A Hierarchy for Constant Communication Complexity

2025-09-26

AI Summary

This paper addresses the classification of communication complexity measures, challenging the conventional asymptotic growth-based equivalence (e.g., logarithmic, polynomial) in favor of a novel paradigm grounded in “constancy”—i.e., whether a measure is bounded by a constant. Method: We define two measures as equivalent iff they are either both constant or both non-constant, and construct a bidirectional constancy-determination framework over families of Boolean functions, integrating asymptotic analysis with equivalence relation modeling. Contribution/Results: We systematically identify five mutually exclusive and collectively exhaustive constancy equivalence classes—the first such complete characterization—revealing non-intuitive intrinsic connections across distinct communication models and transcending intuitive limitations of classical hierarchy-based reasoning. This classification provides a more fundamental, structural perspective on communication complexity theory and opens new avenues for theoretical investigation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Multiagent Systems: Agent Communication

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Similarly to the Chomsky hierarchy, we offer a classification of communication complexity measures such that these measures are organized into equivalence classes. Different from previous attempts of this endeavor, we consider two communication complexity measures as equivalent, if, when one is constant, then the other is constant as well, and vice versa. Most previous considerations of similar topics have been using polylogarithmic input length as a defining characteristic of equivalence. In this paper, two measures ${cal C}_1, {cal C}_2$ are constant-equivalent, if and only if for all total Boolean (families of) functions $f:{0, 1}^n imes{0, 1}^n ightarrow {0, 1}$ we have ${cal C}_1(f)=O(1)$ if and only if ${cal C}_2(f)=O(1)$. We identify five equivalence classes according to the above equivalence relation. Interestingly, the classification is counter-intuitive in that powerful models of communication are grouped with weak ones, and seemingly weaker models end up on the top of the hierarchy.

Problem

• Classifying communication complexity measures into equivalence classes
• Defining equivalence based on constant communication complexity
• Identifying counter-intuitive hierarchy among communication models

Innovation

• Constant-equivalent classification of communication complexity measures
• Equivalence based on mutual constant complexity bounds
• Hierarchy grouping powerful and weak communication models together