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

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 (ESM) property. Whether it can be determinized 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 two simple algorithms for it with run time 𝑂 (𝛿 𝑛 0 ) and 𝑂 (𝛿 𝑛 1 ), where 𝛿 0 = 1.4423 ≈ 3 √ 3 and 𝛿 1 = 1.6181 ≈ ( √ 5 + 1)/2 and 𝑛 is the number of vertices in the graph. We also show that unless the Strong Exponential Time Hypothesis (SETH) fails, there is no algorithm for CIS with run time 𝛿 𝑛 2 , where 𝛿 2 = √ 2 ≈ 1.4142. We then show that without the equality predicate SAT(FO 2 ) is in fact equivalent to CIS in succinct representation. This yields two algorithms for SAT(FO 2 ) without the equality predicate with run time 𝑂 (𝛿

), where 𝑛 is the number of predicates in the input formula. To the best of our knowledge, these are the first exact algorithms for an NEXP-complete decidable logic with run time 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. Similar to CIS, unless SETH fails, there is no algorithm for SAT(FO 2 ) with run time 𝑂 (𝛿

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. Finally, we also perform some small experiments which show that our approach is indeed more promising than the existing method (based on the ESM property). The experiments also show that although theoretically it has the worse run time, the second algorithm in general performs better than the first one.