Propositional Model Counting Research Articles

An Approximate Skolem Function Counter

One approach to probabilistic inference involves counting the number of models of a given Boolean formula. Here, we are interested in inferences involving higher-order objects, i.e., functions. We study the following task: Given a Boolean specification between a set of inputs and outputs, count the number of functions of inputs such that the specification is met. Such functions are called Skolem functions. We are motivated by the recent development of scalable approaches to Boolean function synthesis. This stands in relation to our problem analogously to the relationship between Boolean satisfiability and the model counting problem. Yet, counting Skolem functions poses considerable new challenges. From the complexity-theoretic standpoint, counting Skolem functions is not only #P-hard; it is quite unlikely to have an FPRAS (Fully Polynomial Randomized Approximation Scheme) as the problem of synthesizing a Skolem function remains challenging, even given access to an NP oracle. The primary contribution of this work is the first algorithm, SkolemFC, that computes the number of Skolem functions. SkolemFC relies on technical connections between counting functions and propositional model counting: our algorithm makes a linear number of calls to an approximate model counter and computes an estimate of the number of Skolem functions with theoretical guarantees. Our prototype displays impressive scalability, handling benchmarks comparably to state-of-the-art Skolem function synthesis engines, even though counting all such functions ostensibly poses a greater challenge than synthesizing a single function.

Exact ASP Counting with Compact Encodings

Answer Set Programming (ASP) has emerged as a promising paradigm in knowledge representation and automated reason- ing owing to its ability to model hard combinatorial problems from diverse domains in a natural way. Building on advances in propositional SAT solving, the past two decades have wit- nessed the emergence of well-engineered systems for solv- ing the answer set satisfiability problem, i.e., finding mod- els or answer sets for a given answer set program. In re- cent years, there has been growing interest in problems be- yond satisfiability, such as model counting, in the context of ASP. Akin to the early days of propositional model count- ing, state-of-the-art exact answer set counters do not scale well beyond small instances. Exact ASP counters struggle with handling larger input formulas. The primary contribu- tion of this paper is a new ASP counting framework, called sharpASP, which counts answer sets avoiding larger input formulas. This relies on an alternative way of defining answer sets that allows lifting of key techniques developed in the con- text of propositional model counting. Our extensive empirical analysis over 1470 benchmarks demonstrates significant per- formance gain over current state-of-the-art exact answer set counters. Specifically, by using sharpASP, we were able to solve 1062 benchmarks with PAR2 score of 3082 whereas using prior state-of-the-art, we could only solve 895 bench- marks with PAR2 score of 4205, all other experimental con- ditions being the same.

Combining bounded solving and controllable randomization for approximate model counting

ABSTRACT Propositional model counting is the problem of computing the satisfying assignments count of a given CNF formula. Due to the weak solving ability of the existing exact model counters in large-scale problems, approximate model counting has been proposed as a practical alternative to exact model counting. Most of the best current approaches are based on XOR constraints for approximate model counting, also known as the hashing-based method. In this paper, a new approximate model counter using short XOR constraints is proposed, which integrates bounded solving and controllable randomisation. By constantly adding short XOR constants, the solution space has been reduced to a smaller one. However, a smaller scale of solution space may result in worse result accuracy. Therefore, by limiting the scale of solution space that applies XOR constraints, bounded solving effectively increases the accuracy of a model counter. Controllable randomisation makes use of backbone variables and the constraint level among variables, such that the short XOR constraints can reach the same reduction effect on solution space as the long ones. It improves the quality of XOR constraints as well as the SAT solving efficiency. Experimentally, we demonstrate that in almost every benchmark used in well-known ApproxMC2 and STAC_CNF, our approach outperforms the existing approximate model counters in both accuracy and efficiency.

ApproxASP – a Scalable Approximate Answer Set Counter

Answer Set Programming (ASP) is a framework in artificial intelligence and knowledge representation for declarative modeling and problem solving. Modern ASP solvers focus on the computation or enumeration of answer sets. However, a variety of probabilistic applications in reasoning or logic programming require counting answer sets. While counting can be done by enumeration, simple enumeration becomes immediately infeasible if the number of solutions is high. On the other hand, approaches to exact counting are of high worst-case complexity. In fact, in propositional model counting, exact counting becomes impractical. In this work, we present a scalable approach to approximate counting for answer set programming. Our approach is based on systematically adding XOR constraints to ASP programs, which divide the search space. We prove that adding random XOR constraints partitions the answer sets of an ASP program. In practice, we use a Gaussian elimination-based approach by lifting ideas from SAT to ASP and integrating it into a state of the art ASP solver, which we call ApproxASP. Finally, our experimental evaluation shows the scalability of our approach over the existing ASP systems.

Learning Branching Heuristics for Propositional Model Counting

Propositional model counting, or #SAT, is the problem of computing the number of satisfying assignments of a Boolean formula. Many problems from different application areas, including many discrete probabilistic inference problems, can be translated into model counting problems to be solved by #SAT solvers. Exact #SAT solvers, however, are often not scalable to industrial size instances. In this paper, we present Neuro#, an approach for learning branching heuristics to improve the performance of exact #SAT solvers on instances from a given family of problems. We experimentally show that our method reduces the step count on similarly distributed held-out instances and generalizes to much larger instances from the same problem family. It is able to achieve these results on a number of different problem families having very different structures. In addition to step count improvements, Neuro# can also achieve orders of magnitude wall-clock speedups over the vanilla solver on larger instances in some problem families, despite the runtime overhead of querying the model.

Definability for model counting

We define and evaluate a new preprocessing technique for propositional model counting. This technique leverages definability, i.e., the ability to determine that some gates are implied by the input formula Σ. Such gates can be exploited to simplify Σ without modifying its number of models. Unlike previous techniques based on gate detection and replacement, gates do not need to be made explicit in our approach. Our preprocessing technique thus consists of two phases: computing a bipartition 〈I,O〉 of the variables of Σ where the variables from O are defined in Σ in terms of I, then eliminating some variables of O in Σ. Our experiments show the computational benefits which can be achieved by taking advantage of our preprocessing technique for model counting.

On Preprocessing Techniques and Their Impact on Propositional Model Counting

This paper is concerned with preprocessing techniques for propositional model counting. We have considered several elementary preprocessing techniques: backbone identification, occurrence reduction, vivification, as well as equivalence, AND and XOR gate identification and replacement. All those techniques have been implemented in a preprocessor pmc, freely available on the Web. In order to assess the benefits which can be gained by taking advantage of pmc, we performed many experiments, based on benchmarks coming from several data sets. More precisely, we made a differential evaluation of each elementary preprocessing technique in order to evaluate its impact on the number of variables of the instance, its size, as well as the treewidth of its primal graph. We also considered two combinations of preprocessings: $$ eq $$eq, based on equivalence-preserving techniques only, and $${\#eq}$&#eq, which additionally exploits techniques preserving only the number of models. Several approaches to model counting have also been considered downstream in our experiments: "direct" model counters, including the exact ones Cachet, sharpSAT, and an approximate one SampleCount, as well as the compilation-based model counters C2D, Dsharp, SDD and cnf2obdd have been used. The experimental results we have obtained show that each elementary preprocessing technique is useful, and that some synergetic effects can be achieved by combining them.

Solving #SAT and MAXSAT by Dynamic Programming

We look at dynamic programming algorithms for propositional model counting, also called #SAT, and MaxSAT. Tools from graph structure theory, in particular treewidth, have been used to successfully identify tractable cases in many subfields of AI, including SAT, Constraint Satisfaction Problems (CSP), Bayesian reasoning, and planning. In this paper we attack #SAT and MaxSAT using similar, but more modern, graph structure tools. The tractable cases will include formulas whose class of incidence graphs have not only unbounded treewidth but also unbounded clique-width. We show that our algorithms extend all previous results for MaxSAT and #SAT achieved by dynamic programming along structural decompositions of the incidence graph of the input formula. We present some limited experimental results, comparing implementations of our algorithms to state-of-the-art #SAT and MaxSAT solvers, as a proof of concept that warrants further research.

Solving #SAT and MaxSAT by dynamic programming

We look at dynamic programming algorithms for propositional model counting, also called #SAT, and MaxSAT. Tools from graph structure theory, in particular treewidth, have been used to successfully ...

Preprocessing for Propositional Model Counting

This paper is concerned with preprocessing techniques for propositional model counting. We have implemented a preprocessor which includes many elementary preprocessing techniques, including occurrence reduction, vivification, backbone identification, as well as equivalence, AND and XOR gate identification and replacement. We performed intensive experiments, using a huge number of benchmarks coming from a large number of families. Two approaches to model counting have been considered downstream: ”direct” model counting using Cachet and compilation-based model counting, based on the C2D compiler. The experimental results we have obtained show that our preprocessor is both efficient and robust.

Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width

We provide a parameterized algorithm for the propositional model counting problem #SAT, the runtime of which has a single-exponential dependency on the rank-width of the signed graph of a formula. That is, our algorithm runs in time $\cal{O}t^3 \cdot 2^{3tt+1/2} \cdot \vert\phi\vert$ for a width-t rank-decomposition of the input φ, and can be of practical interest for small values of rank-width. Previously, analogical algorithms have been known --e.g. [Fischer, Makowsky, and Ravve] --with a single-exponential dependency on the clique-width k of the signed graph of a formula with a given k-expression. Our algorithm presents an exponential runtime improvement over the worst-case scenario of the previous one, since clique-width reaches up to exponentially higher values than rankwidth. We also provide an algorithm for the MAX-SAT problem along the same lines.

Model Counting in Product Configuration

We describe how to use propositional model counting for a quantitative analysis of product configuration data. Our approach computes valuable meta information such as the total number of valid configurations or the relative frequency of components. This information can be used to assess the severity of documentation errors or to measure documentation quality. As an application example we show how we apply these methods to product documentation formulas of the Mercedes-Benz line of vehicles. In order to process these large formulas we developed and implemented a new model counter for non-CNF formulas. Our model counter can process formulas, whose CNF representations could not be processed up till now.

Algorithms for propositional model counting

We present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comparison and with sufficient detail for making an actual implementation reasonably easy. We discuss several aspects of the algorithms including worst-case time and space requirements.