THE TEMPORAL LOGIC OF TWO DIMENSIONAL MINKOWSKI SPACETIME IS DECIDABLE

We consider Minkowski spacetime, the set of all point-events of spacetime under the relation of causal accessibility. That is, x can access y if an electromagnetic or (slower than light) mechanical signal could be sent from x to y. We use Prior’s tense language of F and P representing causal accessibility and its converse relation. We consider two versions, one where the accessibility relation is reflexive and one where it is irreflexive. In either case it has been an open problem, for decades, whether the logic is decidable or axiomatisable. We make a small step forward by proving, in each case, that the set of valid formulas over two-dimensional Minkowski spacetime is decidable and that the complexity of each problem is PSPACE-complete.A consequence is that the temporal logic of intervals with real endpoints under either the containment relation or the strict containment relation is PSPACE-complete, the same is true if the interval accessibility relation is “each endpoint is not earlier”, or its irreflexive restriction.We provide a temporal formula that distinguishes between three-dimensional and two-dimensional Minkowski spacetime and another temporal formula that distinguishes the two-dimensional case where the underlying field is the real numbers from the case where instead we use the rational numbers.

R2U2: Tool Overview

R2U2 (Realizable, Responsive, Unobtrusive Unit) is an extensible framework for runtime System Health Management (SHM) of cyber-physical systems. R2U2 can be run in hardware (e.g., FPGAs), or software; can monitor hardware, software, or a combination of the two; and can analyze a range of different types of system requirements during runtime. An R2U2 requirement is specified utilizing a hierarchical combination of building blocks: temporal formula runtime observers (in LTL or MTL), Bayesian networks, sensor filters, and Boolean testers. Importantly, the framework is extensible; it is designed to enable definitions of new building blocks in combination with the core structure. Originally deployed on Unmanned Aerial Systems (UAS), R2U2 is designed to run on a wide range of embedded platforms, from autonomous systems like rovers, satellites, and robots, to human-assistive ground systems and cockpits.R2U2 is named after the requirements it satisfies; while the exact requirements vary by platform and mission, the ability to formally reason about Realizability, Responsiveness, and Unobtrusiveness is necessary for flight certifiability, safety-critical system assurance, and achievement of technology readiness levels for target systems. Realizability ensures that R2U2 is sufficiently expressive to encapsulate meaningful runtime requirements while maintaining adaptability to run on different platforms, transition be- tween different mission stages, and update quickly between missions. Responsiveness entails continuously monitoring the system under test, real-time reasoning, reporting intermediate status, and as-early-as-possible requirements evaluations. Unobtrusiveness ensures compliance with the crucial properties of the target architecture: functionality, certifiability, timing, tolerances, cost, or other constraints.

An Efficient Specification for System Verification

In design of complex and large scale systems, system verification has played an important role. In this article, we focus on specification process of model checking in system verifications. Modeled systems are in general specified by temporal formulas of computation tree logic, and users must know well about temporal specification because the specification might be complex. We propose a method by which specifications with temporal formulas are obtained inductively. We will show verification results using the proposed temporal formula specification method, and show that amount of memory, OBDD nodes, and execution time are reduced.

Inductive Temporal Formula Specifications for System Verification

Design verification has played an important role in the design of large scale and complex systems. In this article, we focus on model checking methods. Behaviors of modeled systems are in general specified by temporal formulas of computation tree logic, and users must know well about temporal specification because the specification might be complex. We propose a method temporal formulas are obtained inductively, and amounts of memory and time are reduced. We will show verification results using the proposed method.

FINITE STATE PROCESSES, Z-TEMPORAL LOGIC AND THE MONADIC THEORY OF THE INTEGERS

We introduce four classes of Z-regular grammars for generating bi-infinite words (i.e. Z-words) and prove that they generate exactly Z-regular languages. We extend the second order monadic theory of one successor to the set of the integers (i.e. Z) and give some characterizations of this theory in terms of Z-regular grammars and Z-regular languages. We prove that this theory is decidable and equivalent to the weak theory. We also extend the linear temporal logic to Z-temporal logic and then prove that each Z-temporal formula is equivalent to a first order monadic formula. We prove that the correctness problem for finite state processes is decidable.