Abstract We introduce a family of norms on the $n \times n$ complex matrices. These arise from probabilistic framework, and their construction validation involve probability theory, partition combinatorics, trace polynomials in noncommuting variables. As consequence, we obtain generalization Hunter’s positivity theorem for complete homogeneous symmetric polynomials.

This paper provides one more step towards bridging the gap between the formal and computational approaches to the verification of cryptographic protocols. We extend the well-known Abadi-Rogaway logic with probabilistic hashes and give a precise semantic interpretation to it using Canetti’s oracle hashes. These are probabilistic polynomialtime hashes that hide all partial information. Finally, w...

In this paper we study several properties of the Exogenous Probabilistic Propositional Logic (EPPL), a logic for reasoning about probabilities, with the purpose of introducing a temporal version Exogenous Probabilistic Linear Temporal Logic (EPLTL). In detail, we give a small model theorem for EPPL and introduce a satisfaction and a model checking algorithm for both EPPL and EPLTL. We are also ...

We generalise the classical notion of stationary distributions of Markov processes to a model of probabilistic programs which includes demonic nondeterminism. As well as removing some of the conditions normally required for stationarity, our generalisation allows the development of a complete theory linking stationary behaviour to long-term average behaviour — the latter being an important prop...

A topos of presheaves can be seen as an extension of classical set theory, where sets vary over informational states, therefore it is a powerful and expressive mathematical framework. I introduce a suitable topos of presheaves where probabilities and probabilistic reasoning can be represented. In this way we obtain a mathematical definition of probabilistic sets. A valid and complete proof syst...

A topos of presheaves can be seen as an extension of classical set theory, where sets vary over informational states, therefore it is a powerful and expressive mathematical framework. I introduce a suitable topos of presheaves where imprecise probabilities and imprecise probabilistic reasoning can be represented. In this way we obtain a mathematical definition of imprecise-probabilistic sets. A...

This paper gives an overview of recent results concerning the modular derivation of (i) modal specification logics, (ii) notions of simulation together with logical characterisations, and (iii) sound and complete axiomatisations, for systems modelled as coalgebras of functors on Set. Our approach applies directly to an inductivelydefined class of coalgebraic types, which subsumes several types ...

This paper presents a probabilistic extension of Neighbourhood Logic (NL,[14, 1]). The study of such an extension is motivated by the need to supply the Probabilistic Duration Calculus (PDC, [10, 4]) with a proof system. The relation between the new logic and PDC is similar to that between DC [15] and ITL [12, 3]. We present a complete proof system for the new logic.

A branching bisimulation for probabilistic systems that is preserved under parallel composition has been defined recently for the alternating model. We show that besides being compositional, it is decidable in polynomial time and it preserves the properties expressible in probabilistic Computation Tree Logic (pCTL). In the ground-complete axiomatization, only a single axiom is added to the axio...

We introduce and demonstrate a new approach to inference in expressive probabilistic programming languages based on particle Markov chain Monte Carlo. Our approach is simple to implement and easy to parallelize. It applies to Turing-complete probabilistic programming languages and supports accurate inference in models that make use of complex control flow, including stochastic recursion. It als...