Distributed and concurrent applications often have subtle bugs that only get exposed under specific schedules. While these schedules may be found by systematic model checking techniques, in practice, model checkers do not scale to large systems. On the other hand, naive random exploration techniques often require a very large number of runs to find the specific interactions needed to expose a bug. In recent years, several random testing algorithms have been proposed that, on the one hand, exploit state-space reduction strategies from model checking and, on the other, provide guarantees on the probability of hitting bugs of certain kinds. These existing techniques exploit two orthogonal strategies to reduce the state space: partial-order reduction and bug depth. Testing algorithms based on partial order techniques, such as RAPOS or POS, ensure nonredundant exploration of independent interleavings among system events by imposing an equivalence relation on schedules and ideally exploring only one schedule from each equivalence class. Techniques based on bug depth, such as PCT, exploit the empirical observation that many bugs are exposed by the clever scheduling of a small number of key events. They bias the sample space of schedules to only cover all executions of small depth, rather than the much larger space of all schedules. At this point, there is no random testing algorithm that combines the power of both approaches. In this paper, we provide such an algorithm. Our algorithm, trace-aware PCT (taPCT), extends and unifies several different algorithms in the random testing literature. It samples the space of low-depth executions by constructing a schedule online, while taking dependencies among events into account. Moreover, the algorithm comes with a theoretical guarantee on the probability of sampling a trace of low depthÐthe probability grows exponentially with the depth but only polynomially with the number of racy events explored. We further show that the guarantee is optimal among a large class of techniques. We empirically compare our algorithm with several state-of-the-art random testing approaches for concurrent software on two large-scale distributed systems, Zookeeper and Cassandra, and show that our approach is effective in uncovering subtle bugs and usually outperforms related random testing algorithms. CCS Concepts: • Mathematics of computing → Combinatorics; • Software and its engineering → Software testing and debugging; • Theory of computation → Generating random combinatorial structures.

Given a model and a specification, the fundamental model-checking problem asks for algorithmic verification of whether the model satisfies the specification. We consider graphs and Markov decision processes (MDPs), which are fundamental models for reactive systems. One of the very basic specifications that arise in verification of reactive systems is the strong fairness (aka Streett) objective. Given different types of requests and corresponding grants, the objective requires that for each type, if the request event happens infinitely often, then the corresponding grant event must also happen infinitely often. All ω-regular objectives can be expressed as Streett objectives and hence they are canonical in verification. To handle the state-space explosion, symbolic algorithms are required that operate on a succinct implicit representation of the system rather than explicitly accessing the system. While explicit algorithms for graphs and MDPs with Streett objectives have been widely studied, there has been no improvement of the basic symbolic algorithms. The worst-case numbers of symbolic steps required for the basic symbolic algorithms are as follows: quadratic for graphs and cubic for MDPs. In this work we present the first sub-quadratic symbolic algorithm for graphs with Streett objectives, and our algorithm is sub-quadratic even for MDPs. Based on our algorithmic insights we present an implementation of the new symbolic approach and show that it improves the existing approach on several academic benchmark examples.

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