Introduction At the heart of the growth of a multi-cellular organism is the process of cellular division… … aka (in computing) self-replication

Introduction In the 50s, John von Neumann wanted to build a machine capable of self-replication Mark II Aiken Relay Calculator (Harvard, 1947)

Introduction In the 50s, John von Neumann wanted to build a machine capable of self-replication … but HOW?

Introduction In the 50s, John von Neumann wanted to build a machine capable of self-replication At the same time, Stanislaw Ulam was working on the computer-based realization of recursive patterns: geometric objects defined recursively. Ulam suggested to Von Neumann to build an “abstract world”, controlled by well-defined rules, to analyze the logical principles of self- replication: this world is the world of cellular automata.

Cellular Automata (CA) Conceived by S.M. Ulam and J. von Neumann Framework for the study of complex systems Organized as a two-dimensional array of cells Each cell can be in a finite number of states Updated synchronously in discrete time steps The state at the next time step depends of the current states of the neighbourhood The transitions are specified in a rule table

Environment states 0 = 1 = 2 = 3 = 4 = etc… Cellular Automata (CA)

Environment states neighbourhood Wolfram (1-D) Von Neumann Moore (Life)

Cellular Automata (CA) Environment states neighbourhood transition rules == ==

Cellular Automata (CA) Environment states neighbourhood transition rules Configuration Initial state of the array

Wolfram’s Elementary CA The simplest class of 1-D CA: two states (0 or 1), and rules that depend only on nearest neighbour values. Since there are 8 possible states for the three cells in a neighbourhood, there are a total of 256 elementary CA, each of which can be indexed with an 8-bit binary number. Rule 30

Wolfram’s Elementary CA Rule 30

Invented by John M. Conway (University of Cambridge) Popularised by Martin Gardner (Scientific American, october 1970, february 1971) Two-dimensional CA Two states per cell: dead and alive Eight neighbours (Moore) 2D CA: Game of Life

Birth of a cell Death of a cell Survival of a cell More than three neighbors Less than three neighbors Two or three neighbors Three neighbors

2D CA: Game of Life

Gliders: Glider gun: Game of Life: the glider

Game of Life

Von Neumann’s CA Environment states = 29 neighborhood = von Neumann transition rules = 29 5 ~ 20M Configuration Initial state of the array ~ 200k cells for the constructor, > 1M for the memory tape

Von Neumann’s Constructor Von Neumann’s Universal Constructor (Uconst) can build any finite machine (Ucomp), given its description D(Ucomp).

Von Neumann’s Constructor Von Neumann’s Universal Constructor (Uconst) can build a copy of itself (Uconst’), given its own description D(Uconst).

Von Neumann’s Constructor Von Neumann’s Universal Constructor (Uconst) can build a copy of itself (Uconst’) and of any finite machine (Ucomp’), given the description of both D(Uconst+Ucomp). The universal constructor is a unicellular organism. MOTHER CELL DAUGHTER CELL GENOME

Von Neumann’s Constructor Ordinary transmission states Standard signal transmission paths (wires) Non-excited: Excited: Input Output

Von Neumann’s Constructor Ordinary transmission states Property 1: Transmission of excitations with a unit delay

Von Neumann’s Constructor Ordinary transmission states Property 2: OR logic gate

Von Neumann’s Constructor Confluent states Signal synchronization Non-directional (depends on neighbor’s direction)

Von Neumann’s Constructor Confluent states Property 1: Introduction of double unit delay

Von Neumann’s Constructor Confluent states Property 2: AND gate

Von Neumann’s Constructor Confluent states Property 4: Fan-out

Von Neumann’s Constructor The XOR gate

Von Neumann’s Constructor The SR flip-flop

Von Neumann’s Constructor Sensitive states Construction Ordinary or special excitation No excitation

Demonstration