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

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.