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 $\mathbf{1 4, 2 0 2}$ instances, pact successfully finished on 603 instances, while Baseline could only finish on 13 instances.

Satisfiability Modulo Theory (SMT) has recently emerged as a powerful tool for solving various automated reasoning problems across diverse domains. Unlike traditional satisfiability methods confined to Boolean variables, SMT can reason on real-life variables like bitvectors, integers, and reals. A natural extension in this context is to ask quantitative questions. One such query in the SMT theory of Linear Real Arithmetic (LRA) is computing the volume of the entire satisfiable region defined by SMT formulas. This problem is important in solving different quantitative verification queries in software verification, cyber-physical systems, and neural networks, to mention a few. We introduce ttc, an efficient algorithm that extends the capabilities of SMT solvers to volume computation. Our method decomposes the solution space of SMT Linear Real Arithmetic formulas into a union of overlapping convex polytopes, then computes their volumes and calculates their union. Our algorithm builds on recent developments in streaming-mode set unions, volume computation algorithms, and AllSAT techniques. Experimental evaluations demonstrate significant performance improvements over existing state-of-the-art approaches.

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.

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.

The past three decades have witnessed notable success in designing efficient SAT solvers, with modern solvers capable of solving industrial benchmarks containing millions of variables in just a few seconds. The success of modern SAT solvers owes to the widely-used CDCL algorithm, which lacks comprehensive theoretical investigation. Furthermore, it has been observed that CDCL solvers still struggle to deal with specific classes of benchmarks comprising only hundreds of variables, which contrasts with their widespread use in real-world applications. Consequently, there is an urgent need to uncover the inner workings of these seemingly weak yet powerful black boxes. In this paper, we present a first step towards this goal by introducing an approach called CausalSAT, which employs causal reasoning to gain insights into the functioning of modern SAT solvers. CausalSAT initially generates observational data from the execution of SAT solvers and learns a structured graph representing the causal relationships between the components of a SAT solver. Subsequently, given a query such as whether a clause with low literals blocks distance (LBD) has a higher clause utility, CausalSAT calculates the causal effect of LBD on clause utility and provides an answer to the question. We use CausalSAT to quantitatively verify hypotheses previously regarded as "rules of thumb" or empirical findings such as the query above. Moreover, CausalSAT can address previously unexplored questions, like which branching heuristic leads to greater clause utility in order to study the relationship between branching and clause management. Experimental evaluations using practical benchmarks demonstrate that CausalSAT effectively fits the data, verifies four "rules of thumb", and provides answers to three questions closely related to implementing modern solvers.

Real-time systems are increasingly being implemented on heterogeneous multi-core platforms to efficiently cater to their diverse and high computation demands. Over the years, researchers have developed mechanisms to efficiently schedule tasks on homogeneous multi-cores such that all tasks meet their execution and deadline requirements. However, devising an efficient scheduling strategy for real-time tasks on heterogeneous platforms has proved to be a challenging as well as computationally expensive problem. Today, there is a severe dearth of low-overhead techniques towards real-time scheduling on heterogeneous platforms. Hence, we propose an effective low-overhead heuristic approach for scheduling a set of periodic tasks executing on a heterogeneous multi-core platform. Employing the concept of deadline partitioning to obtain a set of discrete time slices, we propose a scheme to efficiently schedule tasks over these time slices while incurring low and bounded number of migrations. Conducted experiments have shown promising results and indicate to the practical efficacy of our approach.