Computational complexity of quantifier-free negationless theory of field of rational numbers

Abstract

The following result is an approximation to the answer of the question of Kokorin (Logical Notebook, Unsolved Problems of Mathematics, Novosibirsk, 1986, 41pp; in Russian) about decidability of a quantifier-free theory of field of rational numbers. Let Q 0 be a subset of the set of all rational numbers which contains integers 1 and −1. Let Q 0 be a set containing Q 0 and closed by the functions of addition, subtraction and multiplication. For example Q 0 coincides with Q 0 if Q 0 is the set of all binary rational numbers or the set of all decimal rational numbers. It is clear that Z ⊆ Q 0 ⊆ Q , where Z is the set of all integers and Q is the set of all rational numbers. A negationless theory uses only conjunction and disjunction as logical connectives. Let T be a quantifier-free (universal or free-variable) negationless elementary theory of signature 〈 Q 0 ;| Pol,⩽〉 , where |Pol is the list of all polynomials with rational coefficients from Q 0 in which exponent of the polynomials and rational coefficients are written in binary number system. Theorem 1. Theory T is NP - hard and if Q 0 is everywhere dense then the theory T is decidable by an algorithm which belongs to EXPTIME. The set of all rational numbers or the set of all binary rational numbers may be taken instead of Q 0 . The quantifier-free negationless theory of the field of rational numbers may be regarded as a base of constructive mathematics (Section Mathematical Logic and Foundations of Mathematics, ICM, Warszawa, 1982, p. 21).