A New Kind of Science by Stephen Wolfram

1
A New Kind of Scienceby Stephen Wolfram

• Principle of Computational Equivalence
• - Ting Yan, ty4k_at_cs.virginia.edu

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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/ dont cares
• able to emulate all 256 rules

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Universality

• More colors or more neighbors dont 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!

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Emulating 90 with Universal CA
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An Example Prime Numbers
16 color - and Wolfram argues if you get some
radio signal representing prime numbers, it may
not be intelligence
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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

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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

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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

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Whats 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

10
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?

11
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

12
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

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Will it die out? When?
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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

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Proof Searching
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Gödels Theorem

• Completeness, consistency and universality can
  not coexist
• - if the system includes standard arithmetic
• Standard arithmetic serves as a threshold for
  universality

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Nature or Articrafts?
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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 didnt Church or Turing claim this?

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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