Towards a more efficient approach for the satisfiability of two-variable logic - Lin, Ting-Wei; Lu, Chia-Hsuan; Tan, Tony (Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, 2021)
We revisit the satisfiability problem for two-variable logic, denoted by SAT(FO 2 ), 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(1.4423 n ), O(1.6181 n ) and O(1.3661 n ), respectively.We then show that without the equality predicate SAT(FO 2 ) is in fact equivalent to CIS in succinct representation. This yields the same three simple algorithms as above for SAT(FO 2 ) 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 FO 2 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.