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Clausal proofs, particularly those based on the deletion resolution asymmetric tautology (DRAT) proof system, are widely used by Boolean satisfiability solvers for expressing proofs of unsatisfiability. Their success stems from their simplicity and scalability. When solvers go beyond pure propositional reasoning, however, generating clausal proofs becomes more difficult. Solvers that employ pseudo-Boolean reasoning, including cutting-planes operations, can express proofs in the VeriPB proof system, but its adoption is not widespread. We introduce PBIP (Pseudo-Boolean Implication Proof), a framework that provides an intermediate representation between VeriPB and clausal proofs. We also introduce a toolchain comprising 1) a VeriPB-to-PBIP translator that performs proof trimming and optimization, and 2) a PBIP-to-LRAT translator that makes use of proof-generating operations on ordered binary decision diagrams (BDDs) to generate clausal proofs in LRAT format, a variant of the DRAT that allows efficient checking. We demonstrate the viability of our approach, the effectiveness of our trimming, and the performance of our clausal proof generator on a set of native PB benchmarks and compare our approach to direct checking of VeriPB proofs. 

Deep reinforcement learning (DRL) is a powerful machine learning paradigm for generating agents that control autonomous systems. However, the “black box” nature of DRL agents limits their deployment in real-world safety-critical applications. A promising approach for providing strong guarantees on an agent's behavior is to use Neural Lyapunov Barrier (NLB) certifcates, which are learned functions over the system whose properties indirectly imply that an agent behaves as desired. However, NLB-based certifcates are typically diffcult to learn and even more diffcult to verify, especially for complex systems. In this work, we present a novel method for training and verifying NLB-based certifcates for discrete-time systems. Specifcally, we introduce a technique for certifcate composition, which simplifes the verifcation of highly-complex systems by strategically designing a sequence of certifcates. When jointly verifed with neural network verifcation engines, these certifcates provide a formal guarantee that a DRL agent both achieves its goals and avoids unsafe behavior. Furthermore, we introduce a technique for certifcate fltering, which signifcantly simplifes the process of producing formally verifed certifcates. We demonstrate the merits of our approach with a case study on providing safety and liveness guarantees for a DRL-controlled spacecraft. 

Quantization replaces floating point arithmetic with integer arithmetic in deep neural networks, enabling more efficient on-device inference with less power and memory. However, it also brings in loss of generalization and even potential errors to the models. In this work, we propose a parallelization technique for formally verifying the equivalence between quantized models and their original real-valued counterparts. In order to guarantee both soundness and completeness, mixed integer linear programming (MILP) is deployed as the baseline technique. Nevertheless, the incorporation of two networks as well as the mixture of integer and real number arithmetic make the problem much more challenging than verifying a single network, and thus using MILP alone is inadequate for the non-trivial cases. To tackle this, we design a distributed verification technique that can leverage hundreds of CPUs on high-performance computing clusters. We develop a two-tier parallel framework and propose property- and output-based partition strategies. Evaluated on perception networks quantized with PyTorch, our approach outperforms existing methods in successfully verifying many cases that are otherwise considered infeasible.