A New Kind of Scienceby Stephen Wolfram

Principle of Computational Equivalence

- Ting Yan, ty4k@cs.virginia.edu

Universality (Ch 11)

• Ability for one computational system to emulate another one - “as powerful as” relationship

• Not a new metric, used a lot in the theory of computation

• An Universal Cellular Automaton– 19 colors, 140 rules w/ “don’t cares”– able to emulate all 256 rules

Universality

• More colors or more neighbors don’t increase the computational power

• Rule 110 is also universal …

• CAs can emulate mobile automata, Turing Machines, substitution systems, register machines, number systems, logic circuits, RAMs, …– therefore CAs can emulate a general purpose computer,

with simple rules and complex initial conditions!

Emulating 90 with Universal CA

An Example: Prime Numbers

16 color - and Wolfram argues if you get some radio signal representing prime numbers, it may not be “intelligence”

More on Rule 110

• The complexity shifts from rules to initial conditions - how to program the initial conditions for some certain purpose?

• If we emulate a TM with a rule 110 CA, how efficient will it be?

• Wolfram “strongly suspects” all class 4 CAs are universal

A little more on CA

• Wolfram claims CAs are simple in order to surprise you with complex behaviors and universality

• But actually CAs are powerful - it has infinite heads compared with TMs!

• Initial Conditions are complex if you really want to do something as you want

Principle of Computational Equivalence

• “Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication”

• Almost all? - exceptions?

• Obviously simple? - not a good definition– pseudorandom number generator, not obviously

simple, nor really sophisticated

• Are DFA, PDA and TM equivalent?– Hierarchy of computational sophistication

What’s new?

• CAs with simple rules can be universal• How to exclude “obviously simple” processes?

– Run the process, and use your intuition?– A system is universal or not? - undecidable

• Upper limit on complexity?– “… introduces a new law of nature to the effect that

no system can ever carry out explicit computations that are more sophisticated than those carried out by systems like CAs or TMs”

Intuition Shift

• Wolfram argues that modeling or idealizing as actual system may lose key features of the system. True, but what he suggests is just to use his paradigm and run the simulation.

• Does it really work?

Continuity as Idealization

• “my strong suspicion is that at a fundamental level absolutely every aspect of our universe will in the end turn out to be discrete”

• Even if it were true, or continuous computations were not more “sophisticated” than discrete computations, there might be practical reasons for continuity

Irreducibility

• Reducing computation work plays an important role in “traditional science”

• Even with all information and rules there is irreducible amount of work to do.

• “There are many common systems whose behavior cannot in the end be determined at all except by something like an explicit simulation” ???

• “the only way … just to run them”

Will it die out? When?

Free Will

• Inconsistency between “free will” and definite laws

• Even you know the initial conditions, rules, irreducible work means if you want to know the result, what you can do is just to run it - which may be the source for superficial freedom

• Francis Crick: The Astonishing Hypothesis - due to sensitivity of initial conditions

Proof Searching

Gödel’s Theorem

• Completeness, consistency and universality can not coexist

• - if the system includes standard arithmetic

• Standard arithmetic serves as a threshold for universality

Nature or Articrafts?

True?

• “The Principle of Computational Equivalence now makes the fairly dramatic statement that even in these ways there is nothing fundamentally special about us”

• Why didn’t Church or Turing claim this?

Technology

• “it is perfectly possible for systems even with extreme simple underlying rules to produce behavior that has immense complexity” - but with extremely complex initial conditions, maybe very inefficiently