1LAAA Turing’s Machine A mechanical formalism (1937) –State (memory), rules (program) –Tape (data) Evolutionarily successful –Beat out Church’s mathematical formalism Mental perspective –A mapping of Gödel’s proof of incompleteness to programmable devices –In the war effort, effective computation was often accomplished by arrays of humans. © Sir John Dermont Turing

2LAAA Turing’s Mental Perspective “Computable numbers are those… calculable by finite means… the justification lies in the fact that the human memory is necessarily limited.” The machine is “directly aware” of symbols. “We may compare a man in the process of computing … to a machine…”

3LAAA Cellular Automata (Stan Ulam, et al.) A mass of cells, each of which –Has one of a (finite) number of states –Communicates to other “local” cells Cells compute their next state based on –Their current state –The states of neighboring cells Effectively massively parallel –Distinguished from Turing’s serial machine © Univ. Frankfurt

4LAAA von Neumann Could machines self-reproduce? First attempt –Robot in a “pool of parts” Second attempt (after work with Ulam) –Use cellular automaton to describe a universal Turing machine, programmed to write itself out

5LAAA von Neumann’s Automaton Arthur Burks, Essays on Cellular Automata, 1970.

6LAAA Digital Logic Computing Model Church’s Thesis: –Turing Machines, Cellular Automata, desktop machines (with sufficient memory) are all equivalent (“Turing equivalent”) Real machines constructed from digital components. –Small number of types computing boolean true/false values –One is sufficient: the NAND gate Output true exactly when neither input is true

7LAAA Lionel and Roger Penrose’s Self-Reproducing Analogue Two types of molecules White & Black Molecule are interlocked forming machines –White + Black –Black + White Machine-less universe does nothing Machine in universe generates others Machine flavor is preserved Godfrey Argent St. Andrews Universeity

8LAAA Conway’s “Game of Life” Each cell is alive or dead Population-motivated rules: –Life appears when exactly 3 neighbors –Life survives exactly when 2-3 neighbors dies offstable“spins”“glides”complex alive dead © St. Andrews

9LAAA Infinite generation and Turing Equivalence Certain configurations generate unbounded numbers of new individuals –E.g. “guns” generate streams of gliders Thinking of glider streams as wires –Carrying false values –Gliders interact to generate NAND gates

10LAAA Langton’s Ant Each cell is white or black An oriented “ant” sits atop one square Each step: –Ant inverts square –Moves forward –Turns left/right if new square is white/black –Effectively, ant forces any followers to stray Emergent behavior: mulls about based on input, then shoots off in direction Believed to be Turing equivalent © LENS Ventures

11LAAA Things to Think About Suppose the matrix of these cells is non- uniform; similar results? –E.g. Suppose we place Ant on aperiodic lattice –4-Connected –Not self-similar under shifts Not well understood

12LAAA More Recent Efforts Fredkin (MIT) & Toffoli’s (BU) CAM & programmable matter –Billiard balls with inelastic collision: Turing equivalent –Appropriate shaped bottles containing idealized gas molecules: Turing equivalent Fredkin’s Digital Philosophy –Possible Model of Everything –Believes cellular automata suitable model for physical laws DigitialPhilosophy.org © Tommaso Toffoli

13LAAA Wolfram’s “New” Kind of Science Popularized 1D cellular automata –Each row of cells generates the next –Cells are determined by small neighborhood above Approach to science is unfortunate Some rules (e.g. “rule 110”) Turing equivalent (Matthew Cook) © Wolfram Research

14LAAA Algorithmic Beauty of Sea-Shells Work of Hans Meinhart –Simulation of sea shell growth –Local enhancement; long-range inhibition –Study of periodicity and aperiodicity Follows in footsteps of Aristid Lindenmayer –Father of L-Systems © Scott Camazine © P. Prusinkeiwicz

15LAAA Turing’s 1952 Paper Alan Turing wrote a paper on “chemical basis for morphogenesis,” arguably the first paper in computational biology While many of the original ideas of that paper have been supplanted by better theory, it demonstrates a natural inclination to bring these two disciplines together Only a matter of time before some form of obvious biological computation is established The basis for ethical concern?

16LAAA Biology of Self-Organization 2001 work of Scott Camazine et al. –Ocular dominance in visual cortex (monkey) –Animal coat patterns –Behaviors of social insects –Flocking of birds & schooling of fish © Hubel, Harvard © Camazine © PIXAR