9-Duration Calculus

9-Duration Calculus - Duration Calculus Michael R. Hansen...

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Duration Calculus Michael R. Hansen Department of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark. [email protected] Duration Calculus (DC) is an interval logic which was introduced to express and reason about models of real-time systems. DC was introduced by Zhou Chaochen, Tony Hoare and A.P. Ravn during the ProCoS I Project (ESPRIT BRA 3104, 1989–1991) [6]. Formal techniques for the construction of safety- critical systems were investigated in this project, and in an early case study of gas burner systems, conducted by E.V. Sørensen, A.P. Ravn and H. Rischel, it turned out that certain requirements for such systems were not expressible in the real-time formalisms which were available at that time. A key issue in the design of gas burners is the need to restrict the dura- tion of the undesired state where gas is leaking. This state of the system is unavoidable, as gas must flow for a little while before it can be ignited. But the accumulated time periods in which gas is leaking over a time interval of given size. DC was introduced as a logical approach that supported modelling and reasoning about durational constraints on the states of safety-critical real-time systems [147]. DC was developed as an extension of Interval Temporal Logic (ITL) [32, 85] because many timing properties occurring in the case studies considered were, in fact, interval properties, and if one had a modal logic for intervals as a basis, these requirements could be formalized in a succinct manner. We shall present the basic concepts of DC in this chapter, with an eye to new applications within the area of security protocols. We shall give some background on the introduction of DC and a brief survey of some work done on DC. In [140], there is an overview of early research on DC, in [38], there is a detailed account of the logical foundations of DC, and in the monograph [145], there is a detailed account of DC. A comprehensive survey of interval logics can be found in [27]. A comprehensive introduction to modal logics is presented in [7]. This work is partially funded by The Danish Council for Strategic Research under project MoDES .
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300 Michael R. Hansen Furthermore, in the monograph [90], there is a account of temporal log- ics from a historical perspective, where logics based on notions of intervals are traced back to studies of the meaning of natural language by medieval logicians. John Buridan 2 , for example, regarded the present as a duration and not as a point in time, and he considered the truth of propositions relative to the choice of the present. A proposition p is true during the present if and only if (i±) there is a part of the present during which the truth of p is given . As an example (from [90]), the proposition “Socrates is alive” is true in the entire present (now) if there is a subinterval where it is given that “Socrates is alive”.
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This note was uploaded on 03/31/2010 for the course CAS 707 taught by Professor Ridhakhedri during the Spring '10 term at McMaster University.

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9-Duration Calculus - Duration Calculus Michael R. Hansen...

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