Towards a Formal Description of Understandability (Causality, Pre-Requisites): From Prosorov’s Phonocentric Topology to More General Interior (Closure) Structures

In many real life situations, a text consists of related parts; so, to understand a part, we need to first understand some (or all) preceding parts: e.g., to understand Chapter 3, we first need to understand Chapters 1 and 2. In many cases, this dependence is described by a partial order. For this case, O. Prosorov proposed a natural description of the dependence structure as a topology (satisfying the separation axiom T0). In some practical situations, dependence is more general than partial order: e.g., to understand Chapter 3, we may need to understand either Chapter 1 or Chapter 2, but it is not necessary to understand both. We show that such a general dependence can be naturally described by a known generalization of topology: the notion of an interior (or, equivalently, closure) structure (provided, of course, that this structure satisfies a natural analog of T0-separability). 1 Prosorov’s Topology-Based Description of Understandability: Reminder Understandability: formulation of the general problem. A sufficiently long text usually consists of several parts. For example, a technical book consists of chapters. The parts are arranged in such a way that to understand a part, one needs to first understand some (or even all) previous parts. For example, to understand Chapter 3, one needs to understand Chapter 1. In some cases, in order to understand a certain part, one needs to first understand all the preceding parts. However, in many other cases, not all preceding parts are needed. For example, in a book, Chapter 1 may contain preliminaries