We apply multivariate Lagrange interpolation to synthesizing polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of a quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments.

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Abstract. Recent studies have suggested the applicability of learning to automated compositional verification. However, current learning algorithms fall short when it comes to learning liveness properties. We extend the automaton synthesis paradigm for the infinitary languages by presenting an algorithm to learn an arbitrary regular set of infinite sequences (an ω-regular language) over an alphabet Σ. Our main result is an algorithm to learn a nondeterministic Büchi automaton that recognizes an unknown ω-regular language. This is done by learning a unique projection of it on Σ * using the framework suggested by Angluin for learning regular subsets of Σ * .

Abstract. Algorithms for learning a minimal separating DFA of two disjoint regular languages have been proposed and adapted for different applications. One of the most important applications is learning minimal contextual assumptions in automated compositional verification. We propose in this paper an efficient learning algorithm, called L Sep , that learns and generates a minimal separating DFA. Our algorithm has a quadratic query complexity in the product of sizes of the minimal DFA's for the two input languages. In contrast, the most recent algorithm of Gupta et al. has an exponential query complexity in the sizes of the two DFA's. Moreover, experimental results show that our learning algorithm significantly outperforms all existing algorithms on randomly-generated example problems. We describe how our algorithm can be adapted for automated compositional verification. The adapted version is evaluated on the LTSA benchmarks and compared with other automated compositional verification approaches. The result shows that our algorithm surpasses others in 30 of 49 benchmark problems.

Abstract. This paper presents results on formal verification of high-speed cryptographic software. We consider speed-record-setting hand-optimized assembly software for Curve25519 ellipticcurve key exchange in [11]. Two versions for different microarchitectures are available. We successfully verify the core part of computation, and reproduce a bug in a previously published edition. An SMT solver for the bit-vector theory is used to establish almost all properties. Remaining properties are verified in a proof assistant. We also exploit the compositionality of Hoare logic to address the scalability issue. Essential differences between both versions of the software are discussed from a formal-verification perspective.