Algorithmic analysis of qualitative and quantitative termination problems for affine probabilistic 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; *[Copyright notice will appear here once 'preprint' option is removed.] 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.

Stochastic invariants for probabilistic termination

“…Since synthesis of ǫ-ranking supermartingales supported by a linear predicate map was already addressed in the previous work [15,19], we focus on algorithms related to those aspects of probabilistic reachability which are new, i.e. those related to stochastic invariants and repulsing supermartingales.…”

“…We adapt a well known constrained-based method for generating linear ranking functions and (non-stochastic) invariants in nonprobabilistic programs [22,24,60], which was adapted for synthesizing ǫ-LRSMs in probabilistic programs [15,19]. We briefly recall this approach and explain its adaptation.…”

“…The algorithm of [19] constructs a system of linear inequalities y · Z ≥ d (here Z is a matrix) that is adjusted to P , C, and I, which means that each term in each inequality of the system has one of the following forms:…”

“…The qualitative problem of almost-sure termination has been considered in several works such as [9,15,18,19,34]. The termination for concurrent probabilistic programs under fairness was considered in [67].…”

Constraint-Based Synthesis of Coupling Proofs

“…We assume all programs terminate with probability 1 for any initial state; there are numerous systems for verifying this basic property automatically (see, e.g., [15][16][17]). To extend our fcoupled postconditions, we let cpost(P, P , Q, f) be the smallest set I satisfying:…”