Instantiations, Zippers and EPR Interpolation

Bjørner, Nikolaj; Gurfinkel, Arie; Korovin, Konstantin; Lahav, Ori

doi:10.29007/xt3j

Cited by 3 publications

(5 citation statements)

References 14 publications

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“…Specifically, Alg. 4 is based on (the time-dual of) a model-based ITP algorithm [Bjùrner et al 2013;Chockler et al 2012] whose generalization procedure was inspired by PDR.…”

Section: Forward Reachability In λ-Pdr and Othersmentioning

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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“…Specifically, Alg. 4 is based on (the time-dual of) a model-based ITP algorithm [Bjùrner et al 2013;Chockler et al 2012] whose generalization procedure was inspired by PDR.…”

Section: Forward Reachability In λ-Pdr and Othersmentioning

confidence: 99%

Property-directed reachability as abstract interpretation in the monotone theory

Feldman

Sagiv

Shoham

et al. 2022

Proc. ACM Program. Lang.

View full text Add to dashboard Cite

show abstract

“…Specifically, Alg. 4 is based on (the time-dual of) a model-based ITP algorithm [Bjørner et al 2013;Chockler et al 2012] whose generalization procedure was inspired by PDR.…”

Section: Dual Interpolation the Essence Of Interpolation-based Infere...mentioning

confidence: 99%

“…Algorithm 4 Dual Model-Based ITP, based on [Bjørner et al 2013;Chockler et al 2012] In each iteration, the algorithm samples a prestate 𝜎 𝑏 of a counterexample to the induction of 𝜑, a state in 𝜑 that in one step reaches states outside 𝜑 (line 4). Instead of excluding just the counterexample-similarly to PDR-the algorithm seeks a minimal clause 𝑐 over the literals that are falsified in 𝜎 𝑏 that does not exclude a state from R 𝑠 , the set of states that the system can reach in 𝑠 steps (line 7), and conjoins 𝑐 to the candidate (line 8).…”

Section: Dual Interpolation the Essence Of Interpolation-based Infere...mentioning

confidence: 99%

Property-Directed Reachability as Abstract Interpretation in the Monotone Theory

Feldman¹,

Sagiv²,

Shoham³

et al. 2021

Preprint

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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. CCS Concepts: • Theory of computation → Theory and algorithms for application domains; Program verification; • Software and its engineering → Formal methods.

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“…As a by-product of this research, we are also translating AIGER benchmarks into first-order TPTP benchmarks which can be used to evaluate first-order theorem provers on hardware verification problems. As a future work we are planning to compare and combine our approach with other property directed methods such as PDR/IC3 [19] and interpolation [18,22,29].…”

Section: Resultsmentioning

confidence: 99%

“…Partial interpretations can be represented as sets of ground literals. Let us note that there are also more advanced model representations, e.g., based on cubes used in Z3 [29] or dismatching constraints [30,31] used in iProver, that allow more compact representation of ground sets of literals. In the Inst-Gen framework in place of an explicit model representation we can enforce partial interpretations by fixing a literal selection in clauses using propositional assumptions, this was used in our implementation.…”

Section: Partial Modelsmentioning

confidence: 99%

EPR-based k-induction with Counterexample Guided Abstraction Refinement

Khasidashvili¹,

Korovin²,

Tsarkov³

EPiC Series in Computing

Self Cite

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In recent years it was proposed to encode bounded model checking (BMC) into the effectively propositional fragment of first-order logic (EPR). The EPR fragment can provide for a succinct representation of the problem and facilitate reasoning at a higher level. In this paper we present an extension of the EPR-based bounded model checking with k-induction which can be used to prove safety properties of systems over unbounded runs. We present a novel abstraction-refinement approach based on unsatisfiable cores and models (UCM) for BMC and k-induction in the EPR setting. We have implemented UCM refinements for EPR-based BMC and k-induction in a first-order automated theorem prover iProver. We also extended iProver with the AIGER format and evaluated it over the HWMCC'14 competition benchmarks. The experimental results are encouraging. We show that a number of AIG problems can be verified until deeper bounds with the EPR-based model checking.

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Instantiations, Zippers and EPR Interpolation

Cited by 3 publications

References 14 publications

Property-directed reachability as abstract interpretation in the monotone theory

Property-directed reachability as abstract interpretation in the monotone theory

Property-Directed Reachability as Abstract Interpretation in the Monotone Theory

EPR-based k-induction with Counterexample Guided Abstraction Refinement

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