Logical Form Equivalence:
the Case of Referring Expressions Generation
Kees van Deemter
ITRI
University of Brighton
Brighton BN2 4GJ
United Kingdom
Kees.van.Deemter@itri.brighton.ac.uk
Magn´us M. Halld´orsson
Computer Science Dept.
University of Iceland,
Taeknigardur, 107 Reykjavik, Iceland
and Iceland Genomics Corp., Reykjavik
mmh@hi.is
Abstract
We examine the principle of co-
extensivity which underlies current al-
gorithms for the generation of referring
expressions, and investigate to what ex-
tent the principle allows these algo-
rithms to be generalized. The discus-
sion focusses on the generation of com-
plex Boolean descriptions and sentence
aggregation.
1 Logic in GRE
A key question regarding the foundations of Nat-
ural Language Generation (NLG) is the problem
of logical form equivalence (Appelt 1987). The
problem goes as follows. NLG systems take se-
mantic expressions as input, usually formulated
in some logical language. These expressions are
governed by rules determining which of them
count as ‘equivalent’. If two expressions are
equivalent then, ideally, the NLG program should
verbalize them in the same ways. (Being equiv-
alent, the generator would not be warranted in
distinguishing between the two.) The question
is: what is the proper relation of equivalence?
Appelt argued that classical logical equivalence
(i.e., having the same truth conditions) is not a
good candidate. For example, a0a2a1 a3 is logi-
cally equivalent with a4
a3a5a1
a4
a0 , yet – so the argu-
ment goes – an NLG system should word the two
formulas differently. Shieber (1993) suggested
that some more sophisticated notion of equiva-
lence is needed, which would count fewer seman-
tic expressions as equivalent.1 In the present pa-
per, a different response to the problem is ex-
plored, which keeps the notion of equivalence
classical and prevents the generator from distin-
guishing between inputs that are logically equiva-
lent (i.e., inputs that have the same truth condi-
tions). Pragmatic constraints determine which
of all the logically equivalent semantic expres-
sions is put into words by the NLG program.
Whereas this programme, which might be called
‘logic-oriented’ generation, would constitute a
fairly radical departure from current practice if
applied to all of NLG (Krahmer & van Deemter
(forthcoming); Power 2000 for related work), the
main aim of the present paper is modest: to show
that logic-oriented generation is standard prac-
tice in connection with the generation of referring
expressions (GRE). More specifically, we show
the semantics of current GRE algorithms to be
guided by a surprisingly simple principle of co-
extensivity, while their pragmatics is guided by
Gricean Brevity.
Our game plan is as follows. In section 2, we
illustrate the collaboration between Brevity and
co-extensivity, focussing on ‘simple’ referring ex-
pressions, which intersect atomic properties (e.g.,
‘dog’ and ‘black’). Section 3 proceeds by show-
ing how other algorithms use the principle to le-
gitimize the creation of more elaborate structures
involving, for example, complex Boolean combi-
nations (e.g., the union of some properties, each
of which is the intersection of some atomic prop-
1See also van Deemter (1990) where, on identical
grounds, a variant of Carnap-style intensional isomorphism
was proposed as an alternative notion of equivalence.
erties). This part of the paper will borrow from
van Deemter (2001), which focusses on compu-
tational aspects of GRE. Section 4 asks how the
principle of co-extensivity may be generalized be-
yond GRE and questions its validity.
2 Intersective reference to sets of
domain objects
The Knowledge Base (KB) forming the input to
the generator will often designate objects using
the jargon of computerized databases, which is
not always meaningful for the reader/hearer. This
is true, for example, for an artificial name (i.e.,
a database key) like ‘a0a2a1a4a3a6a5a8a7a10a9a12a11a14a13a16a15 ’ when a per-
son’s proper name is not uniquely distinguishing;
it is also true for objects (e.g., furniture, trees,
atomic particles) for which no proper names are
in common usage. In all such cases, the NLG pro-
gram has to ‘invent’ a description that enables the
hearer to identify the target object. The program
transforms the original semantic structure in the
KB into some other structure.
Let us examine simple references first. Assume
that the information used for interpreting a de-
scription is stored in a KB representing what
properties are true of each given object. In ad-
dition to these properties, whose extensions are
shared between speaker and hearer, there are
other properties, which are being conveyed from
speaker to hearer. For example, the speaker may
say ‘The white poodle is pregnant’, to convey the
new information that the referent of ‘the white
poodle’ is pregnant. GRE ‘sees’ the first, shared
KB only. We will restrict attention to the prob-
lem of determining the semantic content of a de-
scription, leaving linguistic realization aside. (Cf.
Stone and Webber 1998, Krahmer and Theune
1999, which interleave linguistic realization and
generation.) Accordingly, ‘Generation of Refer-
ring Expressions’ (GRE) will refer specifically to
content determination. We will call a GRE algo-
rithm complete if it is successful whenever an in-
dividuating description exists. Most GRE algo-
rithms are limited to individual target objects (for
an exception, Stone 2000), but we will present
ones that refer to sets of objects (Van Deemter
2000); reference to an individual a17 will equal ref-
erence to the singleton set a18a19a17a21a20 .
2.1 The Incremental Algorithm
Dale and Reiter (1995) proposed an algorithm
that takes a shared KB as its input and delivers a
set of properties which jointly identify the target.
Descriptions produced by the algorithm fullfill
the criterion of co-extensivity. According to this
principle, a description is semantically correct if
it has the target as its referent (i.e., its extension).
The authors observed that a semantically correct
description can still be unnatural, but that natural-
ness is not always easy to achieve. In particular,
the problem of finding a (‘Full Brevity’) descrip-
tion that contains the minimum number of prop-
erties is computationally intractable, and human
speakers often produce non-minimal descriptions.
Accordingly, they proposed an algorithm that ap-
proximates Full Brevity, while being of only lin-
ear complexity. The algorithm produces a finite
set a22 of properties a23a25a24a19a26a28a27a29a27a29a27a29a26a30a23a8a31 such that the inter-
section of their denotations a32a29a32a23 a24a34a33a29a33a36a35 a27a29a27a29a27 a35 a32a29a32a23 a31a37a33a29a33
equals the target set a38 , causing a22 to be a ‘dis-
tinguishing description’ of a38 . The properties in
a22 are selected one by one, and there is no back-
tracking, which is why the algorithm is called In-
cremental. As a result, some of the properties in
a22 may be logically superfluous.
For simplicity, we will focus here on properties,
without separating them into Attributes and Val-
ues (see also Reiter and Dale 2000, section 5.4.5).
Accordingly, reflecting the fact that not all prop-
erties are equally ‘preferred’, they are ordered lin-
early in a list IP, with more preferred ones preced-
ing less preferred ones. We also simplify by not
taking the special treatment of head nouns into ac-
count. Suppose a38 is the target set, and a39 is the
set of elements from which a38 is to be selected.2
The algorithm iterates through IP; for each prop-
erty, it checks whether it would rule out at least
one member of a39 that has not already been ruled
out; if so, the property is added to a22 . Members
that are ruled out are removed from a39 . The pro-
cess of expanding a22 and contracting a39 continues
until a39a41a40a41a38 ; if and when this condition is met, a22
is a distinguishing set of properties.
2We have chosen a formulation in which
a42 is a superset
of a43 , rather than a ‘contrast set’, from whose elements those
of a43 must be distinguished (Dale and Reiter 1995). The dif-
ference is purely presentational.
a0a2a1a3a5a4a7a6a8a0 is initialized to the empty set
a9
For eacha10a12a11a14a13 IP do
If a43a16a15a18a17a19a17a10a20a11a22a21a19a21a24a23 a42a26a25a15a27a17a19a17a10a12a11a28a21a29a21 a6a10a12a11 removes dis-
tractors from a42 but keeps all elements of a43a24a9
Then do
a0a30a1a3a27a0a32a31a33a6
a10a12a11a34a9
a6 Property
a10a12a11 is added to
a0
a9
a42
a1a3
a42a30a35a36a17a29a17a10 a11a21a19a21
a6 All elements outside
a17a19a17a10 a11a21a29a21 are removed from a42a37a9
If a42 a3 a43 then Returna0a38a6 Successa9
Return Failure a6 All properties in IP have been
tested, yet a42a18a25a3 a43a39a9
This algorithm, D&RPlur, constructs better and
better approximations of the target set a38 . Assum-
ing (cf. Dale and Reiter 1995) that the tests in the
body of the loop take some constant amount of
time, the worst-case running time is in the order
of a5a8a40 (i.e., a41a43a42 a5a8a40a45a44 ) where a5a46a40 is the total number
of properties.
3 Reference using Boolean descriptions
Based on co-extensivity, the algorithms discussed
construct an intersective Boolean expression (i.e.,
an expression of the form a23 a24 a35 a27a29a27a29a27 a35 a23 a31 , where
a23 a24a19a26a28a27a29a27a29a26a30a23 a31 are atomic) that has the target set as its
extension. But, intersection is not the only oper-
ation on sets. Consider a KB whose domain is a
set of dogs (a47 a26a49a48a12a26a51a50a6a26a51a52 a26
a7 ) and whose only Attributes
are TYPE and COLOUR:
TYPE: dog a18a53a47 a26a49a48a12a26a51a50a12a26a51a52 a26 a7 a20 , poodle a18a53a47 a26a49a48a10a20
COLOUR: black a18a53a47 a26a49a48a12a26a51a50a12a20 , white a18a53a52 a26
a7
a20
In this situation, D&Ra54a56a55a29a57a59a58 does not allow us to
individuate any of the dogs. In fact, however, the
KB should enable one to refer to dog a50 , since it is
the only black dog that is not a poodle:
a18a53a50a12a20 a40a60a48a62a61a63a47a20a50a65a64 a35
a0
a3 a3
a52a66a61
a7
A similar gap exists where disjunctions might be
used. For example, D&Ra54a56a55a19a57a53a58 does not make the
set of dogs that are either white or poodles refer-
able, whereas it is referable in English, e.g., ‘The
white dogs and the poodles’.
Presently, we will investigate how GRE can take
negation and disjunction into account. Section
3.1 will ask how GRE algorithms can achieve
Full Boolean Completeness; section 3.2, which
follows Van Deemter (2001), adds Brevity as a
requirement. Boolean descriptions do the same
thing that intersective descriptions do, except in
a more dramatic way: they ‘create’ even more
structure. As a result, the problem of optimizing
these structures with respect to constraints such as
Brevity becomes harder as well.
As a first step, we show how one can tell which
targets are identifiable given a set of properties
and set intersection. We calculate, for each el-
ement a52 in the domain, the ‘Satellite set’ of a52 ,
that is, the intersection of the extensions of all the
properties that are true ofa52 . Taking all extensions
from our example,
a38a67a47a66a68
a7
a61a63a61a34a69a70a68
a7a10a9
a42a63a47
a44
a40
a52
a3a45a71
a35
a0
a3a6a3
a52a72a61
a7
a35 a48a73a61a63a47a72a50a65a64 a40 a18a53a47 a26a49a48a10a20
a38a67a47a66a68
a7
a61a63a61a34a69a70a68
a7a10a9
a42a74a48
a44
a40
a52
a3a45a71
a35
a0
a3a6a3
a52a72a61
a7
a35 a48a73a61a63a47a72a50a65a64 a40 a18a53a47 a26a49a48a10a20
a38a67a47a66a68
a7
a61a63a61a34a69a70a68
a7a10a9
a42a63a50
a44
a40
a52
a3a45a71
a35 a48a73a61a63a47a20a50a75a64a2a40a41a18a53a47 a26a49a48a12a26a51a50a12a20
a38a67a47a66a68
a7
a61a63a61a34a69a70a68
a7a10a9
a42a63a52
a44
a40
a52
a3a45a71
a35a77a76a79a78 a69a70a68
a7
a40 a18a53a52 a26
a7
a20
a38a67a47a66a68
a7
a61a63a61a34a69a70a68
a7a10a9
a42
a7a45a44
a40
a52
a3a45a71
a35a77a76a79a78 a69a70a68
a7
a40 a18a53a52 a26
a7
a20
If two objects occur in the same Satellite set then
no intersective description can separate them: any
description true of one must be true of the other. It
follows, for example, that no object in the domain
is uniquely identifiable, since none of them occurs
in a Satellite set that is a singleton.
3.1 Boolean descriptions (i): generate and
simplify
Boolean completeness is fairly easy to achieve
until further constraints are added. Suppose the
task is to construct a description of a target set a38 ,
given a set IP of atomic properties, without any fur-
ther constraints. We will discuss an algorithm that
starts by calculating a generalized type of Satel-
lite sets, based on all atomic properties and their
negations.
Construction of Satellite sets:
IPa31a81a80a83a82a79a84a40 IP a85 a18 a23a87a86a67a84a14a23a8a86a89a88 IP a20
For each a52a90a88 a38 do
a38a92a91a79a84a40 a18a53a93a94a84a66a93a94a88 IPa31a81a80a83a82a95a84a96a52a77a88 a32a29a32a93 a33a29a33 a20
a38a67a47a72a68
a7
a61a63a61a97a69a98a68
a7a12a9
a42a63a52
a44
a40a100a99a102a101a46a103a105a104a59a106a66a42a34a32a29a32a93 a33a29a33
a44
First, the algorithm adds to IP the properties whose
extensions are the complements of those in IP.
Then it calculates, for each element a52 in a38 ,
a38a67a47a72a68
a7
a61a34a61a34a69a70a68
a7a10a9
a42a63a52
a44 by lining up all the properties in
IPa31 a80a98a82 that are true of a52 , then taking the intersec-
tion of their extensions. Satellite sets may be
exploited for the construction of descriptions by
forming the union of a number of expressions,
each of which is the intersection of the elements
of a38a56a91 (for some a52a90a88 a38 ).
Description By Satellite sets (DBS):
Descriptiona84a40 a18a6a38a56a91a95a84a96a52a77a88 a38 a20
Meaninga84a40 a0
a91
a103a105a104a46a42 a38a67a47a72a68
a7
a61a63a61a97a69a98a68
a7a12a9
a42a63a52
a44a44
If Meaning = a38
then Return Description
else Fail
(Note that Description is returned instead of
Meaning, since the latter is just the set a38 .) De-
scriptionis a set of sets of sets of domain ob-
jects. As is made explicit in Meaning, this third-
order set is interpreted as a union of intersections.
A Description is successful if it evaluates to
the target set a38 ; otherwise the algorithm returns
Fail. If Fail is returned, no Boolean descrip-
tion of a38 is possible:
Full Boolean Completeness: For any set a43 ,
a43 is obtainable by a sequence of boolean op-
erations on the properties in IP if and only if
a0
a106a2a1a4a3a6a5
a43a8a7a4a9a11a10a2a12a13a12a15a14a16a9a17a10a19a18
a5a21a20a23a22a11a22 equals
a43 .
Proof: The implication from right to left is ob-
vious. For the reverse direction, suppose a43 a25a3
a0
a106a2a1a4a3a6a5
a43a8a7a4a9a11a10a2a12a13a12a15a14a16a9a17a10a19a18
a5a21a20a23a22a11a22 . Then for some
a10a14a13 a43 ,
Satellitesa5 a10 a22 contains an element a10a2a24 that is not in
a43 . But a10a26a25a11a10 a24 a13 Satellites
a5
a10
a22 implies that every
set in IP must either contain both of a10 and a10a2a24 , or
neither. It follows that a43 , which contains only
one of a10a26a25a27a10 a24 , cannot be obtained by a combina-
tion of Boolean operations on the sets in IP.
DBS is computationally cheap: it has a worst-
case running time of a41a43a42
a5
a27
a0
a44 , where a5 is the num-
ber of objects in a38 , and a0 the number of atomic
properties. Rather than searching among all the
possible unions of some large set of sets, a set
a38 a40 a18
a9
a24a19a26a28a27a29a27a29a26
a9
a31 a20 is described as the union of
a5
Satellites sets, each of which equals the intersec-
tion of those (at most a28 a5 ) sets in IPa31a81a80a83a82 that contain
a9
a86 . Descriptions can make use of the Satellite sets
computed for earlier descriptions, causing a fur-
ther reduction of time. Satellites sets can even
be calculated off-line, for all the elements in the
domain, before the need for specific referring ex-
pressions has arisen.3
Unfortunately, the descriptions produced by DBS
tend to be far from brief:
a38a30a29 a40 a18a53a52
a3a45a71
a26a49a48a73a61a34a47a20a50a65a64 a26
a0
a3 a3
a52a66a61
a7
a26 a76 a78 a69a70a68
a7
a20 .
a38a67a47a66a68
a7
a61a63a61a34a69a70a68
a7a10a9
a42a63a50
a44
a40 a18a53a50a12a20
a38a92a91 a40 a18a53a52
a3a45a71
a26 a76a79a78 a69a70a68
a7
a26
a0
a3a6a3
a52a72a61
a7
a26 a48a62a61a63a47a20a50a65a64 a20 .
a38a67a47a66a68
a7
a61a63a61a34a69a70a68
a7a10a9
a42a63a52
a44
a40 a18a53a52 a26
a7
a20
a38a92a80 a40 a18a53a52
a3a45a71
a26 a76 a78 a69a70a68
a7
a26
a0
a3 a3
a52a66a61
a7
a26 a48a73a61a63a47a72a50a65a64 a20 .
a38a67a47a66a68
a7
a61a63a61a34a69a70a68
a7a10a9
a42
a7a45a44
a40 a18a53a52 a26
a7
a20
To describe the target set a38 a40 a18a53a50a12a26a51a52 a26 a7 a20 , for exam-
ple, the algorithm generates the Description
a18a6a38a30a29 a26 a38a56a91 a26 a38a92a80a28a20 . Consequently, the boolean expres-
sion generated is
a42a63a52
a3a45a71
a35 a48a73a61a63a47a20a50a75a64 a35
a0
a3 a3
a52a66a61
a7
a35 a76a79a78 a69a70a68
a7 a44
a85
a42a63a52
a3a45a71
a35a77a76a79a78 a69a70a68
a7
a35
a0
a3 a3
a52a66a61
a7
a35 a48a62a61a63a47a20a50a65a64
a44
a85
a42a63a52
a3a45a71
a35a77a76a79a78 a69a70a68
a7
a35
a0
a3 a3
a52a66a61
a7
a35 a48a62a61a63a47a20a50a65a64
a44 .
But of course, a much shorter description,a0 a3 a3 a52a66a61 a7 ,
would have sufficed. What are the prospects for
simplifying the output of DBS? Unfortunately,
perfect simplification (i.e., Full Brevity) is incom-
patible with computational tractibility. Suppose
brevity of descriptions is defined as follows: a52 a24
is less brief than a52a32a31 if either a52 a24 contains only
atomic properties while a52a8a31 contains non-atomic
properties as well, or a52 a24 contains more Boolean
operators than a52 a31 . Then the intractability of Full
Brevity for intersections of atomic properties log-
ically implies that of the new algorithm:
Proof: Suppose an algorithm, BOOL, produced
a maximally brief Boolean description when-
ever one exists. Then whenever a target set a43
can be described as an intersection of atomic
properties, BOOL(a43 ) would be a maximally
brief intersection of atomic properties, and this
is inconsistent with the intractability of Full
Brevity for intersections of atomic properties.
3Compare Bateman (1999), where a KB is compiled into
a format that brings out the commonalities between objects
before the content of a referring expression is determined.
This negative result gives rise to the question
whether Full Brevity may be approximated, per-
haps in the spirit of Reiter (1990)’s ‘Local
Brevity’ algorithm which takes a given intersec-
tive description and tests whether any set of prop-
erties in it may be replaced by one other property.
Unfortunately, however, simplification is much
harder in the Boolean setting. Suppose, for exam-
ple, one wanted to use the Quine-McCluskey al-
gorithm (McCluskey 1965), known from its appli-
cations to electronic circuits, to reduce the num-
ber of Boolean operators in the description. This
would go only a small part of the way, since
Quine-McCluskey assumes logical independence
of all the properties involved. Arbitrarily com-
plex information about the extensions of prop-
erties can affect the simplification task, and this
reintroduces the spectre of computationally in-
tractability.4 Moreover, the ‘generate and sim-
plify’ approach has other disadvantages in addi-
tion. In particular, the division into two phases,
the first of which generates an unwieldy descrip-
tion while the second simplifies it, makes it psy-
chologically unrealistic, at least as a model for
speaking. Also, unlike the Incremental algorithm,
it treats all properties alike, regardless of their
degree of preferedness. For these reasons, it is
worthwhile to look for an alternative approach,
which takes the Incremental algorithm as its point
of departure. This does not mean that DBS is use-
less: we suggest that it is used for determining
whether a Boolean description exists; if not, the
program returns Fail; if a Boolean description
is possible, the computationally more expensive
algorithm of the following section is called.
3.2 Boolean descriptions (ii): extending the
Incremental algorithm
In this section, we will explore how the Incre-
mental algorithm may be generalized to take all
Boolean combinations into account. Given that
the Incremental algorithm deals with intersections
4For example, the automatic simplificator at
http://logik.phl.univie.ac.at/chris/qmo-
uk.html.O5 can only reduce our description to
a20
a0a2a1
a35 a3
a0a4a0
a20
a12a13a10 if it ‘knows’ that being black, in this
KB, is tantamount to not being white. To reduce even
further, the program needs to know that all elements in the
domain are dogs. In more complex cases, equalities between
complex intersections and/or unions can be relevant.
between sets, Full Boolean Completeness can be
achieved by the addition of set difference. Set
difference may be added to D&RPlur as follows.
First we add negations to the list of atomic proper-
ties (much like the earlier DBS algorithm). Then
D&RPlur runs a number of times: first, in phase
1, the algorithm is performed using all positive
and negative literals; if this algorithm ends before
a39 a40 a38 , phase 2 is entered in which further dis-
tractors are removed from a39 using negations of
intersections of two literals, and so on, until ei-
ther a39 a40 a38 (Success) or all combinations have
been tried (Failure). Observe that the nega-
tion of an intersection comes down to set union,
because of De Morgan’s Law: a23a25a24 a35 a27a29a27a29a27 a35 a23 a31 a40
a23 a24a20a85 a27a29a27a29a27a83a85 a23 a31 . Thus, phase 2 of the algorithm deals
with disjunctions of length 2, phase 3 deals with
disjunctions of length 3, etcetera.
A schematic presentation will be useful, in which
a23a6a5a8a7a10a9 stands for any positive or negative literal.
The length of a property will equal the number
of literals occurring in it. We will say that a D&R
phase uses a set of properties a11 if it loops through
the properties in a11 (i.e., a11 takes the place of IP in
the original D&RPlur).
D&Ra12a8a13a14a13 a55a80a40 a31 :
1. Perform a D&R phase using all prop-
erties of the form a23 a5a8a7a10a9 ;
if this phase is successful then stop, oth-
erwise go to phase (2).
2. Based on the Values of a22 and a39 com-
ing out of phase (1),
perform a D&R phase using all proper-
ties of the form a23 a5a8a7a10a9 a35 a23 a5a8a7a10a9 ;
if this phase is successful then stop, oth-
erwise go to phase (3).
3. Based on the Values of a22 and a39 com-
ing out of phase (2),
perform a D&R phase using all proper-
ties of the form a23a15a5a8a7a10a9 a35 a23a6a5a8a7a10a9 a35 a23a6a5a8a7a10a9 ;
if this phase is successful then stop, oth-
erwise go to phase (4).
Etcetera.
One can require without loss of generality that
no property, considered at any phase, may have
different occurrences of the same atom.5 Since,
therefore, at phase a5 , there is room for properties
of length a5 , the maximal number of phases equals
the total number of atomic properties in the lan-
guage.
Note that D&Ra12a8a13a14a13 a55a80a40 a31 is incremental in two dif-
ferent ways: within a phase, and from one phase
to the next. The latter guarantees that shorter dis-
junctions are favoured over longer ones. Once a
property has been selected, it will not be aban-
doned even if properties selected during later
phases make it superfluous. As a result, one may
generate descriptions like a76 a78 a69a70a68
a7
a35 a42a50a75a47a72a68 a35 a52
a3a45a71 a44
(i.e., ‘white (cats and dogs)’) when a50a73a47a72a68 a35 a52 a3a45a71
(i.e., ‘cats and dogs’) would have sufficed. The
empirical correctness of this type of incremen-
tality is debatable, but repairs can be made if
needed.6 Unfortunately, however, the algorithm
is not tractable as it stands. To estimate its run-
ning time as a function of the number of proper-
ties (a5 a40 ) in the KB and the number of properties
used in the description (a5 a55), note that the maximal
number of properties to be considered equals
a31
a0
a1
a86a3a2 a24
a28
a4
a5 a40
a69a6a5
a40
a31
a0
a1
a86a3a2 a24
a28
a5 a40a8a7
a69
a7
a42
a5 a40a10a9
a69
a44 a7
(The factor of a28 derives from inspecting both each
atom and its negation.) If a5 a55a12a11a13a11 a5 a40 then this
is in the order of a5 a31 a0a40 .7 To avoid intractability,
the algorithm can be pruned. No matter where
this is done, the result is polynomial. By cut-
ting off after phase (1), for example, we gener-
ate negations of atomic properties only, produc-
ing such descriptions as ‘the black dog that is
not a poodle’, while disregarding more complex
descriptions. As a result, Boolean completeness
is lost, but only for references to non-singleton
sets.8 The number of properties to be considered
5For example, it is useless to consider the property
a10a15a14a12a35a102a10a17a16a24a35 a10a15a14 , which must be true of any element in the do-
main, or the property a10 a14 a35 a10 a16 a35 a10 a14 , which is equivalent to
the earlier-considered propertya10a18a14 a35 a10a19a16 .
6E.g., phases might run separately before running in
combination: first phase 1, then 2, 1&2, 3, 1&3, 2&3,
1&2&3, etc. (Suggested by Richard Power.)
7Compare an analogous argument in Dale and Reiter
(1995, section 3.1.1).
8If
a10a15a14
a5a3a20
a31a22a21
a22
a35a33a10a19a16 individuates the individual a23 then
eithera10 a14 a35 a20 a35 a10 a16 ora10 a14 a35 a21 a35a102a10 a16 does. Where singletons
are concerned, set union does not add descriptive power.
by this simpler algorithm equals a42 a5 a40 a44 a31a25a24 a28 a5 a40a26a9a28a27 .
If one wanted to produce more complex descrip-
tions like a76a79a78 a69a70a68 a7 a35 a52 a3a45a71 a35 a0 a3a6a3 a52a72a61 a7 (‘the white dogs
and the poodles’), the algorithm might be cut off
one phase later, leading to a worst-case running
time of a41a2a42 a5a30a29a40 a44 .
4 Discussion
Hybrid algorithms, which make use of elements
of both algorithms, are possible. In particular,
the idea of incrementality can be injected into the
generate and simplify algorithm of section 3.1,
firstly, at the level of the construction of Satel-
lite sets (i.e., by letting a38a8a91 take into account only
those properties from IPa31 a80a83a82 that are necessary for
singling out a52 ) and, secondly, where the union of
the a38a89a47a66a68 a7 a61a63a61a34a69a70a68 a7a10a9 is formed in DBS (i.e., by taking
only those a38a67a47a72a68 a7 a61a34a61a34a69a70a68 a7a10a9 into account that change the
resulting Meaning). Instead of offering any de-
tails on this, we choose to discuss a more general
problem relating to the problem of Logical Form
Equivalence that was noted in section 1.
GRE algorithms exploit a principle of coexten-
sivity for determining what are semantically cor-
rect ways of referring to an entity. Thus, consis-
tent with the idea of logic-oriented generation, the
structure of the description is not prejudged by
the syntactic form of the input to the generator
(i.e., by the fact that the input contains an indi-
vidual constant rather than a description). As a
result, GRE can ‘create’ substantial amounts of
new semantic structure containing, for example,
any number of Boolean operators. In section 1,
it was suggested that the processes of structure
transformation used in GRE might have wider ap-
plicability. The present section questions the va-
lidity of coextensivity as a general principle, first
for GRE (section 4.1), then for sentence aggrega-
tion (section 4.2).
4.1 Descriptions in intensional contexts
The principle of co-extensivity is not valid in in-
tensional contexts. For example, consider
(a) John knows that [the red button] is
dangerous
(b) John knows that [the rocket launch-
ing button] is dangerous.
(a) and (b) have different truth conditions even if
speaker and hearer share the information that the
red button is the rocket launching button. In other
words, the two descriptions are not interchange-
able, even if reader and hearer know them to be
coextensive; what would be necessary is for John
to know that they are coextensive. Extending cur-
rent GRE algorithms to the generation of referring
expressions in intensional contexts is likely to be
a difficult enterprise.
Failiure of substitutivity in intensional contexts
is, of course, a well-known problem, for which
various solutions are available on the theoreti-
cal market (e.g., Montague 1973, Barwise and
Perry 1983). But one has to wonder whether co-
extensivity is ever really sufficient. Consider ex-
tensional truncations of (a) and (b), such as may
be generated from an input I(1) (where the seman-
tic predicate ‘dangerous’ is abbreviated as a0 and
a47 is a constant referring to the button):
I(1) a0 a42a63a47 a44
(aa1 ) [The red button] is dangerous
(ba1 ) [The rocket launching button] is
dangerous
Suppose (a) and (b) are semantically interchange-
able (e.g., when said to someone who knows the
colours and functions of all objects in the do-
main), so a choice between them can only be mo-
tivated by an appeal to pragmatic principles. Even
then, it is difficult to accept that the same choice
must be made regardless whether the input to the
generator is I(1), I(2) or I(3): (Here a17a105a61a42a3a2 a44 says
that a2 is for launching rockets; a4 is the Russellian
description operator.)
I(2) a32a28a42a5a4a6a2
a44
a42 a17
a7
a52a24a42a3a2
a44a8a7
a48a10a9a24a68a83a68
a3a6a5
a42a3a2
a44a44
a33 a84
a0
a42a3a2
a44
I(3) a32a28a42a5a4a6a2 a44 a42 a17a105a61a42a3a2 a44a8a7 a48a10a9a24a68a83a68 a3a6a5 a42a3a2 a44a44 a33 a84 a0 a42a3a2 a44 .
Co-extensivity, after all, does not allow the gen-
erator to distinguish between I(1), I(2) and I(3),
because these three have the same extension!
Perhaps a weakened version of co-extensivity is
needed which allows the generator to add new
structure (e.g., when the input is I(1)), but not to
destroy existing structure (e.g., when the input is
I(2) or I(3)). It is, however, unclear what the the-
oretical justification for such a limitation of co-
extensivity might be.
Note that these problems become more dramatic
as GRE is able to ‘invent’ more structure (e.g.,
elaborate Boolean structure, as discussed in sec-
tion 3). Crucially, we have to assume that, in
an ideal generator, there are many other prag-
matic constraints than Brevity. One description
can be chosen over another, for example, because
it fullfills some additional communicative goal
(Dale and Reiter 1995, section 2.4; also Stone
and Webber 1998). Depending on the commu-
nicative goal, for example, (b) might be chosen
over (a) because the properties that identify the
button also explain why it is dangerous. Brevity
will then have to be interpreted as ‘Brevity pro-
vided all the other constraints are fullfilled’.
4.2 Logic in sentence aggregation
GRE algorithms are sometimes presented as if
the principles underlying them were unrelated to
those underlying other components of an NLG
system.9 This is especially true for the logic-
based structure transformations on which this pa-
per has focused. In what follows, however, we
will suggest that analogous transformations moti-
vate some of the key operations in sentence aggre-
gation (Reiter and Dale 2000, p.133-144). To ex-
emplify, (and limiting the discussion to distribu-
tive readings only) the choice between the (a) and
(b) variants of (1)-(3) involves a decision as to
whether information is expressed in one or more
sentences:
1a. John is eating; Mary is eating; Car-
los is eating.
1b. John, Mary and Carlos are eating.
2a. John is eating; John is drinking;
John is taking a rest
2b. John is eating and drinking and tak-
ing a rest.
3a. If John is eating then Mary is eat-
ing; If Bill is eating then Mary is eating.
3b. If either John or Bill is eating then
Mary is eating.
Writing a11a33a47a72a68a13a12 a42 a38 a44 for a14a15a2 a88 a38 a42a6a11a14a47a66a68a65a42a3a2 a44a44 (Kamp
and Reyle 1993), the linguistic equivalence of
(1a) and (1b) rests on the logical equivalence
9But see Bateman (2000), where GRE and aggregation
are linked.
1a1 . a42a6a11a14a47a66a68a65a42a0 a44a8a7 a11a33a47a72a68a75a42a2a1 a44a8a7 a11a14a47a66a68a65a42a63a50 a44a44a4a3
a11a14a47a66a68a13a12 a42 a18
a0
a26a5a1 a26a51a50a12a20
a44
Analogous to uses of Brevity in GRE, a prefer-
ence for (1b) over (1a) might be motivated by
a preference for a semantic structure with fewer
logical operations. Examples (2)-(3) are not dis-
similar to what we see in (1). For example, the
following logical equivalences support the lin-
guistic equivalence of (2a)/(2b) and (3a)/(3b):
2a1 . a42a7a6 a47
a7a9a8
a47
a7a11a10
a42a63a47
a44a44a12a3
a42a14a13 a2a5a84a15a6 a2
a7a9a8
a2
a7a11a10
a2
a44
a42a63a47
a44
3a1 . a42a42a0 a24 a1 a3 a44a8a7 a42a0 a31 a1 a3 a44a44a16a3
a42a42
a0
a24a16a17
a0
a31
a44
a1 a3
a44
In (a28 a1 ), three properties, a6 a26 a8 and a10 , are aggre-
gated into a13 a2a27a84a18a6 a2 a7a9a8 a2 a7a19a10 a2 (i.e., to have each
of the three properties a6 a26 a8 and a10 ). In (a15 a1 ),
two antecedents a0 a24 and a0 a31 are aggregated into
a0
a24a20a17
a0
a31 .
10 As before, a generator might prefer
the (b) versions because they are structurally sim-
pler than the logically equivalent (a) versions. In
sentence aggregation, however, co-extensivity is
not enough. For example, we expect ‘Eat(j)’ to
be worded differently from ‘Eat(m)’, even if both
propositions are true and consequently have the
same extension. Unlike GRE, therefore, aggrega-
tion requires at least logical equivalence.11
5 Acknowledgment
Thanks are due to Emiel Krahmer for discussion
and comments.

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