Many verification and synthesis approaches rely on solving techniques for quantified Boolean formulas (QBF). Consequently, solution witnesses, in the form of Boolean functions, become more and more important as they represent implementations or counterexamples. We present a recursive counterexample guided abstraction and refinement algorithm (CEGAR) for solving and certifying QBFs that exploits structural reasoning on the formula level. The algorithm decomposes the given QBF into one propositional formula for every block of quantifiers that abstracts from assignments of variables not bound by this quantifier block. Further, we show how to derive an efficient certification extraction method on top of the algorithm. We report on experimental evaluation of this algorithm in the solver QuAbS (Quantified Abstraction Solver) which won the most recent QBF competition (QBFEVAL'18). Further, we show the effectiveness of the certification approach using synthesis benchmarks and a case study for synthesizing winning strategies in Petri Games.
We consider the problem of incrementally solving a sequence of quantified Boolean formulae (QBF). Incremental solving aims at using information learned from one formula in the process of solving the next formulae in the sequence. Based on a general overview of the problem and related challenges, we present an approach to incremental QBF solving which is application-independent and hence applicable to QBF encodings of arbitrary problems. We implemented this approach in our incremental search-based QBF solver DepQBF and report on implementation details. Experimental results illustrate the potent…
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) and pass the obtained formula to a SAT solver for deciding the QBF. State-of-the-art expansion-based solvers process the given formula quantifier-block wise and recursively apply expansion until a solution is found. In this paper, we present a novel algor…
A quantified Boolean formula (QBF) is a propositional formula extended with universal and existential quantification over propositions. There are two methodologies in CEGAR based QBF solving techniques, one that is based on a refinement loop that builds partial expansions and a more recent one that is based on the communication of satisfied clauses. Despite their algorithmic similarity, their performance characteristics in experimental evaluations are very different and in many cases orthogonal. We compare those CEGAR approaches using proof theory developed around QBF solving and present a uni…
We present an experimental study of the effects of quantifier alternations on the evaluation of quantified Boolean formula (QBF) solvers. The number of quantifier alternations in a QBF in prenex conjunctive normal form (PCNF) is directly related to the theoretical hardness of the respective QBF satisfiability problem in the polynomial hierarchy. We show empirically that the performance of solvers based on different solving paradigms substantially varies depending on the numbers of alternations in PCNFs. In related theoretical work, quantifier alternations have become the focus of understanding…
Dependency quantified Boolean formulas (DQBF) is a logic admitting existential quantification over Boolean functions, which allows us to elegantly state synthesis problems in verification such as the search for invariants, programs, or winning regions of games. In this paper, we lift the clausal abstraction algorithm for quantified Boolean formulas (QBF) to DQBF. Clausal abstraction for QBF is an abstraction refinement algorithm that operates on a sequence of abstractions that represent the different quantifier levels. For DQBF we need to generalize this principle to partial orders of abstract…
Vienna University of Technology（维也纳工业大学）；University of Siegen（锡根大学）
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 sol…
The QRAT (quantified resolution asymmetric tautology) proof system simulates virtually all inference rules applied in state of the art quantified Boolean formula (QBF) reasoning tools. It consists of rules to rewrite a QBF by adding and deleting clauses and universal literals that have a certain redundancy property. To check for this redundancy property in QRAT, propositional unit propagation (UP) is applied to the quantifier free, i.e., propositional part of the QBF. We generalize the redundancy property in the QRAT system by QBF specific UP (QUP). QUP extends UP by the universal reduction op…
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 t…
We introduce a novel generalization of Counterexample-Guided Inductive Synthesis (CEGIS) and instantiate it to yield a novel, competitive algorithm for solving Quantified Boolean Formulas (QBF). Current QBF solvers based on counterexample-guided expansion use a recursive approach which scales poorly with the number of quantifier alternations. Our generalization of CEGIS removes the need for this recursive approach, and we instantiate it to yield a simple and efficient algorithm for QBF solving. Lastly, this research is supported by a competitive, though straightforward, implementation of the a…
Incremental SAT and QBF solving potentially yields improvements when sequences of related formulas are solved. An incremental application is usually tailored towards some specific solver and decomposes a problem into incremental solver calls. This hinders the independent comparison of different solvers, particularly when the application program is not available. As a remedy, we present an approach to automated benchmarking of incremental SAT and QBF solvers. Given a collection of formulas in (Q)DIMACS format generated incrementally by an application program, our approach automatically translat…
The aim of this PhD project is to develop fast and robust reasoning tools for dependency quantified Boolean formulas (DQBF). In this paper, we outline two properties, autarkies and symmetries, that potentially can be exploited for pre- and in-processing in the DQBF solving process. DQBF extend quantified Boolean formulas (QBF) with non-linear dependencies between the quantified variables. Automated testing and debugging techniques are an essential part of the solvers tool-chain. For rigorous DQBF solver development, we are working on novel automated testing and debugging techniques as successf…
While symmetries are well understood for Boolean formulas and successfully exploited in practical SAT solving, less is known about symmetries in quantified Boolean formulas (QBF). There are some works introducing adaptions of propositional symmetry breaking techniques, with a theory covering only very specific parts of QBF symmetries. We present a general framework that gives a concise characterization of symmetries of QBF. Our framework naturally incorporates the duality of universal and existential symmetries resulting in a general basis for QBF symmetry breaking.
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. Driven by the needs of various applications which can be encoded by DQBFs in a natural, compact, and elegant way, research on DQBF solving has emerged in the past few years. However, research focused on closed DQBFs in prenex form (where all quantifiers are placed in front of a propositional formula), while non-prenex DQBFs have almost not been studied in the literature. I…
The minimization of propositional formulae is a classical problem in logic, whose first algorithms date back at least to the 1950s with the works of Quine and Karnaugh. Most previous work in the area has focused on obtaining minimal, or quasi-minimal, formulae in conjunctive normal form (CNF) or disjunctive normal form (DNF), with applications in hardware design. In this paper, we are interested in the problem of obtaining an equivalent formula in any format, also allowing connectives that are not present in the original formula. We are primarily motivated in applying minimization algorithms t…
The reactive synthesis problem is to compute a system satisfying a given specification in temporal logic. Bounded synthesis is the approach to bound the maximum size of the system that we accept as a solution to the reactive synthesis problem. As a result, bounded synthesis is decidable whenever the corresponding verification problem is decidable, and can be applied in settings where classic synthesis fails, such as in the synthesis of distributed systems. In this paper, we study the constraint solving problem behind bounded synthesis. We consider different reductions of the bounded synthesis …
Decision procedures for SMT problems based on the theory of bit-vectors are a fundamental component in state-of-the-art software and hardware verifiers. While very efficient in general, certain SMT instances are still challenging for state-of-the-art solvers (especially when such instances include computationally costly functions). In this work, we present an approach for the quantifier-free bit-vector theory (QF_BV in SMT-LIB) based on incremental SMT solving and abstraction refinement. We define four concrete approximation steps for the multiplication, division and remainder operators and co…
Confluence in abstract parallel category systems is established for net class-rewriting in iterative closed multilevel quotient graph structures with uncountable node arities by multi-dimensional transducer operations in topological metrics defined by alphabetically abstracting net block homomorphism. We obtain minimum prerequisites for the comprehensive connector pairs in a multitude dimensional rewriting closure generating confluence in Participatory algebra for different horizontal and vertical level projections modulo abstraction relations constituting formal semantics for confluence in in…
Given a propositional formula F(x,y), a Skolem function for x is a function Ψ(y), such that substituting Ψ(y) for x in F gives a formula semantically equivalent to \exists F. Automatically generating Skolem functions is of significant interest in several applications including certified QBF solving, finding strategies of players in games, synthesising circuits and bit-vector programs from specifications, disjunctive decomposition of sequential circuits etc. In many such applications, F is given as a conjunction of factors, each of which depends on a small subset of variables. Existing algorith…
We propose a new method that takes advantage of structural reductions to accelerate the verification of reachability properties on Petri nets. Our approach relies on a state space abstraction, called polyhedral abstraction, which involves a combination between structural reductions and sets of linear arithmetic constraints between the marking of places. We propose a new data-structure, called a Token Flow Graph (TFG), that captures the particular structure of constraints occurring in polyhedral abstractions. We leverage TFGs to efficiently solve two reachability problems: first to check the re…
Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）；National University of Singapore（新加坡国立大学）
Quantified Boolean Formulas (QBF) extend propositional logic with quantification $\forall, \exists$. In QBF, an existentially quantified variable is allowed to depend on all universally quantified variables in its scope. Dependency Quantified Boolean Formulas (DQBF) restrict the dependencies of existentially quantified variables. In DQBF, existentially quantified variables have explicit dependencies on a subset of universally quantified variables called Henkin dependencies. Given a Boolean specification between the set of inputs and outputs, the problem of Henkin synthesis is to synthesize eac…
The intention of the present study is to establish the mathematical fundamentals for automated problem solving essentially targeted for robotics by approaching the task universal algebraically introducing knowledge as realizations of generalized free algebra based nets, graphs with gluing forms connecting in- and out-edges to nodes. Nets are caused to undergo transformations in conceptual level by type wise differentiated intervening net rewriting systems dispersing problems to abstract parts, matching being determined by substitution relations. Achieved sets of conceptual nets constitute cong…
ASP programs are a convenient tool for problem solving, whereas with large problem instances the size of the state space can be prohibitive. We consider abstraction as a means of over-approximation and introduce a method to automatically abstract (possibly non-ground) ASP programs that preserves their structure, while reducing the size of the problem. One particular application case is the problem of defining declarative policies for reactive agents and reasoning about them, which we illustrate on examples.
Computer programs, so-called solvers, for solving the well-known Boolean satisfiability problem (Sat) have been improving for decades. Among the reasons, why these solvers are so fast, is the implicit usage of the formula's structural properties during solving. One of such structural indicators is the so-called treewidth, which tries to measure how close a formula instance is to being easy (tree-like). This work focuses on logic-based problems and treewidth-based methods and tools for solving them. Many of these problems are also relevant for knowledge representation and reasoning (KR) as well…
With the race to build large-scale quantum computers and efforts to exploit quantum algorithms for efficient problem solving in science and engineering disciplines, the requirement to have efficient and scalable verification methods are of vital importance. We propose a novel formal verification method that is targeted at Quantum Fourier Transform (QFT) circuits. QFT is a fundamental quantum algorithm that forms the basis of many quantum computing applications. The verification method employs abstractions of quantum gates used in QFT that leads to a reduction of the verification problem from H…
Modern program verifiers use logic-based encodings of the verification problem that are discharged by a back end reasoning engine. However, instances of such encodings for large programs can quickly overwhelm these back end solvers. Hence, we need techniques to make the solving process scale to large systems, such as partitioning (divide-and-conquer) and abstraction. In recent work, we showed how decomposing the formula encoding of a termination analysis can significantly increase efficiency. The analysis generates a sequence of logical formulas with existentially quantified predicates that ar…
