By Metropolis N., et al. (eds.)

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**Example text**

R (N ) := – The r-Dirac measure on N is deﬁned: δN On Quantitative Security Policies 31 – Given a ∈ A the function [a κ ] : A → Δ(H≡ ) is such that [a κ ](x) = if x = a then κ else ω In the following we will use this operator instantiated in [a δα ], with δ α αDirac delta measure on . k ≡ l · h [ h μ](a)(H) = μ(a)( ) if k ≡ h, k ≡ δ ⎪ k∈H ⎩0 otherwise – The binary operator ⊕ : Δ(H≡ )A ×Δ(H≡ )A → Δ(H≡ )A is deﬁned as follows: (μ ⊕ μ )(a)(H) = μ(a)(H) + μ (a)(H) The operators are well-deﬁned and enjoy the following properties.

3. Hence, we obtain that ψ is violated because the probability that φ is violated is about 8%. 08. This analysis suggests to deploy the system equipped with a monitoring mechanism that abort an execution when it is about to violate φ. More details are in Section 3. 1 Preliminaries We review the main notions and notations about measure theory and we refer the reader to [3,2] for more details. Given the support set M = ∅, a σ-algebra Σ over M is a set of subsets of M , the measurable sets, containing ∅ and closed under complement and countable 26 P.

These express in a ﬁnite way potentially inﬁnite timed histories and enable us to model check quantitative policies using wellknown techniques. Indeed, the semantics of a HEμ process is given in terms of a CTMC, that implicitly describes both the timed histories and the long run behaviour of a program. 4). 1 Syntax The building blocks of HEμ are stochastic events. e. the parameter of the exponential random variable modelling its duration. 1. A stochastic history expressions ( HEμ ) h ∈ H is a term generated by the following grammar: h1 , h2 ::= (a, α) ψ[e1 ] (stochastic event) | (policy framing) | h1 + h2 (stochastic choice) | δ (deadlock) | h1 · h2 (sequentialization) | (h1 ∗ h2 ) (binary Kleene star) A stochastic event (a, α) performs action a and then successfully terminates.