K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5(1):19-30, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Theta Zand E = f( xt; i) ty; i 1) x; y 2 Sigma; t 2 Sigma ; i 2 Zg. To our knowledge this type of graph has been first used by Sutner and Maas [16] to show that a particular robot motion planning problem in the presence of moving obstacles is Pspace hard. It was used again by Sutner [15] to prove that the predecessor of every recursive configuration is also recursive. A non empty sequence (possibly infinite to the left, to the right or in both directions) of vertices ( w i ; i) in G1 is a path if and only if for at at most a finite number of indices i we have w ....

K. Sutner, De Bruijn graphs and linear cellular automata, Complex Systems, 5 (1991), pp. 19--30.

....of co NP complete problems if the size of the considered finite configurations are supposed bounded by the size of the representation of the considered CA. It is not the case for one dimensional CA for which the inverse can be computed in polynomial time in the size of the transition table (see [12] for another point of view on this problem) Our result is proved by introducing a very adequate set of tiles having an ad hoc property. J. Kari has proved the undecidability of the surjectivity problem for two dimensional CA [6] by introducing a very complicated set of tiles. We keep the ideas ....

K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5:19--30, 1991. 19

....ffi 0 defined by ffi 0 (su; ffi(sut) ut for s; t 2 S, u 2 S n Gamma1 , the underlying graph of which is a De Bruijn graph. The properties of De Bruijn graphs (as, for example, being connected and hamiltonian) can be successfully exploited to get decidability results on CA as in [36] [71]. See also [72] Representation 2. De Bruijn graphs In the case of dimension 1, we also get description of ffi aid of a graph defined as follows: its vertices correspond to, and are labeled by, the words of length n on S, and there is an arrow from su to ut, labeled by ffi(sut) 1 0 0 1 10 ....

....[48] as it is the case for the examples of Figure 6. Incidentally, let us note here that additive cellular automata have been much studied : their arithmetic nature allows to think them easier to understand although they may have arbitrary complex behaviors (along with [48] see [4] 43] [71], 2] 3] and relevant references in these papers) Some attempts to generalize linear cellular automata are set about, with bilinear ones for example [6] 7] The above mentioned difficulty leads to pay attention to the canonical topological structure (evoked in 2:4:3: of the configurations ....

Sutner K. De Bruijn graphs and linear cellular automata. Complex Systems Vol.5 no. 1: 19--30, 1991.

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K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5(1):19-30, 1991.

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Sutner, K.: De Bruijn Graphs and linear cellular automata, Complex Systems, 5(1), 1991, 19--30.

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K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5(1):19-30, 1991.

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K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5(1):19-30, 1991.

....in general have exponential space complexity. Note, though, that B(ae) can be used to determine surjectivity (as well as openness and injectivity) in quadratic time. The fast algorithm uses products of semiautomata rather than the power automaton construction needed for minimization, see [1] and [22]. On the other hand, there is no hope of obtaining a strict classification of cellular automata along the lines of Wolfram s classes by means of easily computable parameters: questions relating to the Wolfram hierarchy are in general highly undecidable, even if we restrict our attention to finite ....

K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5(1):19--30, 1991.

....However, the Garden of Eden Problem is co NLOGcomplete for one dimensional cellular automata, and co NP complete in the two or higher dimensional case, see [9] Another surprising feature of the Garden of Eden problem can be observed with respect to recursive configurations. It is shown in [8] that for one dimensional cellular automata, it is reasonable to consider the subspace of recursive configurations with respect to Gardens ofEden: any recursive configuration that has a predecessor already has a recursive predecessor. However, this predecessor cannot in general be computed ....

K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5(1):19--30, 1991. 10

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