Falsification-Aware Semantics for Temporal Logics and Their Inconsistency-Tolerant Subsystems: Theoretical Foundations of Falsification-Aware Model Checking

Model checking is well known to be a computer-aided method for verifying concurrent systems. Temporal logics and their Kripke-style semantics have been widely used in model checking. Falsification-aware Kripke-style semantics for temporal logics have been required for the theoretical basis of model checking because falsification plays a critical role in obtaining counter-example traces for the underlying object specifications in model checking. However, a useful falsification-aware Kripke-style semantics has yet to be developed for standard temporal logics. Hence, this study introduces two types of falsification-aware Kripke-style semantics for standard temporal logics that have been typically used in model checking. The equivalences among the proposed falsification-aware and standard Kripke-style semantics for standard temporal logics are proved. Furthermore, some inconsistency-tolerant subsystems of standard temporal logics are semantically obtained from the proposed falsification-aware Kripke-style semantics for standard temporal logics by deleting a characteristic condition on the labeling function of the semantics. The proposed semantic framework for standard and inconsistency-tolerant temporal logics is regarded as a unified framework for generalizing and combining the existing standard, inconsistency-tolerant, and many-valued semantic frameworks. This unified semantic framework is useful for obtaining a theoretical basis for generalized (inconsistency-tolerant) model checking, referred to here as falsification-aware model checking.