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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).
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. [1]
Arithmetic-based Turing-complete machines use an arithmetic operation and a conditional jump. Like the two previous universal computers, this class is also Turing-complete. The instruction operates on integers which may also be addresses in memory. Currently there are several known OISCs of this class, based on different arithmetic operations:
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.
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 Z3 was demonstrated in 1998 to be, in principle, Turing-complete. [13] However, because it lacked conditional branching, the Z3 only meets this definition by speculatively computing all possible outcomes of a calculation. Thanks to this machine and its predecessors, Konrad Zuse has often been suggested as the inventor of the computer.
Such a machine is called queue machine and is Turing-complete. Tape memory: The inputs and outputs of automata are often described as input and output tapes. Some machines have additional working tapes, including the Turing machine, linear bounded automaton, and log-space transducer. Transition function
For each m > 1, the set of m-tag systems is Turing-complete; i.e., for each m > 1, it is the case that for any given Turing machine T, there is an m-tag system that emulates T. In particular, a 2-tag system can be constructed to emulate a Universal Turing machine, as was done by Wang (1963) and by Cocke & Minsky (1964).