Linear Temporal Logic and Z Refinement

Abstract

Abstract. Since Z, being a state-based language, describes a system in terms of its state and potential state changes, it is natural to want to describe properties of a specified system also in terms of its state. One means of doing this is to use Linear Temporal Logic (LTL) in which properties about the state of a system over time can be captured. This, however, raises the question of whether these properties are preserved under refinement. Refinement is observation preserving and the state of a specified system is regarded as internal and, hence, non-observable. In this paper, we investigate this issue by addressing the following questions. Given that a Z specification A is refined by a Z specification C , and that P is a temporal logic property which holds for A, what temporal logic property Q can we deduce holds for C ? Furthermore, under what circumstances does the property Q preserve the intended meaning of the property P ? The paper answers these questions for LTL, but the approach could also be applied to other temporal logics over states such as CTL and the µ-calculus.

Slicing Object-Z Specifications for Verification

“…The technique can substantially facilitate verification of specifications since the preparatory construction of the program dependence graph is only linear in the size of the original specification while its state space is usually much larger (or infinite) and might therefore not be amenable for an analysis. Slicing can thus be seen as one method for fighting the state explosion problem in verification, along with other techniques like abstraction (for Z for instance by combining the work of [14] and [4]), symmetry reduction, compositional verification (like e.g. [20]) and partial order reductions.…”

Property transformation under specification change

“…It allows us to readily state temporal properties under the assumption that operations continue to occur [15]. Furthermore, the preservation of LTL properties under blocking semantics refinement has been investigated in [20]. The results show that most LTL properties are preserved by data refinement and all are preserved under simple restrictions on the refinement retrieve relation.…”

Invariant-driven specifications in Maude

“…They describe a system in terms of its state and the changes of this state by giving invariants and pre-and post-conditions of operations, and support specification, verification, refinement, and analysis of programs at early stages of development and at high level of abstraction. Derrick and Smith [18] explore the use of temporal logics to express properties about the state of systems over time, and how those properties are preserved under refinement. They point out how understanding the relation of temporal logics and Z was a first step toward developing model-checking support for the language, as later realized, e.g., in [17].…”