Memoryless Systems Generate the Class of all Discrete Systems

Abstract

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.

A multiplicity of statistical models for modeling the degeneracy of living communicating holons

“…In Section 3, holons are abstracted as automata, also called Discrete Dynamical Systems in [16,17]. Among all possible automata, memoryless Boolean automata are then our main focus because they can be interpreted as basic stimulus-response holons with the capacity to detect a signal in noise and possible interference.…”

“…Therefore, we rather focus on automata and memoryless Boolean automata, particularly suitable for a direct implementation on a computer. Mathematically, we follow [16,17] -where automata are called Discrete Dynamical Systems -and pose the following definition: Definition 1 (Automaton) Given two sets X in and X out , an automaton, of which input (resp. output) signals are the elements of X in (resp.…”