Quantitative Automata under Probabilistic Semantics

Automata with monitor counters, where the transitions do not depend on counter values, and nested weighted automata are two expressive automata-theoretic frameworks for quantitative properties. For a well-studied and wide class of quantitative functions, we establish that automata with monitor counters and nested weighted automata are equivalent. We study for the first time such quantitative automata under probabilistic semantics. We show that several problems that are undecidable for the classical questions of emptiness and universality become decidable under the probabilistic semantics. We present a complete picture of decidability for such automata, and even an almost-complete picture of computational complexity, for the probabilistic questions we consider

An Integrated Approach for the Specification and Analysis of Stochastic Real-Time Systems

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Modelling, reduction and analysis of Markov automata (extended version)

Markov automata (MA) constitute an expressive continuous-time compositional modelling formalism. They appear as semantic backbones for engineering frameworks including dynamic fault trees, Generalised Stochastic Petri Nets, and AADL. Their expressive power has thus far precluded them from effective analysis by probabilistic (and statistical) model checkers, stochastic game solvers, or analysis tools for Petri net-like formalisms. This paper presents the foundations and underlying algorithms for efficient MA modelling, reduction using static analysis, and most importantly, quantitative analysis. We also discuss implementation pragmatics of supporting tools and present several case studies demonstrating feasibility and usability of MA in practice

Variance and Covariance of Several Simultaneous Outputs of a Markov Chain

The partial sum of the states of a Markov chain or more generally a Markov source is asymptotically normally distributed under suitable conditions. One of these conditions is that the variance is unbounded. A simple combinatorial characterization of Markov sources which satisfy this condition is given in terms of cycles of the underlying graph of the Markov chain. Also Markov sources with higher dimensional alphabets are considered. Furthermore, the case of an unbounded covariance between two coordinates of the Markov source is combinatorically characterized. If the covariance is bounded, then the two coordinates are asymptotically independent. The results are illustrated by several examples, like the number of specific blocks in $0$-$1$-sequences and the Hamming weight of the width-$w$ non-adjacent form

A Monte Carlo model checker for probabilistic LTL with numerical constraints

We define the syntax and semantics of a new temporal logic called probabilistic LTL with numerical constraints (PLTLc). We introduce an efficient model checker for PLTLc properties. The efficiency of the model checker is through approximation using Monte Carlo sampling of finite paths through the model’s state space (simulation outputs) and parallel model checking of the paths. Our model checking method can be applied to any model producing quantitative output – continuous or stochastic, including those with complex dynamics and those with an infinite state space. Furthermore, our offline approach allows the analysis of observed (real-life) behaviour traces. We find in this paper that PLTLc properties with constraints over free variables can replace full model checking experiments, resulting in a significant gain in efficiency. This overcomes one disadvantage of model checking experiments which is that the complexity depends on system granularity and number of variables, and quickly becomes infeasible. We focus on models of biochemical networks, and specifically in this paper on intracellular signalling pathways; however our method can be applied to a wide range of biological as well as technical systems and their models. Our work contributes to the emerging field of synthetic biology by proposing a rigourous approach for the structured formal engineering of biological systems