Introduction

This post aims to formalise the models and model changes/model splintering described in this post. As explained there, the idea is to have a meta-model sufficiently general to be able to directly capture the process of moving from one imperfect model to another.

A note on infinity

For simplicity of exposition, I'll not talk about issues of infinite sets, continuity, convergence, etc... Just assume that any infinite set that comes up is just a finite set, large enough for whatever practical purpose we need it for.

Features, worlds, environments

A model $M$ is defined by three object, the set $F$ of features, the set $E$ of environments, and a probability distribution $Q$ . We'll define the first two in this section.

Features

Features are things that might be true or not about worlds, or might take certain values in worlds. For example, "the universe is open" is a possible feature about our universe, "the temperature is $250 K$ " is another possible feature, but instead of returning true or false, it returns the temperature value. Adding more details, such as "the temperature is $250 K$ in room 3, at 12:01" show that features should also be able to take inputs: features are functions.

But what about "the frequency of white light"? That's something that makes sense in many models - white light is used extensively in many contexts, and light has a frequency. The problem with that statement is that light has multiple frequencies; so we should allow features to be, at least in some cases, multivalued functions.

To top that off, sometimes there will be no correct value for a function; "the height of white light" is something that doesn't mean anything. So features have to include partial functions as well.

Fortunately, multivalued and partial functions are even simpler than functions at the formal level: they are just relations. And since the sets in the relations can consist of a single element, in even more generality, a feature is a predicate on a set. We just need to know which set.

So, formally, a feature $F \in F$ consists of an (implicit) label defining what $F$ is (eg "open universe", "temperature in some location") and a set on which it is a predicate. Thus, for example, the features above could be:

$F_{open universe} = {0}$ (features which are simply true or false are functions of a single element).
$F_{temperature} = {R^{+}}$ .
$F_{temperature at location and time} = {L \times T \times R^{+}}$ , for some set $L$

...

Definitions