated by (27): 
(27) a. (The paper was) ago long for Sezy to read. 
b. (The paper was) too long for Susy to read 
quickly. 
c. (The paper was) too long for Suzy to read 
witlmut understanding. 
d. (The paper was) too long for Suzy to con- 
centrate. 
We shall ~ssume for simplicity that ato~-infinitives 
are single lexical item~ of typ~ Vp, that ~for-to" clauses 
have a special atomic type ForP (so that Yfor ~ has 
the type (ForP/VP)/NP), and that predicate phrases 
have a special atomic type PredP. Given these assign- 
ments, the type PredP/(ForP/NP:) for "too long s 
would allow (27a-c), but not (27d). In order to gener- 
al.e all four examples, we shall a~sume that %00 long ~ 
liceuses extraction from any number of positions in 
the embedded cl/xuse greater than or equal to zero, 
aud thus give it the type PredP/(PorP/NPIl I. Again, 
g has all the inference r~les of I generating (2?a-c), 
and the Wkn It rule allows (27d) to be derived as in 
(28) (see Figure 1), giviug theterm in (291: 
(29) too-long (Az\[fo~ (to-concentrate SuzYl\] I 
CONCLUSIONS 
We have introduced a scheme of p~oof figure~ for Lam- 
bek c~tegorial gr~mmax in the style of ~atural de. 
duction, and proposed structural modalities which we 
suggest axe suitable for the capture of linguistic sen. 
eralisations. It zemxins to extend the ~em~mtic treat- 
ment of the structural moda\]ities,, to refine the proof 
theory, and hence to develo p more efficient parsing at, 
gorithms. For the p~eae~t, we. hope that the proposal~ 
made can be seen as gaining linguistic practicaJity in 
the c~tegoria~ description of ~atural Iq~gu~ge, with- 
out losing mathematical elegance. 
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