A new order theory of set systems and better quasi-orderings

TL;DR

This paper introduces a new order type for set systems based on a game-theoretic reformulation, linking it to well and better quasi-orderings, and characterizes set systems with respect to unions and order types.

Contribution

It develops a novel order type for set systems, connects it to BQOs, and characterizes set systems whose unions preserve order type, advancing the theory of quasi-orderings in set systems.

Findings

Defined the order type dim L via a game between Teacher and Learner.

Established the correspondence between the new order type and BQOs.

Characterized set systems where unions preserve the order type.

Abstract

By reformulating a learning process of a set system L as a game between Teacher (presenter of data) and Learner (updater of the abstract independent set), we define the order type dim L of L to be the order type of the game tree. The theory of this new order type and continuous, monotone function between set systems corresponds to the theory of well quasi-orderings (WQOs). As Nash-Williams developed the theory of WQOs to the theory of better quasi-orderings (BQOs), we introduce a set system that has order type and corresponds to a BQO. We prove that the class of set systems corresponding to BQOs is closed by any monotone function. In (Shinohara and Arimura. "Inductive inference of unbounded unions of pattern languages from positive data." Theoretical Computer Science, pp. 191-209, 2000), for any set system L, they considered the class of arbitrary (finite) unions of members of L. From viewpoint of WQOs and BQOs, we characterize the set systems L such that the class of arbitrary (finite) unions of members of L has order type. The characterization shows that the order structure of the set system L with respect to the set-inclusion is not important for the resulting set system having order type. We point out continuous, monotone function of set systems is similar to positive reduction to Jockusch-Owings' weakly semirecursive sets.