Abstract
Kozen introduced probabilistic propositional dynamic logic (PPDL) in 1985 as a compositional framework to reason about probabilistic programs. In this paper we study expressiveness for PPDL and provide a series of results analogues to the classical Hennessy-Milner theorem for modal logic. First, we show that PPDL charaterises probabilistic trace equivalence of probabilistic automata (with outputs). Second, we show that PPDL can be mildly extended to yield a characterisation of probabilistic state bisimulation for PPDL models. Third, we provide a different extension of PPDL, this time characterising probabilistic event bisimulation.
Highlights
Probabilistic programming is an extension of imperative programming that enables the specification and implementation of randomized network and security protocols, machine learning and quantum algorithms
We introduce a suitable extension of propositional dynamic logic (PPDL), called PPDL+, and we show that it characterises state bisimilarity over PPDL models with analytic state spaces
Remark 4.1 Note that this new function constructor apparently resembles the shape of the constructor of PPDL formulas, but it is of a different nature: (−) > r is fixed for each rational number r ∈ Q≥0, and it yields a function, whereas (−) ≤ (−) in the PPDL syntax acts on two variable arguments, and it yields a formula
Summary
Probabilistic programming is an extension of imperative programming that enables the specification and implementation of randomized network and security protocols, machine learning and quantum algorithms. Hennessy and Milner first noticed a relationship between bisimulation of labelled transition systems (LTS) and a simple modal logic, subsequently referred to as Hennessy-Milner logic (HML) [10] They proved that HML characterises bisimilarity (the largest bisimulation) within the class of image-finite LTS: two states in an image-finite LTS are bisimilar if and only if they satisfy exactly the same HML formulas. (i) First, we show that PPDL functions (or, more precisely, PPDL with wellstructured programs) characterise probabilistic trace equivalence of PPDL models. These are the probabilistic analogues of Kripke models — probabilistic automata with a continuous state space and multiple output functions (Section 3). Differently from state bisimilation, this result does not require the state spaces to be analytical
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