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Motivation of discourse models.
One big advantage of tableaux is that they come with an in-built possibility to keep around knowledge that's assumed to be ``already there''. Be it world knowledge, the content of previous sentences, or a new sentence - simply put it on the tableaux, and you can use it. Putting this formally, let's generalize the notion of an initial tableau. Instead of allowing only one initial signed formula at the root node, let's allow a linear tree whose nodes are labeled with signed formulae. We will for instance start model generation for the sentence ``If Mutz is a cat, Mutz likes Hansi.'' with the following tableaux, already containing the world knowledge that Hansi is a canary and Mutz is a cat (we've enclosed the formula for the actual input sentence in a box):
Now what exactly are we going to do with tableaux to model discourse understanding? We think of the hearer (or reader) in a discourse as internally maintaining a tableaux that represents the current set of alternative models for that discourse. Now the hearer's understanding is an online process that receives as input the logical forms of the sentences of the discourse one by one. The hearer has a mechanism for choosing a preferred model (i.e. branch) from the open branches of his internal tableaux. He also maintains a set of deferred branches that can be re-visited if the preferred branch gets closed in the course of further processing.
Append, Saturate, Choose
Upon input, the given logical form is appended to the tableaux as a leaf to all branches. The hearer then saturates the current (i.e. the preferred) tableaux branch, exploring the set of possible models for the sequence of input sentences. If the subtableaux generated by this saturation process contains open branches, then he chooses one of them as the new preferred model, marks some of the other open branches as deferred, and waits for further input. If the saturation yields a closed sub-tableaux, he backtracks. This means that he selects a new preferred branch from the deferred ones, and continues the tableaux expansion on this branch. Backtracking may be repeated until successful, or until some termination criterion is met. In this case discourse processing fails altogether.
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