On the Termination Problem for Probabilistic Higher-Order Recursive Programs

In the last two decades, there has been much progress on model checking of both probabilistic systems and higher-order programs. In spite of the emergence of higher-order probabilistic programming languages, not much has been done to combine those two approaches. In this paper, we initiate a study on the probabilistic higher-order model checking problem, by giving some first theoretical and experimental results. As a first step towards our goal, we introduce PHORS, a probabilistic extension of higher-order recursion schemes (HORS), as a model of probabilistic higher-order programs. The model of PHORS may alternatively be viewed as a higher-order extension of recursive Markov chains. We then investigate the probabilistic termination problem -- or, equivalently, the probabilistic reachability problem. We prove that almost sure termination of order-2 PHORS is undecidable. We also provide a fixpoint characterization of the termination probability of PHORS, and develop a sound (but possibly incomplete) procedure for approximately computing the termination probability. We have implemented the procedure for order-2 PHORSs, and confirmed that the procedure works well through preliminary experiments that are reported at the end of the article

Behavioural Preorders on Stochastic Systems - Logical, Topological, and Computational Aspects

Computer systems can be found everywhere: in space, in our homes, in our cars, in our pockets, and sometimes even in our own bodies. For concerns of safety, economy, and convenience, it is important that such systems work correctly. However, it is a notoriously difficult task to ensure that the software running on computers behaves correctly. One approach to ease this task is that of model checking, where a model of the system is made using some mathematical formalism. Requirements expressed in a formal language can then be verified against the model in order to give guarantees that the model satisfies the requirements. For many computer systems, time is an important factor. As such, we need our formalisms and requirement languages to be able to incorporate real time. We therefore develop formalisms and algorithms that allow us to compare and express properties about real-time systems. We first introduce a logical formalism for reasoning about upper and lower bounds on time, and study the properties of this formalism, including axiomatisation and algorithms for checking when a formula is satisfied. We then consider the question of when a system is faster than another system. We show that this is a difficult question which can not be answered in general, but we identify special cases where this question can be answered. We also show that under this notion of faster-than, a local increase in speed may lead to a global decrease in speed, and we take step towards avoiding this. Finally, we consider how to compare the real-time behaviour of systems not just qualitatively, but also quantitatively. Thus, we are interested in knowing how much one system is faster or slower than another system. This is done by introducing a distance between systems. We show how to compute this distance and that it behaves well with respect to certain properties.Comment: PhD dissertation from Aalborg Universit

Performance evaluation and model checking of probabilistic real-time actors

This dissertation is composed of two parts. In the first part, performance evaluation and verification of safety properties are provided for real-time actors. Recently, the actor-based language, Timed Rebeca, was introduced to model distributed and asynchronous systems with timing constraints and message passing communication. A toolset was developed for automated translation of Timed Rebeca models to Erlang. The translated code can be executed using a timed extension of McErlang for model checking and simulation. In the first part of this dissertation, we induce a new toolset that provides statistical model checking of Timed Rebeca models. Using statistical model checking, we are now able to verify larger models against safety properties comparing to McErlang model checking. We examine the typical case studies of elevators and ticket service to show the efficiency of statistical model checking and applicability of our toolset. In the second part of this dissertation, we enhance our modeling ability and cover more properties by performance evaluation and model checking of probabilistic real-time actors. Distributed systems exhibit probabilistic and nondeterministic behaviors and may have time constraints. Probabilistic Timed Rebeca (PTRebeca) is introduced as a timed and probabilistic actor-based language for modeling distributed real-time systems with asynchronous message passing. The semantics of PTRebeca is a Timed Markov Decision Process (TMDP). We provide SOS rules for PTRebeca, and develop two toolsets for analyzing PTRebeca models. The first toolset automatically generates a TMDP model from a PTRebeca model in the form of the input language of the PRISM model checker. We use PRISM for performance analysis of PTRebeca models against expected reachability and probabilistic reachability properties. Additionally, we develop another toolset to automatically generate a Markov Automaton from a PTRebeca model in the form of the input language of the Interactive Markov Chain Analyzer (IMCA). The IMCA can be used as the back-end model checker for performance analysis of PTRebeca models against expected reachability and probabilistic reachability properties. We present the needed time for the analysis of different case studies using PRISM-based and IMCA-based approaches. The IMCA-based approach needs considerably less time, and so has the ability of analyzing significantly larger models. We show the applicability of both approaches and the efficiency of our tools by analyzing a few case studies and experimental results.Þessi ritgerð er tvískipt. Í fyrri hlutanum er farið í mat og sannprófun á eiginleikum öryggis í rauntímalíkönum. Fyrir stuttu síðan var leikendabyggða málið, Timed Rebeca, notað við líkana dreifingu og ósamstillt kerfi með tímastillingu og samskipti í skilaboðum. Búið var til verkfærasett fyrir sjálfvirka þýðingu á Timed Rebeca líkön yfir í Erlang. Hægt er að nota þýdda kóðann með því að nota tímastillta framlengingu af McErlang fyrir líkanaprófun og hermun. Í fyrri hluta þessarar ritgerðar, ætlum við að kynna verkfærasettið sem veitir tölfræðilega prófun á líkön á Timed Rebeca líkön. Með því að nota tölfræðileg próf á líkön er núna hægt að sannreyna stærri líkön eins og í öryggiskröfum McErlang. Við rannsökum dæmigerðar ferilsathuganir af lyftum og miðasölu til að sýna fram á skilvirkni tölfræðilegra líkana og beitingu verkfærasettsins okkar. Í seinni hluta þessarar ritgerðar aukum við við getu líkanagerðarinnar og við náum yfir fleiri eiginleika með mati á framkvæmd og prófunum á líkönum á líkinda rauntíma leikara. Dreifð kerfi sýna líkindi og brigðgenga hegðun sem kunna að hafa tímamörk. Probabilistic Timed Rebeca (PTRebeca) er kynnt sem tímastillt og líkinda leikarabyggt mál líkindadreifðra rauntímakerfa með ósamstillta sendingu skilaboða. Merkingarfræði PTRebeca er Timed Markov Decision Process (TMDP). Við verðum með SOS reglur fyrir PTRebeca, og þróum tvö verkfærasett til að greina PTRebeca líkön.The work on this dissertation was supported by the project "Timed Asynchronous Reactive Objects in Distributed Systems: TARO" (nr.110020021) of the Icelandic Research Fund

Formal modelling and approximation-based analysis for mode-switching population dynamics

This thesis explores aspects of model specification and analysis for population dynamics which arise when modelling complex interactions and communication structures in agent or component collectives. The motivating examples come from the design of man-made systems where the optimal parametrisations for the behaviours of agents or components are not known a priori. In particular, we introduce a formal modelling framework to support the specification of control problems for collective dynamics in a high-level process algebraic language. A natural choice for the underlying semantics is to consider continuous time Markov decision processes due to their close relation to continuous time Markov chains that have traditionally been used as the mathematical model in numerous high-level modelling languages for stochastic dynamics. Although the theory of the resulting decision processes has a long history, the practical considerations, like computation time, present challenges due to the problem of state space explosion when considering large systems with complex behaviours. State space explosion problems are especially apparent in formal modelling paradigms where the specification of models usually happens at a component or an agent level in terms of a discrete set of states with defined rules for composing the specified behaviours into the dynamics of a system. Such specifications often give rise to very large models which are costly to analyse in full detail. However, when analysing models of collectives we are usually interested in the resulting macro-scale dynamics in terms of some aggregate measures. With that in mind, the second aspect of analysing collective dynamics that is considered in this thesis relates to fluid, linear noise and moment closure-based approximation methods which aim to give a good representation of the macro-scale dynamics of the models while being computationally less costly to analyse. We address a class of models where the population structure results from the assumption that components or agents can only be distinguished from each other based on the state they are in and focus on the particular cases where the population dynamics can be separated into a discrete set of modes. Our study of these models is motivated by considering information propagation via broadcast communication where the behaviour of components can change drastically when new information is received from the rest of the population. We consider existing approximation methods for resulting stochastic processes and propose a novel approach for applying these methods to models incorporating broadcast communication where each level of information available to the collective corresponds to a discrete dynamic mode. The resulting approximations combine continuous dynamics with discrete stochastic jumps and are not immediately simple to treat numerically. To that end we propose further approximations that allow for a computationally efficient analysis. Finally, we demonstrate how the formal modelling framework in conjunction with the developed approximation methods can be used for an example in policy synthesis