Towards a mathematical theory of machine discovery from facts

Abstract

This paper intends to give a theoretical foundation of machine discovery from facts. We point out that the essence of a computational logic of scientific discovery or a logic of machine discovery is the refutability of the entire spaces of hypotheses. We discuss this issue in the framework of inductive inference of length-bounded elementary formal systems (EFSs), which are a kind of logic programs over strings of characters and correspond to context-sensitive grammars in Chomsky hierarchy. First we present some characterization theorems on inductive inference machines that can refute hypothesis spaces. Then we show differences between our inductive inference and some other related inferences such as in the criteria of reliable identification, finite identification and identification in the limit. Finally we show that for any n, the class, i.e. hypothesis space, of length-bounded EFSs with at most n axioms is inferable in our sense, that is, the class is refutable by a consistently working inductive inference machine. This means that sufficiently large hypothesis spaces are identifiable and refutable.