The well-known Kirschner-Panetta model for Tumour-Immune System interplay [Kirschner and Panetta, J. Math. Biol 37 (3), 1998] reproduces a number of features of this essential interaction, but it excludes the possibility of tumour suppression by the immune system in the absence of therapy. Here we present a hybrid-stochastic version of that model. In this new framework, we show that in reality the model is also able to reproduce the suppression, through stochastic extinction after the first spike of an oscillation.

In model checking for temporal logic, the correctness of a system with respect to a desired behavior is verified by checking whether a structure that models the system satisfies a formula describing the behavior. Most existing verification techniques are based on a representation of the system by means of a labeled transition system. In this approach to verification, the efficiency of the model checking is essentially influenced by the number of states of the transition system. In this paper we present a new temporal logic, the selective mu-calculus, and an equivalence between transition systems based on the formulae of this logic. This property preserving equivalence can be used to reduce the size of transition systems. The equivalence (called \-equivalence) is based on the set, \, of actions occurring inside the modal operators of a selective mu-calculus formula. We prove that the \-equivalence coincides with the equivalence induced by the set of the selective mu-calculus formulae with occurring actions in \. Thus, a formula can be more efficiently checked on a transition system \-equivalent to the standard one, but smaller than it, since all the actions not in \ are``discarded.'' 1999 Academic Press

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The theory of abstract interpretation provides a formal framework to develop advanced dataflow analysis tools. The idea is to define a nonstandard semantics which is able to compute, in finite time, an approximated model of the program. In this paper, we define an abstract interpretation framework based on a fixpoint approach to the semantics. This leads to the definition, by means of a suitable set of operators, of an abstract fixpoint characterization of a model associated with the program. Thus, we obtain a specializable abstract framework for bottom-up abstract interpretations of definite logic programs. The specialization of the framework is shown on two examples, namely, ground-dependence analysis and depth-k analysis.

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