Probabilistic and Team PFIN-type Learning: General Properties

TL;DR

This paper investigates the structure of the probability hierarchy in Popperian finite learning, proving its decidability and complex well-ordered nature, and shows that team and probabilistic learning are equally powerful within this framework.

Contribution

It establishes the decidability of the probability hierarchy and demonstrates its intricate structure, also proving the equivalence of team and probabilistic learning in PFIN-type learning.

Findings

The probability hierarchy is decidable.

The hierarchy is well-ordered and order-equivalent to epsilon_0.

Team and probabilistic learning have the same power in PFIN-type learning.

Abstract

We consider the probability hierarchy for Popperian FINite learning and study the general properties of this hierarchy. We prove that the probability hierarchy is decidable, i.e. there exists an algorithm that receives p_1 and p_2 and answers whether PFIN-type learning with the probability of success p_1 is equivalent to PFIN-type learning with the probability of success p_2. To prove our result, we analyze the topological structure of the probability hierarchy. We prove that it is well-ordered in descending ordering and order-equivalent to ordinal epsilon_0. This shows that the structure of the hierarchy is very complicated. Using similar methods, we also prove that, for PFIN-type learning, team learning and probabilistic learning are of the same power.