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

Author(s): Chatterjee, Krishnendu; Fu, Hongfei; Novotný, Petr; Hasheminezhad, Rouzbeh
Title: Algorithmic analysis of qualitative and quantitative termination problems for affine probabilistic programs
Title Series: POPL
Affiliation IST Austria
Abstract: In this paper, we consider termination of probabilistic programs with real-valued variables. The questions concerned are: (a) 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); (b) 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.
Keywords: termination; Concentration; Probabilistic programs; Ranking supermartingale
Conference Title: POPL: Principles of Programming Languages
Volume: 20-22
Conference Dates: January 20 - 22, 2016
Conference Location: St. Petersburg, FL, USA
Publisher: ACM  
Date Published: 2016-01-11
Start Page: 327
End Page: 342
DOI: 10.1145/2837614.2837639
Notes: The research was partly supported by Austrian Science Fund (FWF) Grant No P23499- N23, FWF NFN Grant No S11407-N23 (RiSE/SHiNE), and ERC Start grant (279307: Graph Games). The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement No [291734]. Supported by the Natural Science Foundation of China (NSFC) under Grant No. 61532019
Open access: yes (repository)
IST Austria Authors
  1. Hongfei Fu
    2 Fu
  2. Petr Novotny
    11 Novotny
Related IST Austria Work