Towards a more efficient approach for the satisfiability of two-variable logic

Abstract

We revisit the satisfiability problem for two-variable logic, denoted by SAT(FO2), which is known to be NEXP-complete. The upper bound is usually derived from its well known exponential size model property. Whether it can be determinized/randomized efficiently is still an open question.In this paper we present a different approach by reducing it to a novel graph-theoretic problem that we call Conditional Independent Set (CIS). We show that CIS is NP-complete and present three simple algorithms for it: Deterministic, randomized with zero error and randomized with small one-sided error, with run time O(1.4423n), O(1.6181n) and O(1.3661n), respectively.We then show that without the equality predicate SAT(FO2) is in fact equivalent to CIS in succinct representation. This yields the same three simple algorithms as above for SAT(FO2) without the the equality predicate with run time $O({1.4423^{({2^n})}})$, $O({1.6181^{({2^n})}})$ and $O({1.3661^{({2^n})}})$, respectively, where n is the number of predicates in the input formula. To the best of our knowledge, these are the first deterministic/randomized algorithms for an NEXP-complete decidable logic with time complexity significantly lower than $O({2^{({2^n})}})$. We also identify a few lower complexity fragments of FO2 which correspond to the tractable fragments of CIS.For the fragment with the equality predicate, we present a linear time many-one reduction to the fragment without the equality predicate. The reduction yields equi-satisfiable formulas and incurs a small constant blow-up in the number of predicates.