Denoted as Cb(Un). It represents the current salient entity, the one being focused on in the discourse after Un is interpreted.

Backward-looking Center of Un

Denoted as Cf(Un), are a set forward-looking center of potential future salient entities, the discourse entities evoked by Un any of which could serve as Cb (the salient entity) of the following utterance.

 Forward-looking Center of Un

Centering theory is a theory of both discourse salience and discourse coherence. As a model of discourse salience, it suggests that at any given point in the discourse, one of the entities in the discourse model is salient; As a model of discourse coherence, it proposes that discourses in which adjacent sentences continue to maintain the same salient entity are more coherent than those which shift back and forth between multiple entities.

Hamilton College

Entity-based models link each mention to a previous discourse entity (rather than previous mentions). 

Entity-based Models

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

Centering Theory

Entity grid model is an alternative way to capture entity-based coherence: instead of having a top-down theory, the entity-grid model using machine learning to induce the patterns of entity mentioning that make a discourse more coherent.

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