By Bruno Durand (auth.), Eric Goles, Servet Martínez (eds.)

This booklet includes the classes given on the 5th tuition on complicated structures held at Santiago, Chile, from ninth .to thirteenth December 1996. At this faculty met researchers engaged on parts similar with fresh developments in complicated platforms, which come with dynamical platforms, mobile automata, symbolic dynamics, spatial structures, statistical physics and thermodynamics. Scientists operating in those matters come from numerous components: natural and utilized arithmetic, physics, biology, laptop technology and electric engineering. each one contribution is dedicated to at least one of the above topics. quite often they're dependent as surveys, proposing whilst an unique perspective concerning the subject and displaying as a rule new effects. The paper of Bruno Durand provides the cutting-edge at the relationships among the notions of surjectivity, injectivity and reversibility in mobile automata whilst finite, countless or periodic configurations are thought of, additionally he discusses decidability difficulties similar with the category of mobile automata in addition to worldwide homes pointed out above. The paper of Eric Goles and Martin Matamala supplies a uniform presentation of simulations of Turing machines by means of mobile automata. the most factor is the encoding functionality which needs to be fastened for all Turing laptop. during this context identified effects are revised and new effects are presented.

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L) and time vertically (indices t = 1, 2, ... ,T). We call M the number of cells in the vertical direction of our space-time diagrams; in general T # M. In the figures, Land M where chosen with regard to visualization convenience. Difference patterns (DP) are determined by changing the value at the centre of the first line, and substracting the spacetime pattern with the change from that without the change. In all cases, we assume cyclic boundary conditions. All radii of circular neighbourhoods are made dimensionless after dividing them by the length of a cell edge.

In the first work it was shown the existence of a universal cellular automaton F which limit set A(F) = niEN Fi(AN) is regular in Chomsky's hierarchy of languages. In the second one the author study the symbolic dynamics of the limit set. Two behaviors are distinguished: (1) the limit set of F is attained after a finite number of iterations of the map: A(F) = FP(AN) for some p E Nj in this case we say that F is a stable cellular automaton. (2) A(F) is an infinite intersection of forward images: for any pEN, FP(AN) FP+l(AN).

ORDER The value of g(i) is 2(i)1(i) and the value of h(i,j) = 2(i)(j)(i). When M moves rightward the evolution on the cellular automaton is: .. .. .. .. .. 4 2 ko 1 ko 1 ko 1 ko 1 ko 1 ko 2 4 2 4 2 1 ko 1 ko u 3 u' 1 u' v u' 1 u' 1 u v' 2 4 k2 3 2 u' v' k2 1 2 k2 1 v' 1 k2 1 k2 v' k2 1 k2 1 k2 2 4 2 4 2 ... ... ... On the other hand, when M moves rightward the evolution on the cellular automaton is: .. .. .. " . . 2 4 2 4 2 ko 1 ko 1 ko 1 ko 1 v' ko ko 1 2 4 ko v' u v' 3 ko 1 2 u' 1 ko v' 2 v u' 1 u' 1 u 3 u' 1 u' k2 2 4 2 4 2 k2 1 k2 1 k2 1 1 k2 1 k2 1 k2 1 k2 1 k2 The evolution of the part " .