TY - JOUR
T1 - Property-directed reachability as abstract interpretation in the monotone theory
AU - Feldman, Yotam M.Y.
AU - Sagiv, Mooly
AU - Shoham, Sharon
AU - Wilcox, James R.
N1 - Publisher Copyright:
© 2022 Owner/Author.
PY - 2022/1
Y1 - 2022/1
N2 - 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-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.,-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.
AB - 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-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.,-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.
KW - abstract interpretation
KW - invariant inference
KW - monotone theory
KW - property-directed reachability
KW - reachability diameter
UR - http://www.scopus.com/inward/record.url?scp=85123214135&partnerID=8YFLogxK
U2 - 10.1145/3498676
DO - 10.1145/3498676
M3 - מאמר
AN - SCOPUS:85123214135
VL - 6
JO - Proceedings of the ACM on Programming Languages
JF - Proceedings of the ACM on Programming Languages
SN - 2475-1421
IS - POPL
M1 - 3498676
ER -