Non-monotonic Pre-fixed Points and Learning

TL;DR

This paper introduces a generalized framework for learning processes modeled by non-monotonic pre-fixed points in knowledge spaces, accommodating non-deterministic operators and higher-level structures.

Contribution

It extends previous work to omega-level knowledge spaces and non-deterministic operators, broadening the applicability of the learning process model.

Findings

Established equivalence between pre-fixed points and termination states.

Generalized the model to non-deterministic operators.

Extended the framework to higher-level knowledge spaces.

Abstract

We consider the problem of finding pre-fixed points of interactive realizers over arbitrary knowledge spaces, obtaining a relative recursive procedure. Knowledge spaces and interactive realizers are an abstract setting to represent learning processes, that can interpret non-constructive proofs. Atomic pieces of information of a knowledge space are stratified into levels, and evaluated into truth values depending on knowledge states. Realizers are then used to define operators that extend a given state by adding and possibly removing atoms: in a learning process states of knowledge change nonmonotonically. Existence of a pre-fixed point of a realizer is equivalent to the termination of the learning process with some state of knowledge which is free of patent contradictions and such that there is nothing to add. In this paper we generalize our previous results in the case of level 2 knowledge spaces and deterministic operators to the case of omega-level knowledge spaces and of non-deterministic operators.