Radu Iulian Mardare

Taking it to the limit: approximate reasoning for markov processes

We develop a fusion of logical and metrical principles for reasoning about Markov processes. More precisely, we lift metrics from processes to sets of processes satisfying a formula and explore how the satisfaction relation behaves as sequences of processes and sequences of formulas approach limits. A key new concept is dynamically-continuous metric bisimulation which is a property of (pseudo)metrics. We prove theorems about satisfaction in the limit, robustness theorems as well as giving a topological characterization of various classes of formulas. This work is aimed at providing approximate reasoning principles for Markov processes.

Model checking biological systems described using ambient calculus

We propose a way of performing model checking analysis for biological systems. The technics were developed for a CTL* logic built upon Ambient Calculus. We introduce labeled syntax trees for ambient processes and use them as possible worlds in a Kripke structure developed for a propositional branching temporal logic. The accessibility relation over labeled syntax trees is generated by the reduction over corresponding Ambient Calculus processes. Providing the algorithms for calculating the accessibility relation between states, we open the perspective of using model checking algorithms developed for temporal logics in analyzing any phenomena described in Ambient Calculus.

Modular Markovian logic

We introduce Modular Markovian Logic (MML) for compositional continuous-time and continuous-space Markov processes. MML combines operators specific to stochastic logics with operators reflecting the modular structure of the models, similar to those used by spatial and separation logics. We present a complete Hilbert-style axiomatization for MML, prove the small model property and analyze the relation between stochastic bisimulation and logical equivalence.

Computing Behavioral Distances, Compositionally

We propose a general definition of composition operator on Markov Decision Processes with rewards (MDPs) and identify a well behaved class of operators, called safe, that are guaranteed to be non-extensive w.r.t. the bisimilarity pseudometrics of Ferns et al. [10], which measure behavioral similarities between MDPs. For MDPs built using safe/non-extensive operators, we present the first method that exploits the structure of the system for (exactly) computing the bisimilarity distance on MDPs. Experimental results show significant improvements upon the non-compositional technique.

Stone Duality for Markov Processes

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 Stone-type duality theorem between countable Aumann algebras and countably-generated continuous-space Markov processes. Our results subsume existing results on completeness of probabilistic modal logics for Markov processes.

Decidable extensions of hennessy-milner logic

We propose a new class of logics for specifying and model-checking properties of distributed systems – Dynamic Epistemic Spatial Logics. They have been designed as extensions of Hennessy-Milner logic with spatial operators (inspired by Cardelli-Gordon-Caires spatial logic) and epistemic operators (inspired by dynamic-epistemic logics). Our logics focus on observers, agents placed in different locations of the system having access to some subsystems. Treating them as epistemic agents, we develop completely axiomatized and decidable logics that express the information flow between them in a dynamic and distributed environment. The knowledge of an epistemic agent, is understood as the information, locally available to our observer, about the overall-global system.

Strong Completeness for Markovian Logics

In this paper we present Hilbert-style axiomatizations for three logics for reasoning about continuous-space Markov processes (MPs): (i) a logic for MPs defined for probability distributions on measurable state spaces, (ii) a logic for MPs defined for sub-probability distributions and (iii) a logic defined for arbitrary distributions. These logics are not compact so one needs infinitary rules in order to obtain strong completeness results.

Continuous Markovian Logic - From Complete Axiomatization to the Metric Space of Formulas

Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuous-space and continuous-time labelled Markov processes (CMPs). The modalities of CML approximate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions. In this paper we present a sound and complete Hilbert-style axiomatization of CML for the CMP-semantics and prove some metaproperties including the small model property. CML characterizes stochastic bisimulation and supports the definition of a quantified extension of satisfiability relation that measures the compatibility of a model and a property. Relying on the small model property, we prove that this measure can be approximated, within a given error, by using a distance between logical formulas.

The Measurable Space of Stochastic Processes

We introduce a stochastic extension of CCS endowed with structural operational semantics expressed in terms of measure theory. The set of processes is organised as a measurable space by the sigma-algebra generated by structural congruence. The structural operational semantics associates to each process a set of measures over the space of processes. The measures encode the rates of the transitions from a process state of a system to a measurable set of processes. We prove that the stochastic bisimilarity is a congruence, which extends the structural congruence. In addition to an elegant operational semantics, our calculus provides a canonic way to define metrics on processes that measure how similar two processes are in terms of behaviour.

Observing distributed computation: a dynamic-epistemic approach

We propose a new logic designed for modelling and reasoning about information flow and information exchange between spatially located interconnected agents witnessing a distributed computation. The intention is to trace the process of knowledge acquisition and its dynamics in the context of distributed systems. Underpinning on the dual algebraical-coalgebraical characteristics of process calculi, we design a decidable and completely axiomatized logic that combines the process-algebraical/equational and the modal/coequational features and is developed for process-algebraical semantics.