Recent studies on self - replicating structures implemented on cellular automata

Cellular automata (CA for short) proposed by J. von Neumann and S. Ulam in the middle of 1960's are dynamical systems where time, space and states of sites are all discrete. They consist of a d-dimensional array of identical automata (cells), each of which has one of the nite states and changes its state according to the state-transition rule sets mapping from states of its neighbor cells (usually including itself) to its next state. CA with d = 1; 2; 3 have been practically studied so far. This note reports only 2-dimensional CA. Von Neumann devised CA for the mathematical modelling of self-reproduction 1 of life forms, and contrived a universal constructor-computer automaton (Fig. 1) on 29-state 5-neighbor CA with weak rotational symmetry [23] 2 , in order to realize the idea he previously considered [22]. It actually has the capability of universal construction and computation faithful to the given instruction tape, and if set up with an instruction tape describing how to construct the automaton itself, it reproduces itself (Fig. 2). It was believed to be the rst to model biological self-reproduction by theoretical measures (but Arbib's work [1] presented in the same year as von Neumann's was also notable). After von Neumann's work, many variations were invented. For example, Codd simpli ed von Neumann's universal automaton into that on much simpler 8-state 5-neighbor CA with strong rotational symmetry [6]. Vit anyi created sexually reproducing universal automaton (Fig. 3) [21]. However, all of these models including von Neumann's were in prohibitive size, due mainly to their constructional universality, so it was not possible at that time to actually simulate them on computer 3 . They merely remained theoretical work.