List coloring extends graph coloring by assigning each vertex a list of allowed colors. A graph is k-choosable if it can be properly colored for any choice of lists with k colors each. Deciding k-choosability is π²ₚ-complete, bipartite graphs have unbounded list chromatic number, and planar graphs (famously 4-colorable) are all 5-choosable but not all 4-choosable. To search for graphs of given choosability, we extend SAT Modulo Symmetries (SMS) with custom propagators for list coloring pruning techniques and propose a quantified Boolean (QBF) encoding for choosability. We employ a hybrid approach: pen-and-paper reasoning to optimize our formulas followed by automated case distinction by QBF solvers and SMS. Our methods yield two significant results: (1) a 27-vertex planar graph that is 4-choosable yet cannot be proven so using the combinatorial Nullstellensatz widely applied in previous work (we show this is a smallest graph with that property), and (2) the smallest graph exhibiting a gap between chromatic and list chromatic numbers for chromatic number 3.

Parallel solving via cube-and-conquer is a key method for solving hard instances with SAT. While cube-and-conquer has proven successful for pure SAT problems, notably the Pythagorean triples conjecture, its application to SAT solvers augmented with propagators presents unique challenges as propagators learn constraints dynamically during the search. We study this problem using SAT Modulo Symmetries (SMS) as our primary test case. In our setting, the SMS symmetry-breaking propagator is an ordinary IPASIR-UP propagator; the techniques below do not rely on properties specific to symmetry breaking, except in the benchmark instantiations. Through extensive experimentation comprising over 20,000 CPU hours, we systematically evaluate different cube-and-conquer variants on three well-studied combinatorial problems. Our methodology combines prerun phases to collect learned constraints, various cubing strategies, and parameter tuning via algorithm configuration. The comprehensive empirical evaluation provides new insights into effective cubing strategies for propagator-based SAT solving. Our best method reduces total solving time by factors of 2-10x from improved cubing, and reduces the time for the hardest cubes by factors of 2-50x.

Certification for Quantified Boolean Formulas (QBF) and Dependency Quantified Boolean Formulas (DQBF) is an ongoing challenge. Recent proof complexity work has shown that the majority of QBF and DQBF techniques can be p-simulated by using the independent extension rule. In propositional logic, extension rules are supported by proof checkers using a more general RAT (Resolution Asymmetric Tautology) rule. The next step in (D)QBF certification would be to update these modern RAT formats to match the strength of this independent extension rule. In this paper we first introduce a new dependency scheme called 𝒟^{∀pure}. This rule is the missing ingredient that when added to Blinkhorn’s proof system DQRAT allows it to be provably p-equivalent to the Independent Extended QU-Res, the most powerful of the known QBF and DQBF proof systems. Up until now, DQRAT has only existed in theory, so we implement a prototype checker DQRAT-check which includes our extra rule. In addition to its inclusion in our proof checker we show 𝒟^{∀pure} has other properties that have been found for previous dependency schemes, and each of these observations has potential in solving/checking including the sound integration into the dependency learning solver Qute.

Graph generation and enumeration problems often require handling equivalent graphs---those that differ only in vertex labeling. We study how to extend SAT Modulo Symmetries (SMS), a framework for eliminating such redundant graphs, to handle more complex constraints. While SMS was originally designed for constraints in propositional logic (in NP), we now extend it to handle quantified Boolean formulas (QBF), allowing for more expressive specifications like non-3-colorability (a coNP-complete property). We develop two approaches: a static QBF encoding and a dynamic method integrating SMS into QBF solvers. Our analysis reveals that while specialized approaches can be faster, QBF-based methods offer easier implementation and formal verification capabilities.

We show that extension variables in (D)QBF can be generalised by conditioning on universal assignments. The benefit of this is that the dependency sets of such conditioned extension variables can be made smaller to allow easier refutations. This simple modification instantly solves many challenges in p-simulating the QBF expansion rule, which cannot be p-simulated in proof systems that have strategy extraction. Simulating expansion is even more crucial in DQBF, where other methods are incomplete. In this paper we provide an overview of the strength of this new independent extension rule. We find that a new version of Extended Frege called IndExtFrege+Red can p-simulate a multitude of difficult QBF and DQBF techniques, even techniques that are difficult to approach with ExtFrege+Red. We show six p-simulations, that IndExtFrege+Red p-simulates QRAT, IR(D)-Calc, Q(Drrs)-Res, Fork Resolution, DQRAT and G, which together underpin most DQBF solving and preprocessing techniques. The p-simulations work despite these systems using complicated rules and our new extension rule being relatively simple. Moreover, unlike recent p-simulations by ExtFrege+Red we can simulate the proof rules line by line, which allows us to mix QBF rules more easily with other inference steps.

We obtain the smallest unsatisfiable formulas in subclasses of $k$-CNF (exactly $k$ distinct literals per clause) with bounded variable or literal occurrences. Smaller unsatisfiable formulas of this type translate into stronger inapproximability results for MaxSAT in the considered formula class. Our results cover subclasses of 3-CNF and 4-CNF; in all subclasses of 3-CNF we considered we were able to determine the smallest size of an unsatisfiable formula; in the case of 4-CNF with at most 5 occurrences per variable we decreased the size of the smallest known unsatisfiable formula. Our methods combine theoretical arguments and symmetry-breaking exhaustive search based on SAT Modulo Symmetries (SMS), a recent framework for isomorph-free SAT-based graph generation. To this end, and as a standalone result of independent interest, we show how to encode formulas as graphs efficiently for SMS.

Abstract Quantified conflict-driven clause learning (QCDCL) is one of the main solving approaches for quantified Boolean formulas (QBF). One of the differences between QCDCL and propositional CDCL is that QCDCL typically follows the prefix order of the QBF for making decisions. We investigate an alternative model for QCDCL solving where decisions can be made in arbitrary order. The resulting system $$\textsf{QCDCL}^\textsf {{A\tiny {\MakeUppercase {ny}}}}$$ QCDCLANY is still sound and terminating, but does not necessarily allow to always learn asserting clauses or cubes. To address this potential drawback, we additionally introduce two subsystems that guarantee to always learn asserting clauses ( $$\textsf{QCDCL}^\textsf {{U\tiny {\MakeUppercase {ni}}-A\tiny {\MakeUppercase {ny}}}}$$ QCDCLUNI-ANY ) and asserting cubes ( $$\textsf{QCDCL}^\textsf {{E\tiny {\MakeUppercase {xi}}-A\tiny {\MakeUppercase {ny}}}}$$ QCDCLEXI-ANY ), respectively. We model all four approaches by formal proof systems and show that $$\textsf{QCDCL}^\textsf {{U\tiny {\MakeUppercase {ni}}-A\tiny {\MakeUppercase {ny}}}}$$ QCDCLUNI-ANY is exponentially better than $$\mathsf{{QCDCL}} $$ QCDCL on false formulas, whereas $$\textsf{QCDCL}^\textsf {{E\tiny {\MakeUppercase {xi}}-A\tiny {\MakeUppercase {ny}}}}$$ QCDCLEXI-ANY is exponentially better than $$\mathsf{{QCDCL}} $$ QCDCL on true QBFs. Technically, this involves constructing specific QBF families and showing lower and upper bounds in the respective proof systems. We complement our theoretical study with some initial experiments that confirm our theoretical findings.

We present a new SAT-based method for generating all graphs up to isomorphism that satisfy a given co-NP property. Our method extends the SAT Modulo Symmetry (SMS) framework with a technique that we call co-certificate learning. If SMS generates a candidate graph that violates the given co-NP property, we obtain a certificate for this violation, i.e., `co-certificate' for the co-NP property. The co-certificate gives rise to a clause that the SAT solver, serving as SMS's backend, learns as part of its CDCL procedure. We demonstrate that SMS plus co-certificate learning is a powerful method that allows us to improve the best-known lower bound on the size of Kochen-Specker vector systems, a problem that is central to the foundations of quantum mechanics and has been studied for over half a century. Our approach is orders of magnitude faster and scales significantly better than a recently proposed SAT-based method.

Hitting formulas, introduced by Iwama, are an unusual class of propositional CNF formulas. Not only is their satisfiability decidable in polynomial time, but even their models can be counted in closed form. This stands in stark contrast with other polynomial-time decidable classes, which usually have algorithms based on backtracking and resolution and for which model counting remains hard, like 2-SAT and Horn-SAT. However, those resolution-based algorithms usually easily imply an upper bound on resolution complexity, which is missing for hitting formulas. Are hitting formulas hard for resolution? In this paper we take the first steps towards answering this question. We show that the resolution complexity of hitting formulas is dominated by so-called irreducible hitting formulas, first studied by Kullmann and Zhao, that cannot be composed of smaller hitting formulas. However, by definition, large irreducible unsatisfiable hitting formulas are difficult to construct; it is not even known whether infinitely many exist. Building upon our theoretical results, we implement an efficient algorithm on top of the Nauty software package to enumerate all irreducible unsatisfiable hitting formulas with up to 14 clauses. We also determine the exact resolution complexity of the generated hitting formulas with up to 13 clauses by extending a known SAT encoding for our purposes. Our experimental results suggest that hitting formulas are indeed hard for resolution.

We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) via circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems [ 16 ], but leaving open the most important case of QBF resolution. Different from the Frege case, our characterisation uses a new version of decision lists as its circuit model, which is stronger than the CNFs the system works with. Our decision list model is well suited to compute countermodels for QBFs. Our characterisation works for both Q-Resolution and QU-Resolution. Using our characterisation, we obtain a size-width relation for QBF resolution in the spirit of the celebrated result for propositional resolution [ 4 ]. However, our result is not just a replication of the propositional relation—intriguingly ruled out for QBF in previous research [ 12 ]—but shows a different dependence between size, width, and quantifier complexity. An essential ingredient is an improved relation between the size and width of term decision lists; this may be of independent interest. We demonstrate that our new technique elegantly reproves known QBF hardness results and unifies previous lower-bound techniques in the QBF domain.

Quantified conflict-driven clause learning (QCDCL) is one of the main approaches for solving quantified Boolean formulas (QBF). We formalise and investigate several versions of QCDCL that include cube learning and/or pure-literal elimination, and formally compare the resulting solving models via proof complexity techniques. Our results show that almost all of the QCDCL models are exponentially incomparable with respect to proof size (and hence solver running time), pointing towards different orthogonal ways how to practically implement QCDCL.

A CNF formula is harder than another CNF formula with the same number of clauses if it requires a longer resolution proof. We introduce resolution hardness numbers; they give for m=1,2,... the length of a shortest proof of a hardest formula on m clauses. We compute the first ten resolution hardness numbers, along with the corresponding hardest formulas. To achieve this, we devise a candidate filtering and symmetry breaking search scheme for limiting the number of potential candidates for hardest formulas, and an efficient SAT encoding for computing a shortest resolution proof of a given candidate formula.

A CNF formula is harder than another CNF formula with the same number of clauses if it requires a longer resolution proof. In this paper we introduce resolution hardness numbers; they give for m=1,2,... the length of a shortest proof of a hardest formula on m clauses. We compute the first ten resolution hardness numbers, along with the corresponding hardest formulas. To achieve this, we devise a candidate filtering and symmetry breaking search scheme for limiting the number of potential candidates for hardest for- mulas, and an efficient SAT encoding for computing a shortest resolution proof of a given candidate formula.

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.

We study dependency quantified Boolean formulas (DQBF), an extension of QBF in which dependencies of existential variables are listed explicitly rather than being implicit in the order of quantifiers. DQBF evaluation is a canonical NEXPTIME-complete problem, a complexity class containing many prominent problems that arise in Knowledge Representation and Reasoning. One approach for solving such hard problems is to identify and exploit structural properties captured by numerical parameters such that bounding these parameters gives rise to an efficient algorithm. This idea is captured by the notion of fixed-parameter tractability (FPT). We initiate the study of DQBF through the lens of fixed-parameter tractability and show that the evaluation problem becomes FPT under two natural parameterizations: the treewidth of the primal graph of the DQBF instance combined with a restriction on the interactions between the dependency sets, and also the treedepth of the primal graph augmented by edges representing dependency sets.

Quantified Boolean Formulas (QBFs) can be used to succinctly encode problems from domains such as formal verification, planning, and synthesis. One of the main approaches to QBF solving is Quantified Conflict Driven Clause Learning (QCDCL). By default, QCDCL assigns variables in the order of their appearance in the quantifier prefix so as to account for dependencies among variables. Dependency schemes can be used to relax this restriction and exploit independence among variables in certain cases, but only at the cost of nontrivial interferences with the proof system underlying QCDCL. We introduce dependency learning, a new technique for exploiting variable independence within QCDCL that allows solvers to learn variable dependencies on the fly. The resulting version of QCDCL enjoys improved propagation and increased flexibility in choosing variables for branching while retaining ordinary (long-distance) Q-resolution as its underlying proof system. We show that dependency learning can achieve exponential speedups over ordinary QCDCL. Experiments on standard benchmark sets demonstrate the effectiveness of this technique.

Resolution proof systems for quantified Boolean formulas (QBFs) provide a formal model for studying the limitations of state-of-the-art search-based QBF solvers that use these systems to generate proofs. We study a combination of two proof systems supported by the solver DepQBF: Q-resolution with generalized universal reduction according to a dependency scheme and long distance Q-resolution. We show that the resulting proof system-which we call long-distance Q(D)-resolution-is sound for the reflexive resolution-path dependency scheme. In fact, we prove that it admits strategy extraction in polynomial time. This comes as an application of a general result, by which we identify a whole class of dependency schemes for which long-distance Q(D)-resolution admits polynomial-time strategy extraction. As a special case, we obtain soundness and polynomial-time strategy extraction for long distance Q(D)-resolution with the standard dependency scheme. We further show that search-based QBF solvers using a dependency scheme D and learning with long-distance Q-resolution generate long-distance Q(D)-resolution proofs. The above soundness results thus translate to partial soundness results for such solvers: they declare an input QBF to be false only if it is indeed false. Finally, we report on experiments with a configuration of DepQBF that uses the standard dependency scheme and learning based on long-distance Q-resolution.