Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs

TL;DR

This paper develops algorithmic methods for analyzing the termination behavior of affine probabilistic programs, providing complexity results and practical algorithms for qualitative and quantitative termination questions.

Contribution

It introduces polynomial-time algorithms for membership in LRAPP with demonic non-determinism and complexity bounds for other cases, advancing the analysis of probabilistic program termination.

Findings

Membership in LRAPP is polynomial-time with demonic non-determinism.

NP-hardness and PSPACE membership for angelic non-determinism.

Expectation problem is solvable in 2EXPTIME, PSPACE-hard even for deterministic programs.

Abstract

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 (APP's) with both angelic and demonic non-determinism. An important subclass of APP's is LRAPP which is defined as the class of all APP's 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 APP's with at most demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with angelic non-determinism; moreover, the NP-hardness result holds already for APP's 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 APP's 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 APP's with at most demonic non-determinism.