Quantified Boolean Formulas Solvers Research Articles

Model Checking for Rectangular Hybrid Systems: A Quantified Encoding Approach

Satisfiability Modulo Theories (SMT) solvers have been successfully applied to solve many problems in formal verification such as bounded model checking (BMC) for many classes of systems from integrated circuits to cyber-physical systems. Typically, BMC is performed by checking satisfiability of a possibly long, but quantifier-free formula. However, BMC problems can naturally be encoded as quantified formulas over the number of BMC steps. In this approach, we then use decision procedures supporting quantifiers to check satisfiability of these quantified formulas. This approach has previously been applied to perform BMC using a Quantified Boolean Formula (QBF) encoding for purely discrete systems, and then discharges the QBF checks using QBF solvers. In this paper, we present a new quantified encoding of BMC for rectangular hybrid automata (RHA), which requires using more general logics due to the real (dense) time and real-valued state variables modeling continuous states. We have implemented a preliminary experimental prototype of the method using the HyST model transformation tool to generate the quantified BMC (QBMC) queries for the Z3 SMT solver. We describe experimental results on several timed and hybrid automata benchmarks, such as the Fischer and Lynch-Shavit mutual exclusion algorithms. We compare our approach to quantifier-free BMC approaches, such as those in the dReach tool that uses the dReal SMT solver, and the HyComp tool built on top of nuXmv that uses the MathSAT SMT solver. Based on our promising experimental results, QBMC may in the future be an effective and scalable analysis approach for RHA and other classes of hybrid automata as further improvements are made in quantifier handling in SMT solvers such as Z3.

Planning with Incomplete Information in Quantified Answer Set Programming

AbstractWe present a general approach to planning with incomplete information in Answer Set Programming (ASP). More precisely, we consider the problems of conformant and conditional planning with sensing actions and assumptions. We represent planning problems using a simple formalism where logic programs describe the transition function between states, the initial states and the goal states. For solving planning problems, we use Quantified Answer Set Programming (QASP), an extension of ASP with existential and universal quantifiers over atoms that is analogous to Quantified Boolean Formulas (QBFs). We define the language of quantified logic programs and use it to represent the solutions different variants of conformant and conditional planning. On the practical side, we present a translation-based QASP solver that converts quantified logic programs into QBFs and then executes a QBF solver, and we evaluate experimentally the approach on conformant and conditional planning benchmarks.

Two SAT solvers for solving quantified Boolean formulas with an arbitrary number of quantifier alternations

In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over Boolean variables. Such approaches partially expand one type of variable (either existential or universal) for obtaining a propositional abstraction of the QBF. If this formula is false, the truth value of the QBF is decided, otherwise further refinement steps are necessary. Classically, expansion-based solvers process the given formula quantifier-block wise and use one SAT solver per quantifier block. In this paper, we present a novel algorithm for expansion-based QBF solving that deals with the whole quantifier prefix at once. Hence recursive applications of the expansion principle are avoided and only two incremental SAT solvers are required. While our algorithm is naturally based on the forall Exp+Res calculus that is the formal foundation of expansion-based solving, it is conceptually simpler than present recursive approaches. Experiments indicate that the performance of our simple approach is comparable with the state of the art of QBF solving, especially in combination with other solving techniques.

A Sharp Leap from Quantified Boolean Formula to Stochastic Boolean Satisfiability Solving

Stochastic Boolean Satisfiability (SSAT) is a powerful representation for the concise encoding of quantified decision problems with uncertainty. While it shares commonalities with quantified Boolean formula (QBF) satisfiability and has the same PSPACE-complete complexity, SSAT solving tends to be more challenging as it involves expensive model counting, a.k.a. Sharp-SAT. To date, SSAT solvers, especially those imposing no restrictions on quantification levels, remain much lacking. In this paper, we present a new SSAT solver based on the framework of clause selection and cube distribution previously proposed for QBF solving. With model counting integrated and learning techniques strengthened, our solver is general and effective. Experimental results demonstrate the overall superiority of the proposed algorithm in both solving performance and memory usage compared to the state-of-the-art solvers on a number of benchmark formulas.

CAQE and QuAbS: Abstraction Based QBF Solvers

We present a detailed description, analysis, and evaluation of the clausal abstraction approach for solving quantified Boolean formulas (QBF). The clausal abstraction algorithm started as a solving algorithm for QBFs in prenex conjunctive normal form

QBFRelay, QRATPre+, and DepQBF: Incremental Preprocessing Meets Search-Based QBF Solving

DepQBF is a search-based quantified Boolean formula (QBF) solver implementing the QCDCL paradigm. We submitted DepQBF as part of several tool packages to the QBFEVAL’18 competition, which was part of the FLoC Olympic Games 2018. These packages integr

A PSPACE Subclass of Dependency Quantified Boolean Formulas and Its Effective Solving

Dependency quantified Boolean formulas (DQBFs) are a powerful formalism, which subsumes quantified Boolean formulas (QBFs) and allows an explicit specification of dependencies of existential variables on universal variables. This enables a succinct encoding of decision problems in the NEXPTIME complexity class. As solving general DQBFs is NEXPTIME complete, in contrast to the PSPACE completeness of QBF solving, characterizing DQBF subclasses of lower computational complexity allows their effective solving and is of practical importance.Recently a DQBF proof calculus based on a notion of fork extension, in addition to resolution and universal reduction, was proposed by Rabe in 2017. We show that this calculus is in fact incomplete for general DQBFs, but complete for a subclass of DQBFs, where any two existential variables have either identical or disjoint dependency sets over the universal variables. We further characterize this DQBF subclass to be ΣP3 complete in the polynomial time hierarchy. Essentially using fork extension, a DQBF in this subclass can be converted to an equisatisfiable 3QBF with only a linear increase in formula size. We exploit this conversion for effective solving of this DQBF subclass and point out its potential as a general strategy for DQBF quantifier localization. Experimental results show that the method outperforms state-of-the-art DQBF solvers on a number of benchmarks, including the 2018 DQBF evaluation benchmarks.

Dependency Learning for QBF

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.

The 2016 and 2017 QBF solvers evaluations (QBFEVAL'16 and QBFEVAL'17)

After a break of about five years, in 2016 the classical QBFEVAL has been revived. QBFEVAL is a competitive evaluation of solvers for quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over the propositional variables.Due to the enormous interest in QBFEVAL'16, more recently, QBFEVAL'17 was organized. Both competitions were affiliated to the respective editions of the International Conference on Theory and Applications of Satisfiability Testing (SAT'16 and SAT'17), the major conference in research on SAT and related areas.In this paper we report about the 2016 and 2017 competitive evaluations of QBF solvers (QBFEVAL'16 and QBFEVAL'17), the two most recent events in a series of competitions established with the aim of assessing the advancements in reasoning about QBFs. This report gives an overview of the setup of these two events, on their participants and on the results of the experiments that were performed for evaluating the participating systems.

Design variability verification in Software Product Lines

This paper proposes a novel notion called variability verification applicable to Software Product Lines (SPL). Variability is central to SPL and we have observed that variability is expressed differently at different levels of abstraction in the development flow of SPL. A natural problem in this context is the conformance of variability information expressed at different levels. Design variability verification, in particular, checks whether the variability expressed at the design level conforms to that at the requirement level. Unlike many existing approaches to SPL modelling, our work does not assume a single global view of variation points, even within the same level of abstraction. In our view, an SPL is a concurrent composition of features, where each feature exhibits independent variability. This enables incremental addition of variability. The procedure is compositional in the sense that the verification of an entire SPL consisting of multiple features is reduced to the verification of the individual features. Feature level verification essentially involves standard model checking while, in the second step, a Quantified Boolean Formula (QBF) is synthesized and solved. The method has been implemented and demonstrated in a tool SPL Engine for Design Verification (SPLEnD) on a couple of fairly large case studies. SPLEnD uses SPIN tool for the feature level conformance, while the state-of-the-art QBF solver CirQit is used for the SPL level conformance. SPLEnD easily handles the evolution of SPL by addition of new features and modification of existing features. Experimental results with SPLEnD look very promising.

Long-Distance Q-Resolution with Dependency Schemes

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.

The QBF Gallery: Behind the scenes

Over the last few years, much progress has been made in the theory and practice of solving quantified Boolean formulas (QBF). Novel solvers have been presented that either successfully enhance established techniques or implement novel solving paradigms. Powerful preprocessors have been realized that tune the encoding of a formula to make it easier to solve. Frameworks for certification and solution extraction emerged that allow for a detailed interpretation of a QBF solver's results, and new types of QBF encodings were presented for various application problems.To capture these developments the QBF Gallery was established in 2013. The QBF Gallery aims at providing a forum to assess QBF tools and to collect new, expressive benchmarks that allow for documenting the status quo and that indicate promising research directions. These benchmarks became the basis for the experiments conducted in the context of the QBF Gallery 2013 and follow-up evaluations. In this paper, we report on the setup of the QBF Gallery. To this end, we conducted numerous experiments which allowed us not only to assess the quality of the tools, but also the quality of the benchmarks.

Conformant planning as a case study of incremental QBF solving

We consider planning with uncertainty in the initial state as a case study of incremental quantified Boolean formula (QBF) solving. We report on experiments with a workflow to incrementally encode a planning instance into a sequence of QBFs. To solve this sequence of successively constructed QBFs, we use our general-purpose incremental QBF solver DepQBF. Since the generated QBFs have many clauses and variables in common, our approach avoids redundancy both in the encoding phase as well as in the solving phase. We also present experiments with incremental preprocessing techniques that are based on blocked clause elimination (QBCE). QBCE allows to eliminate certain clauses from a QBF in a satisfiability preserving way. We implemented the QBCE-based techniques in DepQBF in three variants: as preprocessing, as inprocessing (which extends preprocessing by taking into account variable assignments that were fixed by the QBF solver), and as a novel dynamic approach where QBCE is tightly integrated in the solving process. For DepQBF, experimental results show that incremental QBF solving with incremental QBCE outperforms incremental QBF solving without QBCE, which in turn outperforms nonincremental QBF solving. For the first time we report on incremental QBF solving with incremental QBCE as inprocessing. Our results are the first empirical study of incremental QBF solving in the context of planning and motivate its use in other application domains.

Solving QBF with counterexample guided refinement

This article puts forward the application of Counterexample Guided Abstraction Refinement (CEGAR) in solving the well-known PSPACE-complete problem of quantified Boolean formulas (QBF). The article studies the application of CEGAR in two scenarios. In the first scenario, CEGAR is used to expand quantifiers of the formula and subsequently a satisfiability (SAT) solver is applied. First it is shown how to do that for two levels of quantification and then it is generalized for arbitrary number of levels by recursion. It is also shown that these ideas can be generalized to non-prenex and non-CNF QBF solvers. In the second scenario, CEGAR is employed as an additional learning technique in an existing DPLL-based QBF solver. Experimental evaluation of the implemented prototypes shows that the CEGAR-driven solver outperforms existing solvers on a number of benchmark families and that the DPLL solver benefits from the additional type of learning.

QBF Gallery 2014: The QBF Competition at the FLoC 2014 Olympic Games

The QBFGallery 2014 was a competitive evaluation for QBF solvers organized in the context of the FLoC 2014 Olympic Games of the Vienna Summer of Logic. Quantified Boolean formulas (QBF) extend formulas of propositional logic by existential and universal quantifiers over the propositional variables and, thus, lift the satisfiablity problem from NP to PSPACE. In consequence, there are many application problems from fields like verification, synthesis, and planning that can be encoded efficiently in QBF. This motivates the quest for efficient QBF solvers. The QBFGallery 2014 invited the developers of QBF solvers to submit their tools to a competitive evaluation in a uniform environment. Gold, silver, and bronze track medals were awarded to the solvers that solved the most formulas in each of three different tracks covering various kinds of benchmarks. Additionally, the three participants that were most successful overall were decorated with Kurt G odel medals, the official prizes of the Olympic Games. In this paper, we give an overview of the setup of the competition, we briefly review the participating solvers, and finally report on the results of the QBFGallery 2014.

Efficient Extraction of QBF (Counter)models from Long-Distance Resolution Proofs

Many computer science problems can be naturally and compactly expressed using quantified Boolean formulas (QBFs). Evaluating thetruth or falsity of a QBF is an important task, and constructing the corresponding model or countermodel can be as important and sometimes even more useful in practice. Modern search and learning based QBF solvers rely fundamentally on resolution and can be instrumented to produce resolution proofs, from which in turn Skolem-function models and Herbrand-function countermodels can be extracted. These (counter)models are the key enabler of various applications. Not until recently the superiority of long-distanceresolution (LQ-resolution) to short-distance resolution(Q-resolution) was demonstrated. While a polynomial algorithm exists for (counter)model extraction from Q-resolution proofs, it remains open whether it exists forLQ-resolution proofs. This paper settles this open problem affirmatively by constructing a linear-time extraction procedure. Experimental results show the distinct benefits of the proposed method in extracting high quality certificates from some LQ-resolution proofs that are not obtainable from Q-resolution proofs.

Expansion-based QBF solving versus Q-resolution

This article introduces and studies a proof system ∀Exp+Res that enables us to refute quantified Boolean formulas (QBFs). The system ∀Exp+Res operates in two stages: it expands all universal variables through conjunctions and refutes the result by propositional resolution. This approach contrasts with the Q-resolution calculus, which enables refuting QBFs by rules similar to propositional resolution. In practice, Q-resolution enables producing proofs from conflict-driven DPLL-based QBF solvers. The system ∀Exp+Res can on the other hand certify certain expansion-based solvers. So a natural question is to ask which of the systems, Q-resolution and ∀Exp+Res, is more powerful? The article gives several partial responses to this question. On the positive side, we show that ∀Exp+Res can p-simulate tree Q-resolution. On the negative side, we show that ∀Exp+Res does not p-simulate unrestricted Q-resolution. In the favor of ∀Exp+Res we show that ∀Exp+Res is more powerful than a certain fragment of Q-resolution, which is important for DPLL-based QBF solving.

Verification of partial designs using incremental QBF

SAT solving is an indispensable core component of numerous formal verification tools and has found widespread use in industry, in particular when using it in an incremental fashion, e.g., in Bounded Model Checking (BMC). On the other hand, for some applications SAT formulas are not expressive enough, whereas a description via Quantified Boolean Formulas (QBF) is much more adequate, for instance when dealing with partial designs.Motivated by the success of incremental SAT, in this paper we explore various approaches to solve QBF problems in an incremental fashion and thereby make this technology usable as a core component of BMC. Firstly, we realized an incremental QBF solver based on the state-of-the-art QBF solver QuBE: Taking profit from the reuse of some information from previous iterations, the search space can be pruned, in some cases, to even less than a quarter.However, the need for preprocessing QBF formulas prior to the solving phase, that in general cannot be paired with incremental solving because of the non-predictable elimination of variables in the future incremental steps, posed the question of incremental QBF preprocessing. In this context we present an approach for retaining the QBF formula being preprocessed while extending its clauses and prefix incrementally. This procedure results in a significant size reduction of the QBF formulas, hence leading to a reduced solving time.As this may come together with a high preprocessing time, we analyze various heuristics to dynamically disable incremental preprocessing when its overhead raises over a certain threshold and is not compensated by the reduced solving time anymore.For proving the efficacy of our methods experimentally, as an application we consider BMC for partial designs (i.e., designs containing so-called blackboxes which represent unknown parts). Here, we disprove realizability, that is, we prove that an unsafe state is reachable no matter how the blackboxes are implemented. We examine all these incremental approaches from both the point of view of the effectiveness of the single procedure and the benefits that a range of QBF solvers can take from it. On a domain of partial design benchmarks, engaging incremental QBF methods significant performance gains over non incremental BMC can be achieved.

Unified QBF certification and its applications

Quantified Boolean formulae (QBF) allow compact encoding of many decision problems. Their importance motivated the development of fast QBF solvers. Certifying the results of a QBF solver not only ensures correctness, but also enables certain synthesis and verification tasks. To date the certificate of a true formula can be in the form of either a syntactic cube-resolution proof or a semantic Skolem-function model whereas that of a false formula is only in the form of a syntactic clause-resolution proof. The semantic certificate for a false QBF is missing, and the syntactic and semantic certificates are somewhat unrelated. This paper identifies the missing Herbrand-function countermodel for false QBF, and strengthens the connection between syntactic and semantic certificates by showing that, given a true QBF, its Skolem-function model is derivable from its cube-resolution proof of satisfiability as well as from its clause-resolution proof of unsatisfiability under formula negation. Consequently Skolem-function derivation can be decoupled from special Skolemization-based solvers and computed from standard search-based ones. Experimental results show strong benefits of the new method.

Equivalences et forme prénexe pour les formules booléennes quantifiées

Many problems in artificial intelligence and in formal verification can be reduced to a validity test of a quantified boolean formula (QBF). To perform this test, existing QBF solvers need the formula to be in a restrictive prenex form, as conjunctive normal form or negation normal form. This prior prenexing process can lead to an exponential growth of the formula in presence of bi-implications and exclusive disjunctions. In this work, we describe a new method of prenexing with a particular pattern. Our experimental results show the interest of such a method with actual QBF solvers.