Need examples for the following definitions

I am currently reading the paper "Similarity of Narratives" by Loizos Michael (link below) and I am having a hard time figuring out the definitions listed (p.107 - p.109).

Could someone please give me a practical example for each of the definitions?

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The 'vocabulary' used is:

• $$\mathscr{F}$$ - fluents, conditions that change over time. The predicate holds(<fluent>,<state>) defines whether a condition applies in a given state.

• $$\mathscr{A}$$ - actions, describe, erm, actions that happen. The predicate occurs(<action>,<state>) specifies that a given action happens at a particular state (presumably triggering a change to a new state, and changing the value of one or more fluents)

• $$\mathscr{T}$$ - time points, which specify a chronological ordering.

Now we can look at the actual definitions: a discourse combines events (actions) and facts (fluents) and provides a (partial) ordering, ie a sequence in which they occur. This is represented in an acyclic graph. So the discourse describes a sequence of events that happen in a particular order and have an effect on facts. The ordering is relative, as there are no absolute time reference points give. Actions and facts are expressed in predicate calculus by the predicates mentioned above. Actions can involve a character moving from location A to location B; the fluent describing the location of the character will have been changed through that action.

An embedding assigns a given point in time to each state in the discourse. Each state is assigned a time point in a way that is consistent with the relative timings of the states in the discourse. So you have a general story, and you fix each action in time. The absolute time of an action in the embedding has to be before/after other actions according to their order in the discourse.

A domain adds two further predicates to a discourse, namely static($$\phi$$) and causes($$\phi$$,<fluent>). These formulas $$\phi$$ express logical constraints, that add consistency to the discourse. For example, if an action involves killing a character, the consequence has to be that the character is dead afterwards. So that action mandates a particular change, which would be encoded by such a formula. If the character was then to do some action after having been killed, that would invalidate the discourse in that domain. Different domains might have different constraints etc.

A model then works out if the domain is consistent, by evaluating all the actions and events using the domain constraints, and assigning it a truth value depending on whether it is possible or not. I might be wrong there, as I'm not sure I have interpreted the definition correctly — please let me know in a comment if that is the case.

Basically, if a model for a domain exists, it means that the actions and facts are consistent, and hence the narrative works (it is "a discourse compatible with a given domain"): a narrative is a discourse in a domain which, when embedded in a timeline, is consistent.

A default domain is one domain out of a set of other domains which seems preferable over the others.

(I have to stop here for lack of time — I might be able to expand this answer later)