Turing's contribution was in proving the concept regarding

computability (with regard to computing) and presenting it in algorithmic form, while Church and Godel presented their conclusions as mathematical theorems.

Mathematicians and computer scientists explore various aspects of how

computability and theoretical computer science enable scientists and philosophers to deal with mathematical and real-world issues in such areas as logic, mathematics, physical processes, real computation, and learning theory.

It includes the study of

Computability theory and Complexity theory.

Although the previously mentioned scalarizations are correct and linear, there is a conflict between completeness and

computability.

For an overview concerning left-computable real numbers and other classes of real numbers defined by

computability properties, see Zheng (2002).

The literature on

computability and decidability goes back to Kurt Godel's famous incompleteness proof of 1931.

Feigenbaum Parameter g Fixed point and dynamics double period Cellular Stephen Wolfram Evolved Class I and II automata behaviour Cellular Chris Langton Lambda Order automata parameter [lambda] < 1/3 Boolean nets Stuart Kauffman, Variable K Order K = 1 Sandpile Per Bak Shape of Flat and stable experiment sandpile Number theory Alan Turing Set of number

Computability Organization R.

For example, the ACM high school computer science curriculum (Merritt, 1994) includes very few references to few of the topics of the CM unit, only as optional topics, and the more recent ACM K-12 computer science curriculum (Tucker, Deek, Jones, McCowan, Sthepenson, & Verno, 2003) mentions only limits of

computability, in one of its five units, as one topic among 10.

The certainty of this statement was shown by fundamental research in

computability from the legendary code-breaker Alan Turing.

As a consequence,

computability, decidability, verification, program generation and search for solution can be exercised on "top level model" and supply designer with valuable data.

Chaitin [6] developed some insights into the nature of complex self-referential information systems: combining Shannon's information theory and Turing's

computability theory resulted in the development of Algorithmic Information Theory (AIT).

The

computability and the calculability are not interested in the efficiency of the algorithms.

Theory of Recursive Function and Effective

Computability.

Students come across the fact that Turing, Kleene and Church's ideas in the 30's, Chomsky's grammar hierarchy in the 50's, Petri nets, as well as Shepardson and Sturgis' register machines in the 60's were different formalisms to give an answer to the same question of what

computability is.

2) Nature does not find the problem of

computability so difficult, and indeed it would seem clear that the self-organised states allowed by non-equilibrium physics are produced with probability one.