DISCONTINUITY AND THE LAMBEK @ALCULUS 
Mark Hepple 
DeI)artment of Computer Science, University of Shellield, lLegents Court, 
Portohello Street, Sheflield, UK. gmai\]: hepple0dcs, sheffield.at .uk 
Introduction 
This paper is concerned with tile treatment of dis- 
continuous constituency within Categorial Grammar. 
In particular, I address the problem of providing an 
adequate formalisation of categorial commctives l)ro- 
l)osed by Moortgat (1988), which are useful for han- 
dling certain forms of diseontimmus eonstitnency, l)e- 
spite some interesting proposals, a satisfactory logic 
for these e{mnectives has so far remained dnsive. I 
will provide such a h)gie, using an approach that falls 
within the general framework of labelled deductive sys- 
tems (Gabbay, 1991), employing novel methods for 
reasoning ahout linear order in resource nsage. The 
approach is illusl;rated by linguistic al}plications for 
extraction, pied-piping and quanti\[ieation. 
The Lambek calctflus 
Our general fran,ework is the associative I,ambelc cal- 
culus (L: l,mnbek, 1958), a system which falls within 
the class of formalisms known as Categorial Gram- 
mars. The set of types is freely generated firom a 
set of primitiw; (atomic) types (e.g. {s, np .... }), us- 
ing binary infix el)craters \, /, .. The 'meaning' of 
these connectives in L is fixed hy a senlanties for the 
logic, based on a (se,nigroup)algehra of strings (Z:,.), 
i.e. where • is an associative, notl-coulmtltative bi- 
nary operator, with two-sided identity e, and E is the 
set Of non-eHlpty (-7 t:- g) strings over some vocabulary. 
An interpretation funcl;ion \[\[~ assigns some subset of 
Z; to each type, satisfying the conditions be.low for 
conlplex types and gyp(', sequence.s. A type comhi-. 
liar/oil X 1 ..... Xtz --> X0 h.ohls in a model ((£,.), \[\[\]l ), 
if {\[x~ .... ,x.\]\]_cl\[Xol\], a.d is ~lid if it is true in all 
models. '1'here are several formulations of L that all 
realise this same meaning for the connectives.t 
~x,v\]\] = {.~,v < z. I .~ e \[\[xll A ~ < {\[v\]\] } 
\[\[X/Y\]J = {,: e Z; I V; ~: \[~Y~. :,:.y e ~\[X\]\] } 
I\[vxx\]\] -- {~. </: I V,,~ e IIY\]\].,>,. e gx\]}} 
\[\[x~ ..... x,,\]} = {:,:~ ......... ec I< e \[Ix\]~\] A,.,A .... C EX,,I\]} 
\]The alternatlve formulations include e.g. seqnent (l,aml}ek 
1958), proof net (l~.oor(la 199:1 ), and natural deduct.ion systems 
(Morrill e! aL 1990, Barry eg al. 1991). Alternative formula- 
tions carry different advantages, e.g. natural dcductlon is well 
suil.cd for lh,guisl.ic prcsental.ion, whereas proof nets haw~ ben- 
efits for automated theorem proving. 
l)iscontinuous type constructors 
The I,ambek calculus is a purely concatenative system: 
where any two types are combined, the string of the 
result is arrived at by concatenating the strings of the 
types combined. This point is illustrated graphically 
in (la,b), for the Lambek tractors, where (follow- 
ing Moortgat, 1991) each triangle represents a result 
string, and unshaded and shaded areas represent fnne- 
tor and argutlre.nt stril/gs, respectively. 
(1) (a) X/Y (b) YXX (c) XIY (d) XIY 
Y Y Y y 
l'rclixa~ion Suffixal.ion Extraction Infixation 
Various linguistic phenomena, however, suggest the 
existence of discontinous constituency, i.e. situations 
the result string from combining two constituents is 
not produced by concatenating the component strings. 
(See e.g. Bitch, 1981.) Moortgat (1988) suggests aug- 
men£ing \]T, with two discontinuous type construct, ors. 
An exh'aclion fimctor X\]Y is one whose argument cor- 
responds to a non-peripheral (or more precisely, nor 
necessaribj peripheral) suhstring of the result of con> 
binaries, as it, (lc). An infixation fimetor XIY itself 
corresponds to a non-l)eril)heral substring of the re- 
sult of combination, as in (ld). Given these intuitiw'~ 
characterisations, two options arise for the meaning of 
each comlecLive as to whether the point of insertion 
of one striug into the other is free (universal) for lixed 
(existential). In this paper, I will focus on the exis- 
tential variants of the commctives, which appear to he 
the most linguistically useful, and whose interpretive 
conditions arc as follows: 
Previous proposals 
F, ach connective should have two inference rules: a rule 
of proof (showing how to derive a type containing the 
connective), and it rule of nse (showing how to employ 
such a type). This indicates a possible eight infer- 
ence rules that we might hope to state (i.e. proof/use 
x universal/existential x infixation/extraction). V~ri~ 
ous attempts have I~eeu made to provide a logic for tile 
discoutiuuous type constructors, but all have proved 
ullsu('cessflll or unsatisI'actory in some way or another. 
1235 
Moortgat (1988), for example, uses an ordered se- 
quent calculus framework, which allows only two of the 
possible eight rules to be stated: a rule of proof for ex- 
istential T, and a rule of nse for universal ~. Moortgat 
(1.991) nses a proof method in which types are not or- 
dered in proof representations, where linear order con- 
straints and consequences are instead handled using a 
system of string labelling, i.e. types are associated 
with string terms, which are explicitly manipulated 
by inference rules. This approach allows two further 
rules to be stated, but the four expressible rules are 
distribnted one per connective, i.e. a complete logic is 
not given for even any one connective. As Versmissen 
(1991) notes, Moortgat's string label system does not 
allow the recording of a specific position for inserting 
one string into another, as would seem to be required. 
Morrill & Solias (1993) avoid this latter problem by 
augmenting the string \]al~elling algebra with a non- 
associa(,ive pairing operator (., .), allowing labels such 
as {st, s2), indicating an insertion point in between sl 
and s2. 'lPhis system allows versions of T and ~ oper- 
ators to be specified, hut ones whose interpretive def- 
initions differ from Moortgat's. The non-associativity 
of pairing gives rise to limited flexibility for the system 
in terms of the type combinations that can be derived, 
and even the types that can be constructed, e.g. no 
flmctor (X~Y)/Z, where a \] argument is not the first 
sought, is allowed. 
Labelled deduction &: Lambek calculus 
I next develop a formulation of L which can be e×- 
tended to allow for the (existential) discontinuity con- 
nectives. Our starting point is a lambda term se- 
mantics for implicational L due to Bnszkowski (1987), 
ba.sed on the well kuown Cnrry-Howard interpretal;ion 
of proofs (Iloward, 1969)3 This uses a bidirectional 
variant of the lambda calculus whose basic terms are 
directionally typed variables. If t is a term of type 
Y\X (resp. X/Y), and u one of type Y, then (at) t 
(resp. (tu)") is a term of type X. If v is a variable of 
type Y, and t a term of type X, then klv.t (resp. 2"v.t) 
is a term of type Y\X (resp. X/Y). A semantics for 
implicational L is given by the class of terms which 
2Under the Curry-lloward interpretation (lloward, 1969), 
logical formulas are regarded as types of expressions in typed 
lalnbda calculus, with atomic formulas corresponding to basic 
types, and a formula A--~B to the type of functions from A to tl. 
It is dmnonstrable that the set of formulas for which there exists 
stone correspondingly typed lambda term is precisely the theo- 
rems of the impfieatlonal fragment of intuitlonistlc logic. Thus, 
typed lambda calculus provides a s emantlcs for implicational in- 
tuitlonlstic logic, i.e. an independent, characterlsation of 'valid 
deductlon',just as the algebralc semantics of L provides an inde- 
pendent characterisatlon of validity for that system. Semantics 
for vm'ious other logics can be given in terms of classes of typed 
lambda terms, i.e. subsets of the typed lambda terms which 
satisfy certain stated criteria, van Benthem (1983) provides a 
lambda semantics for the system LP, a eonmmtative variant 
of L. Wansing (1990) provides lambda semantics for a range of 
subloglcs of intultlonistie logic. The Curry-tloward interpreta- 
tion so permeates categorlal work that the terms "formula" and 
"type" have become almost interchangeable. Note that I have 
slightly modified BuszkowsM's notation. 
satisfy the conditions: (lll) each subterm contains a 
flee variable, (132) no subterm contains > 1 free occur- 
renee of any variable, (133) each A t (resp.)?') binds the 
leftmost (resp. rightmost) free variable in its scope. 
This semantics can be used in formulating (implica- 
tionM) L as a labelled deductive system (LDS: Gabbay, 
1991). a LM)els are terms of the directionM lambdasys- 
tern, and propagation of labels is via application and 
abstraction in the standard manner. Natural deduc- 
tion rules labelled in this way are as follows: 
(2) a/B:a B:b \[B:v\] 
/E A:a a 
: (~b) ~ --/I 
A/B : X:v.a 
B:b B\A :(~ \[B :v\] 
A : (b~) l \E~ ___A :a \~ 
/3\A : ,~v.a 
We can eusure that only deductions appropriate to 
(implicational) L are made by requiring that the la- 
bel that results with any inference is a term satisfy- 
ing Buszkowski's three conditions. To facilitate test- 
ing this requirement, I use a flmction E, which maps 
from label terms to the string of their free variables 
occurring in the left-right order that follows from type 
directionality (giving what I call a marker term). A 
notion of 'string equivalence' ( ~- ) for marker terms is 
definecl by the axioms: 
(-~.1) *,(>z)±(x.y).z 
(- 2) .-~.. 
(-.3) .~-x.c 
E is recursively specified by the following clauses (where 
PV returns the set of fi'ee variables in a term), but 
it is defined for all and only those terms that satisfy 
Bnszkowski's three conditions. 4 Thus, we can ensure 
correct deduction by requiring of the label that results 
with each inference that there exists some marker term 
m such that )_'(a) = m. 
(~;.I) }2(v) = v where v E Vats 
(r.~) ~((~b) ~) = ~((,).~(1,) 
where l,'v((0 n lW(b) = 0 
(~.a) ~((~,~)~) = ~(.).~(~) 
where FV(a) ffl l!'V(b) = 13 
(~.4) ~(~%.,) =/3 
where l~'V(2~..) ¢ O, ~(a)=' v.~ 
(~.~) >~(Xv..) = 9 
where l"V(~"v.( 0 ¢ ~, ~(a) - p,v 
The followiug proofs illustrate this LDS (nsing t~m 
3111 labelled deduction, each fornulla is associated with a la- 
bel, which records information of the use of resources (i.e. as- 
smnptions) in proving that formula. Inference rules indicate 
how labels are propagated, and may have side conditions which 
refer to labels, using the information recorded to ensure col 
recL infcrencing. Evidently, the Moortgat (1991) and Morrill & 
Solias (1993) formalisms arc LDSs. 
4Condition B2 is enforced by the requirement on the ap- 
plication cases of E. Conditions B1 and B3 are enforced by 
the first and second rcquirement on the ahstraction cases of E, 
respectively. 
1236 
as shorthand for E(t)-m, to indicate a significant 
niarker equivalence): 
x/v:~. Y/Z:~ \[Z:z\] /i,: 
Y : (yz)" /i,; 
>: 
x: (~(vO")" ,+ ~.v.~ 
x/z: yz.(~(vzy)" 
X/(Y/(Z\Y)) : m \[Z:z\] \[Z\Y:y\] \1;: r, 
Y : (z~) ~ ~ z.y 
Y/(z\v) : a';>(<D ~ /~ /1," ~: 
x:(~ x';>(zv)|) " /I ..... 
XlZ.. x"~.(~ X',v.(~v)|) " 
This system can be extended to cover product using 
the inference rules (3), and the additional )2 elanses 
shown following (with the obvious implicit extensions 
of the directional lambda system, and of Buszkowski's 
semantics). Labelling of \[.I\] inferences is via pairing, 
and that of \[,.El inferences llses all operator ada.pged 
from \]lenton el al, (1992), where a term \[b/v.w\].a 
implicitly represents tim suhsl, itiitiol| of b for v+w iii 
a. This rule is nsed hi (4). 
(a) D:,\]\[c:,I,\] a:, i~:~ 
,1 A : a t3.0 : b AIB : (a, b) 
iE 
(s.u) >:((<,, O) = ::(a).>~(~,) 
where l!'V(a) rl FV(b) :: 0 
07.7) >_,(\[b/v.w\].a) = fll')',(b)"fl2 
whe,.~ 1,'v((,)r: ~.'vo) = 0 
(4) x/v/z:,: \[z:q \[v:,v\] z.v:~ / E 
XlV : (~,)' 
x: ((*'z)"vy ~ .... v 
• I') 
x .. \[wl~.v\].((.~O"v)' 5+ x.~,, 
x/(z.v) : ~'~.(\[,,,/~.v\].((=)":D ~)/: 
Labelled deduction gz discontinuity 
'\]'his al)proach can be exi;eilded l;o allow for exist.enl, ial 
i a.nd ,\[. These conneci;iw;s have sLatidard iinl)lica- 
tional inference rules, |lsing additional distinguished 
operators for labelling (with superscript e for extrac- 
tion and i tbr lull×at|on): 
(5) AtI~:~, ~:b \[1~:~} 
k : (oby -- t* 
A'~B : ,Vv.a 
AIB:a B:b \[B:u\] 
A:(ab) i \]E A:a ----ll 
A \[ B : ,\iv.c, 
(lens|tier /irsl;ly how >; Iiuist; I)e extended for the 
abstract.ion cases of |;he new introduction rules. For 
a \[II\] terin such as Aev.a, l.\]|e relevant E case allows 
v to appear non-peripherally in the marker term of 
a. For a \[Ill term such as ,~v.a, v is allowed to he 
discontinuous in the marker of a (we shall see shortly 
how this is possible), bug requires its components to 
appear peripherally. 
where ~W(.\%..) ¢ ~, ~:(a)-' ,q~.v.,e~ 
To allow for the new application operators, the marker 
system must be extended. Recall that the linear order 
information implicit in lahcls is projected onto the left,- 
right dimension in markers. With 1" and 1, however, 
|lie possibility exists that either fiuletor or argument is 
discontirmous in the result of their combination. For 
strings x G \[\[XTY\]\] and y G \[\[Y\]~, for example, we 
know there is some way of wrapping x around y to 
give a resnll, in X, but we do no~ in general know how 
the division of .v should be made. This problem of un- 
certainly ix handled by using operators L and R, where 
1,(rn) and It(m)represent indefinile but complemen~ 
lary left and right suhcomponents of the marker term 
m. (L arlcl 1{ are not projection time|ions.) This idea 
of the significance of :1, and II, in given content via the 
additional axiom ( =' .d), which allows that if the eom- 
plemenl;ary left; and right snhcomponents of a marker 
appear in ,q)propl:iate lel't-rightjuxtaposition, then the 
l~larker~s resotirees Irmy be treated as continuous. 5 
( = .d) I,(a:).\]{,(x) :: :c 
The remainii,g clauses for L; then are: 
(>2.10) >:((,.t,)' 0 : V~:(~O).~(b).lt(~(a)) 
wh..,.o l,'v(.) n l?V(~) = 0 
(::.l I) ~((<,~y) = l,(>~(:)).~(,).l~(~(~)) 
where I~V(~)n vv(b) = 
Some example deriw~tions follow: 
x/v:~. \[v:,v\] x/v:,~- \[v:v\] /1.', ~: \]1,; z 
x: (~:,v)" ..... ,~.~ x: 0,~)" ...... u 
t: JA X1Y : ,\%.(~:y)" X IY '. ,~iy.(zy) r 
\[xlv:×\] Y:y \[xtv:x\] Y:y If': 
IE x: (<D' x: (~y)~ 
l~ ,U Xl(X IY): ,\q,:.(z.~D ~' X I(XIY): A%'.(:vy)' 
(X/Y)lZ:x \[Z:z\] \[Y:y\] 
l \]" x/Y: (~:~)~ 
x: ((:~.~yy)" ,~, J:&Od~(~).v 
I| )\ xtz:,\'~,.((~.~V.v)' I,(x).aO).v - ~..v 
/I 
(xlz)/v ,\>v~.((:,:~)tD" 
5rl'his axiom may I)e seen as stating the lhnlt of what cml be 
said cmtccrldilg 'uncerl.ainly divided' resources, i.e. only where 
tile unccrt.aint.y is elimina{.cd by juxtaposition can the l,,t{ oper- 
~k\[ ol's })e ielllov(!(Ii iiiitkii/~ r bOll/(l o\[.herwise qiidden ~ l'esource vis- 
ible. I,'m'ther reasonable axioms (not in practice required here) 
are l,(e) -::e and l/(e) ~ e, i.e. |.he only possible left and right 
subconlponents of ml 'mnpty' marker are likewise empty. 
1237 
x/Y:~ \[YlZ:v\] \[z:~\] 
Y : (yz) ~ 
x: (~(w)q" ~ =.L(v).~.R(y) 
xtz :~.(~(v,)~) ~ ~ ~.~ /i 
(xtz)/(v,z): ::v~o~.(,4vz)*y 
Word order and NL semantics 
Labels encode both the functional structure and lin- 
ear order information of proofs, and hence are used 
in identifying both the NL semantic and word order 
consequences of combinations. Label terms, however, 
encode distinctions not needed for NL semantics, but 
can easily be simplified to terms involving only a single 
abstraetor (A) and with application notated by simple 
left-right juxtaposition, e.g. 
x~A%.(~(w)")' ~ ~z~.((vz)~). 
To determine the linear order consequences of a 
proof with label a, we might seek a marker m consist- 
ing only of concatenated variables, where E(a) - m. 
These variables would be the labels of the proof's undis- 
charged assumptions, and their order in m would pro- 
vide an order for the types combined under the proof. 
Alternatively, for linguistic derivatlons, we mlght: sub- 
stitute lexical string atoms in place of variables, and 
seeker a marker consisting only of concatenated string 
atoms, i.e. a word string. This method is adeqnate 
for basic L, but problems potentially arise in relation 
to the discontinuity connectives. 
Consider the transformation X/Y => Xi"Y. The con- 
nective of the result type does not record all the lin- 
ear order import of the input type's connective, and 
neither consequently will the application label opera- 
tor for a subsequent \[\]E\]. Itowever, fl-normalisation 
yields a simpler label term whose operators record the 
linear order information originally encoded in the con- 
nectives of the types combined. For exarnple, the fol- 
lowing proof includes a subderivation of X/Y =~ X\]Y. 
The overall proof term does not simply order the proof's 
assumptions under )3 (giving marker L(z).y.l/,(x)),' but 
its t-normal form (xy)" does (giving x.y). 
X/Y:\ \[V:v\] Y:y /l,: 
x :(~)~ II 
XIY : ):v.(~:v) r 
x: ((~%.(~),) ~:)~ 
Of course, normalisation can only bring out ordering 
information that is implicit in the types combined. 1:'or 
example, the combination XTY:~,Y:y => X:(xy) ~ is 
a, theorem, but the label (xy) ~ does not simply of 
der x and y. However, if we require that lexical sub- 
categorisation is stated only using the standard Lam- 
bek connectives, then adequate ordering information 
will always be encoded in labels to allow simple order- 
ing for linguistic derivations. Alternatively, we could 
allow discontinuity connectives to be used in stating 
lexical subcategorisation, and farther allow that lex- 
ical types be associated with complex sh'ing terms, 
constrncted using label operators, which encode the 
requisite ordering information. For example, a word 
w with lexical type XTY might have a string term 
,Vv.(wv)", which does encode the relative ordering of 
w and its argument. A more radical idea is that de- 
duction be made over lexical types together with their 
(possibly complex) lexical string terms, and that the 
testing of side conditions on inferences be done on 
the /3-normal form of the end label, so that the im- 
plicit ordering information of the lexical string term 
is 'brought out', extending proof possibilities. Then, 
tile lexical units of the approach are in effect partial 
proofs or derivationsfi Such a change would greatly 
extend the power of tile approach. (We shall meet a 
linguistic usage for this extension shortly.) 
Linguistic applications 
We shall next briefly consider some linguistic uses of 
the discontinuity connectives in the new approach. The 
most obvkms role for \] is in handling extraction (hence 
its name). Adapting a standard approach, a rela- 
tive pronoun might have type rel/(sTnp), i.e. giving 
a relative clause (rel) if combined with s\]np (a 'sen- 
tence missinga NP somewhere'). Note that standard 
L allows only types rel/(s/np) and rel/(np\s), which 
are appropriate for extraction from, respectively, right 
and left peripheral positions only. For example, whom 
Mary considers _ foolish can be derived under the 
following proof. The atom string (6a) results via 
substitution of lexical string terms in the proof label, 
and )3. Substitntion of lexical semantics and deletion 
of directional distinctions gives (6b). 
(wl ..... ) ( ...... .y) (considers) (foolish) 
rel/(slnp) ...... p:~: ((nl>\S)/adi)/np:y \[np:u\] adj:z 
(nl~\S)/adj : (Vu)" /S 
,,p\s: ((v,,)'~)" 
s: (x((!:O"~y) * 
sJ,,p : a<~,,.(~,( (~:,)' ~)" ) i T~ 
/E 
rel: (,~ .\°,.(~((V,,)~)")b ~ 
(6) a. whom.mm'y.considers.foolish 
b. whom' (,~u.considers / u foolish I mary I) 
Moortgat (1991) suggests that a (for example) sen- 
tentially scoped NP quantifier could be typed s~(s~np), 
if inlixation and extraction could be linked so that 
infixation wan to the position of the 'missing np' of 
sTnp. r Such linkage does not follow from tile defini- 
tions of the connectives but can be implemented in the 
6Thls idea invi~.es cmnparlsons to formalisms such as lea:i- 
ealised tree adjoining 9':aramar (see Joshi et al, 1991), where 
the basic lexical and derivatiomfl units are partial phrase struc- 
ture trees associated with h:xical items. 
7In the approach of Morrill & Solias (1993) such linkage fol- 
lows automatically given the int~erpretive definitions of their con- 
nectives. Moorgat (1990,1991) proposes special purpose quan- 
tification I,ype const.rnetors. 
1238 
present approach by assigning a complex lexical string 
term, as in the lexical entry (<'rYeE,STmNG,SEM>): 
<st(sTnp), A,,.,, ~,°so,,~eone, someone'> 
Such a string term would result under a 'type raising' 
transformation sud, as: np => s~(s\[np). 'Phe example 
goh.u gave someone money can be derived am follows, 
with string and semantic results in (7). 
(~ot~,~on,,) (john) (g,~) (money) 
stCslm,):q ,w:~ ((m,\~)lm,)/m,:v \[m,:~\] m,:~ 
(np\s)/nl) : (yv) r /F 
m,\~: ((w)~) " \l.: 
s: (~((yv)'~)')' II 
sit, p: Z~v.(x((yv)~ z)") ~ 
s: (q ,\%.(,~((wl"~)U q ~ 
(7) a. john.gave.someone.money 
b. sorneone' (Av.gave' v money' john') 
Tllere is a sense in which this view of quantifiers 
seems very natural. Quantifiers behave distribution- 
ally like simple NPs, but semantically are of a higher 
type. I{aising the string component under tile trans- 
formation np => s.\[(slnp) resolves this incompatibility 
without imposing additional word order constraints. 
This aeCOllnt as stated does not allow for multi- 
ple quantitication, 8 bu~, would if lexical string terms 
were l;reated as part;\el proofs used in assembling larger 
deriwfl, ions, as suggested in the previous section. 
In interesting test case, combining both movement 
and scope issues, arises with pied piping construcl, ions, 
where a whdtem moving to clause initial position is 
accompanied by (or 'pied pipes') some larger phrase 
that conl, ains it, as in e.g. the relative clause to whom 
John spoke, where the PP to whom is fronted. Fol- 
lowing Morrill & Sol\as (1993), and ultimately Mor- 
rill (1992), a treatment of pied piping can be given 
using \]" and 1. Again, linkage of in\[ixation and ex- 
traction is achieved via complex lexical string assign- 
mont. A l)P lfied-piping relative t~rono,m might b(' 
(rel/(s\]pp))l(l)plnp) allowing it to infix to a NP I,O- 
sit\on within a Pl', giving a functor rel/(s\]pp), i.e. 
which prefixes to a 'sentence missing I'P' to give a 
relative clause. Ilence, for example, lo whom wonld 
have type rel/(slpp), and so Io whom ,/oh~z spoke is 
~ relative clause. The lexical semantics of whom will 
ensure that the resulting meaning is eqnivalent to the 
nonq)ied piping w~riant whom John spoke to. 
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