Concept Theory: A paradigm for navigation design
April 14th, 2017
Manifest Statement: Computational Programming as a discipline deserves a theoretical inventory of navigably complex spaces which allow for the expressions, manipulations, and solutions to its computational problems.
The interpretation of a navigably complex space is unpacked as follows:
- navigably complex: There is a density of elements, subspaces, objects, structures, types, for which to navigate, as well as a rich variety of ways to navigate this density.
- navigably computable: The navigational interface—the means to navigating to the elements, subspaces, etc.—is effectively computable.
- navigably expressive: The navigational interface is a flexible and accessible namespace for referring to the elements, subspaces, etc.~within.
A space such as this is highly navigable, and allows us to optimize the generality-specificity trade off within predicate logic. It acts as the infrastructure for designing and interpreting detailed models within many different and diverse contexts.
I assert this unpacked interpretation is not absurd, and demonstrate its existence with a template implementation of a navigably complex space. I show its narrative focus to be that of namespaces, and proclaim as its guiding principle: That which cannot be named cannot be accessed.
Namespaces
The idea of a navigably complex namespace is as follows:
- We choose to mitigate the inherent complexity of this space. We do so by stratifying it into layers of names.
- Lower layers of names are used to construct higher layers of names.
- Lower layers have names possessing increased constructive entropy.
- Higher layers have names possessing decreased constructive entropy.
- Names which are completely determined have zero entropic ability, and are known as types.
- The act of resolving a name for translation into another namespace is known as identity resolution.
As we use names in a referential sense, I submit that a referential name has higher entropy when it has a higher ability to delay its reference. The state of a delayed referral is known as a deferral, as such a delayed name is a deferred name. I declare that a navigably complex space of deferred names is the concept space we seek.
I invoke the Curry-Howard Isomorphism as analogy to offer the following claim: A concept space is isomorphic to a given model of specification semantics. This model of specification semantics states that a specification acts as a semantic filter. Weak specifications lead to stronger specifications, but can still have their own algebras. A classical example in mathematics is the definition of a limit in calculus. It is constructed from weaker topological specifications, which are used to define and prove its algebra. As a filter it applies to many functions and sequences, but can be narrowed further to effective contextual instances during application.
The isomorphism exists between concepts and filters, offering dual ways to express, manipulate, and solve computational problems. I offer a three part article giving an exact example of a concept space:
- Stratified Powerset (Part One): Constructs a mathematically oriented rigorously proven, infrastructural layer for our concept space.
- Stratified Powertuple (Part Two): Translates the previously constructed infrastructural layer into a computationally oriented universal data structure.
- Stratified Namespace (Part Three): Explicates the previously translated universal data structure into an implementation of a concept space, with intent for use as an alternative model of programming language compilers.