on discrete time reversibility modulo state renaming and its applications

Time reversibility plays an important role in the analysis of continuous and discrete time Markov chains (DTMCs). Specifically, the computation of the stationary distribution of a reversible Markov chain has been proved to be very efficient and does not require the solution of the system of global balance equations. A DTMC is reversible when the processes at forward and reversed time are probabilistically indistinguishable. In this paper we introduce the concept of ρ-reversibility, i.e., a notion of reversibility modulo a renaming of the states, and we contrast it with the previous definition of dynamic reversibility especially with respect to the assumptions on the state renaming function. We discuss the applications of discrete time reversibility in the embedded and uniformized chains of continuous time processes

Efficient Computation of Renaming Functions for ρ-reversible Discrete and Continuous Time Markov Chains

With the introduction of ρ-reversibility, the basic notion of reversible Markov chain has been relaxed by allowing a wider range of scenarios. Specifically, the reversibility properties are not just sought on the chain itself, but also on all the possible topology-preserving renamings of its state space. Such renamings, called Renaming Functions, exhibit many interesting properties which can be exploited in different contexts. Unfortunately, finding a renaming function for a Markov chain is a very computationally intensive task. Using a naive approach it could require to check for all the possible state space permutations, which is unfeasible for all but the most trivial chains. As a matter of fact, we prove that the corresponding decision problem is polynomially equivalent to Graph Isomorphism. Nevertheless, we introduce an algorithm that, exploiting some necessary conditions for ρ-reversibility, is able to efficiently prune the search space and then verify the remaining renaming candidates. The correctness of the method is theoretically demonstrated and its practical effectiveness is shown over a significant set of discrete and continuous ρ-reversible Markov chains

Applying Reversibility Theory for the Performance Evaluation of Reversible Computations

Reversible computations have been widely studied from the functional point of view and energy consumption. In the literature, several authors have proposed various formalisms (mainly based on process algebras) for assessing the correctness or the equivalence among reversible computations. In this paper we propose the adoption of Markovian stochastic models to assess the quantitative properties of reversible computations. Under some conditions, we show that the notion of time-reversibility for Markov chains can be used to efficiently derive some performance measures of reversible computations. The importance of time-reversibly relies on the fact that, in general, the process’s stationary distribution can be derived efficiently by using numerically stable algorithms. This paper reviews the main results about time-reversible Markov processes and discusses how to apply them to tackle the problem of the quantitative evaluation of reversible computationsReversible computations have been widely studied from the functional point of view and energy consumption. In the literature, several authors have proposed various formalisms (mainly based on process algebras) for assessing the correctness or the equivalence among reversible computations. In this paper we propose the adoption of Markovian stochastic models to assess the quantitative properties of reversible computations. Under some conditions, we show that the notion of time-reversibility for Markov chains can be used to efficiently derive some performance measures of reversible computations. The importance of time-reversibly relies on the fact that, in general, the process's stationary distribution can be derived efficiently by using numerically stable algorithms. This paper reviews the main results about time-reversible Markov processes and discusses how to apply them to tackle the problem of the quantitative evaluation of reversible computations

On the relations between Markov chain lumpability and reversibility

In the literature, the notions of lumpability and time reversibility for large Markov chains have been widely used to efficiently study the functional and non-functional properties of computer systems. In this paper we explore the relations among different definitions of lumpability (strong, exact and strict) and the notion of time-reversed Markov chain. Specifically, we prove that an exact lumping induces a strong lumping on the reversed Markov chain and a strict lumping holds both for the forward and the reversed processes. Based on these results we introduce the class of λρ-reversible Markov chains which combines the notions of lumping and time reversibility modulo state renaming. We show that the class of autoreversible processes, previously introduced in Marin and Rossi (Proceedings of the IEEE 21st international symposium on modeling, analysis and simulation of computer and telecommunication systems MASCOTS, pp. 151–160, 2013), is strictly contained in the class of λρ-reversible chains

A Product-Form Model for the Performance Evaluation of a Bandwidth Allocation Strategy in WSNs

Wireless Sensor Networks (WSNs) are important examples of Collective Adaptive System, which consist of a set of motes that are spatially distributed in an indoor or outdoor space. Each mote monitors its surrounding conditions, such as humidity, intensity of light, temperature, and vibrations, but also collects complex information, such as images or small videos, and cooperates with the whole set of motes forming the WSN to allow the routing process. The traffic in the WSN consists of packets that contain the data harvested by the motes and can be classified according to the type of information that they carry. One pivotal problem in WSNs is the bandwidth allocation among the motes. The problem is known to be challenging due to the reduced computational capacity of the motes, their energy consumption constraints, and the fully decentralised network architecture. In this article, we study a novel algorithm to allocate the WSN bandwidth among the motes by taking into account the type of traffic they aim to send. Under the assumption of a mesh network and Poisson distributed harvested packets, we propose an analytical model for its performance evaluation that allows a designer to study the optimal configuration parameters. Although the Markov chain underlying the model is not reversible, we show it to be.-reversible under a certain renaming of states. By an extensive set of simulations, we show that the analytical model accurately approximates the performance of networks that do not satisfy the assumptions. The algorithm is studied with respect to the achieved throughput and fairness. We show that it provides a good approximation of the max-min fairness requirements

Static versus dynamic reversibility in CCS

The notion of reversible computing is attracting interest because of its applications in diverse fields, in particular the study of programming abstractions for fault tolerant systems. Most computational models are not naturally reversible since computation causes loss of information, and history information must be stored to enable reversibility. In the literature, two approaches to reverse the CCS process calculus exist, differing on how history information is kept. Reversible CCS (RCCS), proposed by Danos and Krivine, exploits dedicated stacks of memories attached to each thread. CCS with Keys (CCSK), proposed by Phillips and Ulidowski, makes CCS operators static so that computation does not cause information loss. In this paper we show that RCCS and CCSK are equivalent in terms of LTS isomorphism

Bulking II: Classifications of Cellular Automata

This paper is the second part of a series of two papers dealing with bulking: a way to define quasi-order on cellular automata by comparing space-time diagrams up to rescaling. In the present paper, we introduce three notions of simulation between cellular automata and study the quasi-order structures induced by these simulation relations on the whole set of cellular automata. Various aspects of these quasi-orders are considered (induced equivalence relations, maximum elements, induced orders, etc) providing several formal tools allowing to classify cellular automata