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12
Continuous capacities on continuous state spaces
 In ICALP’2007. SpringerVerlag LNCS
, 2007
"... Abstract. We propose axiomatizing some stochastic games, in a continuous state space setting, using continuous belief functions, resp. plausibilities, instead of measures. Then, stochastic games are just variations on continuous Markov chains. We argue that drawing at random along a belief function ..."
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Cited by 11 (4 self)
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Abstract. We propose axiomatizing some stochastic games, in a continuous state space setting, using continuous belief functions, resp. plausibilities, instead of measures. Then, stochastic games are just variations on continuous Markov chains. We argue that drawing at random along a belief function is the same as letting the probabilistic player P play first, then letting the nondeterministic player C play demonically. The same holds for an angelic C, using plausibilities instead. We then define a simple modal logic, and characterize simulation in terms of formulae of this logic. Finally, we show that (discounted) payoffs are defined and unique, where in the demonic case, P maximizes payoff, while C minimizes it. 1
Exemplaric Expressivity of Modal Logics
, 2008
"... This paper investigates expressivity of modal logics for transition systems, multitransition systems, Markov chains, and Markov processes, as coalgebras of the powerset, finitely supported multiset, finitely supported distribution, and measure functor, respectively. Expressivity means that logically ..."
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Cited by 3 (0 self)
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This paper investigates expressivity of modal logics for transition systems, multitransition systems, Markov chains, and Markov processes, as coalgebras of the powerset, finitely supported multiset, finitely supported distribution, and measure functor, respectively. Expressivity means that logically indistinguishable states, satisfying the same formulas, are behaviourally indistinguishable. The investigation is based on the framework of dual adjunctions between spaces and logics and focuses on a crucial injectivity property. The approach is generic both in the choice of systems and modalities, and in the choice of a “base logic”. Most of these expressivity results are already known, but the applicability of the uniform setting of dual adjunctions to these particular examples is what constitutes the contribution of the paper.
CONTINUOUS MARKOVIAN LOGICS AXIOMATIZATION AND QUANTIFIED METATHEORY
, 2011
"... Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuoustime labelled Markov processes with arbitrary (analytic) statespaces, henceforth called continuous Markov processes (CMPs). The modalities of CML evaluate the rates of the exp ..."
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Cited by 2 (2 self)
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Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuoustime labelled Markov processes with arbitrary (analytic) statespaces, henceforth called continuous Markov processes (CMPs). The modalities of CML evaluate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions of a CMP. In this paper we present weak and strong complete axiomatizations for CML and prove a series of metaproperties, including the finite model property and the construction of canonical models. CML characterizes stochastic bisimilarity and it supports the definition of a quantified extension of the satisfiability relation that measures the “compatibility ” between a model and a property. In this context, the metaproperties allows us to prove two robustness theorems for the logic stating that one can perturb formulas and maintain “approximate satisfaction”.
Approximating Markov Processes by Averaging
"... Abstract. We take a dual view of Markov processes – advocated by Kozen – as transformers of bounded measurable functions. 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 defi ..."
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Cited by 1 (0 self)
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Abstract. We take a dual view of Markov processes – advocated by Kozen – as transformers of bounded measurable functions. 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 approximation. (iii) It is possible to show that there is a minimal bisimulation equivalent to a process obtained as the limit of the finite approximants. 1
Similarity Quotients as Final Coalgebras
"... Abstract. We give a general framework relating a branching time relation on nodes of a transition system to a final coalgebra for a suitable endofunctor. Examples of relations treated by our theory include bisimilarity (a well known example), similarity, upper and lower similarity for transition sys ..."
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Abstract. We give a general framework relating a branching time relation on nodes of a transition system to a final coalgebra for a suitable endofunctor. Examples of relations treated by our theory include bisimilarity (a well known example), similarity, upper and lower similarity for transition systems with divergence, and nested similarity. Our results describe firstly how to characterize the relation in terms of a given final coalgebra, and secondly how to construct a final coalgebra using the relation. Our theory uses a notion of “relator ” based on earlier work of Thijs. But whereas a relator must preserve binary composition in Thijs ’ framework, it only laxly preserves composition in ours. It is this weaker requirement that allows nested similarity to be an example. 1
Tracebased Semantics for Probabilistic Timed I/O Automata Submitted for review. Full version http://theory.lcs.mit.edu/ ∼mitras/ research/PTIOA06full.pdf
"... Abstract. We propose the Probabilistic Timed I/O Automaton (PTIOA) framework for modelling and analyzing discretely communicating probabilistic hybrid systems. State transition of a PTIOA can be nondeterministic or probabilistic. Probabilistic choices can be based on continuous distributions. Contin ..."
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Abstract. We propose the Probabilistic Timed I/O Automaton (PTIOA) framework for modelling and analyzing discretely communicating probabilistic hybrid systems. State transition of a PTIOA can be nondeterministic or probabilistic. Probabilistic choices can be based on continuous distributions. Continuous evolution of a PTIOA is purely nondeterministic. PTIOAs can communicate through shared actions. By supporting external nondeterminism, the framework allows us to model arbitrary interleaving of concurrently executing automata. The framework generalizes several previously studied automata models of its class. We develop the tracebased semantics for PTIOAs which involves measure theoretic constructions on the space of executions of the automata. We introduce a new notion of external behavior for PTIOAs and show that PTIOAs have simple compositionality properties with respect this external behavior. 1
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.
Approximating Labelled Markov Processes Again!
"... Abstract. Labelled Markov processes are continuousstate fully probabilistic labelled transition systems. They can be seen as coalgebras of a suitable monad on the category of measurable space. The theory as developed so far included a treatment of bisimulation, logical characterization of bisimula ..."
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Abstract. Labelled Markov processes are continuousstate fully probabilistic labelled transition systems. They can be seen as coalgebras of a suitable monad on the category of measurable space. The theory as developed so far included a treatment of bisimulation, logical characterization of bisimulation, weak bisimulation, metrics, universal domains for LMPs and approximations. Much of the theory involved delicate properties of analytic spaces. Recently a new kind of averaging procedure was used to construct approximations. Remarkably, this version of the theory uses a dual view of LMPs and greatly simplifies the theory eliminating the need to consider aanlytic spaces. In this talk I will survey some of the ideas that led to this work. 1