A New Kind of Science
by 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