On Probabilistic Parallel Programs with Process Creation and Synchronisation

We initiate the study of probabilistic parallel programs with dynamic process creation and synchronisation. To this end, we introduce probabilistic split-join systems (pSJSs), a model for parallel programs, generalising both probabilistic pushdown systems (a model for sequential probabilistic procedural programs which is equivalent to recursive Markov chains) and stochastic branching processes (a classical mathematical model with applications in various areas such as biology, physics, and language processing). Our pSJS model allows for a possibly recursive spawning of parallel processes; the spawned processes can synchronise and return values. We study the basic performance measures of pSJSs, especially the distribution and expectation of space, work and time. Our results extend and improve previously known results on the subsumed models. We also show how to do performance analysis in practice, and present two case studies illustrating the modelling power of pSJSs.Comment: This is a technical report accompanying a TACAS'11 pape

Decisive Markov Chains

We consider qualitative and quantitative verification problems for infinite-state Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Markov chains are trivially decisive (for every set F), this also holds for many classes of infinite Markov chains. Infinite Markov chains which contain a finite attractor are decisive w.r.t. every set F. In particular, this holds for probabilistic lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains are decisive. This class includes probabilistic vector addition systems (PVASS) and probabilistic noisy Turing machines (PNTM). We consider both safety and liveness problems for decisive Markov chains, i.e., the probabilities that a given set of states F is eventually reached or reached infinitely often, respectively. 1. We express the qualitative problems in abstract terms for decisive Markov chains, and show an almost complete picture of its decidability for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm of Iyer and Narasimha terminates for decisive Markov chains and can thus be used to solve the approximate quantitative safety problem. A modified variant of this algorithm solves the approximate quantitative liveness problem. 3. Finally, we show that the exact probability of (repeatedly) reaching F cannot be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS, PVASS or (P)NTM.Comment: 32 pages, 0 figure

Probabilistic Bisimulation for Parameterized Systems (Technical Report)

Probabilistic bisimulation is a fundamental notion of process equivalence for probabilistic systems. Among others, it has important applications including formalizing the anonymity property of several communication protocols. There is a lot of work on verifying probabilistic bisimulation for finite systems. This is however not the case for parameterized systems, where the problem is in general undecidable. In this paper we provide a generic framework for reasoning about probabilistic bisimulation for parameterized systems. Our approach is in the spirit of software verification, wherein we encode proof rules for probabilistic bisimulation and use a decidable first-order theory to specify systems and candidate bisimulation relations, which can then be checked automatically against the proof rules. As a case study, we show that our framework is sufficiently expressive for proving the anonymity property of the parameterized dining cryptographers protocol and the parameterized grades protocol, when supplied with a candidate regular bisimulation relation. Both of these protocols hitherto could not be verified by existing automatic methods. Moreover, with the help of standard automata learning algorithms, we show that the candidate relations can be synthesized fully automatically, making the verification fully automated

Eager Markov Chains

Abstract. We consider infinite-state discrete Markov chains which are eager: the probability of avoiding a defined set of final states for more thanÒsteps is bounded by some exponentially decreasing function�(Ò). We prove that eager Markov chains include those induced by Probabilistic Lossy Channel Systems, Probabilistic Vector Addition Systems with States, and Noisy Turing Machines, and that the bounding function�(Ò) can be effectively constructed for them. Furthermore, we study the problem of computing the expected reward (or cost) of runs until reaching the final states, where rewards are assigned to individual runs by computable reward functions. For eager Markov chains, an effective path exploration scheme, based on forward reachability analysis, can be used to approximate the expected reward up-to an arbitrarily small error.

Doctor of Philosophy

dissertationOver the past few decades, synthetic biology has generated great interest to biologists and engineers alike. Synthetic biology combines the research of biology with the engineering principles of standards, abstraction, and automated construction with the ultimate goal of being able to design and build useful biological systems. To realize this goal, researchers are actively working on better ways to model and analyze synthetic genetic circuits, groupings of genes that influence the expression of each other through the use of proteins. When designing and analyzing genetic circuits, researchers are often interested in building circuits that exhibit a particular behavior. Usually, this involves simulating their models to produce some time series data and analyzing this data to discern whether or not the circuit behaves appropriately. This method becomes less attractive as circuits grow in complexity because it becomes very time consuming to generate a sufficient amount of runs for analysis. In addition, trying to select representative runs out of a large data set is tedious and error-prone thereby motivating methods of automating this analysis. This has led to the need for design space exploration techniques that allow synthetic biologists to easily explore the effect of varying parameters and efficiently consider alternative designs of their systems. This dissertation attempts to address this need by proposing new analysis and verification techniques for synthetic genetic circuits. In particular, it applies formal methods such as model checking techniques to models of genetic circuits in order to ensure that they behave correctly and are as robust as possible for a variety of different inputs and/or parameter settings. However, model checking stochastic systems is not as simple as model checking deterministic systems where it is always known what the next state of the system will be at any given step. Stochastic systems can exhibit a variety of different behaviors that are chosen randomly with different probabilities at each time step. Therefore, model checking a stochastic system involves calculating the probability that the system will exhibit a desired behavior. Although it is often more difficult to work with the probabilities that stochastic systems introduce, stochastic systems and the models that represent them are becoming commonplace in many disciplines including electronic circuit design where as parts are being made smaller and smaller, they are becoming less reliable. In addition to stochastic model checking, this dissertation proposes a new incremental stochastic simulation algorithm (iSSA) based on Gillespie's stochastic simulation algorithm (SSA) that is capable of presenting a researcher with a simulation trace of the typical behavior of the system. Before the development of this algorithm, discerning this information was extremely error-prone as it involved performing many simulations and attempting to wade through the massive amounts of data. This algorithm greatly aids researchers in designing genetic circuits as it efficiently shows the researcher the most likely behavior of the circuit. Both the iSSA and stochastic model checking can be used in concert to give a researcher the likelihood that the system will exhibit its most typical behavior. Once the typical behavior is known, properties for nontypical behaviors can be constructed and their likelihoods can also be computed. This methodology is applied to several genetic circuits leading to new understanding of the effects of various parameters on the behavior of these circuits

Propagation Models for Biochemical Reaction Networks

In this thesis we investigate different ways of approximating the solution of the chemical master equation (CME). The CME is a system of differential equations that models the stochastic transient behaviour of biochemical reaction networks. It does so by describing the time evolution of probability distribution over the states of a Markov chain that represents a biological network, and thus its stochasticity is only implicit. The transient solution of a CME is the vector of probabilities over the states of the corresponding Markov chain at a certain time point t, and it has traditionally been obtained by applying methods that are general to continuous-time Markov chains: uniformization, Krylov subspace methods, and general ordinary differential equation (ODE) solvers such as the fourth order Runge-Kutta method. Even though biochemical reaction networks are the main application of our work, some of our results are presented in the more general framework of propagation models (PM), a computational formalism that we introduce in the first part of this thesis. Each propagation model N has two associated propagation processes, one in discrete-time and a second one in continuous-time. These propagation processes propagate a generic mass through a discrete state space. For example, in order to model a CME, N propagates probability mass. In the discrete-time case the propagation is done step-wise, while in the continuous-time case it is done in a continuous flow defined by a differential equation. Again, in the case of the chemical master equation, this differential equation is the equivalent of the chemical master equation itself where probability mass is propagated through a discrete state space. Discrete-time propagation processes can encode methods such as the uniformization method and the fourth order Runge-Kutta integration method that we have mentioned above, and thus by optimizing propagation algorithms we optimize both of these methods simultaneously. In the second part of our thesis, we define stochastic hybrid models that approximate the stochastic behaviour of biochemical reaction networks by treating some variables of the system deterministically. This deterministic approximation is done for species with large populations, for which stochasticity does not play an important role. We propose three such hybrid models, which we introduce from the coarsest to the most refined one: (i) the first one replaces some variables of the system with their overall expectations, (ii) the second one replaces some variables of the system with their expectations conditioned on the values of the stochastic variables, (iii) and finally, the third one, splits each variable into a stochastic part (for low valuations) and a deterministic part (for high valuations), while tracking the conditional expectation of the deterministic part. For each of these algorithms we give the corresponding propagation models that propagate not only probabilities but also the respective continuous approximations for the deterministic variables

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

Verifying Probabilistic Procedural Programs

Monolithic nite-state probabilistic programs have been abstractly modeled by nite Markov chains, and the algorithmic veri - cation problems for them have been investigated very extensively. In this paper we survey recent work conducted by the authors together with colleagues on the algorithmic veri cation of probabilistic procedural programs ([BKS,EKM04,EY04]). Probabilistic procedural programs can more naturally be modeled by recursive Markov chains ([EY04]), or equivalently, probabilistic pushdown automata ([EKM04]). A very rich theory emerges for these models. While our recent work solves a number of veri cation problems for these models, many intriguing questions remain open