Timing and stability in cellular automata - tbi. ?· Timing and stability in cellular automata Konstantin…

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<ul><li><p>Timing and stability in cellular automata</p><p>Konstantin Klemm</p><p>Bioinformatics GroupUniversity of Leipzig, Germany</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 1 / 25</p></li><li><p>Definition</p><p>finite set of states S , often |S | = 2</p><p>mapping (rule table) f : Sz S</p><p>a lattice of n sites with coordination number z where site i hasneighbours a(i , 1), a(i , 2), . . . a(i , z)</p><p>time-discrete dynamics of lattice site i</p><p>si (t + 1) = f [sa(i ,1)(t), . . . sa(i ,z)(t)]</p><p>All lattice sites are updated in synchrony.</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 2 / 25</p></li><li><p>Purpose</p><p>Computer Science: Models of computation,e.g. Game of Life and rule 110 are Turing-complete.</p><p>Artificial Life: Study of self-reproducing structures</p><p>Physics: Abstractions of spatio-temporal dynamics, pattern formation</p><p>Motto: Simplest rules may yield most complex patterns / computations.</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 3 / 25</p></li><li><p>Putative CA dynamics on plant leaves</p><p>Peak et al., PNAS (2004)</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 4 / 25</p></li><li><p>CA model for morphogenesis in Hydrasp</p><p>ace </p><p>time</p><p>Ped</p><p>Ped/CnNK2</p><p>1</p><p>rump</p><p>foot</p><p>Rohlf &amp; Bornholdt, JSTAT (2005)</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 5 / 25</p></li><li><p>Seashell</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 6 / 25</p></li><li><p>Seashell and elementary CA rule 22</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 7 / 25</p></li><li><p>Seashell and elementary CA rule 30</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 8 / 25</p></li><li><p>Role of noise?</p><p>Are the observed patterns robust under small stochasticperturbations?</p><p>Stability analysis in continuous dynamical systems: Make a smallperturbation and check if the system returns to the fixed point / limitcycle.</p><p>Here: Discretized state space. What is a small perturbation?</p><p>Typical implementation of noise in CA:</p><p>Stochastic asynchronous update</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 9 / 25</p></li><li><p>Rule 22, comparison sync / async update</p><p>synchronous update asynchronous update async, another realization</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 10 / 25</p></li><li><p>Rule 150, comparison sync / async update</p><p>synchronous update asynchronous update async, another realization</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 11 / 25</p></li><li><p>Rule 90, comparison sync / async update</p><p>synchronous update asynchronous update async, another realization</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 12 / 25</p></li><li><p>Loss of memory</p><p>0 20 40 60 80 100 120 140 160 180 200time</p><p>0.01</p><p>0.1</p><p>1in</p><p>form</p><p>atio</p><p>n [b</p><p>its]</p><p>information of a cells state at time tmutual information between cells state at time tand initial condition</p><p>Evolution of mutual information between one cells state at time t and thefull initial configuration.</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 13 / 25</p></li><li><p>Summary so far</p><p>Deterministic CA:Complex (interesting) spatio-temporal pattern formation</p><p>CA with noise implemented as stochastic update order:Largely irreproducible dynamics</p><p>Devastating effect of stochastic asynchronous update known for long,cf. Ingerson and Buvel, Physica D (1984)</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 14 / 25</p></li><li><p>How to proceed</p><p>Solution 0</p><p>Use more states and more complicated interactions to implementadditional clock signalscf. Nehaniv, Int. J. Alg. Comp. (2004)</p><p>Solution 1</p><p>Stick to simple rules</p><p>Consider a less destructive type of perturbation.</p><p>Get happy.</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 15 / 25</p></li><li><p>Framework for perturbed timing</p><p>Now time t R, no longer discrete.</p><p>Expand initial condition s(0) to unit interval:</p><p>si (t) =</p><p>{</p><p>fi(s(0)) , if t &gt; 1 isi (0) , otherwise</p><p>where i [0, ] is drawn randomly and independently for each node i .</p><p>Let the system evolve according to</p><p>si (t + 1) = </p><p>[</p><p>(2c)1</p><p>t+c</p><p>tc</p><p>fi (s())d 1/2</p><p>]</p><p>where is the step function and 1/2 &gt; s &gt; 0.</p><p>Check, if system autonomously regains synchrony.</p><p>Klemm &amp; Bornholdt, PNAS 102, 18414 (2005); Phys. Rev. E 72, 055101(R) (2005).</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 16 / 25</p></li><li><p>CA rule 22, initial timing perturbation</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 17 / 25</p></li><li><p>CA rule 22, spatial coarsening of time lags</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 18 / 25</p></li><li><p>CA rule 22, timing perturbation heals</p><p>t=240</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 19 / 25</p></li><li><p>Probability of healing</p><p>0 200 400 600 8000</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>1</p><p>prob</p><p>. tha</p><p>t per</p><p>turb</p><p>atio</p><p>n he</p><p>als </p><p>out</p><p>CA rule 22</p><p>0 200 400 600 800number of cells N</p><p>CA rule 150</p><p>0 200 400 600 800</p><p>CA rule 110</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 20 / 25</p></li><li><p>Time until healing</p><p>100 1000number of cells N</p><p>102</p><p>103</p><p>104</p><p>105</p><p>rule 22rule 150slope 2slope 1</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 21 / 25</p></li><li><p>Unstable pattern in rule 110</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 22 / 25</p></li><li><p>Overview of results</p><p>stable elementary rules0, 4, 22, 32, 36, 54, 72, 76, 104, 128, 132, 160, 164, 200, 204, 218,222, 236, 250,254</p><p>partially stable el. rules (strong dependence on n)90, 122, 126, 150</p><p>unstable elementary rules50, 94, 108, 110, 178</p><p>Elementary CA fail to produce sustained synchronous blinking of allcells</p><p>Game of Life: Blinkers, gliders, spaceships etc. are unstable</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 23 / 25</p></li><li><p>Further results</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 24 / 25</p></li><li><p>Summary</p><p>Stable and unstable CA rules can be distinguished by theirresponse to minimal timing perturbations.</p><p>Konstantin Klemm (Leipzig) Timing &amp; Stability in CA 25 / 25</p></li></ul>