The complexity of subtheories of the existential linear theory of reals
The linear theory consists of all sentences of signatur (0,1,+,<,k·_, k is a rational number) true in the model of real numbers. We shall prove that the existential linear theory of real numbers restricted to a quantifier free part of Krom formulas is NP-complete even for some restrictions on the structure of atoms.
In the case that the quantifier free part is a conjunction of atomic formulas we have nothing else than the linear optimation problem, which is P-complete. In the case of two variables per atomic formula the problem is in NC.
Also the case that all atoms are of the form Σ i a i x i ≥c, such that c>0, is considered.
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