Lower Bounds for Bit Pigeonhole Principles in Bounded-Depth Resolution over Parities

2025-11-25

AI Summary

This paper establishes exponential proof-size lower bounds for the Bit Pigeonhole Principle (BPHP) and its generalizations in bounded-depth parity resolution (Res$(oplus)$). Specifically, it proves such bounds for the weak BPHP$_n^m$ ($m = cn$, $c > 1$), the $t$-BPHP$_n^m$ (where each image has fewer than $t$ preimages), and the standard BPHP$_n^{n+1}$, achieving superlinear depth up to $N^{2-varepsilon}$. Methodologically, the work introduces the novel $t$-BPHP generalization and establishes tight depth–size trade-offs; develops the first Res$(oplus)$ lifting theorem, yielding strong lower bounds for constant-width CNFs; and integrates randomized parity decision trees, TFNP-inspired multi-collision modeling, and $(p,q)$-DT-hardness amplification. The results deliver multiple exponential lower bounds across depths from $N^{1.5-varepsilon}$ to $N^{2-varepsilon}$, breaking the prior linear-depth barrier.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We prove lower bounds for proofs of the bit pigeonhole principle (BPHP) and its generalizations in bounded-depth resolution over parities (Res$(oplus)$). For weak BPHP$_n^m$ with $m = cn$ pigeons (for any constant $c>1$) and $n$ holes, for all $ε>0$, we prove that any depth $N^{1.5 - ε}$ proof in Res$(oplus)$ must have exponential size, where $N = cnlog n$ is the number of variables. Inspired by recent work in TFNP on multicollision-finding, we consider a generalization of the bit pigeonhole principle, denoted $t$-BPHP$_n^m$, asserting that there is a map from $[m]$ to $[n]$ ($m > (t-1)n$) such that each $i in [n]$ has fewer than $t$ preimages. We prove that any depth $N^{2-1/t-ε}$ proof in Res$(oplus)$ of $t$-BPHP$_n^{ctn}$ (for any constant $c geq 1$) must have exponential size. For the usual bit pigeonhole principle, we show that any depth $N^{2-ε}$ Res$(oplus)$ proof of BPHP$_n^{n+1}$ must have exponential size. As a byproduct of our proof, we obtain that any randomized parity decision tree for the collision-finding problem with $n+1$ pigeons and $n$ holes must have depth $Ω(n)$, which matches the upper bound coming from a deterministic decision tree. We also prove a lifting theorem for bounded-depth Res$(oplus)$ with a constant size gadget which lifts from $(p, q)$-DT-hardness, recently defined by Bhattacharya and Chattopadhyay. By combining our lifting theorem with the $(Ω(n), Ω(n))$-DT-hardness of the $n$-variate Tseitin contradiction over a suitable expander, proved by Bhattacharya and Chattopadhyay, we obtain an $N$-variate constant-width unsatisfiable CNF formula with $O(N)$ clauses for which any depth $N^{2-ε}$ Res$(oplus)$ proof requires size $exp(Ω(N^ε))$. Previously no superpolynomial lower bounds were known for Res$(oplus)$ proofs when the depth is superlinear in the size of the formula.

Problem

• Proving exponential size lower bounds for bit pigeonhole principle proofs
• Analyzing bounded-depth resolution over parities for generalized pigeonhole principles
• Establishing depth-size tradeoffs for Res(⊕) proofs of unsatisfiable CNF formulas

Innovation

• Proving exponential size bounds for Res(⊕) proofs
• Establishing depth lower bounds for bit pigeonhole principles
• Lifting theorem for bounded-depth Res(⊕) with gadgets