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Propositional model counting is a classic problem that has recently witnessed many technical advances and novel applications. While the basic model counting problem requires computing the number of all solutions to the given formula, in some important application scenarios, the desired count is not of all solutions, but instead, of all unique solutions up to isomorphism. In such a scenario, the user herself must try to either use the full count that the model counter returns to compute the count up to isomorphism, or ensure that the input formula to the model counter adequately captures the symmetry breaking predicates so it can directly report the count she desires. We study the use of CNF-level and domain-level symmetry breaking predicates in the context of the state-of-the-art in model counting, specifically the leading approximate model counter ApproxMC and the recently introduced exact model counter ProjMC. As benchmarks, we use a range of problems, including structurally complex specifications of software systems and constraint satisfaction problems. The results show that while it is sometimes feasible to compute the model counts up to isomorphism using the full counts that are computed by the model counters, doing so suffers from poor scalability. The addition of symmetry breaking predicates substantially assists model counters. Domain-specific predicates are particularly useful, and in many cases can provide full symmetry breaking to enable highly efficient model counting up to isomorphism. We hope our study motivates new research on designing model counters that directly account for symmetries to facilitate further applications of model counting.

Relational properties, e.g., the connectivity structure of nodes in a distributed system, have many applications in software design and analysis. However, such properties often have to be written manually, which can be costly and error-prone. This paper introduces the MCML approach for empirically studying the learnability of a key class of such properties that can be expressed in the well-known software design language Alloy. A key novelty of MCML is quantification of the performance of and semantic differences among trained machine learning (ML) models, specifically decision trees, with respect to entire input spaces (up to a bound on the input size), and not just for given training and test datasets (as is the common practice). MCML reduces the quantification problems to the classic complexity theory problem of model counting, and employs state-of-the-art approximate and exact model counters for high efficiency. The results show that relatively simple ML models can achieve surprisingly high performance (accuracy and F1 score) at learning relational properties when evaluated in the common setting of using training and test datasets -even when the training dataset is much smaller than the test dataset -indicating the seeming simplicity of learning these properties. However, the use of MCML metrics based on model counting shows that the performance can degrade substantially when tested against the whole (bounded) input space, indicating the high complexity of precisely learning these properties, and the usefulness of model counting in quantifying the true accuracy.

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