DICE is a theory which allows us to compute rhetorical links between segments of discourse on the basis of the speaker's background semantic and pragmatic knowledge. Discourse representations produced by DICE are in the form of segmented DRSs (SDRSs) (Asher asher:93a), in which discourses are represented as DRSs plus discourse relations. These discourse relations act as constraints on discourse coherence by constraining the semantic content of the DRSs they connect. For example, if the discourse relation Narration, which conveys that one constituent of the discourse is a consequent of a previous constituent, is to hold, it must be possible to compute a common topic between the two constituents. This explains the incoherence of the discourse in cog4 -- there is no topic shared between the two sentences in the discourse.
*Max came in. Mary's hair is black.
In this section, I will not give a formal account of the analysis of the discourses in cog3, as the precise details of the formalism are less important than the describiption of how lexical information might interact with discourse-level reasoning. Thus, I will instead give an informal outline of the analysis. The reader is referred to Lascarides and Asher (1991,1993), Lascarides and Copestake (1995), and Lascarides, Copestake and Briscoe lasc_cope_brisc:96 for details of the formal application of DICE.
I label the DRSs representing cog3a,b as 
and
the DRSs representing cog3c(i-iii) as 
respectively in Figure 5.7. The DRS 
will
represent the indefeasible interpretation He began reading your
book at 9pm for cog3c(i), because the sentence has the form
begin+NP and so the event read is picked out
from the default portion of the telic role in qualia structure through
the lexical entry of coercing begin and represented in the
logical form as indefeasible. In contrast, 
will
represent the interpretation He began doing something with your
book at 9pm (there is no default interpretation for what the event
done with the book is) and 
represents the defeasible
interpretation He particularly enjoyed reading your book. The *
marks the fact that the telic event is a default specification. It
has remained default after lexicosyntactic processing due to the
nature of enjoy+NP. The process by which these interpretations
are determined will be described in detail below.
Figure: Logical forms associated with the three discourse
continuations in cog3c.
Let us consider how the rules apply to the discourse
cog3a,b,c(i). 
must be attached to 
. Assuming
that DICE calculates that eating everything in his cage is a
subtype of going nuts, we can attach to with the
discourse relation Elaboration, which conveys, as intuitions
would dictate, that the event in is a part of the event in
. Now we must incorporate into the SDRS
. We can attempt to attach
to either or with a discourse relation.
Attaching directly to does not seem to be possible.
Intuitively, there is no generalisation between :going
nuts and :beginning to read your book, and so there
does not seem to be a topic common to the two structures which is
consistent with world knowledge about going nuts. Attaching
to with Narration is therefore not
possible. Furthermore, the latter event is not a subtype of the
former, and so Elaboration is not possible either. Indeed,
the rules in DICE will compute that no discourse relation can
adequately relate them on the basis of semantic and pragmatic
knowledge. Attaching to also fails in a similar
manner. Thus, there is no way to make sense of cog3c(i) given
the preceding context and the indefeasible interpretation resulting
from lexicosyntactic processing, as cog3a,b,c(i) is predicted
in DICE to be an incoherent discourse.
We can contrast this with the discourses cog3a,b,c(ii) and
cog3a,b,c(iii). In both discourses, we have the same first
step as above: is attached to with the relation
Elaboration. Subsequently, the DRSs for the continuation of
the discourses, and respectively, must be
attached. This is done in both cases according to the analysis in
L&C (1995) (for details see that paper). Since provides
only an underspecified interpretation, the context serves to specify
to 
.
This occurs as a result of the constraints imposed by the rhetorical
relations; the strongest coherence for the discourse results when
is in a clear rhetorical relation to the previous discourse.
Here, doing something with your book has no clear relation to
eating everything in his cage whereas eating your book is a
subtype of that event and so the former event is specified to the
latter. Similarly, for the context overrides the default
interpretation 
with , via the axiom Discourse Wins combined
with the preference for a strongly coherent discourse. In this case,
eating your book is in a much stronger rhetorical relation to
eating everything in his cage than reading your book is and
so the former is preferred. For each of these discourses, then, an
interpretation of the continuing sentence is established which would
allow DICE to attach the sentence to the discourse via a clear
rhetorical relation. This explains the felicity of these discourses
in contrast to the discourse cog3a,b,c(i).