Abstraction Super-structuring Normal Forms: Towards a Theory of Structural Induction

TL;DR

This paper explores the fundamental structural operations involved in induction, identifying a minimal set of operations that can generate all theories, and discusses their implications for understanding hidden variables and philosophical principles of connection.

Contribution

It introduces a formal framework of structural super-structuring normal forms, revealing that only four operations are needed for Turing-complete induction.

Findings

Abstraction and super-structuring are essential in structural induction.

Reverse operations of abstraction and super-structuring complete the set of necessary operations.

The framework has implications for understanding hidden variables and philosophical theories of connection.

Abstract

Induction is the process by which we obtain predictive laws or theories or models of the world. We consider the structural aspect of induction. We answer the question as to whether we can find a finite and minmalistic set of operations on structural elements in terms of which any theory can be expressed. We identify abstraction (grouping similar entities) and super-structuring (combining topologically e.g., spatio-temporally close entities) as the essential structural operations in the induction process. We show that only two more structural operations, namely, reverse abstraction and reverse super-structuring (the duals of abstraction and super-structuring respectively) suffice in order to exploit the full power of Turing-equivalent generative grammars in induction. We explore the implications of this theorem with respect to the nature of hidden variables, radical positivism and the 2-century old claim of David Hume about the principles of connexion among ideas.