We demonstrate how to learn efficient heuristics for automated reasoning algorithms for quantified Boolean formulas through deep reinforcement learning. We focus on a backtracking search algorithm, which can already solve formulas of impressive size - up to hundreds of thousands of variables. The main challenge is to find a representation of these formulas that lends itself to making predictions in a scalable way. For a family of challenging problems, we learned a heuristic that solves significantly more formulas compared to the existing handwritten heuristics.
In this paper, we investigate the feasibility of learning GNN (Graph Neural Network) based solvers and GNN-based heuristics for specified QBF (Quantified Boolean Formula) problems. We design and evaluate several GNN architectures for 2QBF formulae, and conjecture that GNN has limitations in learning 2QBF solvers. Then we show how to learn a heuristic CEGAR 2QBF solver. We further explore generalizing GNN-based heuristics to larger unseen instances, and uncover some interesting challenges. In summary, this paper provides a comprehensive surveying view of applying GNN-embeddings to specified QBF…
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.
We exploit symmetries to give short proofs for two prominent formula families of QBF proof complexity. On the one hand, we employ symmetry breakers. On the other hand, we enrich the (relatively weak) QBF resolution calculus Q-Res with the symmetry rule and obtain separations to powerful QBF calculi.
Prolog is a well known declarative programming language based on propositional Horn formulas. It is useful in various areas, including artificial intelligence, automated theorem proving, mathematical logic and so on. An active research area for many years is to extend Prolog to larger classes of logic. Some important extensions of it includes the constraint logic programming, and the object oriented logic programming. However, it cannot solve problems having arbitrary quantified Horn formulas. To be precise, the facts, rules and queries in Prolog are not allowed to have arbitrary quantified va…
In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems. He defined an infinite number of propositional satisfiability problems, showed that all these problems are either in P or NP-complete, and gave a simple criterion to determine which of the two cases holds. This result is surprising in light of Ladner's theorem, which implies that there are an infinite number of complexity classes between P and NP-complete (under the assumption that P is not equal to NP). Schaefer also stated a dichotomy theorem for quantified generalized Boolean formulas, but this th…
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…
We investigate pruning in search trees of so-called quantified integer linear programs (QIPs). QIPs consist of a set of linear inequalities and a minimax objective function, where some variables are existentially and others are universally quantified. They can be interpreted as two-person zero-sum games between an existential and a universal player on the one hand, or multistage optimization problems under uncertainty on the other hand. Solutions are so-called winning strategies for the existential player that specify how to react on moves of the universal player - i.e. certain assignments of …
We present Graph-$Q$-SAT, a branching heuristic for a Boolean SAT solver trained with value-based reinforcement learning (RL) using Graph Neural Networks for function approximation. Solvers using Graph-$Q$-SAT are complete SAT solvers that either provide a satisfying assignment or proof of unsatisfiability, which is required for many SAT applications. The branching heuristics commonly used in SAT solvers make poor decisions during their warm-up period, whereas Graph-$Q$-SAT is trained to examine the structure of the particular problem instance to make better decisions early in the search. Trai…
The following paper presents a heuristic method by which sum-of-product Boolean expressions can be simplified with a specific focus on the removal of redundant and selective prime implicants. Existing methods, such as the Karnaugh map and the Quine-McCluskey method [1, 2], fail to scale since they increase exponentially in complexity as the quantity of literals increases, doing as such to ensure the solution is algorithmically obtained. By employing a heuristic model, nearly all expressions can be simplified at an overall reduction in computational complexity. This new method was derived from …
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…
Signed systems were introduced as a general, syntax-independent framework for paraconsistent reasoning, that is, non-trivialised reasoning from inconsistent information. In this paper, we show how the family of corresponding paraconsistent consequence relations can be axiomatised by means of quantified Boolean formulas. This approach has several benefits. First, it furnishes an axiomatic specification of paraconsistent reasoning within the framework of signed systems. Second, this axiomatisation allows us to identify upper bounds for the complexity of the different signed consequence relations…
Computer Science Department, UCLA（加州大学洛杉矶分校计算机科学系）；Institut Universitaire de France（法国大学研究院）
Quantified Boolean logic results from adding operators to Boolean logic for existentially and universally quantifying variables. This extends the reach of Boolean logic by enabling a variety of applications that have been explored over the decades. The existential quantification of literals (variable states) and its applications have also been studied in the literature. In this paper, we complement this by studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification, discuss the interplay between variable/l…
We study quantified propositional logics from the complexity theoretic point of view. First we introduce alternating dependency quantified boolean formulae (ADQBF) which generalize both quantified and dependency quantified boolean formulae. We show that the truth evaluation for ADQBF is AEXPTIME(poly)-complete. We also identify fragments for which the problem is complete for the levels of the exponential hierarchy. Second we study propositional team-based logics. We show that DQBF formulae correspond naturally to quantified propositional dependence logic and present a general NEXPTIME upper bo…
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. Motivated by research on heuristics and the satisfiability threshold, Gopalan et al. in 2006 studied connectivity properties of the solution graph and related complexity issues for constraint satisfaction problems in Schaefer's framework. They found dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a…
Several effective preprocessing techniques for Boolean formulas with and without quantifiers use unit propagation to simplify the formula. Among these techniques are vivification, unit propagation look-ahead (UPLA), and the identification of redundant clauses as so-called quantified resolution asymmetric tautologies (QRAT). For quantified Boolean formulas (QBFs), these techniques have been extended to allow the application of universal reduction during unit propagation, which makes the techniques more effective. In this paper, we prove that the generalization of QBF to dependency quantified Bo…
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…
Modern solvers for Boolean Satisfiability (SAT) and Answer Set Programming (ASP) are based on sophisticated Boolean constraint solving techniques. In both areas, conflict-driven learning and related techniques constitute key features whose application is enabled by conflict analysis. Although various conflict analysis schemes have been proposed, implemented, and studied both theoretically and practically in the SAT area, the heuristic aspects involved in conflict analysis have not yet received much attention. Assuming a fixed conflict analysis scheme, we address the open question of how to ide…
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. For this implicitly defined graph, we here study the st-connectivity and connectivity problems. Building on the work of Gopalan et al. ("The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies", 2006/2009), we first investigate satisfiability problems given by CSPs, more exactly CNF(S)-formulas with constants (as considered in Schaefer's famous …
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…
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 …
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 …
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connecti…
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）
Given a specification $φ(X,Y)$ over inputs $X$ and output $Y$, defined over a background theory $\mathbb{T}$, the problem of program synthesis is to design a program $f$ such that $Y=f(X)$ satisfies the specification $φ$. Over the past decade, syntax-guided synthesis (SyGuS) has emerged as a dominant approach for program synthesis where in addition to the specification $φ$, the end-user also specifies a grammar $L$ to aid the underlying synthesis engine. This paper investigates the feasibility of synthesis techniques without grammar, a sub-class defined as $\mathbb{T}$-constrained synthesis. W…
We present novel algorithms for design and design space exploration. The designs discovered by these algorithms are compositions of function types specified in component libraries. Our algorithms reduce the design problem to quantified satisfiability and use advanced solvers to find solutions that represent useful systems. The algorithms we present in this paper are sound and complete and are guaranteed to discover correct designs of optimal size, if they exist. We apply our method to the design of Boolean systems and discover new and more optimal classical digital and quantum circuits for com…
We propose two models of random quantified boolean formulas and their natural random disjunctive logic program counterparts. The models extend the standard models of random k-CNF formulas and the Chen-Interian model of random 2QBFs. The first model controls the generation of programs and QSAT formulas by imposing a specific structure on rules and clauses, respectively. The second model is based on a family of QSAT formulas in a non-clausal form. We provide theoretical bounds for the phase transition region in our models, and show experimentally the presence of the easy-hard-easy pattern and it…
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…
Boolean satisfiability ({\SAT}) has played a key role in diverse areas spanning testing, formal verification, planning, optimization, inferencing and the like. Apart from the classical problem of checking boolean satisfiability, the problems of generating satisfying uniformly at random, and of counting the total number of satisfying assignments have also attracted significant theoretical and practical interest over the years. Prior work offered heuristic approaches with very weak or no guarantee of performance, and theoretical approaches with proven guarantees, but poor performance in practice…
We present a constructive SAT-based algorithm to determine the multiplicative complexity of a Boolean function, i.e., the smallest number of AND gates in any logic network that consists of 2-input AND gates, 2-input XOR gates, and inverters. In order to speed-up solving time, we make use of several symmetry breaking constraints; these exploit properties of XAGs that may be useful beyond the proposed SAT-based algorithm. We further propose a heuristic post-optimization algorithm to reduce the number of XOR gates once the optimum number of AND gates has been obtained, which also makes use of SAT…
This paper develops a parallel computational solver for computing all satifying assignments of a Boolean system of equations defined by Boolean functions of several variables. While there are we known solvers for satisfiability of Boolean formulas in CNF form, these are designed primarily for deciding satisfiability of the formula and do not address the problem of finding all satisfying solutions. Moreover development of parallel solvers for satisfiability problems is still an unfinished problem of Computer Science. The solver proposed in this paper is aimed at representing all solutions of Bo…
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 b…
The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory: Can the set N = $\{1, 2, ...\}$ of natural numbers be divided into two parts, such that no part contains a triple $(a,b,c)$ with $a^2 + b^2 = c^2$ ? A prize for the solution was offered by Ronald Graham over two decades ago. We solve this problem, proving in fact the impossibility, by using the Cube-and-Conquer paradigm, a hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers. An important role is played by dedicated look-ahead heuristics, which indeed allowed to solve the p…
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…
Starting from Boolean algebras of languages closed under quotients and using duality theoretic insights, we derive the notion of Boolean spaces with internal monoids as recognisers for arbitrary formal languages of finite words over finite alphabets. This leads to a setting that is well-suited for applying existing tools from Stone duality as applied in semantics. The main focus of the paper is the development of topo-algebraic constructions pertinent to the treatment of languages given by logic formulas. In particular, using the standard semantic view of quantification as projection, we deriv…
We consider the problem of elimination of existential quantifiers from a Boolean CNF formula. Our approach is based on the following observation. One can get rid of dependency on a set of variables of a quantified CNF formula F by adding resolvent clauses of F eliminating boundary points. This approach is similar to the method of quantifier elimination described in [9]. The difference of the method described in the present paper is twofold: {\bullet} branching is performed only on quantified variables, {\bullet} an explicit search for boundary points is performed by calls to a SAT-solver Altho…
