1 LAAA

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

2 LAAA

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…”

3 LAAA

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

4 LAAA

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

5 LAAA

von Neumann’s Automaton

Arthur Burks, Essays on Cellular Automata, 1970.

6 LAAA

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

ab out

7 LAAA

Lionel and Roger Penrose’sSelf-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

8 LAAA

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 off stable “spins” “glides” complex

alivedead

© St. Andrews

9 LAAA

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

10 LAAA

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

11 LAAA

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

12 LAAA

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

13 LAAA

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

14 LAAA

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

15 LAAA

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?

16 LAAA

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

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

© Hubel, Harvard

© Camazine

© PIXAR