The Noncomputability of Immune Reaction Complexity: Algorithmic Information Gaps under Effective Constraints

TL;DR

This paper explores the inherent limits of immune reaction complexity using algorithmic information theory, introducing measures like NAQ to quantify task hardness and linking advice complexity to rate-distortion theory.

Contribution

It develops a novel theoretical framework for analyzing reaction complexity and advice minimality, introducing the NAQ index and establishing bounds and invariance properties.

Findings

NAQ provides a scale-free measure of task hardness.

Advice complexity relates to rate-distortion with error epsilon.

Empirical NAQ estimates converge via DKW bounds.

Abstract

We introduce a validity-filtered, certificate-based view of reactions grounded in Algorithmic Information Theory. A fixed, total, input-blind executor maps a self-delimiting advice string to a candidate response, accepted only if a decidable or semi-decidable validity predicate V(x, r) holds. The minimum feasible realizer complexity M(x) = min_{r: V(x,r)=1} K(r), with K denoting prefix Kolmogorov complexity, measures the minimal information required for a valid outcome. We define the Normalized Advice Quantile (NAQ) as the percentile of M(x) across a reference pool, yielding a scale-free hardness index on [0, 1] robust to the choice of universal machine and comparable across task families. An Exact Realizer Identity shows that the minimal advice for any input-blind executor equals M(x) up to O(1), while a description plus selection upper bound refines it via computable feature maps, separating description cost K(y) from selection cost log i_y(x). In finite-ambiguity regimes M(x) approximately equals min_y K(y); in generic-fiber regimes the bound is tight. NAQ is quasi-invariant under bounded enumeration changes. An operational converse links NAQ to rate-distortion: communicating advice with error epsilon requires average length near the entropy of target features. Extensions include a resource-bounded variant NAQ_t incorporating time-penalized complexity (Levin's Kt) and an NP-style setting showing linear worst-case advice n - O(1). Finally, a DKW bound guarantees convergence of empirical NAQ estimates, enabling data-driven calibration via compressor-based proxies.