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Testing semantics: connecting processes and process logics
 Algebraic Methodology and Software Technology (AMAST 2006), volume 4019 of Lect. Notes Comp. Sci
, 2006
"... Abstract. We propose a methodology based on testing as a framework to capture the interactions of a machine represented in a denotational model and the data it manipulates. Using a duality that models machines on the one hand, and the data they manipulate on the other, testing is used to capture the ..."
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Abstract. We propose a methodology based on testing as a framework to capture the interactions of a machine represented in a denotational model and the data it manipulates. Using a duality that models machines on the one hand, and the data they manipulate on the other, testing is used to capture the interactions of each with the objects on the other side: just as the data that are input into a machine can be viewed as tests that the machine can be subjected to, the machine can be viewed as a test that can be used to distinguish data. While this approach is based on duality theories that now are common in semantics, it accomplishes much more than simply moving from one side of the duality to the other; it faithfully represents the interactions that embody what is happening as the computation proceeds. Our basic philosophy is that tests can be used as a basis for modeling interactions, as well as processes and the data on which they operate. In more abstract terms, tests can be viewed as formulas of process logics, and testing semantics connects processes and process logics, and assigns computational meanings to both. 1 Introduction: The
Operator algebras and the operational semantics of probabilistic languages
 IN: PROCEEDINGS OF MFCSIT04 – THIRD IRISH CONFERENCE ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE AND INFORMATION TECHNOLOGY
, 2004
"... We investigate the construction of linear operators representing the semantics of probabilistic programming languages expressed via probabilistic transition systems. Finite transition relations, corresponding to finite automata, can easily be represented by finite dimensional matrices; for the infin ..."
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We investigate the construction of linear operators representing the semantics of probabilistic programming languages expressed via probabilistic transition systems. Finite transition relations, corresponding to finite automata, can easily be represented by finite dimensional matrices; for the infinite case we need to consider an appropriate generalisation of matrix algebras. We argue that C∗algebras, or more precisely Approximately Finite (or AF) algebras, provide a sufficiently rich mathematical structure for modelling probabilistic processes. We show how to construct for a given probabilistic language a unique AF algebra A and how to represent the operational semantics of processes within this framework: finite computations correspond directly to operators in A, while infinite processes are represented by elements in the socalled strong closure of this algebra.
Proving Approximate Implementations for Probabilistic I/O Automata?? Abstract
, 2006
"... In this paper we introduce the notion of approximate implementations for Probabilistic I/O Automata (PIOA) and develop methods for proving such relationships. We employ a task structure on the locally controlled actions and a task scheduler to resolve nondeterminism. The interaction between a schedu ..."
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In this paper we introduce the notion of approximate implementations for Probabilistic I/O Automata (PIOA) and develop methods for proving such relationships. We employ a task structure on the locally controlled actions and a task scheduler to resolve nondeterminism. The interaction between a scheduler and an automaton gives rise to a trace distribution—a probability distribution over the set of traces. We define a PIOA to be a (discounted) approximate implementation of another PIOA if the set of trace distributions produced by the first is close to that of the latter, where closeness is measured by the (resp. discounted) uniform metric over trace distributions. We propose simulation functions for proving approximate implementations corresponding to each of the above types of approximate implementation relations. Since our notion of similarity of traces is based on a metric on trace distributions, we do not require the state spaces nor the space of external actions of the automata to be metric spaces. We discuss applications of approximate implementations to verification of probabilistic safety and termination.
Verifying Statistical Zero Knowledge with Approximate Implementations ⋆
"... Abstract. Statistical zeroknowledge (SZK) properties play an important role in designing cryptographic protocols that enforce honest behavior while maintaining privacy. This paper presents a novel approach for verifying SZK properties, using recently developed techniques based on approximate simula ..."
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Abstract. Statistical zeroknowledge (SZK) properties play an important role in designing cryptographic protocols that enforce honest behavior while maintaining privacy. This paper presents a novel approach for verifying SZK properties, using recently developed techniques based on approximate simulation relations. We formulate statistical indistinguishability as an implementation relation in the TaskPIOA framework, which allows us to express computational restrictions. The implementation relation is then proven using approximate simulation relations. This technique separates proof obligations into two categories: those requiring probabilistic reasoning, as well as those that do not. The latter is a good candidate for mechanization. We illustrate the general method by verifying the SZK property of the wellknown identification protocol proposed by Girault, Poupard and Stern.
Stone duality for markov processes
 In Proceedings of the 28th Annual IEEE Symposium on Logic in Computer Science: LICS
"... We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions. We prove a Stonetype duality theorem between countable Aumann algebras and countablygenerated continuousspace Markov proc ..."
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We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions. We prove a Stonetype duality theorem between countable Aumann algebras and countablygenerated continuousspace Markov processes. Our results subsume existing results on completeness of probabilistic modal logics for Markov processes. 1.
Approximating Markov Processes By Averaging
"... Normally, one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition. We, however, take a dual view of Markov processes as transformers of bounded measurable function ..."
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Normally, one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition. We, however, take a dual view of Markov processes as transformers of bounded measurable functions. This is very much in the same spirit as a “predicatetransformer ” view, which is dual to the statetransformer view of transition systems. We redevelop the theory of labelled Markov processes from this view point, in particular we explore approximation theory. We obtain three main results: (i) It is possible to define bisimulation on general measure spaces and show that it is an equivalence relation. The logical characterization of bisimulation can be done straightforwardly and generally. (ii) A new and flexible approach to approximation based on averaging can be given. This vastly generalizes and streamlines the idea of using conditional expectations to compute approximations. (iii) We show that there is a minimal process bisimulationequivalent to a given process, and this minimal process is obtained as the limit of the finite approximants.
Labelled Markov Processes as Generalised Stochastic Relations
"... Labelled Markov processes (LMPs) are labelled transition systems in which each transition has an associated probability. In this paper we present a universal LMP as the spectrum of a commutative C ∗algebra consisting of formal linear combinations of labelled trees. This yields a simple tracetree s ..."
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Labelled Markov processes (LMPs) are labelled transition systems in which each transition has an associated probability. In this paper we present a universal LMP as the spectrum of a commutative C ∗algebra consisting of formal linear combinations of labelled trees. This yields a simple tracetree semantics for LMPs that is fully abstract with respect to probabilistic bisimilarity. We also consider LMPs with distinguished entry and exit points as stateful stochastic relations. This allows us to define a category LMP, with measurable spaces as objects and LMPs as morphisms. Our main result in this context is to provide a predicatetransformer duality for
The Duality of State and Observation in Probabilistic Transition Systems
"... Abstract. In this paper we consider the problem of representing and reasoning about systems, especially probabilistic systems, with hidden state. We consider transition systems where the state is not completely visible to an outside observer. Instead, there are observables that partly identify the s ..."
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Abstract. In this paper we consider the problem of representing and reasoning about systems, especially probabilistic systems, with hidden state. We consider transition systems where the state is not completely visible to an outside observer. Instead, there are observables that partly identify the state. We show that one can interchange the notions of state and observation and obtain what we call a dual system. In the case of deterministic systems, the double dual gives a minimal representation of the behaviour of the original system. We extend these ideas to probabilistic transition systems and to partially observable Markov decision processes (POMDPs). 1
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, 2007
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The Duality of State and Observations
, 2007
"... In this paper we consider the problem of representing and reasoning about systems, especially probabilistic systems, with hidden state. We consider transition systems where the state is not completely visible to an outside observer. Instead, there are observables that partly identify the state. We s ..."
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In this paper we consider the problem of representing and reasoning about systems, especially probabilistic systems, with hidden state. We consider transition systems where the state is not completely visible to an outside observer. Instead, there are observables that partly identify the state. We show that one can interchange the notions of state and observation and obtain what we cann a dual system. The double dual gives a minimal representation of the behaviour of the original system. We extend this to nondeterministic systems and to probabilistic transition systems and finally to partially observable Markov decision processes (POMDPs). In the case of finite automata restricted to one observable, we obtain Brzozowski’s algorithm for minimizing finitestate language acceptors. 1