Higher-Order Coloured Unification and Natural Language 
Semantics 
Claire Gardent 
Computational Linguistics 
Universit£t des Saarlandes 
D-Saarbriicken 
claire@coil, uni-sb, de 
Michael Kohlhase 
Computer Science 
Universit~t des Saarlandes 
D-Saarbriicken 
kohlhase¢cs, uni-sb, de 
Abstract 
In this paper, we show that Higher-Order 
Coloured Unification - a form of unification 
developed for automated theorem proving 
- provides a general theory for modeling 
the interface between the interpretation 
process and other sources of linguistic, non 
semantic information. In particular, it pro- 
vides the general theory for the Primary 
Occurrence Restriction which (Dalrymple 
et al., 1991)'s analysis called for. 
1 Introduction 
It is well known that Higher-Order Unification 
(HOU) can be used to construct the semantics of 
Natural Language: (Dalrymple et al., 1991) - hence- 
forth, DSP - show that it allows a treatment of VP- 
Ellipsis which successfully captures the interaction 
of VPE with quantification and nominal anaphora; 
(Pulman, 1995; Gardent and Kohlhase, 1996) use 
HOU to model the interpretation of focus and its 
interaction with focus sensitive operators, adverbial 
quantifiers and second occurrence expressions; (Gar- 
dent et al., 1996) shows that HOU yields a sim- 
ple but precise treatment of corrections; Finally, 
(Pinkal, 1995) uses linear HOU to reconstruct under- 
specified semantic representations. 
However, it is also well known that the HOU 
approach to NL semantics systematically over- 
generates and that some general theory of the in- 
terface between the interpretation process and other 
sources of linguistic information is needed in order 
to avoid this. 
In their treatment of VP-ellipsis, DSP introduce 
an informal restriction to avoid over-generation: the 
Primary Occurrence Restriction (POR). Although 
this restriction is intuitive and linguistically well- 
motivated, it does not provide a general theoretical 
framework for extra-semantic constraints. 
In this paper, we argue that Higher-Order 
Coloured Unification (HOCU, (cf. sections 3,6), a 
restricted form of HOU developed independently for 
theorem proving, provides the needed general frame- 
work. We start out by showing that the HOCU 
approach allows for a precise and intuitive model- 
ing of DSP's Primary Occurrence Restriction (cf. 
section 3.1). We then show that the POR can be 
extended to capture linguistic restrictions on other 
phenomena (focus, second occurrence expressions 
and adverbial quantification) provided that the no- 
tion of primary occurrence is suitably adjusted (cf. 
section 4). Obviously a treatment of the interplay of 
these phenomena and their related notion of primary 
occurrence is only feasible given a precise and well- 
understood theoretical framework. We illustrate this 
by an example in section 4.4. Finally, we illustrate 
the generality of the HOCU framework by using it 
to encode a completely different constraint, namely 
Kratzer's binding principle (cf. section 5). 
2 Higher-Order Unification and NL 
semantics 
The basic idea underlying the use of HOU for NL 
semantics is very simple: the typed A-calculus is 
used as a semantic representation language while se- 
mantically under-specified elements (e.g. anaphors 
and ellipses) are represented by free variables whose 
value is determined by solving higher-order equa- 
tions. For instance, the discourse (la) has (lb) as 
a semantic representation where the value of R is 
given by equation (lc) with solutions (ld) and (le). 
(1) a. Dan likes golf. Peter does too. 
b. like(dan, golf)AR(peter) 
c. like(dan,golf) = R(dan) 
d. R = Ax. like(x, golf) 
e. R = Ax. like(dan,golf) 
The process of solving such equations is tradition- 
ally called unification and can be stated as follows: 
given two terms M and N, find a substitution of 
terms for free variables that will make M and N 
equal. For first order logic, this problem is decidable 
and the set of solutions can be represented by a sin- 
gle most general unifier. For the typed A-calculus, 
the problem is undecidable, but there is an algorithm 
which - given a solvable equation - will enumerate 
a complete set of solutions for this equation (Huet, 
1975). 
Note that in (1), unification yields a linguistically 
valid solution (ld) but also an invalid one: (le). 
To remedy this shortcoming, DSP propose an in- 
formal restriction, the Primary Occurrence Re- 
striction: 
In what follows, we present a unification framework 
which solves both of these problems. 
3 Higher-Order Coloured 
Unification (HOCU) 
There is a restricted form of HOU which allows for 
a natural modeling of DSP's Primary Occurrence 
Restriction: Higher-Order Coloured Unification de- 
veloped independently for theorem proving (Hutter 
and Kohlhase, 1995). This framework uses a variant 
of the simply typed A-calculus where symbol occur- 
rences can be annotated with so-called colours and 
substitutions must obey the following constraint: 
Given a labeling of occurrences as either 
primary or secondary, the POR excludes 
of the set of linguistically valid solutions, 
any solution which contains a primary oc- 
currence. 
For any colour constant c and any 
c-coloured variable V~, a well-formed 
coloured substitution must assign to Vc a c- 
monochrome term i.e., a term whose sym- 
bols are c-coloured. 
Here, a primary occurrence is an occurrence that 
is directly associated with a source parallel element. 
Neither the notion of direct association, nor that of 
parallelism is given a formal definition; but given an 
intuitive understanding of these notions, a source 
parallel element is an element of the source (i.e. 
antecedent) clause which has a parallel counterpart 
in the target (i.e. elliptic or anaphoric) clause. 
To see how this works, consider example (1) again. 
In this case, dan is taken to be a primary occur- 
rence because it represents a source parallel element 
which is neither anaphoric nor controlled i.e. it is 
directly associated with a source parallel element. 
Given this, equation (lc) becomes (2a) with solu- 
tions (2b) and (2c) (primary occurrences are under- 
lined). Since (2c) contains a primary occurrence, it 
is ruled out by the POR and is thus excluded from 
the set of linguistically valid solutions. 
(2) a. like(dan, golf)=R(dan) 
b. R = Ax.like(x, golf) 
c. R = Ax.like(dan, golf) 
Although the intuitions underlying the POR are 
clear, two main objections can be raised. First, the 
restriction is informal and as such provides no good 
basis for a mathematical and computational evalua- 
tion. As DSP themselves note, a general theory for 
the POR is called for. Second, their method is a 
generate-and-test method: all logically valid solu- 
tions are generated before those solutions that vio- 
late the POR and are linguistically invalid are elimi- 
nated. While this is sufficient for a theoretical anal- 
ysis, for actual computation it would be preferable 
never to produce these solutions in the first place. 
3.1 Modeling the Primary Occurrence 
Restriction 
Given this coloured framework, the POR is directly 
modelled as follows: Primary occurrences are pe- 
coloured whilst free variables are -~pe-coloured. For 
the moment we will just consider the colours pe (pri- 
mary for ellipsis) and ~pe (secondary for ellipsis) as 
distinct basic colours to keep the presentation sim- 
ple. Only for the analysis of the interaction of e.g. 
ellipsis with focus phenomena (cf. section 4.4) do we 
need a more elaborate formalization, which we will 
discuss there. 
Given the above restriction for well-formed 
coloured substitutions, such a colouring ensures that 
any solution containing a primary occurrence is 
ruled out: free variables are -~pe-coloured and must 
be assigned a -~pe-monochrome term. Hence no sub- 
stitution will ever contain a primary occurrence (i.e. 
a pe-coloured symbol). For instance, discourse (la) 
above is assigned the semantic representation (3a) 
and the equation (3b) with unique solution (3c). In 
contrast, (3d) is not a possible solution since it as- 
signs to an -~pe-coloured variable, a term containing 
a pe-coloured symbol i.e. a term that is not -~pe- 
monochrome. 
(3) a. like(danpe,gol f) A R~pe(peter) 
b. like(danpe, golf)= R~pe(danpe) 
c. R~pe = Ax.like(x, golf) 
d. R~pe = Ax.like(danpe,gOl f) 
3.2 HOCU theory 
To be more formal, we presuppose a finite set 
g = {a, b, c, pe, -~pe,...) of colour constants and a 
2 
countably infinite supply ~ -- {A, B,...} of colour 
variables. 
As usual in A-calculus, the set wff of well- 
formed formulae consists of (coloured 1) con- 
stants ca,runs~,runsA,..., (possibly uncoloured) 
variables x, xa,yb,... (function) applications of 
the form MN and A-abstractions of the form 
Ax.M. Note that only variables without colours 
can be abstracted over. We call a formula M c- 
monochrome, if all symbols in M are bound or 
tagged with c. 
We will need the so-called colour erasure IMI of 
M, i.e. the formula obtained from M by erasing all 
colour annotations in M. We will also use various 
elementary concepts of the A-calculus, such as free 
and bound occurrences of variables or substitutions 
without defining them explicitly here. In particular 
we assume that free variables are coloured in all for- 
mulae occuring. We will denote the substitution of 
a term N for all free occurrences of x in M with 
\[N/x\]M. 
It is crucial for our system that colours annotate 
symbol occurrences (i.e. colours are not sorts!), in 
particular, it is intended that different occurrences 
of symbols carry different colours (e.g. f(xb, Xa)) 
and that symbols that carry different colours are 
treated differently. This observation leads to the no- 
tion of coloured substitutions, that takes the colour 
information of formulae into account. In contrast 
to traditional (uncoloured) substitutions, a coloured 
substitution a is a pair (at,at), where the term 
substitution a t maps coloured variables (i.e. the 
pair xc of a variable x and the colour c) to formulae 
of the appropriate type and the colour substitu- 
tion a c maps colour variables to colours. In order to 
be legal (a g-substitution) such a mapping a must 
obey the following constraints: 
• If a and b are different colours, then \[a(xa)\[ = 
\[a(xb)\[, i.e. the colour erasures have to be equal. 
• If c E C is a colour constant, then a(x¢) is c- 
monochrome. 
The first condition ensures that the colour erasure 
of a C-substitution is a well-defined classical substi- 
tution of the simply typed A-calculus. The second 
condition formalizes the fact that free variables with 
constant colours stand for monochrome subformu- 
lae, whereas colour variables do not constrain the 
substitutions. This is exactly the trait, that we will 
exploit in our analysis. 
1Colours axe indicated by subscripts labeling term 
occurrences; whenever colours axe irrelevant, we simply 
omit them. 
Note that/37/-reduction in the coloured A-calculus 
is just the classical notion, since the bound vari- 
ables do not carry colour information. Thus we 
have all the known theoretical results, such as the 
fact that/~/-reduction always terminates producing 
unique normal forms and that /3T/-equality can be 
tested by reducing to normal form and comparing 
for syntactic equality. This gives us a decidable test 
for validity of an equation. 
In contrast to this, higher-order unification tests 
for satisfiability by finding a substitution a that 
makes a given equation M = N valid (a(M) =~ 
a(N)), even if the original equation is not (M ~Z, 
N). In the coloured A-calculus the space of (se- 
mantic) solutions is further constrained by requiring 
the solutions to be g-substitutions. Such a substi- 
tution is called a C-unifier of M and N. In par- 
ticular, C-unification will only succeed if compara- 
ble formulae have unifiable colours. For instance, 
introa (Pa, jb, Xa) unifies with introa (Ya, jA, Sa) but 
not with introa (Pa, ja, sa) because of the colour clash 
on j. 
It is well-known, that in first-order logic (and in 
certain related forms of feature structures) there 
is always a most general unifier for any equation 
that is solvable at all. This is not the case for 
higher-order (coloured) unification, where variables 
can range over functions, instead of only individu- 
als. Fortunately, in our case we are not interested 
in general unification, but we can use the fact that 
our formulae belong to very restricted syntactic sub- 
classes, for which much better results are known. In 
particular, the fact that free variables only occur on 
the left hand side of our equations reduces the prob- 
lem of finding solutions to higher-order matching, 
of which decidability has been proven for the sub- 
class of third-order formulae (Dowek, 1992) and is 
conjectured for the general case. This class, (intu- 
itively allowing only nesting functions as arguments 
up to depth two) covers all of our examples in this 
paper. For a discussion of other subclasses of formu- 
lae, where higher-order unification is computation- 
ally feasible see (Prehofer, 1994). 
3 
Some of the equations in the examples have multi- 
ple most general solutions, and indeed this multiplic- 
ity corresponds to the possibility of multiple differ- 
ent interpretations of the focus constructions. The 
role of colours in this is to restrict the logically pos- 
sible solutions to those that are linguistically sound. 
4 Linguistic Applications of the 
POR 
In section 3.1, we have seen that HOCU allowed for 
a simple theoretical rendering of DSP's Primary Oc- 
currence Restriction. But isn't this restriction fairly 
idiosyncratic? In this section, we show that the re- 
striction which was originally proposed by DSP to 
model VP-ellipsis, is in fact a very general constraint 
which far from being idiosyncratic, applies to many 
different phenomena. In particular, we show that it 
is necessary for an adequate analysis of focus, second 
occurrence expressions and adverbial quantification. 
Furthermore, we will see that what counts as a 
primary occurrence differs from one phenomenon to 
the other (for instance, an occurrence directly asso- 
ciated with focus counts as primary w.r.t focus se- 
mantics but not w.r.t to VP-ellipsis interpretation). 
To account for these differences, some machinery is 
needed which turns DSP's intuitive idea into a fully- 
blown theory. Fortunately, the HOCU framework is 
just this: different colours can be used for different 
types of primary occurrences and likewise for differ- 
ent types of free variables. In what follows, we show 
how each phenomenon is dealt with. We then illus- 
trate by an example how their interaction can be 
accounted for. 
4.1 Focus 
Since (Jackendoff, 1972), it is commonly agreed that 
focus affects the semantics and pragmatics of utter- 
ances. Under this perspective, focus is taken to be 
the semantic value of a prosodically prominent ele- 
ment. Furthermore, focus is assumed to trigger the 
formation of an additional semantic value (hence- 
forth, the Focus Semantic Value or FSV) which is 
in essence the set of propositions obtained by making 
a substitution in the focus position (cf. e.g. (Kratzer, 
1991)). For instance, the FSV of (4a) 2 is (4b), the 
set of formulae of the form l(j,x) where x is of type 
e, and the pragmatic effect of focus is to presuppose 
that the denotation of this set is under considera- 
tion. 
(4) a. Jon likes SARAH 
b. {l(j,x) l x e wife} 
In (Gardent and Kohlhase, 1996), we show that 
HOU can successfully be used to compute the FSV 
of an utterance. More specifically, given (part of) an 
utterance U with semantic representation Sere and 
foci F1... F n, we require that the following equa- 
2Focus is indicated using upper-case. 
tion, the FSV equation, be soIved: 
Sem = Gd(F1)... (F ~) 
On the basis of the Gd value, we then define the 
FSV, written Gd, as follows: 
Definition 4.1 (Focus Semantic Value) 
Let Gd be of type ~ = ~k --~ t and n be the number of 
loci (n < k), then the Focus Semantic Value deriv- 
able from Gd, written G---d, is {Gd(tl... t n) I ti e 
wife,}. 
This yields a focus semantic value which is in 
essence Kratzer's presupposition skeleton. For in- 
stance, given (4a) above, the required equation will 
be l(j, s) = Gd(s) with two possible values for Gd: 
Ax.l(j, x) and Ax.l(j, s). Given definition (4.1), (4a) 
is then assigned two FSVs namely 
(5) a. Gd= {l(j,x) l x e Wife} 
b. G'--d = {l(j,s) l x ~ Wife} 
That is, the HOU treatment of focus over- 
generates: (5a) is an appropriate FSV, but not (5b). 
Clearly though, the POR can be used to rule out 
(5b) if we assume that occurrences that are directly 
associated with a focus are primary occurrences. To 
capture the fact that those primary occurrences are 
different from DSP's primary occurrences when deal- 
ing with ellipsis, we colour occurrences that are di- 
rectly associated with focus (rather than a source 
parallel element in the case of ellipsis) pf. Conse- 
quently, we require that the variable representing 
the FSV be -~pf coloured, that is, its value may not 
contain any pf term. Under these assumptions, the 
equation for (4a) will be (6a) which has for unique 
solution (6b). 
(6) a. l(j, Spf) = FSV~pf(Spf) 
b. FSV~pf = Ax.l(j, x) 
4 
4.2 Second Occurrence Expressions 
A second occurrence expression (SOE) is a partial or 
complete repetition of the preceding utterance and 
is characterised by a de-accenting of the repeating 
part (Bartels, 1995). For instance, (Tb) is an SOE 
whose repeating part only likes Mary is deaccented. 
(7) a. Jon only likes MARY. 
b. No, PETER only likes Mary. 
In (Gardent, 1996; Gardent et al., 1996) we show 
that SOEs are advantageously viewed as involving a 
deaccented anaphor whose semantic representation 
must unify with that of its antecedent. Formally, 
this is captured as follows. Let SSem and TSem be 
the semantic representation of the source and target 
clause respectively, and TP 1 ... TP n, SP 1 ... SP n 
be the target and source parallel elements 3, then the 
interpretation of an SOE must respect the following 
equations: 
An(Sp1,..., SP n) = SSem 
An(Tp1,..., TP '~) = TSem 
Given this proposal and some further assumptions 
about the semantics of only, the analysis of (Tb) in- 
volves the following equations: 
(8) An(j)= VP\[P e {)~x.like(x,y) l y • wife} 
A P(j) ~ P = ~x.like(x, m)\] 
An(p) = VP\[P • FSV A P(p) 
--+ P = Ax.like(x, m)\] 
Resolution of the first equation then yields two 
solutions: 
An = )~zVP\[P • {;kx.like(x,y) l Y • wife} 
A P(z) ~ P = )~x.like(x, m)\] 
An = AzVP\[P • {)~x.like(x,y) l Y • wife} 
A P(j) ~ P = )~x.like(x, m)\] 
Since An represents the semantic information 
shared by target and source clause, the second so- 
lution is clearly incorrect given that it contains in- 
formation (j) that is specific to the source clause. 
Again, the POR will rule out the incorrect solutions, 
whereby contrary to the VP-ellipsis case, all occur- 
rences that are directly associated with parallel el- 
ements (i.e. not just source parallel elements) are 
taken to be primary occurrences. The distinction is 
implemented by colouring all occurrences that are 
directly associated with parallel element ps, whereas 
the corresponding free variable (An) is coloured as 
--ps. Given these constraints, the first equation in 
(8) is reformulated as: 
An~ps(jps) = VP\[P • {)~x.like(x,y) l Y • wife} 
A P(Jps) --+ P = Ax.like(x, m)\] 
with the unique well-coloured solution 
An.,s = )~z.VP\[P • {Ax.like(x,y) l y • wife} 
A P(z) --~ P = )~x.like(x, m)\] 
4.3 Adverbial quantification 
Finally, let us briefly examine some cases of adver- 
bial quantification. Consider the following example 
from (von Fintel, 1995): 
Tom always takes SUE to Al's mother. 
Yes, and he always takes Sue to JO's mother. 
In (Gardent and Kohlhase, 1996), we suggest that 
such cases are SOEs, and thus can be treated as 
involving a deaccented anaphor (in this case, the 
anaphor he always takes Sue to _'s mother). Given 
some standard assumptions about the semantics of 
3As in DSP, the identification of parallel elements is 
taken as given. 
5 
always, the equations constraining the interpretation 
An of this anaphor are: 
An(al) = always (Tom take x to al's mother) 
(Tom take Sue to al's mother) 
An(jo) = always FSV 
(Tom take Sue to Jo's mother) 
Consider the first equation. If An is the semantics 
shared by target and source clause, then the only 
possible value for An is 
)~z.always (Tom take x to z's mother) 
(Tom take Sue to z's mother) 
where both occurrences of the parallel element m 
have been abstracted over. In contrast, the following 
solutions for An are incorrect. 
Az.always (Tom take x to al's mother) 
(Tom 
)~z.always (Tom 
(Tom 
Az.always (Tom 
take Sue to z's mother) 
take x to al's mother) 
take Sue to al's mother) 
take x to z's mother.) 
(Tom take Sue to al's mother) 
Once again, we see that the POR is a necessary 
restriction: by labeling as primary, all occurrences 
representing a parallel element, it can be ensured 
that only the first solution is generated. 
4.4 Interaction of constraints 
Perhaps the most convincing way of showing the 
need for a theory of colours (rather than just an in- 
formal constraint) is by looking at the interaction of 
constraints between various phenomena. Consider 
the following discourse 
(9) a. Jon likes SARAH 
b. Peter does too 
Such a discourse presents us with a case of inter- 
action between ellipsis and focus thereby raising the 
question of how DSP' POR for ellipsis should inter- 
act with our POR for focus. 
As remarked in section 3.1, we have to interpret 
the colour -~pe as the concept of being not primary 
for ellipsis, which includes pf (primary for focus). In 
order to make this approach work formally, we have 
to extend the supply of colours by allowing boolean 
combinations of colour constants. The semantics of 
these ground colour formula is that of propositional 
logic, where -~d is taken to be equivalent to the dis- 
junction of all other colour constants. 
Consequently we have to generalize the second 
condition on C-substitutions 
• For all colour annotations d of symbols in a(xc) 
d ~ c in propositional logic. 
Thus X.d can be instantiated with any coloured 
formula that does not contain the colour d. The 
HOCU algorithm is augmented with suitable rules 
for boolean constraint satisfaction for colour equa- 
tions. 
The equations resulting from the interpretation of 
(9b) are: 
l(jpe, 8pf) ~-- R-,pe(jpe) 
R~pe(P) = FSV~pf(F) 
where the first equation determines the interpre- 
tation of the ellipsis whereas the second fixes the 
value of the FSV. Resolution of the first equation 
yields the value Ax.l(x, Spf) for R~pe. As required, 
no other solution is possible given the colour con- 
stralnts; in particular Ax.l(jpe, Spf) is not a valid so- 
lution. The value of R~pe(jpe) is now l(Ppe, 8pf) SO 
that the second equation is4: 
l(p, Spf) = FSV~pf(F) 
Under the indicated colour constraints, three so- 
lutions are possible: 
FSV~pf = Ax.l(p, x), F = spf 
FSV~pf = AO.O(p), F = Ax.l(x, Spf) 
FSV~pf = ~X.X, F = l(p, spf) 
The first solution yields a narrow focus read- 
ing (only SARAH is in focus) whereas the second 
and the third yield wide focus interpretations corre- 
sponding to a VP and an S focus respectively. That 
is, not only do colours allow us to correctly capture 
the interaction of the two PORs restricting the in- 
terpretation of ellipsis of focus, they also permit a 
natural modeling of focus projection (cf. (Jackend- 
off, 1972)). 
5 Another constraint 
An additional argument in favour of a general the- 
ory of colours lies in the fact that constraints that 
are distinct from the POR need to be encoded to 
prevent HOU analyses from over-generating. In this 
section, we present one such constraint (the so-called 
weak-crossover constraint) and show how it can be 
implemented within the HOCU framework. 
In essence, the main function of the POR is to en- 
sure that some occurrence occuring in an equation 
appears as a bound variable in the term assigned 
by substitution to the free variable occurring in this 
equation. However, there are cases where the dual 
4Note that this equation falls out of our formal sys- 
tem in that it is untyped and thus cannot be solved by 
the algorithm described in section 6 (as the solutions will 
show, we have to allow for FSV and F to have different 
types). However, it seems to be a routine exercise to aug- 
ment HOU algorithms that can cope with type variables 
like (Hustadt, 1991; Dougherty, 1993) with the colour 
methods from (Hutter and Kohlhase, 1995). 
6 
constraint must be enforced: a term occurrence ap- 
pearing in an equation must appear unchanged in 
the term assigned by substitution to the free vari- 
able occurring in this equation. The following ex- 
ample illustrates this. 
(Chomsky, 1976) observes that focused NPs 
pattern with quantified and wh-NPs with re- 
spect to pronominal anaphora: when the quanti- 
fied/wh/focused NP precedes and c-commands the 
pronoun, this pronoun yields an ambiguity between 
a co-referential and a bound-variable reading. This 
is illustrated in example 
(10) We only expected HIMi to claim 
that he~ was brilliant 
where the presence of the pronoun hei gives rise 
to two possible FSVs s 
FSV = {Ax.ex(x,y,i) l Y E wife} 
FSV = {Ax.ex(x,y,y) \[ y E Wife} 
thus allowing two different readings: the corefen- 
tial or strict reading 
VP\[P E {Ax.ex(x,y,i) I Y E Wife} 
A P(we) --+ P = Ax.ex(x, i, i)\] 
and the bound-variable or sloppy reading. 
VP\[P E {Ax.ex(x,y,y)) \[ y E wife} 
^ P(we) ~ P = Ax.ex(x, i, i))\] 
In contrast, if the quantified/wh/focused NP does 
not precede and c-command the pronoun, as in 
(11) We only expected himi to claim 
that HEi was brilliant 
there is no ambiguity and the pronoun can only 
give rise to a co-referential interpretation. For in- 
stance, given (11) only one reading arises 
VP\[P E {Ax.ex(x,i,y) l Y E Wife} 
A P(we) ~ P = Ax.ex(x, i, i)\] 
where the FSV is {Ax.ex(x,i,y) l Y E wife}. 
To capture this data, Government and Binding 
analyses postulate first, that the antecedent is raised 
by quantifier raising and second, that pronouns that 
are c-commanded and preceded by their antecedent 
are represented either as a A-bound variable or as 
a constant whereas other pronouns can only be rep- 
resented by a constant (cf. e.g. (Kratzer, 1991)'s 
binding principle). Using HOCU, we can model this 
restriction directly. As before, the focus term is pf- 
and the FSV variable -~pf-coloured. Furthermore, 
we assume that pronouns that are preceded and c- 
commanded by a quantified/wh/focused antecedent 
are variable coloured whereas other pronouns are 
-~pf-coloured. Finally, all other terms are taken to 
5We abbreviate exp( x, cl(y, blt( i) ) ) to ex( x, y, i) to in- 
crease legibility. 
be --pf-coloured. Given these assumptions, the rep- 
resentation for (10) is ex~o~(we~pf,ipf ,iA) and the 
corresponding FSV equation 
R~pf(ipf) -- )~x.eX~pf (x, ipf, in) 
has two possible solutions 
R~0f = )~y.)~x.ex~pf (x, y, i~0f) 
R~of = )~y.)~x.ex~of(x , y, x) 
In contrast, the representation for (11) is 
ex-.pf(We~of, i~0f, ipf) and the equation is 
R-~pf(ipf) = )~x.ex~pf(X, i~of , /0f ) 
with only one well-coloured solution 
R~0f = )~y.Ax.ex~of ( x , i~of , Y) 
Importantly, given the indicated colour con- 
straints, no other solutions are admissible. Intu- 
itively, there are two reasons for this. First, the 
definition of coloured substitutions ensures that the 
term assigned to R~0f is -~pf-monochrome. In par- 
ticular, this forces any occurrences of/of to appear 
as a bound variable in the value assigned to R~pf 
whereas in can appear either as i~0f (a colour vari- 
able unifies with any colour constant) or as a bound 
variable - this in effect models the sloppy/strict am- 
biguity. Second, a colour constant only unifies with 
itself. This in effect rules out the bound variable 
reading in (11): if the i~0f occurrence were to be- 
come a bound variable, the value of R~of would 
then Ay.)~x.ex~of(x, y, y) . But then by ~-reduction, 
R~of(ipf ) would be )~x.ex~of(x, iof,iof ) which does 
not unify with the right hand side of the original 
equation i.e ~x.ex.of(x , i-0f, i0f). 
For a more formal account of how the unifiers are 
calculated see section 6.1. 
6 Calculating Coloured Unifiers 
Since the HOCU is the principal computational de- 
vice of the analysis in this paper, we will now try 
to give an intuition for the functioning of the algo- 
rithm. For a formal account including all details and 
proofs see (Hutter and Kohlhase, 1995). 
Just as in the case of unification for first-order 
terms, the algorithm is a process of recursive decom- 
position and variable elimination that transform sets 
of equations into solved forms. Since C-substitutions 
have two parts, a term- and a colour part, we need 
two kinds (M =t N for term equations and c =c d 
for colour equations). Sets g of equations in solved 
form (i.e. where all equations are of the form x = M 
such that the variable x does not occur anywhere else 
in M or g) have a unique most general C-unifier a~ 
that also C-unifies the initial equation. 
There are several rules that decompose the syntac- 
tic structure of formulae, we will only present two of 
them. The rule for abstractions transforms equa- 
tions of the form )~x.A =t )~y.B to \[c/x\]A =t \[c/y\]B, 
and Ax.A =t B to \[c/x\]A =t Bc where c is a new 
constant, which may not appear in any solution. The 
rule for applications decomposes ha(s1,... ,s n) =t 
hb(tl,...,t '~) to the set {a =c b, sl =t tl,...,s,~ =t 
tn}, provided that h is a constant. Furthermore 
equations are kept in 13~/-normal form. 
The variable elimination process for colour vari- 
ables is very simple, it allows to transform a set 
g U {A =c d} of equations to \[d/A\]g U {A =c d}, 
making the equation {A =c d} solved in the result. 
For the formula case, elimination is not that simple, 
since we have to ensure that la(XA)l = la(xs)l to 
obtain a C-substitution a. Thus we cannot simply 
transform a set gU{Xd =t M} into \[M/Xd\]EU{Xd __t 
M}, since this would (incorrectly) solve the equa- 
tions {Xc = fc,Xd = gd}. The correct variable 
elimination rule transforms $ U {Xd =t M} into 
a(g) U {Xd =1 M, xc, = M1,...,Xc~ =t Mn}, where 
ci are all colours of the variable x occurring in M and 
g, the M i are appropriately coloured variants (same 
colour erasure) of M, and a is the g-substitution 
that eliminates all occurrences of x from g. 
Due to the presence of function variables, sys- 
tematic application of these rules can terminate 
with equations of the form xc(sl,...,s n) =t 
hd(tl,...,tm). Such equations can neither be fur- 
ther decomposed, since this would loose unifiers (if 
G and F are variables, then Ga = Fb as a solution 
Ax.c for F and G, but {F = G,a = b} is unsolv- 
able), nor can the right hand side be substituted for 
x as in a variable elimination rule, since the types 
would clash. Let us consider the uncoloured equa- 
tion x(a) ~t a which has the solutions (Az.a) and 
(Az.z) for x. 
The standard solution for finding a complete set 
of solutions in this so-called flex/rigid situation is 
to substitute a term for x that will enable decompo- 
sition to be applicable afterwards. It turns out that 
for finding all g-unifiers it is sufficient to bind x to 
terms of the same type as x (otherwise the unifier 
would be ill-typed) and compatible colour (other- 
wise the unifier would not be a C-substitution) that 
either 
• have the same head as the right hand side; the 
so-called imitation solution (.kz.a in our exam- 
ple) or 
• where the head is a bound variable that enables 
the head of one of the arguments of x to become 
head; the so-called projection binding ()~z.z). 
In order to get a better understanding of the situ- 
ation let us reconsider our example using colours. 
z(a¢) -- ad. For the imitation solution (~z.ad) we 
"imitate" the right hand side, so the colour on a 
must be d. For the projection solution we instantiate 
($z.z) for x and obtain ()kz.z)ac, which f~-reduces to 
ac. We see that this "lifts" the constant ac from the 
argument position to the top. Incidentally, the pro- 
jection is only a C-unifier of our coloured example, 
if c and d axe identical. 
Fortunately, the choice of instantiations can be 
further restricted to the most general terms in the 
categories above• If Xc has type f~n --+ c~ and hd has 
type ~ -~ a, then these so-called general bind- 
ings have the following form: 
G h = ~kzal... z a".hd(H~l (-5),..., Hem (-5)) 
where the H i are new variables of type f)-~ ~ Vi and 
the ei are either distinct colour variables (if c E CI)) 
or ei = d = c (ifc E C). If his one of the bound 
variables z ~' , then ~h is called an imitation bind- 
ing, and else, (h is a constant or a free variable), a 
projection binding• 
The general rule for flex/rigid equations trans- 
forms {Xc(Sl,...,s n) =t hd(tl,...,tm)} into 
{Xc(S 1 .... , s n) =t hal(t1,..., tin), Xc =t ~h}, which 
in essence only fixes a particular binding for the 
head variable Xc. It turns out (for details and proofs 
see (Hutter and Kohlhase, 1995)) that these general 
bindings suffice to solve all flex/rigid situations, pos- 
sibly at the cost of creating new flex/rigid situations 
after elimination of the variable Xc and decompo- 
sition of the changed equations (the elimination of 
x changes xc(sl,..., s n) to ~h(sl, ..., s n) which has 
head h). 
6.1 Example 
To fortify our intuition on calculating higher-order 
coloured unifiers let us reconsider examples (10) and 
(11) with the equations 
R~pf(ipf) __t ~x.ex~pf(X, ipf, iA) 
R~pf(ipf) =t Ax.ex~pf(X, i-~pf, ipf) 
We will develop the derivation of the solutions for 
the first equations (10) and point out the differences 
for the second (11). As a first step, the first equation 
is decomposed to 
R~pf(ipf, c) :t ex~pf(C, ipf, iA) 
where c is a new constant• Since R~pf is a vari- 
able, we are in a flex/rigid situation and have the 
possibilities of projection and imitation. The pro- 
jection bindings Axy.x and )~xy.y for R~pf would 
lead us to the equations ipf =t eX~pf(C, ipf,iA) and 
c =t eX~pf (c, ipf, iA), which are obviously unsolvable, 
since the head constants ipf (and c resp.) and eX~pf 
8 
clash 6. So we can only bind R~pf to the imitation 
binding ~kyx•ex~pf(H~pf(y, x), H~2pf (y, x), H 3 (y, x)). 
Now, we can directly eliminate the variable R~pf, 
since there are no other variants. The resulting equa- 
tion 
eX~pf(Hlpf(ipf, c), H2pf (ipf, c), g 3 (ipf, c)) 
=t eX~pf (c, ipf, iA) 
can be decomposed to the equations 
(17) Hlpf(ipf,C) __t c 
H~pf(ipf, c) =t ipf 
g3pf(/pf, C) __--t iA 
Let us first look at the first equation; in this 
flex/rigid situation, only the projection binding 
)kzw.w can be applied, since the imitation binding 
Azw.c contains the forbidden constant c and the 
other projection leads to a clash. This solves the 
equation, since (Azw.w)(ipf,c) j3-reduces to c, giv- 
ing the trivial equation c __t c which can be deleted 
by the decomposition rules• 
Similarly, in the second equation, the projection 
binding Azw.z for H 2 solves the equation, while the 
second projection clashes and the imitation binding 
)kzw.ipf is not -~pf-monochrome. Thus we are left 
with the third equation, where both imitation and 
projection bindings yield legal solutions: 
• The imitation binding for H3pf is )kzw.i~pf, and 
not Azw.iA, as one is tempted to believe, since 
it has to be -~pf-monochrome. Thus we are left 
with i~pf =t iA, which can (uniquely) be solved 
by the colour substitution \[-~pf/A\]. 
• If we bind H 3 to Azw.z, then we are left with ~pf 
Zpf. _-t iA, which can (uniquely) be solved by the 
colour substitution \[pf/A\]. 
If we collect all instantiations, we arrive at exactly 
the two possible solutions for R~pf in the original 
equations, which we had claimed in section 5: 
R~pf = ~kyx.ex~pf(X, y, i~pf) 
R~pf = )kyx•ex~pf(X, y, x) 
Obviously both of them solve the equation and 
furthermore, none is more general than the other, 
since i~pf cannot be inserted for the variable x in 
the second unifier (which would make it more general 
than the first), since x is bound• 
In the case of (11) the equations corresponding 
1 __t 2 " __t - and to (17) are H.~pf(e, ipf) - e, H~pf(e, Zpf) - ?,~pf 
H3pf(ipf) __t ipf. Given the discussion above, it is im- 
mediate to see that H 1 has to be instantiated with -~pf 
the projection binding ~kzw.w, H 2 with the imitation 
6For (11) we have the same situation• Here the cor- 
• t responding equation is tpf -- ex~pf(C, i~pf, ipf). 
binding Azw.i~of, since the projection binding leads 
to a colour clash (i~f =t ipf) and finally H~pf has to 
be bound to the projection binding Azw.z, since the 
imitation binding Azw.ipf is not -~pf-monochrome. 
Collecting the bindings, we arrive at the unique so- 
lution R~f = Ayx.ex~pf(x, i~pf, x). 
7 Conclusion 
Higher-Order Unification has been shown to be a 
powerful tool for constructing the interpretation of 
NL. In this paper, we have argued that Higher- 
Order Coloured Unification allows a precise speci- 
fication of the interface between semantic interpre- 
tation and other sources of linguistic information, 
thus preventing over-generation. We have substan- 
tiated this claim by specifying the linguistic, extra- 
semantic constraints regulating the interpretation of 
VP-ellipsis, focus, SOEs, adverbial quantification 
and pronouns whose antecedent is a focused NP. 
Other phenomena for which the HOCU approach 
seems particularly promising are phenomena in 
which the semantic interpretation process is obvi- 
ously constrained by the other sources of linguistic 
information. In particular, it would be interesting to 
see whether coloured unification can appropriately 
model the complex interaction of constraints govern- 
ing the interpretation and acceptability of gapping 
on the one hand, and sloppy/strict ambiguity on the 
other. 
Another interesting research direction would be 
the development and implementation of a monos- 
tratal grammar for anaphors whose interpretation 
are determined by coloured unification. Colours 
are tags which decorate a semantic representation 
thereby constraining the unification process; on the 
other hand, there are also the reflex of linguistic, 
non-semantic (e.g. syntactic or prosodic) informa- 
tion. A full grammar implementation would make 
this connection more precise. 
8 Acknowledgements 
The work reported in this paper was funded by the 
Deutsche Forschungsgemeinschaft (DFG) in Sonder- 
forschungsbereich SFB-378, Project C2 (LISA). 

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