Representing Paraconsistent Reasoning via Quantified Propositional Logic

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

Over the last years, innovative parallel and distributed SAT solving techniques were presented that could impressively exploit the power of modern hardware and cloud systems. Two approaches were particularly successful: (1) search-space splitting in a Divide-and-Conquer (D &C) manner and (2) portfolio-based solving. The latter executes different solvers or configurations of solvers in parallel. For quantified Boolean formulas (QBFs), the extension of propositional logic with quantifiers, there is surprisingly little recent work in this direction compared to SAT.In this paper, we present ParaQooba, a novel framework for parallel and distributed QBF solving which combines D &C parallelization and distribution with portfolio-based solving. Our framework is designed in such a way that it can be easily extended and arbitrary sequential QBF solvers can be integrated out of the box, without any programming effort. We show how ParaQooba orchestrates the collaboration of different solvers for joint problem solving by performing an extensive evaluation on benchmarks from QBFEval’22, the most recent QBF competition.

An open quantified boolean formula (QBF) is a QBF that contains free (unquantified) variables. A solution to such a QBF is a quantifier-free formula that is logically equivalent to the given QBF. Although most recent QBF research has focused on closed QBF, there are a number of interesting applications that require one to consider formulas with free variables. This article shows how clause/cube learning for DPLL-based closed-QBF solvers can be extended to solve QBFs with free variables. We do this by introducing sequents that generalize clauses and cubes and allow learning facts of the form “under a certain class of assignments, the input formula is logically equivalent to a certain quantifier-free formula”.KeywordsInference RuleFree VariableInitial SequentPartial AssignmentCurrent AssignmentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Current algorithms for bounded model checking (BMC) use SAT methods for checking satisfiability of Boolean formulas. These BMC methods suffer from a potential memory explosion problem. Methods based on the validity of Quantified Boolean Formulas (QBF) allow an exponentially more succinct representation of the checked formulas, but have not been widely used, because of the lack of an efficient decision procedure for QBF. We evaluate the usage of QBF in BMC, using general-purpose SAT and QBF solvers. We also present a special-purpose decision procedure for QBF used in BMC, and compare our technique with the methods using general-purpose SAT and QBF solvers on real-life industrial benchmarks. Our procedure performs much better for BMC than the general-purpose QBF solvers, without incurring the space overhead of propositional SAT.

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 incrementally 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 and in the solving phase. Experimental results show that incremental QBF solving outperforms non-incremental QBF solving. Our results are the first empirical study of incremental QBF solving in the context of planning and motivate its use in other application domains.

Skolem and Herbrand functions are important certificates validating the truth and falsity, respectively, of quantified Boolean formulas (QBFs). They are essential in various synthesis and verification applications. Recent advancement established a linear time extraction of Skolem/Herbrand functions from QBF consensus/resolution proofs. However, the obtained functions are often excessively large and improper for practical applications. To overcome this limitation, this paper characterizes various flexibilities of QBF certificates, and exploits them for certificate simplification. Experiments show substantial reduction on QBF certificates in terms of circuit size and depth, which are of primary concerns for synthesis applications.

Term-resolution provides an elegant mechanism to prove that a quantified Boolean formula (QBF) is true. It is a dual to Q-resolution and is practically highly important as it enables certifying answers of DPLL-based QBF solvers. While term-resolution and Q-resolution are very similar, they are not completely symmetrical. In particular, Q-resolution operates on clauses and term-resolution operates on models of the matrix. This paper investigates the impact of this asymmetry. We will see that there is a large class of formulas (formulas with “big models”) whose term-resolution proofs are exponential. As a possible remedy, the paper suggests to prove true QBFs by refuting their negation (negate-refute), rather than proving them by term-resolution. The paper shows that from the theoretical perspective this is indeed a favorable approach. In particular, negation-refutation p-simulates term-resolution and there is an exponential separation between the two calculi. These observations further our understanding of proof systems for QBFs and provide a strong theoretical underpinning for the effort towards non-CNF QBF solvers.

In this paper we investigate the use of preprocessing when solving Quantified Boolean Formulas (QBF). Many different problems can be efficiently encoded as QBF instances, and there has been a great deal of recent interest and progress in solving such instances efficiently. Ideas from QBF have also started to migrate to CSP with the exploration of Quantified CSPs which offer an intriguing increase in representational power over traditional CSPs. Here we show that QBF instances can be simplified using techniques related to those used for preprocessing SAT. These simplifications can be performed in polynomial time, and are used to preprocess the instance prior to invoking a worst case exponential algorithm to solve it. We develop a method for preprocessing QBF instances that is empirically very effective. That is, the preprocessed formulas can be solved significantly faster, even when we account for the time required to perform the preprocessing. Our method significantly improves the efficiency of a range of state-of-the-art QBF solvers. Furthermore, our method is able to completely solve some instances just by preprocessing, including some instances that to our knowledge have never been solved before by any QBF solver.

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.

There has been tremendous progress in the past decade in the field of quantified Boolean formulas (QBF), both in practical solving as well as in creating a theory of corresponding proof systems and their proof complexity analysis. Both for solving and for proof complexity, it is important to have interesting formula families on which we can test solvers and gauge the strength of the proof systems. There are currently few such formula families in the literature. We initiate a general programme how to transform computationally hard problems (located in the polynomial hierarchy) into QBFs hard for the main QBF resolution systems Q-Res and QU-Res that relate to core QBF solvers. We illustrate this general approach on three problems from graph theory and logic. This yields QBF families that are provably hard for Q-Res and QU-Res (without any complexity assumptions).

In this paper we present a procedure for solving quantified boolean formulas (QBF), which uses And-Inverter Graphs (AIGs) as the core data-structure. We make extensive use of structural information extracted from the input formula such as functional definitions of variables and non-linear quantifier structures. We show how this information can directly be exploited by the symbolic, AIG based representation. We implemented a prototype QBF solver based on our ideas and performed a number of experiments proving the effectiveness of our approach, and moreover, showing that our method is able to solve QBF instances on which state-of-the-art QBF solvers known from literature fail.

Quantified Boolean Formulas (QBFs) are powerful expressions to naturally and concisely encode many decision problems in computer science, such as robotic planning, hardware/software synthesis and verification, among others. Their effective solving and certificate (in terms of model and countermodel) generation play crucial roles to enable practical applications. In this work, we give a new view on QBF solving and certificate generation by the cube distribution interpretation. It provides a largely increased flexibility for QBF reasoning and allows compact certificate derivation with don’t cares. Through this interpretation, we develop a QBF solver based on the prior clause selection framework. Experimental results demonstrate the superiority of our solver in both solving performance and certificate size compared to other state-of-the-art solvers with certificate generation ability.

The Quantified Boolean Formulae (QBF) solver Squolem can generate certificates of validity, based on Skolem functions. We present independent checking of these certificates in the HOL4 theorem prover. This enables HOL4 users to benefit from Squolem's automation for valid QBF problems. Detailed performance data shows that LCF-style checking of validity certificates is often (but not always) feasible even for large QBF instances. Additionally, our work provides high correctness assurances for Squolem's claims of validity and uncovered a soundness bug in a previous version of its certificate validator QBV.

Nondeterministic finite automata (NFAs) are an important model of computation. The equivalence problem for NFAs is known to be PSPACE-complete, and several specialized algorithms have been developed to solve this problem. In this paper, we approach the equivalence problem for NFAs by means of another PSPACE-complete problem: quantified satisfiability (QSAT). QSAT asks whether a quantified Boolean formula (QBF) is true. We encode the NFA equivalence problem as a QBF which is true if and only if the input NFAs are not equivalent. In addition to determining the equivalence of NFAs, we are also able to decode strings that witness the inequivalence of two NFAs by looking into the solving certificate. This is a novel application of certified QSAT solving. Lastly, we formally verify key aspects of the encoding in the Isabelle/HOL theorem prover.KeywordsQSATQBFFinite automataInteractive theorem provingIsabelle/HOLFormal methods

Bounded Model Checking (BMC) is a major verification method for finding errors in sequential circuits. BMC accomplishes this by iteratively unfolding a circuit k times, adding the negated property, and finally converting the BMC instance into a sequence of satisfiability (SAT) problems. When considering incomplete designs (i.e. those containing so-called blackboxes), we rather need the logic of Quantified Boolean Formulas (QBF) to obtain a more precise modeling of the unknown behavior of the blackbox. Here, we answer the question of unrealizability of a property, where finding a path of length k proves that the property is violated regardless of the implementation of the blackbox. To boost this task, solving blackbox BMC problems incrementally has been shown to be feasible [3], although the restrictions required in the preprocessing phase reduce its effectiveness. In this paper we enhance the verification procedure when using an off-the-shelf QBF solver, through a stronger preprocessing of the QBF formulas applied in an incremental fashion.KeywordsTransition RelationPreprocessing MethodPartial DesignCumulative TimeSequential CircuitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.