Model Checking CTL is Almost Always Inherently Sequential

Abstract

Abstract. The model checking problem for CTL is known to be P-complete (Clarke, Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of CTL obtained by restricting the use of temporal modalities or the use of negations-restrictions already studied for LTL by Sistla and Clarke (1985) and Markey (2004). For all these fragments, except for the trivial case without any temporal operator, we systematically prove model checking to be either inherently sequential (P-complete) or very efficiently parallelizable (LOGCFL-complete). For most fragments, however, model checking for CTL is already P-complete. Hence our results indicate that, in cases where the combined complexity is of relevance, approaching CTL model checking by parallelism cannot be expected to result in any significant speedup.We also completely determine the complexity of the model checking problem for all fragments of the extensions ECTL, CTL + , and ECTL + .

A Team Based Variant of CTL

“…For the sake of a contradiction, assume that ϕ is such that F a ϕ = F s EFp . Consider the following Kripke model K = (W, R, V ), where W = {1, 2, 3, 4}, R = {(1, 4), (4, 4), (2,3) We will show that F a,k ϕ = F s,k ϕ ′ . Let K be an arbitrary Kripke structure and T be a team of K of size at most k. Then it holds…”

“…Moreover a classification of the computational complexity of fragments of the satisfiability as well as the model checking problem of CTL by means of allowed Boolean operators and/or combinations of allowed temporal operators has been obtained recently [19,3]. A survey on Kripke semantics with connections to several areas of logic, e.g., temporal, dependence, and hybrid logic can be found in a work of Meier et al [20].…”

The Model Checking Fingerprints of CTL Operators

“…How CTL operators compare with respect to the complexity of model checking, was investigated in [3] in the following way. They completely characterize the complexity of CTL-MC(T, ∧, ∨) for every set T of CTL operators.…”

“…In the same way, the model checking problem for LTL was studied in detail [1]. The model checking problem for CTL has been deeper understood in [3] who examined the complexity of CTL fragments that have arbitrary CTL operators that are combined only with all monotone Boolean operators. For model checking, there are seven relevant fragments of Boolean operators [1], but only one of these was considered in [3].…”

Quirky Quantifiers: Optimal Models and Complexity of Computation Tree Logic

“…Many important properties of computations like safety, deadlock-freeness or fairness are expressible in temporal depth two or three. Exceptions are CTL model checking, which is inherently sequential only for unbounded temporal depth [BM+11]; furthermore modal satisfiability (as a sublogic of CTL) drops down to NP for bounded depth, but is otherwise PSPACEcomplete even for only one proposition [Hal95].…”