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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]
The halting problem is a decision problem about properties of computer programs on a fixed Turing-complete model of computation, i.e., all programs that can be written in some given programming language that is general enough to be equivalent to a Turing machine. The problem is to determine, given a program and an input to the program, whether ...
If τ 1 and τ 2 are terms, then (τ 1 τ 2) is a term. Nothing is a term if not required to be so by the first two rules. Derivations : A derivation is a finite sequence of terms defined recursively by the following rules (where α and ι are words over the alphabet { S , K , I , (, )} while β, γ and δ are terms):
to convert a Universal Turing machine "program", and the numerical symbols on the tape (Turing's "figures", symbols "1" and "0"), into a "theorem"—that is, a (monstrously long) string of sentences that define the successive actions of the machine, (all) the figures of the tape, and the location of the "tape head".
If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. [21] Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai, runs in quasi-polynomial ...
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.
Three examples of Turing patterns Six stable states from Turing equations, the last one forms Turing patterns. The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and spots, can arise naturally and autonomously from a homogeneous, uniform state.