We present a method for approximating the semantics of probabilistic programs to the purpose of constructing semantics-based analyses of such programs. The method resembles the one based on Galois connection as developed in the Cousot framework for abstract interpretation. The main difference between our approach and the standard theory of abstract interpretation is the choice of linear space structures instead of order-theoretic ones as semantical (concrete and abstract) domains. We show that our method generates "best approximations" according to an appropriate notion of precision defined in terms of a norm. Moreover, if re-casted in a order-theoretic setting these approximations are correct in the sense of classical abstract interpretation theory. We use Concurrent ...

This paper reports on and discusses three notions of approximation for Labelled Markov Processes that have been developed last year. The three schemes are improvements over former constructions [11,12] in the sense that they define approximants that capture more properties than before and that converge faster to the approximated process. One scheme is constructive and the two others are driven by properties on which one wants to focus. All three constructions involve quotienting the state-space in some way and the last two are quotients with respect to sets of temporal properties expressed in a simple logic with a greatest fixed point operator. This gives the possibility of customizing approximants with respect to properties of interest and is thus an important step towards using automated techniques intended for finite state systems, e.g., model checking, for continuous state systems. Another difference between the schemes is how they relate approximants with the approximated process. The requirement that approximants should be simulated by

F8.928> h1 Gg // G 2 0 h2 G 2 g // G 3 0 G 3 g // h3 ::: 1 F1 f F 2 1 Ff oo F 3 1 F 2 f oo ::: F 3 f oo (1) Here 0 is the initial metric space, 1 is the nal metric space, and, for all n < !, h n+1 := Fh n (G n+1 0 ,! FG n 0). Let G ! 0 be the colimit in Met of the !-chain formed by the G n 0, with colimit injections n<F