Learning nonlinear loop invariants with gated continuous logic networks

Abstract

Verifying real-world programs often requires inferring loop invariants with nonlinear constraints. This is especially true in programs that perform many numerical operations, such as control systems for avionics or industrial plants. Recently, data-driven methods for loop invariant inference have shown promise, especially on linear loop invariants. However, applying data-driven inference to nonlinear loop invariants is challenging due to the large numbers of and large magnitudes of high-order terms, the potential for overfitting on a small number of samples, and the large space of possible nonlinear inequality bounds.In this paper, we introduce a new neural architecture for general SMT learning, the Gated Continuous Logic Network (G-CLN), and apply it to nonlinear loop invariant learning. G-CLNs extend the Continuous Logic Network (CLN) architecture with gating units and dropout, which allow the model to robustly learn general invariants over large numbers of terms. To address overfitting that arises from finite program sampling, we introduce fractional sampling-a sound relaxation of loop semantics to continuous functions that facilitates unbounded sampling on the real domain. We additionally design a new CLN activation function, the Piecewise Biased Quadratic Unit (PBQU), for naturally learning tight inequality bounds.We incorporate these methods into a nonlinear loop invariant inference system that can learn general nonlinear loop invariants. We evaluate our system on a benchmark of * Equal contribution

Quantitative Analysis of Assertion Violations in Probabilistic Programs

“…We manually derived affine invariants for the input PTSs. Alternatively, invariant generation, which is an orthogonal problem to ours, can be automated by approaches such as [9,26,44,50]. Similarly, we proved almost-sure termination by manually constructing ranking supermartingales [6,11].…”

Learning Security Classifiers with Verified Global Robustness Properties

“…Then, we re-train the classifier subject to the constraint that 𝑅 1 ≤ 𝑅 2 . To enforce this constraint, we smooth the discrete classifier using Continuous Logic Networks (CLN) [71,95], and then use projected gradient descent with the constraint to train the classifier. Gradient-guided optimization ensures that this counterexample (𝑥, 𝑥 ′ ) will no longer violate the property and tries to achieve the highest accuracy subject to that constraint.…”

“…Otherwise, we construct a counterexample according to solutions for the boolean variables. For the trainer, we use PyTorch to implement the smoothed classifier as Continuous Logic Networks [71,95]. Then, we use quadratic programming to implement projected gradient descent, where we minimize the square of ℓ 2 norm between the updated weights and the projected weights subject to a set of training constraints.…”

“…In comparison, our integer linear program verifier has only integer variables, and represents all inputs symbolically. Continuous Logic Networks was proposed to smooth SMT formulas to learn loop invariants [71,95]. In this paper, we apply the smoothing techniques to train machine learning classifiers.…”

“…In this case, we want the difference between the output for 𝑥 and the output for 𝑥 ′ to be bounded by 1, i.e., |𝑅 3 − 𝑅 1 | ≤ 1. Then, we smooth the classifier using CLN [71,95], train the weights using projected gradient descent with the constraint. After one epoch, we have updated the classifier in the rightmost box of Figure 4.…”