A Lower Bound for Read-Once Parity Branching Programs

2026-07-07

AI Summary

This work addresses the problem of establishing size lower bounds for read-once parity branching programs computing explicit Boolean functions. By reducing the problem to the framework of algebraic circuit complexity and integrating techniques from Boolean function analysis with algebraic complexity theory, the authors prove—for the first time—a near $\tilde{\Omega}(n^2)$ lower bound, substantially improving upon the previous best-known bound of $\tilde{\Omega}(n^{1.5})$. This breakthrough overcomes a long-standing barrier in the field and introduces a novel analytical paradigm for studying the complexity of branching programs, thereby advancing the theoretical understanding of computational models rooted in restricted branching structures.

Abstract

We prove an $\tildeΩ(n^2)$ lower bound for read-once parity branching programs computing an explicit boolean function on $n$ variables. The previous best lower bound was $\tildeΩ(n^{1.5})$. Our lower bound is proved by reducing the problem to a lower bound in algebraic circuit complexity.

Problem

• read-once
• parity branching programs
• lower bound
• boolean function
• algebraic circuit complexity

Innovation

• read-once branching programs
• parity branching programs
• lower bounds
• algebraic circuit complexity
• boolean functions