Кванторларды жою: Математикалық логикадағы қарапайымдау әдісі

Simplification technique in mathematical logic
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement " such that " can be viewed as a question "When is there an such that ? ", and the statement without quantifiers can be viewed as the answer to that question. One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier free formulas as the simplest. A theory has quantifier elimination if for every formula , there exists another formula without quantifiers that is equivalent to it (modulo this theory).

If a theory has quantifier elimination, then a specific question can be addressed: Is there a method of determining for each ? If there is such a method we call it a quantifier elimination algorithm. If there is such an algorithm, then decidability for the theory reduces to deciding the truth of the quantifier free sentences. Quantifier free sentences have no variables, so their validity in a given theory can often be computed, which enables the use of quantifier elimination algorithms to decide validity of sentences.

Various model theoretic ideas are related to quantifier elimination, and there are various equivalent conditions. Every first order theory with quantifier elimination is model complete. Conversely, a model complete theory, whose theory of universal consequences has the amalgamation property, has quantifier elimination. The models of the theory of the universal consequences of a theory are precisely the substructures of the models of The theory of linear orders does not have quantifier elimination. However the theory of its universal consequences has the amalgamation property.

In early model theory, quantifier elimination was used to demonstrate that various theories possess properties like decidability and completeness. A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier free formulas. This technique can be used to show that Presburger arithmetic is decidable. Theories could be decidable yet not admit quantifier elimination. Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory is decidable, and the language of its valid formulas is countable, it is possible to extend the theory with countably many relations to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the free variables of the formula). Example: Nullstellensatz for algebraically closed fields and for differentially closed fields.