Very simple models, the self-modifying automata and chain of self-modifying automata, can explain self-referential properties of living beings

Very often, living beings seem able to change their functioning when external conditions vary. In order to study this property, we have devised abstract machines whose internal organisation changes whenever the external conditions vary. The internal organisations of these machines (or programs), are as simple as possible, functions of discrete variables. We call such machines self-modifying automata. These machines stabilise after any transient steps when they go indefinitely through a loop called p-cycle or limit cycle of length p. More often than not, the p in the cycle is equal to one and the cycle reduces to a fixed point. In this case the external value (v) can be considered as the index of function f such as: f(v) (v) = v and the machine has the property of self-replication and to be self-referential. Many authors, in computer and natural science, consider that self-referential objects are a main concept in comprehension of perception, behaviour and associations. In the third part, we have studied chains of automata. Only one automaton changes its internal organisation at each step. Chains of automata have better performances than single self-modifying automata: Higher frequency of fixed point occurrence and a shorter transient length. The performances of the chains of automata improve when the value of their internal states increases whereas the performances of single automata decrease.