# Download PDF by E. Goles, Servet Martínez: Neural and Automata Networks: Dynamical Behavior and

By E. Goles, Servet Martínez

"Et moi, ..., si j'avait Sll remark en revenir. One sennce arithmetic has rendered the human race. It has positioned good judgment again je n'y serais aspect alle.' Jules Verne whe," it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded non- The sequence is divergent; consequently we should be smse'. capable of do whatever with it. Eric T. Bell O. Heaviside arithmetic is a device for proposal. A hugely useful software in an international the place either suggestions and non linearities abound. equally, every kind of elements of arithmetic function instruments for different elements and for different sciences. using an easy rewriting rule to the quote at the correct above one reveals such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One carrier common sense has rendered com puter technological know-how .. .'; 'One carrier classification thought has rendered arithmetic .. .'. All arguably actual. And all statements accessible this fashion shape a part of the raison d'!ltre of this sequence.

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0- + +- - - + _+++ .......... 7. O)' of the BNN. The internal sites are weighted with -1 and the external ones with + 1. O) if jEVo otherwise where aj = -1 for internal sites and aj = +1 otherwise. Any site (i,j') E 7h x 7h\I x I is fixed in the quiescent state 0 (boundary condition). B. Synchronous iteration of the BNN for ration converges to a cycle of period 2. iIi = 45. 6. Bounded Majority Network. i') - b) (j,j')EV oN As in the pr~vious example, we suppose that all sites in E x E \ I x I are fixed in state 0 (boundary condition).

41 ALGEBRAIC INVARIANTS ON NEURAL NETWORKS Now call Se the cardinality of G E ~(Y). IT G E ~(") (Y) choose anyone of its elements and call it t,. e. G E dY) \ ~(")(Y), take t, such that t, - k( mod T) i G (being G a k-chain t, is necessarily unique). Then G = it, + Ik( mod T) : ° ~ 1 < sc}. For any G E dY) write te = t, + (s. - l)k( mod T) which is an element of G. 2) == f. + IJ mod T) To illustrate above concepts we shall exhibit two examples. Consider the periodic sequence Y = (0,0,1,1,0,0,1,1,0,0,1,1) with T = 12.

0) = (q', s' , l) then in the next step of time the internal state of T is q', the new symbol in coordinate rno is s' , and the head moves to position rno + l. At each time step the configuration of the Turing Machine T is determined by: its internal state q E Q, the tape coordinate rn E ~ at which the head points, and the string of symbols 8 = (s; : i E ~) E E~n on the tape, where E~n contains the countable set of elements 8 = (s; : i E ~) E E~ with only a finite number of symbols different from o.