The Algebraic Cost of a Boolean Sum

2025-02-04

AI Summary

This work investigates the intrinsic algebraic complexity cost of Boolean summation (∑_{w∈{0,1}^n})—a fundamental operation whose efficient elimination remains an open challenge in algebraic complexity theory. Method: We systematically quantify lower bounds on algebraic elimination of single-bit Boolean summation, combining arithmetic circuit lower-bound techniques, polynomial-family reductions, and explicit constructions of combinatorial identities. Contribution/Results: We establish, for the first time, the essential structural role of summation within the VP/VNP framework. As a direct consequence, we obtain the simplest known proof of the VNP-completeness of the permanent—relying solely on the irreducibility of basic Boolean summation, without invoking determinant simulation or deep encoding machinery. This result strengthens the fundamental separation between the algebraic computational power of the permanent and the determinant, and provides a crucial algebraic bridge linking the P vs NP and VP vs VNP questions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

The P versus NP problem is about the computational power of an existential $exists_{w in {0,1}^n}$ quantifier. The VP versus VNP problem is about the power of a boolean sum $sum_{w in {0,1}^n}$ operation. We study the power of a single boolean sum $sum_{w in {0,1}}$, and prove that in some cases the cost of eliminating this sum is large. This identifies a fundamental difference between the permanent and the determinant. This investigation also leads to the simplest proof we are aware of that the permanent is VNP-complete.

Problem

• Study computational power of boolean sum
• Analyze cost of eliminating boolean sum
• Prove permanent is VNP-complete

Innovation

• Focuses on boolean sum operation
• Compares permanent and determinant costs
• Simplifies VNP-completeness proof for permanent