Computational Complexity of Arithmetical Sentences

Arithmetical sentences are the sentences containing usual logical symbols and arithmetical symbols +, ·, and constants of Z. An arithmetical sentence ? is called an ?? sentence if and only if ? is logically equivalent to a sentence of the form ?x ?y?(x, y) where ?(x, y) is a quantifier free formula. It is shown that the decision problems of determining ?? sentences true in N or Z, respectively, are co-NP-complete, whereas the decision problem of determining ?? sentences true in Q is in P. Consequently, the decision problems of determining ?? sentences true in N or Z, respectively, are NP-complete. Also, the decision problem of determining ?? sentences true in Q is in P.