LTL Path Checking Is Efficiently Parallelizable

Abstract

Abstract. We present an AC 1 (logDCFL) algorithm for checking LTL formulas over finite paths, thus establishing that the problem can be efficiently parallelized. Our construction provides a foundation for the parallelization of various applications in monitoring, testing, and verification.Linear-time temporal logic (LTL) is the standard specification language to describe properties of reactive computation paths. The problem of checking whether a given finite path satisfies an LTL formula plays a key role in monitoring and runtime verification [12,10,6,1,4], where individual paths are checked either online, during the execution of the system, or offline, for example based on an error report. Similarly, path checking occurs in testing [2] and in several static verification techniques, notably in Monte-Carlo-based probabilistic verification, where large numbers of randomly generated sample paths are analyzed [22].Somewhat surprisingly, given the widespread use of LTL, the complexity of the path checking problem is still open [18]. The established upper bound is P: The algorithms in the literature traverse the path sequentially (cf. [10,18,12]); by going backwards from the end of the path, one can ensure that, in each step, the value of each subformula is updated in constant time, which results in bilinear running time. The only known lower bound is NC 1 [8], the complexity of evaluating Boolean expressions. The large gap between the bounds is especially unsatisfying in light of the recent trend to implement path checking algorithms in hardware, which is inherently parallel. For example, the IEEE standard temporal logic PSL [13], an extension of LTL, has become part of the hardware description language VHDL, and several tools [6,4] are available to synthesize hardware-based monitors from assertions written in PSL. Can we improve over the sequential approach by evaluating entire blocks of path positions in parallel?In this paper, we show that LTL path checking can indeed be parallelized efficiently. Our approach is inspired by work in the related area of evaluating monotone Boolean circuits [11,9,15,3,17,5]. Rather than sequentially traversing the path, we consider the circuit that results from unrolling the formula over the path. Figure 1 shows such a circuit for the formula ((a U b) U (c U d)) U e and a path of length 5.

A survey of challenges for runtime verification from advanced application domains (beyond software)

“…Solutions to these research efforts exist (see for instance [66,85,106,149,152,220,221]). We refer to [164] for a recent survey on this topic.…”

“…Impact of utility on monitoring complexity. The existing work on the complexity of monitoring [85,152,220] (called path checking in this context) only considers the problem of providing a single Boolean verdict in an offline manner. Tight complexity bounds for the online monitoring problem or other variants of the problem with different output utility (e.g., a verdict stream) have not yet been established.…”

Path Checking for MTL and TPTL over Data Words

“…For LTL and periodic words without data values, it was shown in [21] that the pathchecking problem can be solved using an efficient parallel algorithm. More precisely, the problem belongs to AC 1 (LogDCFL), a subclass of NC.…”

“…By Theorem 3.6 in [24], infinite path checking for LTL can be reduced in logspace to finite path checking for LTL. Finite path checking for LTL is in AC 1 (LogDCFL) ⊆ P [21]. Proof.…”

MapReduce for parallel trace validation of LTL properties

“…Kuhtz and Finkbeiner showed in 2009 that LTL path checking belongs to the complexity class AC 1 (logDCFL) [29]; this result entails that the process can be efficiently split by evaluating entire blocks of events in parallel. Rather than sequentially traversing the trace, their work considers the circuit that results from "unrolling" the formula over the trace.…”