The first step in formalizing the corpus is to move knowability from a descriptive property to a formal requirement.
In the Corpus, knowability is defined as:
The preserved ability of a system to justify its behavior through enforceable constraints across time and composition.
To formalize this, we must define the relationship between a system’s structure \(S\), its behavior \(B\), and the constraints \(C\) that govern them.
1. The Physical System
We define a system \(S\) as a subset of the universe \(U\) (\(S \subseteq U\)), characterized by its physical constitution and a distinct boundary.
The system possesses a state space \(\Omega\) and a set of constraints \(C\) induced by its nature:
- \(\Omega = \Omega(S)\)
- \(C = C(S)\)
2. Transition Structure and Invariants
The transition structure \(T\) represents all possible state changes permitted by the system’s constraints:
\[ T \subseteq \Omega \times \Omega \] \[ T = { (\omega, \omega’) \in \Omega \times \Omega \mid C(S) \text{ permits } \omega \to \omega’ } \]
From this, we define the Invariant Space \(I\) as the set of states or properties that remain preserved across all valid transitions in \(T\).
3. The Representation Space
A system’s ability to “know” or justify itself depends on its Representation Space \(R\), which is a subsystem of the state space:
\[ R \subseteq \Omega \]
This \(R\) generates its own perceived invariant space, \(I_{rep}\):
\[ I_{rep} = I(R) \]
4. The Knowability Gap
Because any representation \(R\) is finite, while the physical constraints \(C(S)\) can be arbitrarily complex, the represented invariants and actual invariants rarely coincide:
\[ I_{rep} \neq I \]
5. Formal Definition of Knowability
We can now define Knowability (\(K\)) as the degree to which the represented invariant space accurately maps to the actual invariant space. It is proportional to the measure (\(\mu\)) of their intersection relative to the total actual invariant space:
\[ K \propto \frac{\mu(I_{rep} \cap I)}{\mu(I)} \]
This ratio captures the essence of the corpus: a system is knowable only to the extent that its internal representations (documentation, tests, types, models) successfully capture the invariants that actually govern its behavior.
More to follow on the algebra of invariant space.