The study of phase transition phenomenon of NP complete problems plays an important role in understanding the nature of hard problems. In this paper, we follow this line of research by considering the problem of counting solutions of Constraint Satisfaction Problems (#CSP). We consider the random model, i.e. RB model. We prove that phase transition of #CSP does exist as the number of variables approaches infinity and the critical values where phase transitions occur are precisely located. Preliminary experimental results also show that the critical point coincides with the theoretical derivation. Moreover, we propose an approximate algorithm to estimate the expectation value of the solutions number of a given CSP instance of RB model.
United Technologies Research Center（联合技术研究中心）；SRI International（SRI国际）
Given a Boolean formula $ϕ(x)$ in conjunctive normal form (CNF), the density of states counts the number of variable assignments that violate exactly $e$ clauses, for all values of $e$. Thus, the density of states is a histogram of the number of unsatisfied clauses over all possible assignments. This computation generalizes both maximum-satisfiability (MAX-SAT) and model counting problems and not only provides insight into the entire solution space, but also yields a measure for the \emph{hardness} of the problem instance. Consequently, in real-world scenarios, this problem is typically infeas…
Hashing-based model counting has emerged as a promising approach for large-scale probabilistic inference on graphical models. A key component of these techniques is the use of xor-based 2-universal hash functions that operate over Boolean domains. Many counting problems arising in probabilistic inference are, however, naturally encoded over finite discrete domains. Techniques based on bit-level (or Boolean) hash functions require these problems to be propositionalized, making it impossible to leverage the remarkable progress made in SMT (Satisfiability Modulo Theory) solvers that can reason di…
Max Planck Institute for Software Systems (MPI-SWS)（马克斯·普朗克软件系统研究所）
#SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In the realm of satisfiability modulo theories (SMT) there is a growing need for model counting solvers, coming from several application domains (quantitative information flow, static analysis of probabilistic programs). In this paper, we show a reduction from an approximate version of #SMT to SMT. We focus on the theories of integer arithmetic and linear real arithmetic. We propose m…
Approximate model counting for bit-vector SMT formulas (generalizing \#SAT) has many applications such as probabilistic inference and quantitative information-flow security, but it is computationally difficult. Adding random parity constraints (XOR streamlining) and then checking satisfiability is an effective approximation technique, but it requires a prior hypothesis about the model count to produce useful results. We propose an approach inspired by statistical estimation to continually refine a probabilistic estimate of the model count for a formula, so that each XOR-streamlined query yield…
Chennai Mathematical Institute, India IAI, TCG CREST, Kolkata, India（钦奈数学研究所，印度 IAI，TCG CREST，科利尔，印度）；Georgia Institute of Technology, USA University of Toronto, Canada（佐治亚理工学院，美国 蒙特利尔大学，加拿大）
Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）
There are already quite a few tools for solving the Satisfiability Modulo Theories (SMT) problems. In this paper, we present \texttt{VolCE}, a tool for counting the solutions of SMT constraints, or in other words, for computing the volume of the solution space. Its input is essentially a set of Boolean combinations of linear constraints, where the numeric variables are either all integers or all reals, and each variable is bounded. The tool extends SMT solving with integer solution counting and volume computation/estimation for convex polytopes. Effective heuristics are adopted, which enable t…
We propose a new approximate method for counting the number of the solutions for constraint satisfaction problem (CSP). The method derives from the partition function based on introducing the free energy and capturing the relationship of probabilities of variables and constraints, which requires the marginal probabilities. It firstly obtains the marginal probabilities using the belief propagation, and then computes the number of solutions according to the partition function. This allows us to directly plug the marginal probabilities into the partition function and efficiently count the number …
While the solution counting problem for propositional satisfiability (#SAT) has received renewed attention in recent years, this research trend has not affected other AI solving paradigms like answer set programming (ASP). Although ASP solvers are designed to enumerate all solutions, and counting can therefore be easily done, the involved materialization of all solutions is a clear bottleneck for the counting problem of ASP (#ASP). In this paper we propose dynamic programming-based #ASP algorithms that exploit the structure of the underlying (ground) ASP program. Experimental results for a pro…
University of Chinese Academy of Sciences（中国科学院大学）；Peking University（北京大学）
Constrained counting is important in domains ranging from artificial intelligence to software analysis. There are already a few approaches for counting models over various types of constraints. Recently, hashing-based approaches achieve both theoretical guarantees and scalability, but still rely on solution enumeration. In this paper, a new probabilistic polynomial time approximate model counter is proposed, which is also a hashing-based universal framework, but with only satisfiability queries. A variant with a dynamic stopping criterion is also presented. Empirical evaluation over benchmarks…
McGill University（麦吉尔大学）；Mila – Quebec AI Institute（米拉 - 魁北克人工智能研究所）
We present the Neural Satisfiability Network (NSNet), a general neural framework that models satisfiability problems as probabilistic inference and meanwhile exhibits proper explainability. Inspired by the Belief Propagation (BP), NSNet uses a novel graph neural network (GNN) to parameterize BP in the latent space, where its hidden representations maintain the same probabilistic interpretation as BP. NSNet can be flexibly configured to solve both SAT and #SAT problems by applying different learning objectives. For SAT, instead of directly predicting a satisfying assignment, NSNet performs marg…
National ICT Australia（澳大利亚国家信息与通信技术研究院）；The University of Melbourne（墨尔本大学）
Model counting is the task of computing the number of assignments to variables V that satisfy a given propositional theory F. Model counting is an essential tool in probabilistic reasoning. In this paper, we introduce the problem of model counting projected on a subset P of original variables that we call 'priority' variables. The task is to compute the number of assignments to P such that there exists an extension to 'non-priority' variables V¶that satisfies F. Projected model counting arises when some parts of the model are irrelevant to the counts, in particular when we require additional v…
KU Leuven（鲁汶大学）；Czech Technical University in Prague（布拉格捷克理工大学）
We study the symmetric weighted first-order model counting task and present ApproxWFOMC, a novel anytime method for efficiently bounding the weighted first-order model count in the presence of an unweighted first-order model counting oracle. The algorithm has applications to inference in a variety of first-order probabilistic representations, such as Markov logic networks and probabilistic logic programs. Crucially for many applications, we make no assumptions on the form of the input sentence. Instead, our algorithm makes use of the symmetry inherent in the problem by imposing cardinality con…
Department of Computer Science（计算机科学系）；University of Oxford（牛津大学）
Weighted model counting (WMC) has emerged as a prevalent approach for probabilistic inference. In its most general form, WMC is #P-hard. Weighted DNF counting (weighted #DNF) is a special case, where approximations with probabilistic guarantees are obtained in O(nm), where n denotes the number of variables, and m the number of clauses of the input DNF, but this is not scalable in practice. In this paper, we propose a neural model counting approach for weighted #DNF that combines approximate model counting with deep learning, and accurately approximates model counts in linear time when width is…
Quantitative extensions of logic programming often require the solution of so called second level inference tasks, i.e., problems that involve a third operation, such as maximization or normalization, on top of addition and multiplication, and thus go beyond the well-known weighted or algebraic model counting setting of probabilistic logic programming under the distribution semantics. We introduce Second Level Algebraic Model Counting (2AMC) as a generic framework for these kinds of problems. As 2AMC is to (algebraic) model counting what forall-exists-SAT is to propositional satisfiability, it…
College of Computer Science and Technology（计算机科学与技术学院）；Jilin University（吉林大学）
Model counting is the problem of computing the number of satisfying assignments of a given propositional formula. Although exact model counters can be naturally furnished by most of the knowledge compilation (KC) methods, in practice, they fail to generate the compiled results for the exact counting of models for certain formulas due to the explosion in sizes. Decision-DNNF is an important KC language that captures most of the practical compilers. We propose a generalized Decision-DNNF (referred to as partial Decision-DNNF) via introducing a class of new leaf vertices (called unknown vertices)…
Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University（符号计算与知识工程教育部重点实验室，吉林大学）；School of Computing, National University of Singapore（新加坡国立大学计算学院）
Model counting is a fundamental problem which has been influential in many applications, from artificial intelligence to formal verification. Due to the intrinsic hardness of model counting, approximate techniques have been developed to solve real-world instances of model counting. This paper designs a new anytime approach called PartialKC for approximate model counting. The idea is a form of partial knowledge compilation to provide an unbiased estimate of the model count which can converge to the exact count. Our empirical analysis demonstrates that PartialKC achieves significant scalability …
National University of Singapore（新加坡国立大学）；Rice University（莱斯大学）
Propositional model counting is a fundamental problem in artificial intelligence with a wide variety of applications, such as probabilistic inference, decision making under uncertainty, and probabilistic databases. Consequently, the problem is of theoretical as well as practical interest. When the constraints are expressed as DNF formulas, Monte Carlo-based techniques have been shown to provide a fully polynomial randomized approximation scheme (FPRAS). For CNF constraints, hashing-based approximation techniques have been demonstrated to be highly successful. Furthermore, it was shown that has…
Technische Universität Dresden（德累斯顿工业大学）；University of Wrocław（弗罗茨瓦夫大学）
In logic-based knowledge representation, query answering has essentially replaced mere satisfiability checking as the inferencing problem of primary interest. For knowledge bases in the basic description logic ALC, the computational complexity of conjunctive query (CQ) answering is well known to be ExpTime-complete and hence not harder than satisfiability. This does not change when the logic is extended by certain features (such as counting or role hierarchies), whereas adding others (inverses, nominals or transitivity together with role-hierarchies) turns CQ answering exponentially harder. We…
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加州大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
Integrating logical reasoning within deep learning architectures has been a major goal of modern AI systems. In this paper, we propose a new direction toward this goal by introducing a differentiable (smoothed) maximum satisfiability (MAXSAT) solver that can be integrated into the loop of larger deep learning systems. Our (approximate) solver is based upon a fast coordinate descent approach to solving the semidefinite program (SDP) associated with the MAXSAT problem. We show how to analytically differentiate through the solution to this SDP and efficiently solve the associated backward pass. W…
In Peña (2007), MCMC sampling is applied to approximately calculate the ratio of essential graphs (EGs) to directed acyclic graphs (DAGs) for up to 20 nodes. In the present paper, we extend that work from 20 to 31 nodes. We also extend that work by computing the approximate ratio of connected EGs to connected DAGs, of connected EGs to EGs, and of connected DAGs to DAGs. Furthermore, we prove that the latter ratio is asymptotically 1. We also discuss the implications of these results for learning DAGs from data.
