Abstract
An arithmetical sentence ϕ is called an ∀∃ sentence if and only if ϕ is logically equivalent to a sentence of the form ∀ x∃ y ψ( x, y) where ψ( x, y) is a formula containing no quantifiers and no other free variables except x and y. We prove that given an ∀∃ sentence ϕ in conjunctive or disjunctive normal form there is a polynomial time algorithm to decide whether or not ϕ is true in every integral domain with characteristic 0. We then prove that there is an algorithm to decide whether or not an ∀∃ sentence ϕ is true in every integral domain.
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