Quantifier Elimination - History

History

In early model theory, quantifier elimination was used to demonstrate that various theories possess certain model-theoretic 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 is used to show that Presburger arithmetic, i.e. the theory of the additive natural numbers, 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 in a countable language is decidable, it is possible to extend its language with countably many relations to ensure that it admits quantifier elimination (for example, one can introduce a relation symbol for each formula).

Example: Nullstellensatz in ACF and DCF.

Read more about this topic:  Quantifier Elimination

Famous quotes containing the word history:

    Don’t give your opinions about Art and the Purpose of Life. They are of little interest and, anyway, you can’t express them. Don’t analyse yourself. Give the relevant facts and let your readers make their own judgments. Stick to your story. It is not the most important subject in history but it is one about which you are uniquely qualified to speak.
    Evelyn Waugh (1903–1966)

    There are two great unknown forces to-day, electricity and woman, but men can reckon much better on electricity than they can on woman.
    Josephine K. Henry, U.S. suffragist. As quoted in History of Woman Suffrage, vol. 4, ch. 15, by Susan B. Anthony and Ida Husted Harper (1902)

    In history an additional result is commonly produced by human actions beyond that which they aim at and obtain—that which they immediately recognize and desire. They gratify their own interest; but something further is thereby accomplished, latent in the actions in question, though not present to their consciousness, and not included in their design.
    Georg Wilhelm Friedrich Hegel (1770–1831)