Identification capacity and rate-query tradeoffs in classification systems

Abstract: We study a one-shot identification analogue of rate-distortion for discrete classification under three resources: tag rate L (bits of side information stored per entity), identification cost W (attribute-membership queries per identification, excluding global preprocessing and amortized caching), and distortion D (misclassification probability). The question is to characterize achievable triples (L,W,D) when a decoder must recover an entity's class from limited observations. Zero-error barrier. If two distinct classes induce the same attribute profile, then the observation pi(V) is identical for both and no decoder can identify the class from attribute queries alone. Thus, if the profile map pi is not injective on classes, zero-error identification without tags is impossible (a zero-error feasibility threshold). Achievability and converse at D=0. With k classes, nominal tags of L = ceil(log2 k) bits enable O(1) identification cost with D=0. Conversely, any scheme with D=0 must satisfy L >= log2 k bits (tight). Without tags (L=0), identification requires Omega(n) queries in the worst case and may incur D>0. Combinatorial structure. Minimal sufficient query families form the bases of a matroid; the induced distinguishing dimension is well-defined and links to zero-error source coding via graph entropy. We illustrate implications for type systems, databases, and biological taxonomy. All results are mechanized in Lean4 (6000+ lines, 0 sorry).