Language-based Abstractions for Dynamical Systems

Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of effectively performing analyses. This has motivated a large body of research, across many disciplines, into abstraction techniques that provide smaller ODE systems while preserving the original dynamics in some appropriate sense. In this paper we give an overview of a recently proposed computer-science perspective to this problem, where ODE reduction is recast to finding an appropriate equivalence relation over ODE variables, akin to classical models of computation based on labelled transition systems.Comment: In Proceedings QAPL 2017, arXiv:1707.0366

Performance Guarantees for Homomorphisms Beyond Markov Decision Processes

Most real-world problems have huge state and/or action spaces. Therefore, a naive application of existing tabular solution methods is not tractable on such problems. Nonetheless, these solution methods are quite useful if an agent has access to a relatively small state-action space homomorphism of the true environment and near-optimal performance is guaranteed by the map. A plethora of research is focused on the case when the homomorphism is a Markovian representation of the underlying process. However, we show that near-optimal performance is sometimes guaranteed even if the homomorphism is non-Markovian. Moreover, we can aggregate significantly more states by lifting the Markovian requirement without compromising on performance. In this work, we expand Extreme State Aggregation (ESA) framework to joint state-action aggregations. We also lift the policy uniformity condition for aggregation in ESA that allows even coarser modeling of the true environment

Robust satisfaction of temporal logic specifications via reinforcement learning

We consider the problem of steering a system with unknown, stochastic dynamics to satisfy a rich, temporally-layered task given as a signal temporal logic formula. We represent the system as a finite-memory Markov decision process with unknown transition probabilities and whose states are built from a partition of the state space. We present provably convergent reinforcement learning algorithms to maximize the probability of satisfying a given specification and to maximize the average expected robustness, i.e. a measure of how strongly the formula is satisfied. Robustness allows us to quantify progress towards satisfying a given specification. We demonstrate via a pair of robot navigation simulation case studies that, due to the quantification of progress towards satisfaction, reinforcement learning with robustness maximization performs better than probability maximization in terms of both probability of satisfaction and expected robustness with a low number of training examples

Robust Satisfaction of Temporal Logic Specifications via Reinforcement Learning

We consider the problem of steering a system with unknown, stochastic dynamics to satisfy a rich, temporally layered task given as a signal temporal logic formula. We represent the system as a Markov decision process in which the states are built from a partition of the state space and the transition probabilities are unknown. We present provably convergent reinforcement learning algorithms to maximize the probability of satisfying a given formula and to maximize the average expected robustness, i.e., a measure of how strongly the formula is satisfied. We demonstrate via a pair of robot navigation simulation case studies that reinforcement learning with robustness maximization performs better than probability maximization in terms of both probability of satisfaction and expected robustness.Comment: 8 pages, 4 figure

Compositional Verification and Optimization of Interactive Markov Chains

Interactive Markov chains (IMC) are compositional behavioural models extending labelled transition systems and continuous-time Markov chains. We provide a framework and algorithms for compositional verification and optimization of IMC with respect to time-bounded properties. Firstly, we give a specification formalism for IMC. Secondly, given a time-bounded property, an IMC component and the assumption that its unknown environment satisfies a given specification, we synthesize a scheduler for the component optimizing the probability that the property is satisfied in any such environment

When are Stochastic Transition Systems Tameable?

A decade ago, Abdulla, Ben Henda and Mayr introduced the elegant concept of decisiveness for denumerable Markov chains [1]. Roughly speaking, decisiveness allows one to lift most good properties from finite Markov chains to denumerable ones, and therefore to adapt existing verification algorithms to infinite-state models. Decisive Markov chains however do not encompass stochastic real-time systems, and general stochastic transition systems (STSs for short) are needed. In this article, we provide a framework to perform both the qualitative and the quantitative analysis of STSs. First, we define various notions of decisiveness (inherited from [1]), notions of fairness and of attractors for STSs, and make explicit the relationships between them. Then, we define a notion of abstraction, together with natural concepts of soundness and completeness, and we give general transfer properties, which will be central to several verification algorithms on STSs. We further design a generic construction which will be useful for the analysis of {\omega}-regular properties, when a finite attractor exists, either in the system (if it is denumerable), or in a sound denumerable abstraction of the system. We next provide algorithms for qualitative model-checking, and generic approximation procedures for quantitative model-checking. Finally, we instantiate our framework with stochastic timed automata (STA), generalized semi-Markov processes (GSMPs) and stochastic time Petri nets (STPNs), three models combining dense-time and probabilities. This allows us to derive decidability and approximability results for the verification of these models. Some of these results were known from the literature, but our generic approach permits to view them in a unified framework, and to obtain them with less effort. We also derive interesting new approximability results for STA, GSMPs and STPNs.Comment: 77 page

Formal Controller Synthesis for Markov Jump Linear Systems with Uncertain Dynamics

Automated synthesis of provably correct controllers for cyber-physical systems is crucial for deployment in safety-critical scenarios. However, hybrid features and stochastic or unknown behaviours make this problem challenging. We propose a method for synthesising controllers for Markov jump linear systems (MJLSs), a class of discrete-time models for cyber-physical systems, so that they certifiably satisfy probabilistic computation tree logic (PCTL) formulae. An MJLS consists of a finite set of stochastic linear dynamics and discrete jumps between these dynamics that are governed by a Markov decision process (MDP). We consider the cases where the transition probabilities of this MDP are either known up to an interval or completely unknown. Our approach is based on a finite-state abstraction that captures both the discrete (mode-jumping) and continuous (stochastic linear) behaviour of the MJLS. We formalise this abstraction as an interval MDP (iMDP) for which we compute intervals of transition probabilities using sampling techniques from the so-called 'scenario approach', resulting in a probabilistically sound approximation. We apply our method to multiple realistic benchmark problems, in particular, a temperature control and an aerial vehicle delivery problem.Comment: 14 pages, 6 figures, under review at QES