Hongfei Fu

Algorithmic analysis of qualitative and quantitative termination problems for affine probabilistic programs

In this paper, we consider termination of probabilistic programs with real-valued variables. The questions concerned are: 1. qualitative ones that ask (i) whether the program terminates with probability 1 (almost-sure termination) and (ii) whether the expected termination time is finite (finite termination); 2. quantitative ones that ask (i) to approximate the expected termination time (expectation problem) and (ii) to compute a bound B such that the probability to terminate after B steps decreases exponentially (concentration problem). To solve these questions, we utilize the notion of ranking supermartingales which is a powerful approach for proving termination of probabilistic programs. In detail, we focus on algorithmic synthesis of linear ranking-supermartingales over affine probabilistic programs (APPs) with both angelic and demonic non-determinism. An important subclass of APPs is LRAPP which is defined as the class of all APPs over which a linear ranking-supermartingale exists. Our main contributions are as follows. Firstly, we show that the membership problem of LRAPP (i) can be decided in polynomial time for APPs with at most demonic non-determinism, and (ii) is NP-hard and in PSPACE for APPs with angelic non-determinism; moreover, the NP-hardness result holds already for APPs without probability and demonic non-determinism. Secondly, we show that the concentration problem over LRAPP can be solved in the same complexity as for the membership problem of LRAPP. Finally, we show that the expectation problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APPs without probability and non-determinism (i.e., deterministic programs). Our experimental results demonstrate the effectiveness of our approach to answer the qualitative and quantitative questions over APPs with at most demonic non-determinism.

Termination Analysis of Probabilistic Programs Through Positivstellensatz’s

We consider nondeterministic probabilistic programs with the most basic liveness property of termination. We present efficient methods for termination analysis of nondeterministic probabilistic programs with polynomial guards and assignments. Our approach is through synthesis of polynomial ranking supermartingales, that on one hand significantly generalizes linear ranking supermartingales and on the other hand is a counterpart of polynomial ranking-functions for proving termination of nonprobabilistic programs. The approach synthesizes polynomial ranking-supermartingales through Positivstellensatz’s, yielding an efficient method which is not only sound, but also semi-complete over a large subclass of programs. We show experimental results to demonstrate that our approach can handle several classical programs with complex polynomial guards and assignments, and can synthesize efficient quadratic ranking-supermartingales when a linear one does not exist even for simple affine programs.

Approximating acceptance probabilities of CTMC-paths on multi-clock deterministic timed automata

We consider the problem of approximating the probability mass of the set of timed paths under a continuous-time Markov chain (CTMC) that are accepted by a deterministic timed automaton (DTA). As opposed to several existing works on this topic, we consider DTA with multiple clocks. Our key contribution is an algorithm to approximate these probabilities using finite difference methods. An error bound is provided which indicates the approximation error. The stepping stones towards this result include rigorous proofs for the measurability of the set of accepted paths and the integral-equation system characterizing the acceptance probability, and a differential characterization for the acceptance probability.

Maximal Cost-Bounded Reachability Probability on Continuous-Time Markov Decision Processes

In this paper, we consider multi-dimensional maximal cost-bounded reachability probability over continuous-time Markov decision processes (CTMDPs). Our major contributions are as follows. Firstly, we derive an integral characterization which states that the maximal cost-bounded reachability probability function is the least fixed-point of a system of integral equations. Secondly, we prove that the maximal cost-bounded reachability probability can be attained by a measurable deterministic cost-positional scheduler. Thirdly, we provide a numerical approximation algorithm for maximal cost-bounded reachability probability. We present these results under the setting of both early and late schedulers. Besides, we correct a fundamental proof error in the PhD Thesis by Martin Neuhauser on maximal time-bounded reachability probability by completely new proofs for the more general case of multi-dimensional maximal cost-bounded reachability probability.

Computing game metrics on markov decision processes

In this paper we study the complexity of computing the game bisimulation metric defined by de Alfaro et al. on Markov Decision Processes. It is proved by de Alfaro et al. that the undiscounted version of the metric is characterized by a quantitative game μ-calculus defined by de Alfaro and Majumdar, which can express reachability and ω-regular specifications. And by Chatterjee et al. that the discounted version of the metric is characterized by the discounted quantitative game μ-calculus. In the discounted case, we show that the metric can be computed exactly by extending the method for Labelled Markov Chains by Chen et al. And in the undiscounted case, we prove that the problem whether the metric between two states is under a given threshold can be decided in NP∩coNP, which improves the previous PSPACE upperbound by Chatterjee et al.

Automated Recurrence Analysis for Almost-Linear Expected-Runtime Bounds

We consider the problem of developing automated techniques for solving recurrence relations to aid the expected-runtime analysis of programs. Several classical textbook algorithms have quite efficient expected-runtime complexity, whereas the corresponding worst-case bounds are either inefficient (e.g., QUICK-SORT), or completely ineffective (e.g., COUPON-COLLECTOR). Since the main focus of expected-runtime analysis is to obtain efficient bounds, we consider bounds that are either logarithmic, linear, or almost-linear (

Non-polynomial Worst-Case Analysis of Recursive Programs

\mathcal{O}(\log n)

Decidability of behavioral equivalences in process calculi with name scoping

,

Deciding Probabilistic Simulation between Probabilistic Pushdown Automata and Finite-State Systems

\mathcal{O}(n)

Verifying Probabilistic Timed Automata Against Omega-Regular Dense-Time Properties.

,