Binary strings of finite VC dimension

TL;DR

This paper explores the complexity of binary strings through the lens of VC dimension, revealing their topological and logical properties and introducing new bounds and classifications for strings with finite VC dimension.

Contribution

It introduces the concept of VC dimension for binary strings, provides bounds, characterizes low VC dimension strings, and links these ideas to topology and logic.

Findings

Strings of bounded VC dimension are meagre in the real topology.

Bi-infinite strings with VC dimension d form a non-sofic shift space.

Characterization of low VC dimension strings (0, 1, 2) and their logical connections.

Abstract

Any binary string can be associated with a unary predicate $P$ on $\mathbb{N}$. In this paper we investigate subsets named by a predicate $P$ such that the relation $P(x+y)$ has finite VC dimension. This provides a measure of complexity for binary strings with different properties than the standard string complexity function (based on diversity of substrings). We prove that strings of bounded VC dimension are meagre in the topology of the reals, provide simple rules for bounding the VC dimension of a string, and show that the bi-infinite strings of VC dimension $d$ are a non-sofic shift space. Additionally we characterize the irreducible strings of low VC dimension (0,1 and 2), and provide connections to mathematical logic.