Almost-certain eventualities and abstract probabilities in the quantitative temporal logic qTL

Authors:
Annabelle McIver;Carroll Morgan
Affiliations:
Department of Computing, Macquarie University, Sydney NSW 2019 Australia;Department of Engineering and Computer Science, University of New South Wales, Sydney NSW 2052 Australia
Venue:
Theoretical Computer Science
Year:
2003

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Abstract

'Almost-certain eventualities' are liveness properties that hold with probability 1. 'Abstract probabilities' in transition systems are those known only to be bounded away from zero and one.Vardi (Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, Portland, 1985, p. 327) showed that almost-certain properties in linear temporal logic depend only on abstract probabilities rather than on the probabilities' precise values. We discuss the extent to which a similar result holds in the quantitative temporal logic qTL derived from the quantitative modal µ,calculus qMµ (Proceedings the Formal Methods Pacific '97, Springer, Singapore, 1997, also available http://web.comlab.ox.ac.uk/oucl/research/areas/probs/bibliography.html; logic J, IGPL 7(6) (1999) 779, http://www3.oup.co.uk/igpl/Volume_07/Issue_06, http://web.comlab. ox.ac.uk/oucl/research/areas/probs/bibliography.html), and we show how to specialise the logic to these cases. The aim is to provide a simpler calculus than the full logic, one that is in a certain sense complete for proving almost-certain eventualities from abstract-probabilistic assumptions.We concinde by considering the complexity of the specialised logic.