Incremental Processing and Acceptability 
Glyn Morrill* 
Universitat Polit~cnica de Catalunya 
We describe a left-to-right incremental procedure for the processing of Lambek categorial grammar 
by proof net construction. A simple metric of complexity, the profile in time of the number 
of unresolved valencies, correctly predicts a wide variety of performance phenomena including 
garden pathing, the unacceptability of center embedding, preference for lower attachment, left- 
to-right quantifier scope preference, and heavy noun phrase shift. 
1. Introduction 
Contemporary linguistics rests on abstractions and idealizations which, however fruit- 
ful, should eventually be integrated with human computational performance in lan- 
guage use. In this paper we consider the case of  processing on the basis of 
Lambek categorial grammar (Lambek 1958). We argue that an incremental procedure of 
proof net construction affords an account of various processing phenomena, including 
garden pathing, the unacceptability of center embedding, preference for lower attach- 
ment, left-to-right quantifier scope preference, and heavy noun phrase shift. We give 
examples of each of these phenomena below. 
Garden pathing (Bever 1970) is illustrated by the following contrasts: 
(1) a. The horse raced past the barn. 
b. ?The horse raced past the barn fell. 
(2) a. The boat floated down the river. 
b. ?The boat floated down the river sank. 
(3) a. The dog that knew the cat disappeared. 
b. ?The dog that knew the cat disappeared was rescued. 
Typically, although the b sentences are perfectly well formed they are perceived as 
being ungrammatical due to a strong tendency to interpret their initial segments as in 
the a sentences. 
The unacceptability of centre embedding is illustrated by the fact that while the 
nested subject relativizations of (4) exhibit little variation in acceptability, the nested 
* Departament de Llenguatges i Sistemes Informatics, Modul C 5 - Campus Nord, Jordi Girona Salgado 1-3, E-08034 Barcelona. E-maih morrill@lsi.upc.es; http://www-lsi.upc.es/~morrill/ 
(~) 2000 Association for Computational Linguistics 
Computational Linguistics Volume 26, Number 3 
object relativizations (5) exhibit a severe deterioration in acceptability (Chomsky 1965, 
Chap. 1). 
(4) a. The dog that chased the cat barked. 
b. The dog that chased the cat that saw the rat barked. 
c. The dog that chased the cat that saw the rat that ate the cheese barked. 
(5) a. The cheese that the rat ate stank. 
b. ?The cheese that the rat that the cat saw ate stank. 
c. ??The cheese that the rat that the cat that the dog chased saw ate 
stank. 
Discussing such center embedding, Johnson (1998) presents the essential idea devel- 
oped here, noting that processing overload of dependencies invoked in psycholin- 
guistic literature could be rendered in terms of the maximal number of unresolved 
dependencies as represented by proof nets. 
Kimball (1973, 27) considers sentences such as (6), which are three ways am- 
biguous according to the attachment of the adverb. He points out that the lower the 
attachment of the adverb, the higher the preference (he calls this relationship Right 
Association). 
(6) Joe said that Martha believed that Ingrid fell today. 
Left-to-right quantifier scope preference is illustrated by: 
(7) a. Someone loves everyone. 
b. Everyone is loved by someone. 
Both sentences exhibit both quantifier scopings: 
(8) a. ~xVy(love y x) 
b. Vy3x(love y x) 
However, while the dominant reading of (7a) is (8a), that of (7b) is (8b), i.e., the 
preference is for the first quantifier to have wider scope. Note that the same effect is 
observed when the quantifiers are swapped: 
(9) a. Everyone loves someone. 
b. Someone is loved by everyone. 
While both sentences in (9) have both quantifier scopings, the preferred readings give 
the first quantifier wide scope. 
Finally, we will look at heavy noun phrase shift, which is the preference for com- 
plex object noun phrases to "shift" to the end of the sentence. Consider the two sen- 
tences in (10); the second, in which the "heavy" direct object follows the indirect object, 
is more acceptable than the first. 
320 
Morrill Incremental Processing and Acceptability 
(10) a. ?John gave the painting that Mary hated to Bill. 
b. John gave Bill the painting that Mary hated. 
We argue that a simple metric of categorial processing complexity explains these 
and other performance phenomena. 
2. Lambek Calculus 
We shall assume some familiarity with Lambek categorial grammar as presented in, for 
example, Moortgat (1988, 1997), Morrill (1994), or Carpenter (1998), and limit ourselves 
here to reviewing some central technical and computational aspects. 
The types, or (category) formulas, of Lambek calculus are freely generated from 
a set of primitives by the binary infix connectives "/" (over), "V' (under) (directional 
divisions) and "-" (product). With respect to a semigroup algebra (L, +) (i.e., a set 
L closed under an associative binary operation + of adjunction), each formula A is 
interpreted as a subset \[A\] of L by residuation as follows: 
(11) ~\[a.B\]\] ~- ($1q-82\]$1 E ~a\] & s 2 E ~B~} 
~AkB\]\] = {sirs' C \[\[AB, s'+s E HB\]\]} 
~B/A\] -- (sirs' E \[\[A~,s+s' E ~B\]I} 
A sequent, F ~ A, comprises a succedent formula A and an antecedent configuration 
F, which is a a finite sequence of formulas. 1 A sequent is valid if and only if in all inter- 
pretations the ordered adjunction of elements inhabiting the antecedent formulas al- 
ways yields an element inhabiting the succedent formula. The following Gentzen-style 
sequent presentation is sound and complete for this interpretation (Buszkowski 1986, 
Dogen 1992), and indeed for free semigroups (Pentus 1994): hence the Lambek calculus 
can make an impressive claim to be the logic of concatenation; a parenthetical notation 
A(F) represents a configuration containing a distinguished subconfiguration F. 
F~A A(A)~B Cut 
(12) a. A ~ A id " A(F).::~ B 
F ~ A A(B) ~ C \L A,F ~ B \R 
b. A(F,A\B) =~ C F ::~ AkB 
F ~ A A(B) ~ C F,A=~B /R 
c. &(B/A,F) ~ C /L r ~ B/A 
F,A,B,A ~ C F ~ A A =~ B • L .R 
d. F,A.B,A ~ C F,A =~ A.B 
By way of example, "lifting" A ~ B/(A\B) is generated as follows: 
(13) 
A =~ A B=~ B \ L 
A,A\B ~ B 
A ~ B/(A\B) /R 
1 Officially, the antecedent is nonempty, a detail we gloss over. 
321 
Computational Linguistics Volume 26, Number 3 
And "composition" A\B, B\C ~ A\C can be derived thus: 
(14) BoB C ~ C \L 
A ~ A B, B\C ~ C 
A,A\B, B\C ~ C 
A\B,B\C ~ A\C \R 
Every rule, with the exception of Cut, where the Cut formula A does not appear 
in the conclusion, has exactly one connective occurrence less in its premisses than in 
its conclusion. Lambek (1958) proved Cut elimination--that every proof has a Cut- 
free counterpart--hence a decision procedure for theoremhood is given by backward- 
chaining proof search in the Cut-free fragment. The nonatomic instances of the id 
axiom are derivable from atomic instances by the rules for the connectives. But even 
in the Cut-free atomic-id calculus there is spurious ambiguity: equivalent derivations 
differing only in irrelevant rule ordering. For example, composition as above has the 
following alternative derivation: 
(15) A=~A B~B \L 
A,A\B ~ B C ~ C 
A,A\B, B\C =~ C 
A\B,B\C =~ A\C \R 
\C 
One approach to this problem consists in defining, within the Cut-free atomic-id 
space, normal form derivations in which the succession of rule application is regulated 
(K6nig 1989, Hepple 1990, Hendriks 1993). Each sequent has a distinguished category 
formula (underlined) on which rule applications are keyed: 
(16) a. P ~ P id 
a(A) ~ B 
A(.a) ~ B_ 
r~4 a(B)~c \L A,F~B \R 
b. a(F,A\B) ~ C r ~ A\B 
r~A &(~)~C r,A~B 
c. A(B/A,F) ~ C /L F ~ B/~ /R 
A~A F~B 
-- .R d. A,F~A.B 
In the regulated calculus there is no spurious ambiguity, and provided there is 
no explicit or implicit antecedent product, i.e., provided .L is not needed, F ~ A is 
a theorem of the Lambek calculus iff F ~ A is a theorem of the regulated calculus. 
However, apart from the issue regarding .L, there is a general cause for dissatisfaction 
with this approach: it assumes the initial presence of the entire sequent to be proved, 
i.e., it is in principle nonincremental; on the other hand, allowing incrementality on 
the basis of Cut would reinstate with a vengeance the problem of spurious ambigu- 
ity, for then what are to be the Cut formulas? Consequently, the sequent approach is 
ill-equipped to address the basic asymmetry of --the asymmetry of its pro- 
cessing in time---and has never been forwarded in a model of the kind of processing 
phenomena cited in the introduction. 
322 
Morrill Incremental Processing and Acceptability 
An alternative formulation (Ades and Steedman 1982, Steedman 1997), which 
from its inception has emphasized a capacity to produce left-branching, and there- 
fore incrementally processable, analyses, is comprised of combinatory schemata such 
as the following (together with a Cut rule, feeding one rule application into an- 
other): 
(17) a. A,A\B => B B/A,A =~ B 
b. A ~ (B/A)\B A ~ B/(A\B) 
c. A\B,B\C ~ A\C C/B,B/A =~ C/A 
By a result of Zielonka (1981), the Lambek calculus is not axiomatizable by any fi- 
nite set of combinatory schemata, so no such combinatory presentation can constitute 
the logic of concatenation in the sense of Lambek calculus. Combinatory categorial 
grammar does not concern itself with the capture of all (or only) the concatenatively 
valid combinatory schemata, but rather with incrementality, for example, on a shift- 
reduce design. An approach (also based on regulation of the succession of rule ap- 
plication) to the associated problem of spurious ambiguity is given in Hepple and 
Morrill (1989) but again, to our knowledge, there is no predictive relation between 
incremental combinatory processing and the kind of processing phenomena cited in 
the introduction. 
3. Proof Nets 
Lambek categorial derivations are often presented in the style of natural deduction 
or sequent calculus. Here we are concerned with categorial proof nets (Roorda 1991) 
as the fundamental structures of proof in categorial logic, in the same sense that lin- 
ear proof nets were originally introduced by Girard (1987) as the fundamental struc- 
tures of proof in linear logic. (Cut-free) proof nets exhibit no spurious ambiguity and 
play the role in categorial grammar that parse trees play in phrase structure gram- 
mar. 
Surveys and articles on the topic include Lamarche and Retor¢ (1996), de Groote 
and Retor¢ (1996), and Morrill (1999). Still, at the risk of proceeding at a slightly slower 
pace, we aim nonetheless to include here enough details to make the present paper 
self-contained. 
A polar category formula is a Lambek categorial type labeled with input (°) or 
output (o) polarity. A polar category formula tree is a binary ordered tree in which 
the leaves are labeled with polar atoms (literals) and each local tree is one of the 
following (logical) links: 
A ° B" .. B ° A" . 
-- U -- 1 (18) a. A\B ° A\B ° 
B • A o .. A ° B o . 
b. B/A. 11 B/AO 1 
A ° B" . B ° A ° .. 
c. A.B ° 1 A-B ° 11 
Without polarities, a formula tree is a kind of formation tree of the formula at its root: 
323 
Computational Linguistics Volume 26, Number 3 
daughters are labeled with the immediate subformulas of their mothers. The polar- 
ities indicate sequent sidedness, input for antecedent and output for succedent; the 
polarity propagation follows the sidedness of subformulas in the sequent rules: in the 
antecedent (input) rule for A\B the subformula A goes in a succedent (output) and the 
subformula B goes in an antecedent (input); in the succedent (output) rule for A\B the 
subformula A goes in an antecedent (input) and the subformula B goes in a succedent 
(output); etc. The labels i and ii indicate whether the corresponding sequent rule is 
unary or binary. Note that in the output links the order of the subformulas is switched; 
this corresponds to a cyclic reading of sequents: the succedent type is adjacent to the 
first antecedent type. 
A proof frame is a finite sequence of polar category formula trees, exactly one 
of which has a root of output polarity (corresponding to the unique succedent of 
sequents). 
An axiom linking on a set of literal labeled leaves is a partitioning of the set 
into pairs of complementary leaves that is planar in its ordering, i.e., there are no 
two pairs {L1,LB},{La, L4} such that L1 < L2 < L3 < L4. Geometrically, planarity 
means that where the leaves are ordered on a line, paired leaves can be connected in 
the half plane without crossing. Axiom links correspond to id instances in a sequent 
proof. 
A proof structure is a proof frame together with an axiom linking on its leaves. A 
proof net is a proof structure in which every elementary (i.e., visiting vertices at most 
once) cycle crosses the edges of some i-link. 2 Geometrically, an elementary cycle is the 
perimeter of a face or cluster of faces in a planar proof structure. There is a proof net 
with roots A°,AI",... ,An" iff A1,...,A, ~ A is a valid sequent. 
4. Incremental Processing Load and Acceptability 
Let us assume the following lexical assignments: 
(19) 
barn - barn 
:= CN 
horse - horse 
:= CN 
past - )~x)~y)~z(past x (y z)) 
:= ((N\St)\(N\St))/N 
raced - race 
:= N\S+ 
the - the 
:= N/CN 
The feature + on S marks the projection of a tensed verb form; a verb phrase modified 
by past need not be tensed. Let us consider the incremental processing of (la) as proof 
2 This criterion, adapted from that of Lecomte and Retor6 (1995), derives from Girard's (1987) long trip 
condition, which is an involved mathematical result. Danos and Regnier (1989) express it in terms of 
acydicity and connectivity of certain subgraphs. Intuitively, acyclicity assures that the subformulas of 
ii-links (binary rules) occur in different subproofs, whereas connectivity assures that the subformulas of 
i-links (unary rules) occur in the same subproofs (attributed to Jean Gallier by Philippe de Groote, p.c.). 
However the single-succedent (intuitionistic) nature of Cut-free categorial proofs in fact renders the connectivity requirement redundant, hence we have just an acyclicity test. 
324 
Morrill Incremental Processing and Acceptability 
net construction. 3 In the first case, we suppose that one initially expects some target 
category, perhaps (though not necessarily) S. This 'principle of expectation' seems a 
reasonable or obvious principle of communication; as we shall see, it turns out to 
be technically critical. After perception of the word the there is the following partial 
proof net (for simplicity we omit features, included in lexical entries, from proof nets 
themselves): 
(20) 
N" ii GN ° \/ 
S ° N/GN" 
the 
Here there are three unmatched valencies/unresolved dependencies; no axiom links 
can yet be placed, but after horse we can build: 
(21) 
S* 
I I 
I I 
N" ii GN ° I \/ : 
N/GN ° CN" 
the horse 
Now there are only two unmatched valencies. After raced we have, on the correct 
analysis, the following: 
(22) 
I I 
t I 
N* ii CN* I N ° ii S" \/ : \/ 
S ° N/CN" CN" N\S" 
the horse raced 
Note that linking the Ns is possible, but we are interested in the history of 
the correct analysis, and in that, the verb valencies are matched by the adverb that 
3 The procedure is similar in spirit to that in the appendix of Ades and Steedman (1982), but we perform 
"reductions" by axiom links on complementary literal pairs rather than by combinatory schemata on 
category formulas. 
325 
Computational Linguistics Volume 26, Number 3 
follows (henceforth we indicate only the principal connective of a mother node): 
(23) 
S ° 
I I 
I I 
N" ii CN* I N ° ii S ° \/ ', \/ 
/ CN" \ 
I 
I 
I I 
I I 
S ° i N" \/ 
\ 
I 
I 
I 
I 
I 
N ° ii S ° \/ 
ii \ 
! 
N* 
the horse raced past 
Observe that a cycle is created, but as required it crosses the edges of an i-link. At the 
penultimate step we have: 
(24) 
S ° 
I 
I 
I I 
I I 
S ° i N ° \/ 
\ \ _m 
I t I I 
I \[ I I 
N ° ii CN ° I N ° ii S" N ° N" ii CN ° \/ ', \/ \/ 
I CN" \ I 
I 
I 
I 
I 
I 
N* ii S" \/ 
ii \ 
the horse raced past the 
The final proof net analysis is given in Figure 1. 
The semantics associated with a categorial proof net, i.e., the proof as a lambda 
term (intuitionistic natural deduction proof, under the Curry-Howard correspondence) 
is extracted by associating a distinct index with each output division node and travel- 
ing as follows, starting by going up at the unique output root (de Groote and Retor6 
1996): 
(25) • traveling up at the mother of an output division link, perform 
the lambda abstraction with respect to the associated index of 
the result of traveling up at the daughter of output polarity; 
traveling up at the mother of an output product link, form the 
ordered pair of the result of traveling up at the right daughter 
(first component) and the left daughter (second component); 
326 
Morrill Incremental Processing and Acceptability 
I I 
I I 
N" ii CN ° I \/ 
, I 
S ° / CN" 
I 
I 
I I I 
J 
I I I I I 
I I S° i N* N ° ii S* 
, , \/ \/ 
\] \[ \ ii \ 
I I 
N ° ii S ° \ ii 
\ / 
I I I I 
I I I I 
N ° N" ii CN" t \/ 
, I 
/ CN" 
the horse raced past the barn 
Figure 1 
Proof net analysis of (la) the horse raced past the barn. 
traveling up at one end of an axiom link, continue down at the 
other end; 
traveling down at an (input) daughter of an input division link, 
perform the functional application of the result of traveling 
down at the mother to the result of traveling up at the other 
(output) daughter; 
traveling down at the left (respectively, right) daughter of an 
input product link, take the first (respectively, second) projection 
of the result of traveling down at the mother; 
traveling down at the (input) daughter of an output division 
link, return the associated index; 
traveling down at a root, return the associated lexical semantics. 
Thus for our example we obtain (26a), which is logically equivalent to (26b). 
(26) a. (&x&y&z(past x (y z)) (the barn) M(race 1) (the horse)) 
b. (past (the barn) (race (the horse))) 
The analysis of (lb) is less straightforward. Whereas in (la) raced expresses a one- 
place predication ("go quickly"), in (lb) it expresses a two-place predication (there 
was some agent racing the horse); horse is modified by an agentless passive participle, 
but the adverbial past the barn is modifying race. Within the confines of the Lambek 
calculus, the characterization we offer assumes the lexical assignment to the passive 
participle given in the following: 4 
4 In general, grammar requires the expressivity of more powerful categorial logics than just Larnbek 
calculus; however, so far as we are aware, the characterizations we offer within the Lambek calculus 
bear the same properties with regard to our processing considerations as their more sophisticated 
categorial logic refinements, because the latter are principally concerned with generalizations of word 
order, whereas the semantic dependencies on which our complexity metric depends remain the same. 
327 
Computational Linguistics Volume 26, Number 3 
I ......... I I I I -- ...... I 
SIo I I I I ------ 1 N .... 
I I I I \[ I I I I 
' I , ~ / I I , S I. SIo /. C N ° C N ° \ N" I N ° 
\/ \/ ', \/ \/ 
\ g o \ \ \/ \ 
/ \ \ 
I 
I 
I 
I 
N* CN ° \/ 
/ 
I I 
1 I 
I I I 
I I \[ I 
S ° CN" 
I t 
, I 
i SI° N ° 
\/ 
, , 
N ° N • 
/ 
I I 
I I 
C'N° I \/ 
, 
/ CN ° 
N ° S • \/ 
\ 
the horse raced 
Figure 2 
Proof net analysis of (lb) the horse raced past the barn fell. 
p~t the barn fell 
(27) fell - fall 
:= N\S+ 
raced - (Ax)~yAz\[(y z) A ~w(x z w)\],race2) 
:= ((CN\CN) / (N\(N\S-))).(N\(N\S-)) 
Here raced is classified as the product of an untensed transitive verbal type, which can 
be modified by the adverbial past the barn by composition, and an adnominalizer of 
this transitive verbal type. According to this, (lb) has the proof net analysis given in 
Figure 2. The semantics extracted is (28a), equivalent to (28b) 
(28) a. (fall (the (Tra(&xAy&z\[(y z) A 3w(x z w)\], race2) )~29~30 
()~uAv)~w(past u (v w)) (the barn) MI(( 
(Tr2(/~p/~s&t\[(s t) A 3q(p t q)\], race2) 29 41) 30) horse))) 
b. (fall (the)~8\[(horse 8) A 37(past (the barn) (race2 8 7))\])) 
Let us assign to each proof net analysis a complexity profile that indicates, before 
and after each word, the number of unmatched literals, i.e., unresolved valencies or 
dependencies, under process at that point. This is a measure of the course of memory 
load in optimal incremental processing. We are not concerned here with resolution of 
lexical ambiguity or serial backtracking: we are supposing sufficient resources that the 
nondeterminism of selection of lexical entries and their parallel consideration is not 
the critical burden. Rather, the question is: which among parallel competing analyses 
places the least load on memory? 
Since entropy degrades the structure of memory, it requires more energy to pursue 
an analysis that is high cost in memory than to pursue one that is low cost. From these 
simple economic considerations we derive our main claim: 
(29) Principle of Acceptability 
Acceptability is inversely proportional to the sum in time of the memory 
load of unresolved valencies. 
If other factors are constant, the principle makes a quantitative prediction. We can 
distinguish two cases: synonymy and ambiguity. In the case of synonymy, semantics is 
constant. It is then predicted that amongst synonymous forms of expression, the lower 
328 
Morrill Incremental Processing and Acceptability 
~'\S ~° ~°\/ 
....... i i ....... I ........ i t ........ i 
i J i t i 
', I ', 1 1 Ii '~ I slo i , , 
cN~ • ..... • ...... 'N~ ~'N" 7" N~ /" -- / \ / \ / i--', / \ I--i :- ,, 
~o N" c'x o \ \ \ x ° N" ~'~° \ \ \ 
~/ \/ \/ ~/ \/ \/ r <:.x, / I i <N ° / i i C'N* 
Figure 3 
Proof net analysis of (4b) the dog that chased the cat that saw the rat barked. 
the complexity curve, the higher the preference for the form of expression. In the case 
of ambiguity, prosodics is constant. It is then predicted that amongst the readings of an 
ambiguous expression, the lower the complexity curve, the more dominant the reading. 
The complexity profile is easily read off a completed proof net: the complexity 
between two words is the number of axiom links bridging rightwards (forwards in 
time) at that point. Thus for (la) and (lb) analyzed in Figures 1 and 2, the complexity 
profiles are as follows: 
(30) 6 b 
5 
4 b a 
3 ab b b 
2 a b 
1 ab a a 
0 a 
a. the horse raced past the barn 
b. the horse raced past the barn fell 
We see that after the first two words the complexity of the locally ambiguous initial 
segment of (lb) is consistently higher than that of its garden path (la). The areas of 
the a and b curves are 12 and 22 respectively, predicting that in (lb) the less costly 
but incorrect analysis could be salient, as indeed it is. 
Johnson (1998) considers center embedding for subject and object relativization 
from a similar point of view. We assume here the relative pronoun lexical assignments 
shown in (31). s 
(31) that - AxAyAz\[(y z) A (x z)\] 
:= (CN\CN)/(N\S+) 
that - AxAyAz\[(y z) A (x z)\] 
:= (CN\CN)/(S+/N) 
The proof net analysis of sentence (4b) is shown in Figure 3, and that of sentence (5b) 
is shown in Figure 4. Let us compare the complexities: 
(32) 9 b 
8 b 
7 b b 
6 b 
5 b 
4 ab ab a a 
3 ab a a a a 
2 ab 
1 ab 
0 ab 
a. the dog that chased the cat that saw the rat barked 
b. the cheese that the rat that the cat saw ate stank 
5 For better linguistic treatment not affecting the point at hand, see Morrill (1994, Chap. 8). 
329 
Computational Linguistics Volume 26, Number 3 
8 ° 
I 
_ _ f I , 
CN ° CN" N ° S ° \/ \/ 
N" CN ° \ \ 
/ CN" / 
N" CN ° \/ 
! 
..... I ........... 
I I I I 
I i ' '' I ' I SlO I S~° 
CN ° CN ° N* -- ~ N ° . \ / \/ ' , , \/ 
NI° I I \ \ ('N ° \ N ° 
\/ ', \/ 
CN* / / CN ° / . 
N\ ?" 
\ N ° N o S • \/ \/ 
! \ 
the cheese that the rat 
Figure 4 
Proof net analysis of (5b) the cheese that the rat that the cat saw ate stank. 
The profile of (5b) is higher; indeed it rises above 8, thus reaching what is usually taken 
to be the limit of short-term memory. Johnson attributes the increasing ill-formedness 
of centre embedded constructions to the number of incomplete dependencies at the 
"maximal cut" of a proof net. This almost corresponds to the maximum height of 
a complexity profile here, except Johnson includes no target category, whereas we 
will argue in relation to quantifier scope preference that this is critical. However, 
we also differ from Johnson in attributing relative acceptability to the area under the 
complexity curve, not only its maximal height. This is because we believe acceptability 
is to be explained in terms of the energy required to maintain processes in memory 
over time, and not just in terms of peak memory load. Finally, it happens that our 
proposal solves a problem encountered by Johnson. Gibson and Thomas (1996) observe 
that (33a) is easier to comprehend than (33b). 
(33) a. The chance that the nurse who the doctor supervised lost the reports 
bothered the intern. 
b. ?The intern who the chance that the doctor lost the reports bothered 
supervised the nurse. 
Johnson notes that his proposal does not capture this difference since both sentences 
have the same size maximal cut. Under our account, on the other hand, it is the com- 
plexity curves as a whole that account for acceptability. In these sentences, although 
the height is the same, the complexity curves are not: the area of (33a) is less than that 
of (33b). 6 Thus, whereas Johnson must look to other factors to explain this difference, 
our account makes the correct prediction. 
6 To save space we exclude the proof nets; the curves are: 
ab 
ab 
ab 
ab 
b ab ab b b 
a b 
b b a 
a a a a 
ab 
ab ab 
ab 
330 
Morrill Incremental Processing and Acceptability 
I 
I I 
F .................................... I 
i 1 1 I 
I ................................. I-- -- 
I ~ I I I I 
I I I ............... I I l 
I I F I I I I I I I 
I I I r-- ~ - - I I--I I I- -t I 1 I r 
, I I I i I I II I ' I I I i i I I I 
N, (N \ S) / CP, GP / S, N, (N \ S) / CP, CP / S, N, N \ S. (N \ S) \ (N \ $) 
Joe sMd that 
t 
I I I 
' - ~ I I I 
S, N, {N \ / CP / S. 
Martha believed that Ingrid fell today 
................................. 
I 
p ............................. 
I I 
i i 
I ................. i_ _ 
i E i 
I i ....... i i 
i i i I I I 
l i -- i -- i i r 
I i i i i 
(N \ S) / CP, CP / S, N, N \ S, (N \ ) 
I 
I 
I 
t 
I 
I 
I 
\ (N \ s) 
Joe said that Martha believed that 
I i i I , , I \] Ij I II I\[ 
I ~ I ! I I I 
S, X. (N \ S) / CP, CP / S. N. (N \ S) / CP. GP 
Ingrid fell today 
I 
............ I 
I I I 
I -- -- -- I T 
I \[ I 
, I , 1 I i I 
N. N \ S. (N \ S) \ (N \ S) 
Joe said that Martha believed that Ingrid fell today 
Figure 5 
Proof net analyses for (34) Joe said that Martha believed that Ingrid fell today, with lowest (top), 
middle (center), and highest (bottom) attachment of the adverb today. 
5. Preferred Readings 
The examples so far involve comparison of different expressions, having many differ- 
ences to which their relative acceptabilities could be attributed. More factors would 
be held constant by comparing alternative readings of an ambiguous expression, and 
most appropriately of all, a structurally ambiguous expression, where there is no lexi- 
cal alternation. Consider the ambiguity that arises in a sentence such as (34) (repeated 
from (6)) from the possibility of attaching the adverb at different syntactic levels. As 
we saw in the introduction, the lower the attachment of the adverb in such sentences, 
the higher the preference (Kimball 1973, 27). 
(34) Joe said that Martha believed that Ingrid fell today. 
In Figure 5 we give the analyses for the lowest, the middle, and the highest attach- 
ments. We now abbreviate proof nets by flattening formula trees into their linear rep- 
resentations (since this conceals the order switching of output links, the notation belies 
the underlying planarity, but the portrayal of word-by-word complexity is unaltered). 
Accordingly, the complexity profiles are: 
(35) 6 c bc 
5 c c bc bc 
4 abc 
3 
2 abc ab a 
1 abc ab ab a a 
0 abc 
a. lowest attachment 
b. Joe said that Martha believed that Ingrid fell today middle attachment 
c. highest attachment 
331 
Computational Linguistics Volume 26, Number 3 
I 
I 
I I 
I I 
I I L i 
1 i i I I I 
i t t i I I 
N, N / CN, CN, {GN \ CN) 
I I 
/ (N 
I 
I I I 
I 
---- --I-- I i i I I I ) 
i I I I I t I I 
\ s), (N \ s) / N, N. N \ IN / CN), ON 
the book that shocked Mary 's title 
I I i f i t 
i I I i I I I I- I i I I i I I I 
i i I I i I I I I i I I I I I i N. N / CN. CN. (ON \ ON) / (N \ S), (N \ 
s) / N. N, N \ (t," / CN), CN 
the book that shocked Mary 's title 
Figure 6 
Proof net analysis for the sensical (top) and nonsensical (bottom) interpretations of (36) the book that shocked Mary's title. 
The same effect occurs strongly in (36), where the preferred reading is the one 
given by the lowest attachment, even though that one is the nonsensical reading. 
(36) the book that shocked Mary's title 
The analyses are given in Figure 6. The complexities are thus: 
(37) 4 a a 
3 a a 
2 b b ab 
1 ab b b ab 
0 
a. the book that shocked Mary "s title 
b. 
ab 
sensical 
nonsensical 
6. Left-to-right Quantifier Scope Preference 
Let us consider now another instance of ambiguity: quantifier scope preference. A 
rudimentary account of sentence-peripheral quantifier phrase scoping is obtained in 
Lambek calculus by means of lexical assignments such as the following: 7 
(38) everyone - )~xVy(x y) 
:= St/(N\St) 
everyone - )~xVy(x y) 
:= (St/N)\St 
someone - )~x3y(x y) 
:= St/(N\St) 
someone - )~x3y(x y) 
:= (St/N)\St 
Given these assignments, one analysis of (7a) is that given in Figure 7. This is the 
subject wide scope analysis: its extracted and simplified semantics are as in (39). 
7 For a more refined treatment (not requiring multiple lexical categorizations), for which the results of 
this paper stand unchanged, see Morrill (1994, Chap. 4). 
332 
Morrill Incremental Processing and Acceptability 
I I 
I I 
I s ° 
I 
L 
s ° 
I 
J 
f 
I I I I 
I I I I I 
S ° N • N ° S • 
\/ \/ i 
\ \ N ° \/ \/ 
/ / 
I 
I 
I I 
I 
N • S ° \/ 
/ \/ 
\ 
S ° 
someone loves everyone 
Figure 7 
Proof net analysis for the subject wide scope reading of (7a) someone loves everyone (3V). 
(39) a. (Ax3y(x y) M(AxVy(x y) &2(love 2 1))) 
b. 3xVy(love y x) 
A second analysis is that given in Figure 8. This is the object wide scope analysis: its 
extracted and simplified semantics are as in (40). 
(40) a. (.~xVy(x y) .~2(.~x3y(x y) M(love 2 1))) 
b. Vy3x(love y x) 
Let us compare the complexity profiles for the two readings: 
(41) 4 b 
3 
2 a 
1 ab 
0 
ab 
ab 
a. 3V (subject wide scope, Figure 7) 
b. someone loves everyone V3 (object wide scope, Figure 8) 
S ! 
S ° 
I I 
I I 
s ° N" N ° \ / \ 
\ \/ 
/ 
I 
I I 
I 
I I I I 
S" N ° S ° / ' 
, \/ 
\ N ° / \/ \/ 
/ \ 
S ° 
someone loves everyone 
Figure 8 
Proof net analysis for the object wide scope reading of (7a) someone loves everyone (V3). 
333 
Computational Linguistics Volume 26, Number 3 
s'o ".'o s'. ,-~ / \/ 
I~'. \\ 
S ° / 
................................... 
I 
......... I I I I ....... I 
I SIo h 1 I _ _ N • ......... 
I CN o CN" \ N" I N o N" N ° -- -- I 
,' \/ \/ I \/ \/ \/ ~, ', ', 
CN ° ON" \ \ N° \ \ \ I ,.q, S o \/ ~/ \/ \J ', \/ 
\ N o /\ \ , s. \, ~,/ \\/ 
everyone is loved by someone 
Figure 9 
Proof net analysis for the subject wide scope reading of (7b) everyone is loved by someone (V3). 
At the only point of difference, the subject wide scope reading, the preferred reading, 
has the lower complexity. 
We emphasize that some target category is expected at the start of  pro- 
cessing. Lambek proof nets have the property that they can be kept planar under 
any cyclic permutation of the roots; one can view them as a circular list and require 
planarity on the interior (or exterior) of a disc. The same proof net would then have 
different complexity profiles depending on whether one set in axiom links clockwise 
or counterclockwise. Language processing, though, takes place in time, which is not 
cyclic, so we order the roots along a line, starting, according to the principle of ex- 
pectation, with the target category. As a reviewer has pointed out, if, for example, the 
target category were placed at the end of the time line, then the predictions of the 
relative acceptabilities in (41) would be reversed! 
For the passive (7b), let there be assignments as in (42). The preposition by projects 
an agentive adverbial phrase; is is a functor over (post)nominal modifiers (the man 
outside, John is outside, etc.) and passive loved is treated exactly like passive raced 
in (27). 
(42) by - )~x)~yAz\[\[z = x\] A (y z)\] 
:= ((N\S-)\(N\S-))/N 
is - )~x)~y(x )~z\[z = y\] y) 
:-- (N\S+)/(CN\CN) 
loved - (;~x;~y),z\[(y z) A 3w(x z w)\],love) 
:= ((CN\CN) / (N\(N\S-))).(N\(N\S-)) 
A V3 analysis of (7b) is given in Figure 9. This has semantics, after some simplifi- 
cation, as in (43), which is equivalent to (40). 
(43) V163937\[\[9 = 7\] A (love 16 9)\] 
An 3V analysis of (7b) is given in Figure 10. This has semantics, after some sim- 
plification, as in (44), which is equivalent to (39). 
(44) ~16V1437\[\[7 = 16\] A (love 14 7)\] 
334 
Morrill Incremental Processing and Acceptability 
I ......... I 
I I 
I -- I 
j I I t 
S~ I \] Sl ° N* N ° , \/ \/ 
S" \ \/ 
! 
I 
I I 
I I 
......... I I I I ....... J 
s o N • -- -- 
I I I E I I 
t CN o CN • \ N • l N* S ° N" N o S ° F ~ I----I 
, \/ \/ I \/ \/ / , , 
CN ° CN" \ \ No \ \ \ I N" \/ \/ \/ ,, 
\ \ x o / / " • /'~ s° 
everyone is loved by 
Figure 10 
Proof net analysis for the object wide scope reading of (7b) everyone is loved by someone (3V). 
Again, the preferred reading has the lower complexity profile: 
(45) 
6 b 
5 
4 b b a 
3 ab 
2 a a 
1 ab 
0 ab 
a. V3 (Figure 9) b. everyone is loved by someone 3V (Figure 10) 
7. Preferred Forms 
Another good test would be to compare different expressions that are synonymous, 
holding semantics constant. Our account appears to explain the preference for heavy 
noun phrases to appear at the end of the verb phrase (heavy noun phrase shift). Of 
the following two sentences, repeated from (10), the second is more acceptable: 
(46) a. ?John gave the painting that Mary hated to Bill. 
b. John gave Bill the painting that Mary hated. 
The analyses are given in Figure 11. The complexities are thus: 
(47) 4 a 
3 a a b 
2 ab ab a b b 
1 ab b b a 
0 
a. John gave the painting that Mary hated to 
b. John gave Bill the painting that Mary hated 
a 
b a 
Bill 
335 
Computational Linguistics Volume 26, Number 3 
I -- -- I , I I , 
S, N, IN \ S) 
John 
/ 
save 
t 
I 
k I I 
------L ........ ........ I 
I I I I I L I 
- - -I- I - - I I I - - I I 
(N • PP), N \[ CN. CN, \[CN \ CN) \[ (S \[ N), N, (N \ S) / N, PP 
the p. that Mary hated 
/ N, N 
I I 
I -- -- b 
•. (N \ S I) 
I I 
I I I I I I I 
/ (N - N), N, N \[ C'N. ('N. ff-'N \ CN) / 15; / Nt. N, (N \ / N 
John gave Bill the P. that Mary hated 
Figure 11 
Proof net analysis for (46b) John gave Bill the painting that Mary hated (bottom) and (46a) John 
gave the painting that Mary hated to Bill (top). 
8. Conclusion: Valencies versus Category Complexity 
Finally, another dramatic example of unacceptability 
ing: 8 
is provided by the follow- 
(48) a. That two plus two equals four surprised Jack. 
b. ?That that two plus two equals four surprised Jack astonished Ingrid. 
c. ??That that that two plus two equals four surprised Jack astonished 
Ingrid bothered Frank. 
The passive paraphrases, however, seem more or less equally acceptable: 
(49) a. Jack was surprised that two plus two equals four. 
b. Ingrid was astonished that Jack was surprised that two plus two 
equals four. 
c. Frank was bothered that Ingrid was astonished that Jack was 
surprised that two plus two equals four. 
In Figure 12 we give the analysis of (48b) and in Figure 13 that of (49b). It is very 
interesting to observe that the complexity profile of the latter is in general lower even 
though the analysis has more than twice the total number of links. 
8 In some linguistic analyses it is claimed that sentencial subjects are obligatorily extraposed into a 
presentencial topic position, which is not available in nonroot clauses. Under our account no such 
abstract claims are necessary. 
336 
Morrill Incremental Processing and Acceptability 
f 
.............................. I 
I 
I ............... 
I I 
I I ...... I 
I I i r I 
I I ~-- J ---- I I ---- 
I 
t 
I 
I 
I 
l 
I I I I I 
S, CP / S. clip / S. N, (N \ N) / N, N, (N \ S) / N. N, (CP \ S) / N, 
I I 
I I 
N, (CP \ S) / N. N 
that that two plus two equals four surp. Jack ast, Ingrid 
Figure 12 
Proof net analysis of (48b) that that two plus two equals four surprised Jack astonished Ingrid. 
I I 
s, N, (N \ S) 
I ..... I 
I I I I 
.......... i- ...... q. ..... E .... I 
I I I I I ~ I I I 
-- -- -- q- -- ~1 1 -- -- --I .... I ...... r-- -- I P I I 
! 
/ (CN \ GN), ((CN \ CN) / (N ~ (CP \ S))} * (N \ (CP \ S)), ((CP \ S) \ (CP \ S)) \] S 
Ingrld w~ ast. that 
I ..... I 
......... L--~ J ...... 
I P I I I I I i 
', r , 1 , , , I ~ , I I ~ I , , 
N, (N \ S) / (CN \ CN), ((CN \ CN) / (N \ (('P \ S))) . (N \ (CP \ S)), ((CP \ S) \ (CP \ S{)) \[ / S 
Jack w~ surp. that 
I ....... I I 
N, (N \ N} \[ X. N. (N \ S) / X, N, 
two plus two equals four 
Figure 13 
Proof net analysis of (49b) Ingrid was astonished that Jack was surprised that two plus two equals four. 
(50) 
7 a 
6 a a 
5 a a 
4 b ab 
3 a a b 
2 b b b b ab b 
I ab b b a 
0 a 
a. that that two plus two equals four surp. Jack ast. Ingrid 
b. Ingrid was ast. that Jack was surp. that two plus two equals four 
Acceptability is no doubt a product of many factors other than just the structural 
ones considered here. Still, in as much as structural factors may exist, we think this 
example in particular provides good support for the view that it is resolution of atomic 
valencies, rather than complexity of categories, which underlies their contribution to 
acceptability. 
337 
Computational Linguistics Volume 26, Number 3 
Acknowledgments 
For helpful remarks I thank Gabriel Bes, 
anonymous reviewers, and various 
audiences, and especially for recent 
comment, Ewan Klein and Mark Steedman. 
References 
Ades, Anthony E. and Mark J. Steedman. 
1982. On the Order of Words. Linguistics 
and Philosophy, 4:517-558. 
Bever, Thomas. 1970. The cognitive basis for 
linguistic structures. In J. R. Hayes, editor, 
Cognition and the Growth of Language, Wiley, 
New York. 
Buszkowski, Wojciech. 1986. Completeness 
results for Lambek syntactic calculus. 
Zeitschrift f~r mathematische Logik und 
Grundlage der Mathematik, 32:13-28. 
Carpenter, Bob. 1998. Type-Logical Semantics. 
MIT Press, Cambridge, MA. 
Chomsky, Noam. 1965. Aspects of the Theory 
of Syntax. MIT Press, Cambridge, MA. 
Danos, Vincent and Laurent Regnier. 1989. 
The Structure of Multiplicatives. Archive 
for Mathematical Logic, 28:181-203. 
Dogen, Kosta. 1992. A brief survey of 
frames for the Lambek calculus. Zeitschrift 
fiir mathematische Logik und Grundlage der 
Mathematik, 38:179-187. 
Gibson, E. and J. Thomas. 1996. The 
processing complexity of English 
center-embedded and self-embedded 
structures. Proceedings of NELS, University 
of Massachusetts, Amherst. 
Girard, Jean-Yves. 1987. Linear Logic. 
Theoretical Computer Science, 50:1-102. 
de Groote, Philippe and Christian Retort. 
1996. On the Semantic Readings of 
Proof-Nets. In Geert-Jan Kruijff, Glyn 
Morrill and Richard. T. Oehrle, editors, 
Proceedings of Formal Grammar 1996, 
Prague, pages 57-70. 
Hendriks, Herman. 1993. Studied 
Flexibility: Categories and Types in 
Syntax and Semantics. D.Phil. 
dissertation, Universiteit van Amsterdam, 
Amsterdam. 
Hepple, Mark. 1990. Normal form theorem 
proving for the Lambek calculus. In 
H. Karlgren, editor, Proceedings of the 13th 
International Conference on Computational 
Linguistics (COLING 1990), pages 173-178, 
Stockholm. 
Hepple, Mark and Glyn Morrill. 1989. 
Parsing and Derivational Equivalence. 
Proceedings of the Fourth Conference of the 
European Chapter of the Association for 
Computational Linguistics, pages 10-18, 
Manchester. 
Johnson, Mark. 1998. Proof nets and the 
complexity of processing center 
embedded constructions, Journal of Logic, 
Language and Information, 7(4):433-447. 
Kimball, John. 1973. Seven principles of 
surface structure parsing in natural 
. Cognition, 2:15-47. 
KOnig, Esther. 1989. Parsing as natural 
deduction. Proceedings of the 27th Annual 
Meeting, pages 272-279, Vancouver. 
Association for Computational 
Linguistics. 
Lamarche, Frangois and Christian Retor6. 
1996. Proof nets for the Lambek 
calculus--an overview. In V. Michele 
Abrusci and Claudia Casadio, editors, 
Proofs and Linguistic Categories: Applications 
of Logic to the Analysis and Implementation of 
Natural Language. CLUEB, Bologna, 
pages 241-262. 
Lambek, Joachim. 1958. The mathematics of 
sentence structure. American Mathematical 
Monthly, 65:154-170. 
Lecomte, Alain and Christian Retor4. 1995. 
An alternative categorial grammar. Glyn 
Morrill and Richard. T. Oehrle, editors, 
Proceedings of Formal Grammar 1995, 
Barcelona, pages 181-196. 
Moortgat, Michael. 1988. Categorial 
Investigations: Logical and Linguistic Aspects 
of the Lambek Calculus. Foris, Dordrecht. 
Moortgat, Michael. 1997. Categorial Type 
Logics. In Johan van Benthem and Alice 
ter Meulen, editors, Handbook of Logic and 
Language. Elsevier, Amsterdam, 
pages 93-177. 
Morrill, Glyn. 1994. Type Logical Grammar: 
Categorial Logic of Signs. Kluwer Academic 
Publishers, Dordrecht. 
Morrill, Glyn. 1999. Geometry of 
Lexico-Syntactic Interaction. Proceedings of 
the Ninth Conference of the European Chapter 
of the Association for Computational 
Linguistics, Bergen, pages 61-70. 
Pentus, Mati. 1994. Language completeness 
of the Lambek calculus. Proceedings of the 
Eighth Annual IEEE Symposium on Logic in 
Computer Science, Paris, pages 487-496. 
Roorda, Dirk. 1991. Resource Logics; 
Proof-Theoretical Investigations. D.Phil. 
dissertation, Universiteit van Amsterdam, 
Amsterdam. 
Steedman, Mark J. 1997. Surface Structure and 
Interpretation. MIT Press, Cambridge, MA. 
Zielonka, Wojciech. 1981. Axiomatizability 
of Ajdukiewicz-Lambek calculus by 
means of cancellation scheme. Zeitschrift 
fi, ir mathematische Logik und Grundlage der 
Mathematik, 27:215-224. 
