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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]
Counter machines with two counters are Turing complete: they can simulate any appropriately-encoded Turing machine, but there are some simple functions that they cannot compute. Counter machines with only a single counter can recognize a proper superset of the regular languages and a subset of the deterministic context free languages .
A Turing degree is an equivalence class of the relation ≡ T. The notation [X] denotes the equivalence class containing a set X. The entire collection of Turing degrees is denoted . The Turing degrees have a partial order ≤ defined so that [X] ≤ [Y] if and only if X ≤ T Y. There is a unique Turing degree containing all the computable ...
Turing completeness is the ability for a computational model or a system of instructions to simulate a Turing machine. A programming language that is Turing complete is theoretically capable of expressing all tasks accomplishable by computers; nearly all programming languages are Turing complete if the limitations of finite memory are ignored.
Although Horn clause logic programs are Turing complete, [1] [2] for most practical applications, Horn clause programs need to be extended to "normal" logic programs with negative conditions. For example, the definition of sibling uses a negative condition, where the predicate = is defined by the clause X = X :
DETROIT (Reuters) -U.S. automakers Ford Motor and General Motors will donate $1 million each, along with vehicles, to U.S. President-elect Donald Trump's January inauguration, company ...
Among other things, he developed a proof showing that the Rule 110 cellular automaton is Turing-complete. Cook presented his proof at the Santa Fe Institute conference CA98 before the publishing of Wolfram's book—an action that led Wolfram Research to accuse Cook of violating his NDA and resulted in the blocking of the publication of the ...