Quantified Boolean Formulas: Proof Complexity and Models of Solving

Quantified Boolean formulas (QBF), which form the canonical PSPACE-complete decision problem, are a decidable fragment of first-order logic. Any problem that can be solved within a polynomial-size space can be encoded succinctly as a QBF, including many concrete problems in computer science from domains such as verification, synthesis and planning. Automated solvers for QBF are now reaching the point of industrial applicability. In this thesis, we focus on dependency awareness, a dedicated solving paradigm for QBF. We show that dependency schemes can be envisaged in terms of dependency quantified Boolean formulas (DQBF), exposing strong connections between these two previously disparate entities. By introducing new lower-bound techniques for QBF proof systems, we study the relative strengths of models of dependency-aware solving, including the proposal of new, stronger models. Proof Complexity: Using the strategy extraction paradigm, we introduce new lower-bound techniques that apply to resolution-based QBF proof systems. In particular, we use the technique to prove exponential lower bounds for a new family of QBFs called the equality formulas. Our technique also affords considerably simpler, more intuitive proofs of some existing QBF proof-size lower bounds. Models of Solving: We apply our lower bound techniques to show new separations for QBF proof systems parametrised by dependency schemes. We also propose new models of dynamic dependency-aware solving and prove that they are exponentially stronger than the existing static models. Finally, we introduce Merge Resolution, a proof system modelling CDCL-style solving for DQBF, which is the first of its kind

Hard QBFs for Merge Resolution

We prove the first proof size lower bounds for the proof system Merge Resolution (MRes [Olaf Beyersdorff et al., 2020]), a refutational proof system for prenex quantified Boolean formulas (QBF) with a CNF matrix. Unlike most QBF resolution systems in the literature, proofs in MRes consist of resolution steps together with information on countermodels, which are syntactically stored in the proofs as merge maps. As demonstrated in [Olaf Beyersdorff et al., 2020], this makes MRes quite powerful: it has strategy extraction by design and allows short proofs for formulas which are hard for classical QBF resolution systems. Here we show the first exponential lower bounds for MRes, thereby uncovering limitations of MRes. Technically, the results are either transferred from bounds from circuit complexity (for restricted versions of MRes) or directly obtained by combinatorial arguments (for full MRes). Our results imply that the MRes approach is largely orthogonal to other QBF resolution models such as the QCDCL resolution systems QRes and QURes and the expansion systems ?Exp+Res and IR

Reinterpreting Dependency Schemes: Soundness Meets Incompleteness in DQBF

Dependency quantified Boolean formulas (DQBF) and QBF dependency schemes have been treated separately in the literature, even though both treatments extend QBF by replacing the linear order of the quantifier prefix with a partial order. We propose to merge the two, by reinterpreting a dependency scheme as a mapping from QBF into DQBF. Our approach offers a fresh insight on the nature of soundness in proof systems for QBF with dependency schemes, in which a natural property called ‘full exhibition’ is central. We apply our approach to QBF proof systems from two distinct paradigms, termed ‘universal reduction’ and ‘universal expansion’. We show that full exhibition is sufficient (but not necessary) for soundness in universal reduction systems for QBF with dependency schemes, whereas for expansion systems the same property characterises soundness exactly. We prove our results by investigating DQBF proof systems, and then employing our reinterpretation of dependency schemes. Finally, we show that the reflexive resolution path dependency scheme is fully exhibited, thereby proving a conjecture of Slivovsky