Michael R. Hansen

Duration calculus: Logical foundations

The Duration Calculus (abbreviated DC) represents a logical approach for formal design of real-time systems, where real numbers are used to model time and Boolean valued functions over time are used to model states and events of real-time systems. Since its introduction, DC has been applied to many case studies and it has been extended in several directions. The aim of this paper is to provide a thorough presentation of the logic.

An Extended Duration Calculus for Hybrid Real-Time Systems

Duration Calculus is a real-time interval logic which can be used to specify and reason about timing and logical constraints on discrete states in a dynamic system. It has been used to specify and verify designs for a number of real-time systems. This paper extends the Duration Calculus with notations to capture properties of piecewise continuous states. This is useful for reasoning about hybrid systems with a mixture of continuous and discrete states. The proof theory of Duration Calculus is extended such that results proven using mathematical analysis can be used freely in the logic. This provides a flexible interface to conventional control theory.

Provably Correct Systems

As computers increasingly control the systems and services we depend upon within our daily lives like transport, communications, and the media, ensuring these systems function correctly is of utmost importance. This book consists of twelve chapters and one historical account that were presented at a workshop in London in 2015, marking the 25th anniversary of the European ESPRIT Basic Research project ProCoS (Provably Correct Systems). The ProCoS I and II projects pioneered and accelerated the automation of verification techniques, resulting in a wide range of applications within many trades and sectors such as aerospace, electronics, communications, and retail. The following topics are covered: An historical account of the ProCoS projectHybrid Systems Correctness of Concurrent Algorithms Interfaces and Linking Automatic VerificationRun-time Assertions Checking Formal and Semi-Formal Methods Provably Correct Systems provides researchers, designers and engineers with a complete overview of the ProCoS initiative, past and present, and explores current developments and perspectives within the field.

An Adequate First Order Interval Logic

This paper introduces left and right neighbourhoods as primitive interval modalities to define other unary and binary modalities of intervals in a first order logic with interval length. A complete first order logic for the neighbourhood modalities is presented. It is demonstrated how the logic can support formal specification and verification of liveness and fairness, and also of various notions of real analysis.

Semantics and Completeness of Duration Calculus

Duration Calculus was introduced in [1] as a notation to specify real-time systems, and as a calculus to verify theorems about such systems. Its distinctive feature is reasoning about durations of states within any time interval, without explicit mention of absolute time. Duration Calculus, which is an extension of Interval Temporal Logic, was originally designed to reason about real-time requirements for control systems; but it has been used at other levels of abstraction also: for example to give real-time semantics to communicating processes executed on a shared processor configuration and to reason about the correctness of a circuit transformation. The purpose of this paper is to introduce a formal syntax and semantics for Duration Calculus, and to prove its completeness — relative to the completeness of Interval Temporal Logic.

DEHAR: A distributed energy harvesting aware routing algorithm for ad-hoc multi-hop wireless sensor networks

One of the key design goals in Wireless Sensor Networks is long lasting or even continuous operation. Continuous operation is made possible through energy harvesting. Keeping the network operational imposes a demand to prevent network segmentation and power loss in nodes. It is therefore important that the best energy-wise route is found for each data transfer from a source node to the sink node. We present a new adaptive and distributed routing algorithm for finding energy optimised routes in a wireless sensor network with energy harvesting. The algorithm finds an energy efficient route from each source node to a single sink node, taking into account the current energy status of the network. By simulation, the algorithm is shown to be able to adapt to changes in harvested and stored energy. Simulations show that continuous operation is possible.

Model-checking discrete duration calculus

Duration Calculus was introduced in [ZHR91] as a logic to specify and reason about requirements for real-time systems. It is an extension of Interval Temporal Logic [Mos85] where one can reason about integrated constraints over time-dependent and Boolean valued states without explicit mention of absolute time. Several major case studies, e.g. the gas burner system in [RRH93], have shown that Duration Calculus provides a high level of abstraction for both expressing and reasoning about specifications. Using Timed Automata [A1D92] one can express how real-time systems can be constructed at a level of detail which is close to an actual implementation. We consider in the paper the correctness of Timed Automata with respect to Duration Calculus formulae. For a subset of Duration Calculus, we show that one can automatically verify whether a Timed Automaton ℳ is correct with respect to a formulaD, abbreviated ℳ ⊨D, i.e. one can domodel-checking. The subset we consider is expressive enough to formalize the requirements to the gas burner system given in [RRH93]; but only for a discrete time domain. Model-checking is done by reducing the correctness problem ℳ ⊨D to the inclusion problem of regular languages.

Finite divergence

Real-time and hybrid systems have been studied so far under the assumption of finite variability. In this paper, we consider models in which systems exhibiting finite divergence can also be analysed. In such systems the state of the system can change infinitely often in a finite time. This kind of behaviour arises in many representations of hybrid systems, and also in theories of nonlinear systems. The aim, here, is to provide a theory where pathological behaviour such as finite divergence can be analysed if only to pvoue that it does not occur in systems of interest. Finite divergence is studied using the framework of duration calculus. Axioms and proof rules are given. Patterns of occurrence of divergence are classified into dense divergence, accumulative divergence and discrete divergence by appropriate axioms. Induction rules are given for reasoning about discrete divergence.

Deciding an interval logic with accumulated durations

A decidability result and a model-checking procedure for a rich subset of Duration Calculus (DC) [19] is obtained through reductions to first-order logic over the real-closed field and to Multi-Priced Timed Automata (MPTA) [13]. In contrast to other reductions of fragments of DC to reachability problems in timed automata, the reductions do also cover constraints on positive linear combinations of accumulated durations. By being able to handle accumulated durations under chop as well as in arbitrary positive Boolean contexts, the procedures extend the results of Zhou et al. [22] on decidability of linear duration invariants to a much wider fragment of DC.

A timed semantics for SDL

An alternative formal semantics for describing the temporal aspects for the ITU-T specification language SDL is proposed, based on the interval logic Duration Calculus (DC). It is shown how DC can be used to give an SDL semantics with a precise treatment of temporal phenomena. The semantics allows true concurrency. We show how it can be used to address issues such as the verification of temporal properties, process scheduling, and the nature of viewed (shared) variables.