Symmetry breaking is a crucial technique in modern combinatorial solving, but it is difficult to be sure it is implemented correctly. The most successful approach to deal with bugs is to make solvers certifying, so that they output not just a solution, but also a mathematical proof of correctness in a standard format, which can then be checked by a formally verified checker. This requires justifying symmetry reasoning within the proof, but developing efficient methods for this has remained a long-standing open challenge. A fully general approach was recently proposed, but it relies on encoding lexicographic orders with big integers, which quickly becomes infeasible for large symmetries. In this work, we develop a method for instead encoding orders with auxiliary variables. We show that this leads to orders-of-magnitude speed-ups in both theory and practice by running experiments on proof logging and checking for SAT symmetry breaking using the state-of-the-art satsuma symmetry breaker and the VeriPB proof checking toolchain.

Applied combinatorial optimization has witnessed a revolution in performance since the turn of the millennium, but the complexity of modern solvers is making bugs an ever more serious concern. The most promising remedy is to make solvers certifying, so that they use proof logging to generate machine-verifiable proofs of correctness. We present the first example of state-of-the-art certified graph colouring by equipping the solver ZykovColor with VeriPB proof logging. Combined with the formally verified CakePB checker, this provides end-to-end formally certified results. An experimental evaluation shows excellent results with only moderate overhead for proof logging and checking.

Abstract We consider the verification of neural network policies for discrete-time stochastic systems with respect to reach-avoid specifications. We use a learner-verifier procedure that learns a certificate for the specification, represented as a neural network. Verifying that this neural network certificate is a so-called reach-avoid supermartingale (RASM) proves the satisfaction of a reach-avoid specification. Existing approaches for such a verification task rely on computed Lipschitz constants of neural networks. These approaches struggle with large Lipschitz constants, especially for reach-avoid specifications with high threshold probabilities. We present two key contributions to obtain smaller Lipschitz constants than existing approaches. First, we introduce logarithmic RASMs (logRASMs), which take exponentially smaller values than RASMs and hence have lower theoretical Lipschitz constants. Second, we present a fast method to compute tighter upper bounds on Lipschitz constants based on weighted norms. Our empirical evaluation shows we can consistently verify the satisfaction of reach-avoid specifications with probabilities as high as $$99.9999\%$$ 99.9999 % .

We present recursive small-step multi-agent A* (RS-MAA*), an exact algorithm that optimizes the expected reward in decentralized partially observable Markov decision processes (Dec-POMDPs). RS-MAA* builds on multi-agent A* (MAA*), an algorithm that finds policies by exploring a search tree, but tackles two major scalability concerns. First, we employ a modified, small-step variant of the search tree that avoids the double exponential outdegree of the classical formulation. Second, we use a tight and recursive heuristic that we compute on-the-fly, thereby avoiding an expensive precomputation. The resulting algorithm is conceptually simple, yet it shows superior performance on a rich set of standard benchmarks.