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• Decidability_of_first-order_theories_of_the_real_numbers abstract "A first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables. The corresponding first-order theory is the set of sentences that are actually true of the real numbers. There are several different such theories, with different expressive power, depending on the primitive operations that are allowed to be used in the expression. A fundamental question in the study of these theories is whether they are decidable: that is, is there an algorithm that can take a sentence as input and produce as output a yes or no answer to the question of whether the sentence belongs to the theory.The theory of real closed fields is the theory in which the primitives are multiplication and addition; that is, in this theory, comparisons are possible only between polynomials. It is decidable, a fact known as the Tarski–Seidenberg theorem. This was first proved by Alfred Tarski; see Tarski–Seidenberg theorem and Quantifier elimination. Current implementations of decision procedures for the theory of real closed fields are often based on quantifier elimination by cylindrical algebraic decomposition.Tarski's exponential function problem concerns the extension of this theory to another primitive operation, the exponential function. It is an open problem whether it is decidable, but if Schanuel's conjecture holds then the decidability of this theory would follow. In contrast, the extension of the theory of real closed fields with the sine function is undecidable since this allows encoding of the undecidable theory of integers (see Matiyasevich's theorem).Still, one can handle the undecidable case with functions such as sine by using algorithms that do not necessarily terminate always. Especially, one can design algorithms that are only required to terminate for input formulas that are robust, that is, formulas, whose satisfiability does not change if the formula is slightly perturbed. Alternatively, it is also possible to use purely heuristic approaches.".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageID "43578228".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageLength "3251".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageOutDegree "23".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageRevisionID "683497593".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Alfred_Tarski.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Algorithm.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Category:Formal_theories_of_arithmetic.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Category:Real_numbers.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Cylindrical_algebraic_decomposition.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Decidability_(logic).
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Diophantine_set.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Existential_quantification.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Exponential_function.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink First-order_logic.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Polynomial.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Quantifier_elimination.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Real_closed_field.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Real_number.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Robustness.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Schanuels_conjecture.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Sine.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Tarskis_exponential_function_problem.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Tarski–Seidenberg_theorem.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Theory_(mathematical_logic).
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLink Universal_quantification.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLinkText "Complications".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLinkText "Decidability of first-order theories of the real numbers".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLinkText "change the decidability of the theory".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageWikiLinkText "may change decidability".
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageUsesTemplate Template:Refimprove.
• Decidability_of_first-order_theories_of_the_real_numbers wikiPageUsesTemplate Template:Reflist.
• Decidability_of_first-order_theories_of_the_real_numbers subject Category:Formal_theories_of_arithmetic.
• Decidability_of_first-order_theories_of_the_real_numbers subject Category:Real_numbers.
• Decidability_of_first-order_theories_of_the_real_numbers hypernym Set.
• Decidability_of_first-order_theories_of_the_real_numbers comment "A first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables. The corresponding first-order theory is the set of sentences that are actually true of the real numbers.".
• Decidability_of_first-order_theories_of_the_real_numbers label "Decidability of first-order theories of the real numbers".
• Decidability_of_first-order_theories_of_the_real_numbers sameAs Q18206382.
• Decidability_of_first-order_theories_of_the_real_numbers sameAs m.011sn8qy.
• Decidability_of_first-order_theories_of_the_real_numbers sameAs Q18206382.
• Decidability_of_first-order_theories_of_the_real_numbers wasDerivedFrom Decidability_of_first-order_theories_of_the_real_numbers?oldid=683497593.
• Decidability_of_first-order_theories_of_the_real_numbers isPrimaryTopicOf Decidability_of_first-order_theories_of_the_real_numbers.