In Logique mathematics, one calls problem of the decision the fact of determining in a mechanical way, by a algorithm, if a statement is a theorem of first order levelling logic, i.e. if it is derived in a system from deduction (see Système in Hilbert, Calcul of the séquents, natural Déduction), without other axioms that those of the equality. In an equivalent way by the Theorem of complétude, it is a question of knowing if a statement is universally valid, i.e. true in all the models (of the equality). It is about algorithmic Décidabilité. Known as differently, the question is that of the Décidabilité levelling calculation of the predicates first order : is the whole of the universally valid statements of the calculation of the first order predicates décidable?
The problem of the decision depends in fact on the choice on the first order language: its signature, " briques" basic, which allows the construction of the statements, symbols of constants, functions (or operations), and of predicate (or relation), for example, 0, +, ≤,….
One also speaks about the problem of the decision in a axiomatic Théorie given, for example in the Arithmétique of Peano. IT is then a question of determining if a statement is a theorem of the theory in question. In a given language, a positive solution with the problem of the decision finiment provides a positive solution to the problems of the decision for all the theories axiomatisables of this language. Indeed, a statement C results from a finished system of axioms if and only if one can derive in formal logic that the conjunction of these axioms involves C .
The problem was posed by David Hilbert and Wilhelm Ackermann in 1928. Besides one uses sometimes the German term Entscheidungsproblem to indicate the problem of the decision, it is the case very often in English, to avoid confusions.
The question goes back to Gottfried Leibniz which, with the seventeenth century, imagined the construction of a machine which could handle symbols in order to déteminer the values of the mathematical statements. It understood that the first step would be to have a formal Language clearly.
Alonzo Church and Alan Turing gave (independently) in 1936, a negative answer to the problem of the decision for the arithmetic one (for example the Arithmétique of Peano or a stronger coherent theory). They largely use the methods developed by Kurt Gödel to show the first theorem of incomplétude. One can state the result thus:
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a theory recursively axiomatisable, coherent and capable of " to formalize the arithmétique" , is algorithmiquement indécidable.
The precise conditions of the theorem are those of the theorem of Gödel-Rosser. If these conditions are examined, one realizes that they are axiomatisable carried out by a theory finiment axiomatisable, and thus one obtains a negative answer to the problem of the decision in the language of arithmetic (one can take 0,1, +, ×). This result is often called theorem of Church :
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the calculation of the first order predicates levelling in the language of arithmetic is algorithmiquement indécidable.
One from of deduced by coding a negative answer for the problem from the decision as soon as the language contains a binary symbol of predicate (in addition to the equality). On the other hand if the language contains only unary symbols of predicates and symbols of constant (not of symbols of function), then the calculation of the first order predicates levelling corresponding, the calculation of the monadic predicates first order , is décidable.
In addition, there exist theories décidables whose language contains a binary symbol of predicate: the theory of the dense orders (that of Q the whole of rational in the only language of the order) to take a very simple example, or the Arithmetic of Presburger to which one can add without damage the relation of order, which is defined with the addition.
To answer the question, especially to answer it negatively, it was necessary however that the concept of calculable Fonction, i.e. calculable mechanically, by an algorithm, is formalized. That was done in several stages. Several models of calculation, which one would say now Turing-complete, appeared in the years 1930. One can quote the λ-calculable functions of Alonzo Church (1930), the general recursive functions of Herbrand and Gödel (Gödel 1934, by specifying an idea of Herbrand 1931), the machines of Turing (1936), the systems of Post (1936), the recursive functions within the meaning of Kleene (1936)… All these models appeared equivalent, which is an argument for the Thèse of Church (1936): one captured well by one of these models the concept of function calculable.
To show the indecidability of arithmetic, the argumentation of Turing is the following one. Let us suppose that we have an algorithm of decision for the Arithmétique of Peano. The question of knowing if a machine of Turing given stops or not, can be formulated as a first order statement (one uses the methods developed by Gödel), which would then be solved by the algorithm of decision. But Turing had proven previously that there is no general algorithm to decide stop of a Machine of Turing.
See too
Refer
Sources
In French
- Rene Cori, Daniel Lascar, Logical mathematics (volume II) - Masson - 1994 - ISBN 2-225-84080-6
- Jean-Pierre Azra, Bernard Jaulin Récursivité - Gauthier-Villars 1973 - ISBN 2-04-007244-6
In English
- S.C. Kleene - Introduction to metamathematics - Elsevier - 1952 - (republished)
Original articles
- Alonzo Church, “Year unsolvable problem off elementary number theory”, American Newspaper off Mathematics , 58 (1936), pp 345-363
- Alonzo Church, “has note one the Entscheidungsproblem”, Journal off Symbolic Logic , 1 (1936), pp 40-41.
- Alan Turing, “One computable numbers, with year application to the Entscheidungsproblem”, Proceedings off the London Mathematical Society , Series 2, 42 (1936), pp 230-265. Online version. Errata appeared in Series 2, 43 (1937), pp 544-546.
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