Wolfram’s New Kind of Science

12/17/2002
Traditional science would suggest that standard mathematical analysis would provide a basis for thinking about the behavior of cellular automata and other systems with simple underlying rules. But such analysis is only useful when the overall behavior is fairly simple (Chapter 10). Traditional Science, according to Wolfram, has limited applicability when trying to understand complex behavior. Wolfram’s New Kind of Science attempts to develop a science that can address this kind of complexity. While most people seem to get lost in the details of this massive book, Wolfram put forward a few key ideas that should not be missed:

The Notion of Computation
Initial conditions of a cellular automata system can be viewed as corresponding to the input of a computation. The state of the system after a given number of steps can be viewed as corresponding to the output. Even though the internal workings of two systems may have little in common, the computations they perform may nevertheless be very similar.

The Principle of Universality
It is possible to build universal systems whose underlying construction remains the same but which can be made to perform different tasks just by being programmed in different ways. For example the computer hardware remains fixed, but the computer can be programmed for different tasks by loading different pieces of software. Cellular automata can also be used to emulate other types of systems. (p.656)

Implications of Universality
All universal systems are ultimately capable of exactly the same kinds of computations. One might assume that by using more sophisticated underlying rules one could construct a system with greater computational capabilities, but as soon as one has passed the threshold of universality its properties are remarkably independent of the details of its construction. Universality is far more common than previously thought. (i.e., Rule 110, Chapter 3)

The Principle of Computational Equivalence
Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication. (Chapter 12)

The Principle of Computational Irreducibility
Even with complete and accurate rules and initial conditions it would still take an irreducible amount of computational work to reach the result. (p.737)

Implications
The implications of these few axioms are far reaching. Wolfram briefly discusses their application in technology (p. 840) but as he suggests they can be applied in many fields. The one that immediately comes to mind is epistemology.

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~ by severalfourmany on December 17, 2002.

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