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In computability theory, a system of data-manipulation rules (such as a model of computation, a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine [1] [2] (devised by English mathematician and computer scientist Alan Turing).
Among the 88 possible unique elementary cellular automata, Rule 110 is the only one for which Turing completeness has been directly proven, although proofs for several similar rules follow as simple corollaries (e.g. Rule 124, which is the horizontal reflection of Rule 110). Rule 110 is arguably the simplest known Turing complete system. [2] [5]
Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. [3] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function. Lambda calculus may be untyped or typed. In typed lambda calculus ...
Turing's first task had to write a generalized expression using logic symbols to express exactly what his Un(M) would do. Turing's second task is to "Gödelize" this hugely long string-of-string-of-symbols using Gödel's technique of assigning primes to the symbols and raising the primes to prime-powers, per Gödel's method.
Turing did not explicitly state that the Turing test could be used as a measure of "intelligence", or any other human quality. He wanted to provide a clear and understandable alternative to the word "think", which he could then use to reply to criticisms of the possibility of "thinking machines" and to suggest ways that research might move forward.
It is possible to define a complete system using only one (improper) combinator. An example is Chris Barker's iota combinator, which can be expressed in terms of S and K as follows: ιx = xSK. It is possible to reconstruct S, K, and I from the iota combinator. Applying ι to itself gives ιι = ιSK = SSKK = SK(KK) which is functionally ...
In other words, if a language is defined based on some oracle in C, then we can assume that it is defined based on a complete problem for C. Complete problems therefore act as "representatives" of the class for which they are complete. The Sipser–Lautemann theorem states that the class BPP is contained in the second level of the polynomial ...
In the 1980s, Stephen Wolfram engaged in a systematic study of one-dimensional cellular automata, or what he calls elementary cellular automata; his research assistant Matthew Cook showed that one of these rules is Turing-complete. The primary classifications of cellular automata, as outlined by Wolfram, are numbered one to four.