research-article

Efficient Analysis of Probabilistic Programs with an Unbounded Counter

Authors:

Tomás Brázdil,

Stefan Kiefer,

Antonín KŭceraAuthors Info & Claims

Journal of the ACM (JACM), Volume 61, Issue 6

Article No.: 41, Pages 1 - 35

https://doi.org/10.1145/2629599

Published: 17 December 2014 Publication History

Get Access

Abstract

We show that a subclass of infinite-state probabilistic programs that can be modeled by probabilistic one-counter automata (pOC) admits an efficient quantitative analysis. We start by establishing a powerful link between pOC and martingale theory, which leads to fundamental observations about quantitative properties of runs in pOC. In particular, we provide a “divergence gap theorem”, which bounds a positive non-termination probability in pOC away from zero. Using these observations, we show that the expected termination time can be approximated up to an arbitrarily small relative error in polynomial time, and the same holds for the probability of all runs that satisfy a given ω-regular property encoded by a deterministic Rabin automaton.

References

[1]

E. Allender, P. Bürgisser, J. Kjeldgaard-Pedersen, and P. B. Miltersen. 2008. On the complexity of numerical analysis. SIAM J. Comput. 38, 1987--2006.

Digital Library

Google Scholar

[2]

P. Billingsley. 1995. Probability and Measure. Wiley.

Google Scholar

[3]

T. Brázdil, V. Brožek, J. Holeček, and A. Kučera. 2008. Discounted properties of probabilistic pushdown automata. In Proceedings of LPAR'08. Lecture Notes in Computer Science Series, Vol. 5330, Springer, 230--242.

Digital Library

Google Scholar

[4]

T. Brázdil, V. Brožek, and K. Etessami. 2010a. One-counter stochastic games In Proceedings of FST&TCS'10. Leibniz International Proceedings in Informatics, Vol. 8, Schloss Dagstuhl--Leibniz-Zentrum füür Informatik, 108--119.

Google Scholar

[5]

T. Brázdil, V. Brožek, K. Etessami, and A. Kučera. 2011a. Approximating the termination value of one-counter MDPs and stochastic games. In Proceedings of ICALP'11, Part II. Lecture Notes in Computer Science, vol. 6756, Springer, 332--343.

Digital Library

Google Scholar

[6]

T. Brázdil, V. Brožek, K. Etessami, A. Kučera, and D. Wojtczak. 2010b. One-counter Markov decision processes. In Proceedings of SODA'10. SIAM, 863--874.

Digital Library

Google Scholar

[7]

T. Brázdil, J. Esparza, S. Kiefer, and A. Kučera. 2013. Analyzing probabilistic pushdown automata. Formal Methods Syst. Design 43, 2, 124--163. DOI 10.1007/s10703-012-0166-0.

Digital Library

Google Scholar

[8]

T. Brázdil, J. Esparza, and A. Kučera. 2005a. Analysis and prediction of the long-run behavior of probabilistic sequential programs with recursion. In Proceedings of FOCS'05. IEEE Computer Society Press, 521--530.

Digital Library

Google Scholar

[9]

T. Brázdil, S. Kiefer, and A. Kučera. 2011b. Efficient analysis of probabilistic programs with an unbounded counter. In Proceedings of CAV'11. Lecture Notes in Computer Science, vol. 6806, Springer, 208--224.

Digital Library

Google Scholar

[10]

T. Brázdil, A. Kučera, P. Novotný, and D. Wojtczak. 2012. Minimizing expected termination time in one-counter Markov decision processes. In Proceedings of ICALP'12, Part II. Lecture Notes in Computer Science, vol. 7392, Springer, 141--152.

Digital Library

Google Scholar

[11]

T. Brázdil, A. Kučera, and O. Stražovský. 2005. On the decidability of temporal properties of probabilistic pushdown automata, Proceedings of STACS'05. Lecture Notes in Computer Science, Vol. 3404, Springer. 145--157.

Digital Library

Google Scholar

[12]

J. Canny. 1988. Some algebraic and geometric computations in PSPACE. In Proceedings of STOC'88. ACM Press, 460--467.

Digital Library

Google Scholar

[13]

K. Chatterjee and L. Doyen. 2010. Energy parity games. In Proceedings of ICALP'10, Part II. Lecture Notes in Computer Science, vol. 6199, Springer, 599--610.

Digital Library

Google Scholar

[14]

K. Chatterjee, L. Doyen, T. Henzinger, and J.-F. Raskin. 2010. Generalized mean-payoff and energy games Proceedings of FST&TCS'10. Leibniz International Proceedings in Informatics Series, vol. 8, Schloss Dagstuhl--Leibniz-Zentrum füür Informatik, 505--516.

Google Scholar

[15]

J. Esparza, D. Hansel, P. Rossmanith, and S. Schwoon. 2000. Efficient algorithms for model checking pushdown systems. In Proceedings of CAV'00. Lecture Notes in Computer Science, vol. 1855, Springer, 232--247.

Digital Library

Google Scholar

[16]

J. Esparza, A. Kučera, and R. Mayr. 2004. Model-checking probabilistic pushdown automata. In Proceedings of LICS'04 (13). IEEE Computer Society Press, 12--21.

Digital Library

Google Scholar

[17]

J. Esparza, A. Kučera, and R. Mayr. 2005. Quantitative analysis of probabilistic pushdown automata: Expectations and variances. In Proceedings of LICS'05. IEEE Computer Society Press, 117--126.

Digital Library

Google Scholar

[18]

K. Etessami, D. Wojtczak, and M. Yannakakis. 2008. Quasi-birth-death processes, tree-like QBDs, probabilistic 1-counter automata, and pushdown systems. In Proceedings of 5th International Conference on Quantitative Evaluation of Systems (QEST'08). IEEE Computer Society Press.

Digital Library

Google Scholar

[19]

K. Etessami, D. Wojtczak, and M. Yannakakis. 2010. Quasi-birth-death processes, tree-like QBDs, probabilistic 1-counter automata, and pushdown systems. Performance Eval. 67, 9, 837--857.

Digital Library

Google Scholar

[20]

K. Etessami and M. Yannakakis. 2005a. Algorithmic verification of recursive probabilistic systems. In Proceedings of TACAS'05. Lecture Notes in Computer Science, vol. 3440, Springer, 253--270.

Digital Library

Google Scholar

[21]

K. Etessami and M. Yannakakis. 2005b. Checking LTL properties of recursive Markov chains. In Proceedings of 2nd International Conference on Quantitative Evaluation of Systems (QEST'05). IEEE Computer Society Press, 155--165.

Digital Library

Google Scholar

[22]

K. Etessami and M. Yannakakis. 2005c. Recursive Markov chains, stochastic grammars, and monotone systems of non-linear equations. In Proceedings of STACS'05. Lecture Notes in Computer Science Series, vol. 3404, Springery, 340--352.

Digital Library

Google Scholar

[23]

J. Hopcroft and J. Ullman. 1979. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.

Digital Library

Google Scholar

[24]

E. Isaacson and H. B. Keller. 1966. Analysis of Numerical Methods. Wiley.

Google Scholar

[25]

J. Kemeny and J. Snell. 1960. Finite Markov chains. D. Van Nostrand Company.

Google Scholar

[26]

S. Kiefer, M. Luttenberger, and J. Esparza. 2007. On the convergence of Newton's method for monotone systems of polynomial equations. In Proceedings of STOC'07. ACM, 217--226.

Digital Library

Google Scholar

[27]

J. Křetínský and R. Ledesma-Garza. 2013. Rabinizer 2: Small deterministic automata for LTL\GU. In Proceedings of ATVA'13. Lecture Notes in Computer Science, vol. 8172. Springer, 446--450.

Google Scholar

[28]

M. Neuts. 1981. Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Dover Publications.

Google Scholar

[29]

J. Rosenthal. 2006. A First Look at Rigorous Probability Theory. World Scientific Publishing.

Google Scholar

[30]

A. Stewart, K. Etessami, and M. Yannakakis. 2013. Upper bounds for Newton's method on monotone polynomial systems, and P-time model checking of probabilistic one-counter automata. In Proceedings of CAV'13. Lecture Notes in Computer Science, vol. 8044, Springer, 495--510.

Digital Library

Google Scholar

[31]

W. Thomas. 1991. Automata on infinite objects. Handbook of Theoretical Computer Science B, 135--192.

Google Scholar

[32]

D. Williams. 1991. Probability with Martingales. Cambridge University Press.

Google Scholar

Cited By

View all

Finkel AHaddad SYe L(2023)Introducing Divergence for Infinite Probabilistic ModelsReachability Problems10.1007/978-3-031-45286-4_10(127-140)Online publication date: 5-Oct-2023
https://doi.org/10.1007/978-3-031-45286-4_10
Abdulla PAtig MAgarwal RGodbole AS. K(2022)Probabilistic Total Store OrderingProgramming Languages and Systems10.1007/978-3-030-99336-8_12(317-345)Online publication date: 29-Mar-2022
https://doi.org/10.1007/978-3-030-99336-8_12
Brázdil TChatterjee KKučera ANovotný PVelan D(2019)Deciding Fast Termination for Probabilistic VASS with NondeterminismAutomated Technology for Verification and Analysis10.1007/978-3-030-31784-3_27(462-478)Online publication date: 28-Oct-2019
https://dl.acm.org/doi/10.1007/978-3-030-31784-3_27
Show More Cited By

Index Terms

Efficient Analysis of Probabilistic Programs with an Unbounded Counter
1. Theory of computation
  1. Semantics and reasoning
    1. Program reasoning
      1. Program verification

Recommendations

Zero-reachability in probabilistic multi-counter automata
CSL-LICS '14: Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

We study the qualitative and quantitative zero-reachability problem in probabilistic multi-counter systems. We identify the undecidable variants of the problems, and then we concentrate on the remaining two cases. In the first case, when we are ...
The complexity of regular abstractions of one-counter languages
LICS '16: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

We study the computational and descriptional complexity of the following transformation: Given a one-counter automaton (OCA) A, construct a nondeterministic finite automaton (NFA) B that recognizes an abstraction of the language L(A): its (1) downward ...
Pushdown and One-Counter Automata: Constant and Non-constant Memory Usage
Descriptional Complexity of Formal Systems
Abstract
It cannot be decided whether a one-counter automaton accepts each string in its language using a counter whose value is bounded by a constant. Furthermore, when the counter is bounded by a constant, its value cannot be limited by any recursive ...

Reviews

Reviewer: Kamal Lodaya

The authors consider a probabilistic recursive program working on data of unbounded size, for example, with a geometric probability distribution for what the data looks like. This makes sense for common data structures like trees. Is the expected time that the program will terminate finite__?__ Is it almost sure (probability 1) that all runs, finite and infinite, satisfy a property specified by a deterministic Rabin automaton on infinite words__?__ When the program can be modeled as a probabilistic one-counter automaton, the authors show such questions can be answered in polynomial time. An earlier work considered more general probabilistic pushdown systems, but used the decidability of the existential theory of the reals, which has greater complexity. The main idea is that although the system is modeled as an infinite Markov chain, the questions can be answered by analyzing another finite Markov chain with some "good" conditions. The paper is quite technical; the justification that these ideas are correct involves the construction of suitable martingales. Online Computing Reviews Service

Access critical reviews of Computing literature here

Become a reviewer for Computing Reviews.

Comments

Information & Contributors

Information

Published In

Journal of the ACM Volume 61, Issue 6

November 2014

285 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2700084

Editor:
Victor Vianu
University of California, San Diego

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 December 2014

Accepted: 01 April 2014

Revised: 01 September 2013

Received: 01 April 2012

Published in JACM Volume 61, Issue 6

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

Institute for Theoretical Computer Science (ITI)
Royal Society

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

15
Total Citations
View Citations
299
Total Downloads

Downloads (Last 12 months)10
Downloads (Last 6 weeks)2

Reflects downloads up to 02 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Finkel AHaddad SYe L(2023)Introducing Divergence for Infinite Probabilistic ModelsReachability Problems10.1007/978-3-031-45286-4_10(127-140)Online publication date: 5-Oct-2023
Abdulla PAtig MAgarwal RGodbole AS. K(2022)Probabilistic Total Store OrderingProgramming Languages and Systems10.1007/978-3-030-99336-8_12(317-345)Online publication date: 29-Mar-2022
Brázdil TChatterjee KKučera ANovotný PVelan D(2019)Deciding Fast Termination for Probabilistic VASS with NondeterminismAutomated Technology for Verification and Analysis10.1007/978-3-030-31784-3_27(462-478)Online publication date: 28-Oct-2019
Wang DHoffmann JReps T(2018)PMAF: an algebraic framework for static analysis of probabilistic programsACM SIGPLAN Notices10.1145/3296979.319240853:4(513-528)Online publication date: 11-Jun-2018
Wang DHoffmann JReps TFoster JGrossman D(2018)PMAF: an algebraic framework for static analysis of probabilistic programsProceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation10.1145/3192366.3192408(513-528)Online publication date: 11-Jun-2018
Mayr RSchewe STotzke PWojtczak DAceto LIngólfsdóttir A(2017)MDPs with energy-parity objectivesProceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science10.5555/3329995.3330066(1-12)Online publication date: 20-Jun-2017
Chatterjee KNovotný PŽikelić Ð(2017)Stochastic invariants for probabilistic terminationACM SIGPLAN Notices10.1145/3093333.300987352:1(145-160)Online publication date: 1-Jan-2017
Chatterjee KNovotný PŽikelić ÐCastagna GGordon A(2017)Stochastic invariants for probabilistic terminationProceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages10.1145/3009837.3009873(145-160)Online publication date: 1-Jan-2017
Mayr RSchewe STotzke PWojtczak D(2017)MDPs with energy-parity objectives2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS.2017.8005131(1-12)Online publication date: Jun-2017
Castagna GGordon A(2017)Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming LanguagesundefinedOnline publication date: 1-Jan-2017
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Cited By

Index Terms

Recommendations

Zero-reachability in probabilistic multi-counter automata

The complexity of regular abstractions of one-counter languages

Pushdown and One-Counter Automata: Constant and Non-constant Memory Usage

Reviews

Access critical reviews of Computing literature here

Comments

Published In

Publisher

Publication History

Permissions

Check for updates

Author Tags

Qualifiers

Funding Sources

Other Metrics

Article Metrics

Other Metrics

Cited By

Login options

Full Access

PDF

eReader

Abstract

References

Cited By

Index Terms

Recommendations

Zero-reachability in probabilistic multi-counter automata

The complexity of regular abstractions of one-counter languages

Pushdown and One-Counter Automata: Constant and Non-constant Memory Usage

Reviews

Access critical reviews of Computing literature here

Comments

Information

Published In

Publisher

Publication History

Permissions

Check for updates

Author Tags

Qualifiers

Funding Sources

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

Get Access

Login options

Full Access

View options

PDF

eReader

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations