Automatically inferring loop invariants via algorithmic learning

Jung, Young-Seok; Kong, Soonho; David, Cristina; Wang, Bow-Yaw; Yi, Kwangkeun

doi:10.1017/s0960129513000078

Cited by 4 publications

(2 citation statements)

References 29 publications

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“…The monotone theory [Bshouty 1995] has been used for other purposes in invariant inference. Jung et al [2015] use Bshouty's CDNF learning algorithm to infer predicate abstraction invariants, employing over-and under-approximations to resolve membership queries, sometimes relying on random guesses. Feldman et al [2021] use Bshouty's Λ-algorithm for provably-efficient inference of an invariant whose 𝑠-reachable (cf.…”

Section: Related Workmentioning

confidence: 99%

Property-directed reachability as abstract interpretation in the monotone theory

Feldman

Sagiv

Shoham

et al. 2022

Proc. ACM Program. Lang.

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Inferring inductive invariants is one of the main challenges of formal verification. The theory of abstract interpretation provides a rich framework to devise invariant inference algorithms. One of the latest breakthroughs in invariant inference is property-directed reachability (PDR), but the research community views PDR and abstract interpretation as mostly unrelated techniques. This paper shows that, surprisingly, propositional PDR can be formulated as an abstract interpretation algorithm in a logical domain. More precisely, we define a version of PDR, called Λ-PDR, in which all generalizations of counterexamples are used to strengthen a frame. In this way, there is no need to refine frames after their creation, because all the possible supporting facts are included in advance. We analyze this algorithm using notions from Bshouty’s monotone theory, originally developed in the context of exact learning. We show that there is an inherent overapproximation between the algorithm’s frames that is related to the monotone theory. We then define a new abstract domain in which the best abstract transformer performs this overapproximation, and show that it captures the invariant inference process, i.e., Λ-PDR corresponds to Kleene iterations with the best transformer in this abstract domain. We provide some sufficient conditions for when this process converges in a small number of iterations, with sometimes an exponential gap from the number of iterations required for naive exact forward reachability. These results provide a firm theoretical foundation for the benefits of how PDR tackles forward reachability.

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Section: Related Workmentioning

confidence: 99%

Property-directed reachability as abstract interpretation in the monotone theory

Feldman

Sagiv

Shoham

et al. 2022

Proc. ACM Program. Lang.

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“…The result in §7.3.1 applies to decision trees in the purely propositional setting, and leverages the CDNF algorithm [Bshouty 1995], which admits a bound on the number of hypotheses the learner presents before converging to the invariant. The CDNF algorithm [Bshouty 1995] has been applied by Jung et al [2015] to infer quantified invariants through predicate abstraction. They resolve membership queries by over-and under-approximations to some invariants, and use a random guess when the these are not conclusive, potentially leading to failures and restarts.…”

Section: Related Workmentioning

confidence: 99%

Learning the Boundary of Inductive Invariants

Feldman¹,

Sagiv²,

Shoham³

et al. 2020

Preprint

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We study the complexity of invariant inference and its connections to exact concept learning. We define a condition on invariants and their geometry, called the fence condition, which permits applying theoretical results from exact concept learning to answer open problems in invariant inference theory. The condition requires the invariant's boundary-the states whose Hamming distance from the invariant is one-to be backwards reachable from the bad states in a small number of steps. Using this condition, we obtain the first polynomial complexity result for an interpolation-based invariant inference algorithm, efficiently inferring monotone DNF invariants. We further harness Bshouty's seminal result in concept learning to efficiently infer invariants of a larger syntactic class of invariants beyond monotone DNF. Lastly, we consider the robustness of inference under program transformations. We show that some simple transformations preserve the fence condition, and that it is sensitive to more complex transformations.

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Invariant Inference with Provable Complexity from the Monotone Theory

Feldman

Shoham

2022

Static Analysis

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Automatically inferring loop invariants via algorithmic learning

Cited by 4 publications

References 29 publications

Property-directed reachability as abstract interpretation in the monotone theory

Property-directed reachability as abstract interpretation in the monotone theory

Learning the Boundary of Inductive Invariants

Invariant Inference with Provable Complexity from the Monotone Theory

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