A Hierarchy for Constant Communication Complexity
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.