Theory of computation is a separate branch of mathematics that studies the general properties of algorithms. It originated in the 30s of the 20th century.
Computation, however, can be traced in mathematics throughout the period of its existence. The need for accurate mathematical specification of the intuitive notion of algorithm became inevitable after realizing the impossibility of the existence of algorithms for solution of many massive problems, especially related to arithmetic and mathematical logic (the problem of truth and arithmetic formulas and the first order predicate calculus formulas, 10th Hilbert problem solving ‘Coherence Diophantine equations, etc.). To prove the non-existence of an algorithm it is necessary to have its precise mathematical definition, so after the formation of the concept of the algorithm as a new and separate entity, the primary problem was to find an adequate formal model of the algorithm and study their properties. In this, formal models have been proposed for both the original concept of the algorithm and for the original concept algorithmically computable function.
Students writing their research paper on theory of computation have to know that for the first time the concept of the computation appeared in the works of E. Borel (1912) and G. Weil (1921).
The first formal computation models were ?-function (Alonzo Church 1932) and general-recursive function (Kurt Godel, 1934). These classes were defined as functions, which graphs were generated according to ?-conversion computations and Herbrand-Gödel computations. In 1936, Stephen Cole Kleene extended the concept of general-recursive function in case of partial functions by introducing the notion of partial recursive functions, and described a class of such functions in purely functional terms. In 1943, Emily Post suggested a model of computable functions on the basis of a special form calculation he has introduced (canonical systems).
To formalize the concept of the algorithm accurate mathematical descriptions of algorithmic machines and computability have been proposed. The first formal model algorithmic machine was Turing machine (Alan Turing, Emil Post, 1936). From the later models, we note the normal algorithms (Markov, I952) and register machines (D. Sheperdson, G. Sterdzhis, 1963).
In 1936, A. Church and S. Kleene proved the convergence of general recursive functions and ?-conversions classes. Based on this fact and analysis of the ideas that led to these concepts A. Church has proposed a thesis about the convergence of AOF class and general-recursive functions. S. Kleene summarized this idea in the case of partial functions. Driven by A. Turing in 1937 match partial recursive classes of functions and features, calculated on Turing machines, has once again confirmed Church’s thesis. After these matches have been set for all known formal models of AOF. So there is every reason to believe that any of the above formal models adequately clarifies the intuitive notion of AOF.
Theory of computation emerged as part of mathematical logic, the concept of the computation is closely related to the concept of counting. Its first and most numerous application is in mathematical logic. Theory of computation is theoretical foundation of programming, it has application wherever there are algorithmic problems (foundations of mathematics, information theory, control theory, structural analysis, computational mathematics, probability theory, linguistics, economics, etc.).
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