Verifying realworld 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, datadriven methods for loop invariant inference have shown promise, especially on linear loop invariants. However, applying datadriven inference to nonlinear loop invariants is challenging due to the large numbers of and large magnitudes of highorder 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 (GCLN), and apply it to nonlinear loop invariant learning. GCLNs 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 nonlinear loop invariants and show it solves 26 out of 27 problems, 3 more than prior work, with an average runtime of 53.3 seconds. We further demonstrate the generic learning ability of GCLNs by solving all 124 problems in the linear Code2Inv benchmark. We also perform a quantitative stability evaluation and show GCLNs have a convergence rate of 97.5% on quadratic problems, a 39.2% improvement over CLN models.
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DataDriven Invariant Learning for Probabilistic Programs
Morgan and McIver’s weakest preexpectation framework is one of the most wellestablished methods for deductive verification of probabilistic programs. Roughly, the idea is to generalize binary state assertions to realvalued expectations, which can measure expected values of probabilistic program quantities. While loopfree programs can be analyzed by mechanically transforming expectations, verifying loops usually requires finding an invariant expectation, a difficult task.
We propose a new view of invariant expectation synthesis as a regression problem: given an input state, predict the average value of the postexpectation in the output distribution. Guided by this perspective, we develop the first datadriven invariant synthesis method for probabilistic programs. Unlike prior work on probabilistic invariant inference, our approach can learn piecewise continuous invariants without relying on template expectations. We also develop a datadriven approach to learn subinvariants from data, which can be used to upper or lowerbound expected values. We implement our approaches and demonstrate their effectiveness on a variety of benchmarks from the probabilistic programming literature.
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 Award ID(s):
 2153916
 NSFPAR ID:
 10423578
 Editor(s):
 Shoham, Sharon; Vizel, Yakir
 Date Published:
 Journal Name:
 Lecture notes in computer science
 Volume:
 13371
 ISSN:
 03029743
 Page Range / eLocation ID:
 33–54
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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