Set systems: order types, continuous nondeterministic deformations, and quasi-orders

TL;DR

This paper explores the structure of set systems through order types, their relation to well-quasi-orders, and how continuous nondeterministic deformations preserve properties like finite elasticity, with implications for computational learning theory.

Contribution

It introduces a novel game-based definition of order types for set systems, links them to well-quasi-orders, and proves preservation of finite elasticity under continuous monotone functions.

Findings

Any well-quasi-order can be represented by a set system with matching maximal order type.

Finite elasticity is preserved under continuous, monotone functions and various language operators.

An order-type-preserving embedding exists from quasi-orders to subspaces of Cantor spaces.

Abstract

By reformulating a learning process of a set system L as a game between Teacher and Learner, we define the order type of L to be the order type of the game tree, if the tree is well-founded. The features of the order type of L (dim L in symbol) are (1) We can represent any well-quasi-order (wqo for short) by the set system L of the upper-closed sets of the wqo such that the maximal order type of the wqo is equal to dim L. (2) dim L is an upper bound of the mind-change complexity of L. dim L is defined iff L has a finite elasticity (fe for short), where, according to computational learning theory, if an indexed family of recursive languages has fe then it is learnable by an algorithm from positive data. Regarding set systems as subspaces of Cantor spaces, we prove that fe of set systems is preserved by any continuous function which is monotone with respect to the set-inclusion. By it, we prove that finite elasticity is preserved by various (nondeterministic) language operators (Kleene-closure, shuffle-closure, union, product, intersection,. . ..) The monotone continuous functions represent nondeterministic computations. If a monotone continuous function has a computation tree with each node followed by at most n immediate successors and the order type of a set system L is {\alpha}, then the direct image of L is a set system of order type at most n-adic diagonal Ramsey number of {\alpha}. Furthermore, we provide an order-type-preserving contravariant embedding from the category of quasi-orders and finitely branching simulations between them, into the complete category of subspaces of Cantor spaces and monotone continuous functions having Girard's linearity between them. Keyword: finite elasticity, shuffle-closure