Motivated by the concept of degeneracy in biology [3], we establish a first connection between the Multiplicity Principle [4, 5] and mathematical statistics. Specifically, we exhibit two families of tests that satisfy this principle to achieve the detection of a signal in noise.

Automata are machines, which receive inputs, accordingly update their internal state, and produce output, and are a common abstraction for the basic building blocks used in engineering and science to describe and design complex systems. These arbitrarily simple machines can be wired together—so that the output of one is passed to another as its input—to form more complex machines. Indeed, both modern computers and biological systems can be described in this way, as assemblies of transistors or assemblies of simple cells. The complexity is in the network, i.e., the connection patterns between simple machines. The main result of this paper is to show that the range of simplicity for parts as compared to the complexity for wholes is in some sense complete: the most complex automaton can be obtained by wiring together direct-output memoryless components. The model we use—discrete-time automata sending each other messages from a fixed set of possibilities—is certainly more appropriate for computer systems than for biological systems. However, the result leads one to wonder what might be the simplest sort of machines, broadly construed, that can be assembled to produce the behaviour found in biological systems, including the brain.

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Communicating living systems detect and process a multiplicity of events with degeneracy, to continuously cope with environmental aleatoric incertitude. The concept of holon, communicating at various scales of living organizations, is hereafter formalized through dynamical systems driven by the multiplicity of statistical models. Then, the stimulus-response of elementary biological holons can be modeled by memoryless Boolean automata with different signal processing methods, in presence of noise and stochastic interference. Detection of a specified signal, to update the automaton state, can be performed via multiple families of update functions, with differentiated balances between sensitivity and specificity in presence of interference: (i) Neyman-Pearson update functions provide the best possible sensitivity to detect the signal of interest in absence of interference, but cannot guarantee a desired specificity; (ii) by detecting large amplitudes of any signal in noise, update functions based on Random Distortion Testing yield a suboptimal sensitivity to detect the signal in noise, but guarantee a wanted specificity even in presence of interference. Thus, statistical inference theories offer functional and structural redundancy and open prospects to model fractal-like holarchies, via networks of communicating degenerated automata, to feature properties of the immune system.