Property Directed Reachability with Extended Resolution

TL;DR

This paper introduces PdrER, an extension of the Pdr model checking algorithm that uses Extended Resolution to produce shorter proofs and verify more complex safety properties more efficiently.

Contribution

PdrER integrates Extended Resolution into Pdr, enabling shorter proofs and improved performance in safety property verification, with algorithmic enhancements for practical use.

Findings

PdrER solves more verification instances than Pdr.

PdrER outperforms Pdr in time efficiency.

PdrER can solve problems Pdr cannot within time limits.

Abstract

Property Directed Reachability (\textsc{Pdr}), also known as IC3, is a state-of-the-art model checking algorithm widely used for verifying safety properties. While \textsc{Pdr} is effective in finding inductive invariants, its underlying proof system, Resolution, limits its ability to construct short proofs for certain verification problems. This paper introduces \textsc{PdrER}, a novel generalization of \textsc{Pdr} that uses Extended Resolution (ER), a proof system exponentially stronger than Resolution, when constructing a proof of correctness. \PdrEV leverages ER to construct shorter bounded proofs of correctness, enabling it to discover more compact inductive invariants. While \PdrEV is based on \textsc{Pdr}, it includes algorithmic enhancements that had to be made in order to efficiently use ER in the context of model checking. We implemented \textsc{PdrER} in a new open-source verification framework and evaluated it on the Hardware Model Checking Competition benchmarks from 2019, 2020 and 2024. Our experimental evaluation demonstrates that \textsc{PdrER} outperforms \textsc{Pdr}, solving more instances in less time and uniquely solving problems that \textsc{Pdr} cannot solve within a given time limit. We argue that this paper represents a significant step toward making strong proof systems practically usable in model checking.