For what purpose was the dataset created? The dataset was created with two primary purposes. First, it serves as a benchmark for einsum libraries, enabling the assessment of both the efficiency in determining contraction paths and the performance in executing einsum expressions. The dataset instances were created by the authors. Who funded the creation of the dataset?

Discrete integration is a fundamental problem in computer science that concerns the computation of discrete sums over exponentially large sets. Despite intense interest from researchers for over three decades, the design of scalable techniques for computing estimates with rigorous guarantees for discrete integration remains the holy grail. The key contribution of this work addresses this scalability challenge via an efficient reduction of discrete integration to model counting. The proposed reduction is achieved via a significant increase in the dimensionality that, contrary to conventional wisdom, leads to solving an instance of the relatively simpler problem of model counting. Building on the promising approach proposed by Chakraborty et al, our work overcomes the key weakness of their approach: a restriction to dyadic weights. We augment our proposed reduction, called DeWeight, with a state of the art efficient approximate model counter and perform detailed empirical analysis over benchmarks arising from neural network verification domains, an emerging application area of critical importance. DeWeight, to the best of our knowledge, is the first technique to compute estimates with provable guarantees for this class of benchmarks.

In this paper, we make a concrete progress towards answering discrete integration queries.

Satisfiability Modulo Theory (SMT) solvers have advanced automated reasoning, solving complex formulas across discrete and continuous domains. Recent progress in propositional model counting motivates extending SMT capabilities toward model counting, especially for hybrid SMT formulas. Existing approaches, like bit-blasting, are limited to discrete variables, highlighting the challenge of counting solutions projected onto the discrete domain in hybrid formulas. We introduce pact, an SMT model counter for hybrid formulas that uses hashing-based approximate model counting to estimate solutions with theoretical guarantees. pact makes a logarithmic number of SMT solver calls relative to the projection variables, leveraging optimized hash functions. pact achieves significant performance improvements over baselines on a large suite of benchmarks. In particular, out of 14,202 instances, pact successfully finished on 603 instances, while Baseline could only finish on 13 instances.

Modern society is full of computational challenges that rely on probabilistic reasoning, statistics, and combinatorics. Interestingly, many of these questions can be formulated by encoding them into propositional formulas and then asking for its number of models. With a growing interest in practical problem-solving for tasks that involve model counting, the community established the Model Counting (MC) Competition in fall of 2019 with its first iteration in 2020. The competition aims at advancing applications, identifying challenging benchmarks, fostering new solver development, and enhancing existing solvers for model counting problems and their variants. The first iteration, brought together various researchers, identified challenges, and inspired numerous new applications. In this paper, we present a comprehensive overview of the 2021-2023 iterations of the Model Counting Competition. We detail its execution and outcomes. The competition comprised four tracks, each focusing on a different variant of the model counting problem. The first track centered on the model counting problem (MC), which seeks the count of models for a given propositional formula. The second track challenged developers to submit programs capable of solving the weighted model counting problem (WMC). The third track was dedicated to projected model counting (PMC). Finally, we initiated a track that combined projected and weighted model counting (PWMC). The competition continued with a high level of participation, with seven to nine solvers submitted in various different version and based on quite diverging techniques.

Quantitative information flow analyses (QIF) are a class of techniques for measuring the amount of confidential information leaked by a program to its public outputs. Shannon entropy is an important method to quantify the amount of leakage in QIF. This paper focuses on the programs modeled in Boolean constraints and optimizes the two stages of the Shannon entropy computation to implement a scalable precise tool PSE. In the first stage, we design a knowledge compilation language called \ADDAND that combines Algebraic Decision Diagrams and conjunctive decomposition. \ADDAND avoids enumerating possible outputs of a program and supports tractable entropy computation. In the second stage, we optimize the model counting queries that are used to compute the probabilities of outputs. We compare PSE with the state-of-the-art probably approximately correct tool EntropyEstimation, which was shown to significantly outperform the existing precise tools. The experimental results demonstrate that PSE solved 55 more benchmarks compared to EntropyEstimation in a total of 441. For 98% of the benchmarks that both PSE and EntropyEstimation solved, PSE is at least $10\times$ as efficient as EntropyEstimation.

Model counting is a fundamental task that involves determining the number of satisfying assignments to a logical formula, typically in conjunctive normal form (CNF). While CNF model counting has received extensive attention over recent decades, interest in Pseudo-Boolean (PB) model counting is just emerging partly due to the greater flexibility of PB formulas. As such, we observed feature gaps in existing PB counters such as a lack of support for projected and incremental settings, which could hinder adoption. In this work, our main contribution is the introduction of the PB model counter PBCount2, the first exact PB model counter with support for projected and incremental model counting. Our counter, PBCount2, uses our Least Occurrence Weighted Min Degree (LOW-MD) computation ordering heuristic to support projected model counting and a cache mechanism to enable incremental model counting. In our evaluations, PBCount2 completed at least 1.40x the number of benchmarks of competing methods for projected model counting and at least 1.18x of competing methods in incremental model counting.

Model counting is a fundamental problem in automated reasoning with applications in probabilistic inference, network reliability, neural network verification, and more. Although model counting is computationally intractable from a theoretical perspective due to its #P-completeness, the past decade has seen significant progress in developing state-of-the-art model counters to address scalability challenges. In this work, we conduct a rigorous assessment of the scalability of model counters in the wild. To this end, we surveyed 11 application domains and collected an aggregate of 2262 benchmarks from these domains. We then evaluated six state-of-the-art model counters on these instances to assess scalability and runtime performance. Our empirical evaluation demonstrates that the performance of model counters varies significantly across different application domains, underscoring the need for careful selection by the end user. Additionally, we investigated the behavior of different counters with respect to two parameters suggested by the model counting community, finding only a weak correlation. Our analysis highlights the challenges and opportunities for portfolio-based approaches in model counting.

Approximate model counting is the task of approximating the number of solutions to an input Boolean formula. The state-of-the-art approximate model counter for formulas in conjunctive normal form (CNF), ApproxMC, provides a scalable means of obtaining model counts with probably approximately correct (PAC)-style guarantees. Nevertheless, the validity of ApproxMC's approximation relies on a careful theoretical analysis of its randomized algorithm and the correctness of its highly optimized implementation, especially the latter's stateful interactions with an incremental CNF satisfiability solver capable of natively handling parity (XOR) constraints. We present the first certification framework for approximate model counting with formally verified guarantees on the quality of its output approximation. Our approach combines: (i) a static, once-off, formal proof of the algorithm's PAC guarantee in the Isabelle/HOL proof assistant; and (ii) dynamic, per-run, verification of ApproxMC's calls to an external CNF-XOR solver using proof certificates. We detail our general approach to establish a rigorous connection between these two parts of the verification, including our blueprint for turning the formalized, randomized algorithm into a verified proof checker, and our design of proof certificates for both ApproxMC and its internal CNF-XOR solving steps. Experimentally, we show that certificate generation adds little overhead to an approximate counter implementation, and that our certificate checker is able to fully certify $84.7\%$ of instances with generated certificates when given the same time and memory limits as the counter.