On the Complexity of Horn and Krom Fragments of Second-Order Boolean Logic

The canonical complete problem for ${\mathbf{PSPACE}}\xspace$ is the quantified Boolean formula problem (QBF) [Stockmeyer73]. This generalization of the Boolean satisfiability problem (SAT) asks whether a Boolean sentence of the form $Q_{1}p_{1}\ldots Q_{n}p_{n}\psi$, where $Q_{i}\in\{\exists,\forall\}$, is true. Today QBF attracts widespread interest in diverse research communities. In particular, QBF solving techniques are important in application domains such as planning, program synthesis and verification, adversary games, and non-monotonic reasoning, to name a few [8995437]. A further generalization of QBF is the dependency quantified Boolean formula problem (DQBF) [PETERSON2001957, peterson79]. This problem, complete for nondeterministic exponential time (${\mathbf{NEXP}}\xspace$), asks whether a Boolean sentence of the form

with constraints $C_{i}\subseteq\{p_{1},\ldots,p_{n}\}$ is true; here, the selection of truth values for $q_{i}$ may only depend on that of those variables that are in $C_{i}$. In other words, DQBF enriches QBF by allowing nonlinear dependency patterns between variables. DQBF-specifications can be exponentially more succinct compared to that of QBF and have found applications in areas such as non-cooperative games, SMT, and bit-vector logics. Furthermore, the development of DQBF-solvers is also well under way [W18].

Put in different terms, DQBF instances can be seen as Boolean sentences of the form

where each $f_{i}$ is a Boolean function variable whose occurrences in $\psi$ are of the form $f_{i}(p_{i_{1}},\ldots,p_{i_{k}})$, for some fixed sequence of proposition variables $p_{i_{1}},\ldots,p_{i_{k}}$. In previous studies, extensions of DQBF with alternating function quantification have also been considered. The so-called alternating dependency quantified Boolean formula problem (ADQBF) was shown to be complete for alternating exponential time with polynomially many alternations (${\mathbf{AEXP}(\mathrm{poly})}\xspace$) in [HannulaKLV16]. This work was preceded by the works of Lück [Luck16] and Lohrey [Lohrey12] studying second-order Boolean logic with explicit quantification of Boolean functions (denoted $\mathsf{SO}_{2}$ in this work). Their results showed, e.g., that restricting the alternations of function quantification to $k-1$ yields complete problems for the $k$th levels of the exponential hierarchy.

In this article we embark on a systematic study of the complexity of fragments of $\mathsf{SO}_{2}$, defined by combining restrictions on the structure of function terms and the Boolean matrix. A remarkable fact is that, when restricting attention to Horn formulae, all the complexity distinctions between SAT, QBF, and DQBF disappear. Bubeck and Büning [BubeckB06] showed that those DQBF instances whose quantifier-free part is a conjunction of Horn clauses are solvable in polynomial time. Consequently, all the aforementioned problems over Horn formulae are ${\mathbf{P}}\xspace$-complete. This implies that the high complexity of (D)QBF is not a straightforward consequence of its quantification structure; rather, structural complexity from the quantifier-free part is also needed. An immediate question is: How complex quantification is required to neutralize structural limitations, such as the Horn form, on the quantifier-free part? It is exactly this interplay between quantification and quantifier-free formula structure that will be the focus of this paper.

A formula of $\mathsf{SO}_{2}$ is in $\Sigma_{k}$ or in $\Pi_{k}$ if it is in prenex normal form with $k-1$ alternations for function quantification, with the first quantifier block being respectively existential or universal. If the quantifier-free part of a formula is in conjunctive normal form, then it is called (a) Horn if each clause has at most one positive literal, (b) Krom if each clause contains at most two literals, and (c) core if it is both Horn and Krom. A formula is called (i) simple if it contains no nested function terms, and (ii) unique if in it each function variable is associated with a unique argument tuple. These last two criteria, in particular, are meaningful for formulae involving second-order quantification. Uniqueness and simpleness are also the characteristics of function terms introduced in the process of Skolemization, and more importantly, tacitly assumed in the DQBF problem. One of the goals of this paper is to determine the impact of such restrictions. This way we generalize the aforementioned results on DQBF, which can be understood in our terms as unique simple $\Sigma_{1}$.

Our contributions are the following. We show, one the one hand, that the complexity of DQBF over Krom or core formulae collapses to ${\mathbf{NL}}\xspace$, and that this result extends to simple and unique $\Pi_{1}$ and $\Pi_{2}$. On the other hand, we show that almost all other cases are complete for the corresponding, or their neighboring, levels of the exponential hierarchy. Some cases are left open; most intriguing such case is the inverse of the DQBF-Horn problem (i.e.\xspace, simple and unique $\Pi_{1}$ Horn), which is only known to be between ${\mathbf{NL}}\xspace$ and ${{\Uppi}^{\mathrm{E}}_{1}}$. A summary of our results can be found in LABEL:tab:results-fragments.