A Natural NP-Complete Problem with a Nontrivial Lower Bound

Abstract

Let ${\operatorname{SAT}}_ < (\mathbb{N})$ denote the following problem. Instance. A conjunction $\varphi $ of (in)equalities $t_1 = t_2 $ or $t_1 < t_2 $, where $t_1 $, $t_2 $ are terms of the form $f_1 f_2 \cdots f_s (e)$, where $e \in \mathbb{N}$, $s \geqq 0$ and each $f_i $ is a monadic function symbol. Question. Is $\varphi $ satisfiable on $\mathbb{N}$? Let ${\operatorname{SAT}}_ < ^{2,2} (\mathbb{N})$ denote the following subproblem of ${\operatorname{SAT}}_ < (\mathbb{N})$ defined by the following restriction: we assume that $0 \leqq s \leqq 2$ and each $f_i \in \{ {g_1 ,g_2 } \}$. These two problems are NP-complete. We show that they are solved by a Turing machine using a polynomial number of deterministic steps and only n nondeterministic steps. This is nearly optimal since we prove that any problem in ${\operatorname{NTIME}}(n)$ is reducible in deterministic time $O(n)$ to ${\operatorname{SAT}}_ < (\mathbb{N})$ (respectively, ${\operatorname{SAT}}_ < ^{2,2} (\mathbb{N})$). It follows from the result $ \cup _c {\operatorname{DTIME}}(cn) \subsetneqq {\operatorname{NTIME}}(n)$ of Paul et al. [Proc. 24th Annual IEEE Symposium on Foundations of Computer Sciences, 1983, pp. 429–438] that these problems are not in $ \cup _c {\operatorname{DTIME}}(cn)$. Further, we show that ${\operatorname{SAT}}_ < ^{2,2} (\mathbb{N})$ and ${\operatorname{SAT}}_ < (\mathbb{N})$ belong to $\Sigma _2 (n)$, the class of problems solved in time $O(n)$ by alternating Turing machines using one alternation. They are the first natural problems proved to be in $\Sigma _2 (n) - \cup _c {\operatorname{DTIME}}(cn)$.