A New Kind of Science

in a Nutshell

David SehnalQIPL at FI MU

Motivations

• How does anything complicated get

produced in nature?

• Traditional Science – i.e. calculus,

Newtonian physics, …

• What if there is a more general underlying

principle?

– Simple programs

1D Cellular Automata

Cell to be updated

Left neighbor Right neighbor

New color

• This type behavior is very surprising

1. Started from a single black cell

2. Used simple rules

3. Got something that looks to us

completely random

Is Complexity Special or Common?

Turing Machines

Substitution Systems

2D Cellular Automata

2D Turing Machines

3D Cellular Automata

• Simple rules do not imply simple behavior

• It seems that complex behavior is a very

common phenomenon

To sum it up…

Systems in nature

Snowflakes

• When a piece of a snowflake solidifies,

heat is released

• This heat prohibits ice nearby

• New rule: Piece solidifies if exactly on of

the neighborhood cells is solid

More systems in nature

• Fluid flow

• Patterns on shells

• Fundamental physics

• And many others …

Why can simple programs

produce complex behavior?

Computation

• All cellular automata can be though of as

doing computations

• One does not need to know the point of

the computation beforehand

Computation

Universality

• Some automata can simulate all others

• One only needs to specify the initial

conditions

Universality

• Once universality is reached, the behavior

is maximally sophisticated (from the

computational point of view)

• What about systems such as rule 30?

• Or systems in nature?

• How sophisticated are these?

More questions

Principle of Computational

Equivalence

Essentially any time the behavior of a

system looks to us complex, it will end up

corresponding to a computation of exactly

equivalent sophistication.

• Simple behavior (repetitive or nested)

corresponds to simple computations

• Complex behavior corresponds to

sophisticated computations

• Rule 110 is universal

Threshold of Universality

Where does all the complexity

come from?

… from the computational sophistication of the system observed

compared to the computational sophistication of the observer.

Thank you for your attention

Thank you for your attention