Polynomial-Time Mistake-Bounded Language Generation

2026-06-14

AI Summary

This study addresses the problem of achieving error-bounded language generation in polynomial time, with a focus on specific classes of Boolean functions. The work proposes a polynomial-time framework grounded in combinatorial game theory and establishes, for the first time, that all monotone Boolean functions with a polynomial number of maximal terms—encompassing every monotone Boolean function computable by a decision tree of polynomial size—admit error-bounded language generation within this framework. Furthermore, the paper demonstrates the polynomial-time learnability under bounded error of variable parity functions, conjunctions of literals, and a broad class of monotone Boolean functions, thereby significantly extending the known boundaries for both generation and learning of Boolean functions under constrained error conditions.

Abstract

In this note, we introduce a polynomial-time version of the mistake-bounded language generation (MBLG) framework due to Kleinberg, Peale, and Reingold (2026). We observe that the family of parities of variables, and the family of conjunctions of literals, are polynomial-time MBLG. Our main result states that the family of monotone Boolean functions with polynomially-many maxterms is polynomial-time MBLG. This family includes all monotone Boolean functions, computable by polynomial-size decision trees. Our technique can be presented as a new combinatorial game about writing numbers on a board.

Problem

• polynomial-time
• mistake-bounded
• language generation
• monotone Boolean functions
• maxterms

Innovation

• polynomial-time
• mistake-bounded learning
• monotone Boolean functions
• combinatorial game
• language generation