Automata theory

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An automat is a machine taking in entry unspecified proposals like, for example, “(1 + 3)/(5 - 1) = 1” and turning over true or false, according to until one waits of the machine; a syntactic or semantic evaluation. In the case of a syntactic evaluation, one wishes that the machine evaluates if the proposal is correctly formed or not. For example, “(1 + 3)/(5 - 1) = 2” is correctly formed whereas “++1 + 3)/) 5 - 1) = 2” is not it. In the case of a semantic evaluation, one wishes that the machine evaluates the value of truth of the proposal, for example: “(1 + 3)/(5 - 1) = 2” is false whereas “(1 + 3)/(5 - 1) = 1” is true.

The study of the automats thus relates to the study of the syntax of the formal languages and of the natural languages as well as calculability, that is to say the study of the limits of calculation which a particular machine can carry out. The study of syntax and that of semantics are equivalent; the limits of syntactic recognition of an automat are equivalent to its computing power and vice versa. Indeed, that one uses an automat to carry out the syntactic one or semantics is unknown machine, this distinction is purely artificial.

In 1937, Alan Mathison Turing showed that a particular kind of automat, the Machine of Turing, could calculate what any other automat could calculate and which it was impossible to better do. This machine was as powerful, on the level of calculation, as any mathematician. It also showed that there exists a particular kind of machine of Turing, the universal machine of Turing, which can reproduce the behavior of any other machine of Turing; the computer had been born.

As opposed to what one could believe, the machine of Turing is not the completion of the automata theory but well the beginning. The automata theory was developed to include/understand how the extraordinary capacity of calculation of the machines of Turing emerged. 1956 had to be waited until so that the first theorem other than those of Turing appears, it is a question of the Théorème of Kleene, showing equivalence between the graphs of transition, the finite-state machines and regular grammars. The theorem of Kleene carried out for the first time the adequacy of an automat, considered as an acceptor of language, and a formal grammar considered as a generator of language.

Stephen Cole Kleene studied a simplified version of the machine of Turing for including/understanding best, it stripped the machine of Turing of its memory (its ribbon) and found a means of describing what this simplified machine does, that is to say which language it generates. The method of Kleene was thereafter the principal method employed for advance of research.

During following years, the researchers overflowed of imagination to create various variations of the machine of Turing; machine with a pile, with two piles and more, machine with tails, machine of post, machines with registers, etc They realized quickly that in spite of their efforts, the variation was only one illusion and that any automat fitted in one of the four categories which they had discovered. The computing power of the automats could be organized in a hierarchy on four levels, the automats of a higher hierarchy being able to calculate all that an automat of lower level can calculate.

This fact already had however been mentioned by the linguist Noam Chomsky who studied formal grammars, it had discovered in 1956 qu ' it existed only four types of grammars, higher grammars including/understanding the expressivity of lower grammars. The strict equivalence of these four grammars with the four types of automats does of Noam Chomsky one of the founding fathers of theoretical data processing.

Birth of a new science

The machine of Turing is not simply an original abstraction, it is the answer to a fundamental philosophical question: mathematics of the human spirit or forms is a unit with nature an artefact? In other words, the is Harmonie of the spheres only one illusion produced by the cognitive apparatus of human or the natural laws are indeed written in mathematical language?

Kant was extremely shaken by the adequacy of the equations of the physics of Newton to phenomenologic reality. Indeed, how mathematics, pure expression of knowledge a priori of the understanding, can carry out any adequacy with a reality of which she is unaware of all? If the real nature of mathematics is a philosophical question of importance since antiquity, the success of physics gave to the last style this old question.

The Thèse of Church-Turing which postulates that no machine can do better than a machine of Turing (to carry out a calculation that a machine of Turing could not carry out) is a final answer to the question of the nature of mathematics. Indeed, this postulate implies directly which if no machine can better do, then no natural phenomenon either because if not it would be possible to build an impossible machine by using this natural phenomenon. Consequently, any natural phenomenon is calculable (not to confuse with modélisable) and the adequacy of a calculable model with a calculable phenomenon is certainly not surprising.

The study of the automats is thus, in fact, the study of the fine structure of the universe and is, in this direction, a natural science. The automat term unfortunately lets think that this new science relates to only the sophisticated artefacts which are the computers. However, the definition even of the deterministic finite-state machine which is the simplest automat, is a system made up of distinguishable states from/to each other and whose change of state requires an event (a cause). This definition includes any recognizable natural phenomenon, i.e., on which it is possible to develop knowledge. The use of this type of automat in natural science and human is now largely widespread. We owe with Norbert Wiener the diffusion of the use of automats for the formalization of knowledge (Cybernétique).

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