Machine-learnable Sets

2026-06-27

AI Summary

This work proposes a formal framework for characterizing the machine learnability of sets of binary strings through bounded-complexity Boolean autoencoders. The framework unifies recognizability, generatability, and learnability from examples as structural properties of discrete sets, realized via Boolean threshold function networks. By integrating an iterative optimization algorithm, the approach effectively approximates and converges to strictly learnable states. Empirical evaluations demonstrate that diverse discrete structures—including Rorschach patterns and complex “wild” sets—exhibit machine learnability under this formalism, underscoring both its expressive power and practical utility.

Abstract

In this study we present a formal definition of large discrete sets having, informally, three properties: their elements are easily recognized, easily generated, and the latter tasks are easily learned from examples. The formalism is specialized to sets of binary strings and a definition of "machine-learnability" based on the existence of a bounded-complexity Boolean autoencoder that fixes the elements of the set. We present experiments where the autoencoders are implemented by nets of Boolean threshold functions. Machine-learnability is demonstrated for Rorschach patterns (that may have reversed contrast in the mirrored half), and considerably "wilder" sets whose elements are only approximately fixed by admissible autoencoders. In the second case we demonstrate a simple iteration that evolves wild sets to make them properly machine-learnable.

Problem

• machine-learnable sets
• discrete sets
• Boolean autoencoder
• learnability
• binary strings

Innovation

• machine-learnable sets
• Boolean autoencoder
• threshold networks
• set learnability
• iterative refinement