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A New Proof Rule for Almost-Sure Termination [article]

Annabelle McIver, Carroll Morgan, Benjamin Lucien Kaminski, Joost-Pieter Katoen
2017 arXiv   pre-print
An important question for a probabilistic program is whether the probability mass of all its diverging runs is zero, that is that it terminates "almost surely".  ...  Proving that can be hard, and this paper presents a new method for doing so; it is expressed in a program logic, and so applies directly to source code.  ...  We will write AS for "almost sure" and AST for "almost-sure termination" or "almost-surely terminating". But the standard variant rule we mentioned above is too weak for AST in general.  ... 
arXiv:1711.03588v2 fatcat:xkq5trnsazgqpbsgc2ymls7whi

Proving Positive Almost Sure Termination Under Strategies [chapter]

Olivier Bournez, Florent Garnier
2006 Lecture Notes in Computer Science  
We introduce the notion of positive almost sure termination under strategies, and we provide sufficient criteria to prove termination of a given probabilitic rewrite system under strategies.  ...  Termination was considered under arbitrary unrestricted policies. Policies correspond to strategies for non-probabilistic rewrite systems.  ...  Previous hypotheses yield almost sure termination, but not positive almost sure termination. Indeed, the proof build a subsequence of indexes ψ(n) yielding almost surely to termination.  ... 
doi:10.1007/11805618_27 fatcat:rgv4httr4jgx5bsklfh55qdcqa

Automated Termination Analysis of Polynomial Probabilistic Programs [article]

Marcel Moosbrugger, Ezio Bartocci, Joost-Pieter Katoen, Laura Kovács
2021 arXiv   pre-print
Positive almost sure termination (PAST) focuses on termination in a finite expected number of steps.  ...  Almost sure termination (AST) is concerned with the question whether a program terminates with probability one on all possible inputs.  ...  The work of [10] uses martingale theory to characterize positive almost sure termination (PAST).  ... 
arXiv:2010.03444v3 fatcat:u4v3xuydc5bw3eun3h47pwklxy

New Approaches for Almost-Sure Termination of Probabilistic Programs [article]

Mingzhang Huang, Hongfei Fu, Krishnendu Chatterjee
2018 arXiv   pre-print
Second, we present a new approach based on Central Limit Theorem for the almost-sure termination problem, and show that this approach can establish almost-sure termination of programs which none of the  ...  We study the almost-sure termination problem for probabilistic programs.  ...  -Almost-Sure Termination: CLT-Based Approach. We present a new approach based on Central Limit Theorem (CLT) that is sound to establish almost-sure termination.  ... 
arXiv:1806.06683v2 fatcat:m2fzoybhxfbifgoloszldrvvwi

The Probabilistic Termination Tool Amber [article]

Marcel Moosbrugger, Ezio Bartocci, Joost-Pieter Katoen, Laura Kovács
2021 arXiv   pre-print
Amber combines martingale theory with properties of asymptotic bounding functions and implements relaxed versions of existing probabilistic termination proof rules to prove/disprove (positive) almost sure  ...  We describe the Amber tool for proving and refuting the termination of a class of probabilistic while-programs with polynomial arithmetic, in a fully automated manner.  ...  Probabilistic programs are almost-surely terminating (AST) if they terminate with probability 1 on all inputs. They are positively AST (PAST) if their expected runtime is finite [5] .  ... 
arXiv:2107.13072v1 fatcat:glpim3x7infmjf4k6kte4wprli

Induction for positive almost sure termination

Isabelle Gnaedig
2007 Proceedings of the 9th ACM SIGPLAN international conference on Principles and practice of declarative programming - PPDP '07  
The induction principle consists in assuming that terms smaller than the starting terms for an induction ordering are positively almost surely terminating.  ...  We extend to the probabilistic case a technique we proposed for termination of usual rewriting under strategies.  ...  Acknowledgments We would like to thank Olivier Bournez for fruitful discussions on termination of probabilistic rewriting and on this paper, and for giving us motivating examples.  ... 
doi:10.1145/1273920.1273943 dblp:conf/ppdp/Gnaedig07 fatcat:3rdtzrr5g5g4jpcu57475etdaq

A reinforcement learning process in extensive form games

Jean-François Laslier, Bernard Walliser
2005 International Journal of Game Theory  
., 2001), the CPR rule is here adapted for actions in perfect information extensive form games.  ...  The CPR ("cumulative proportional reinforcement") learning rule stipulates that an agent chooses a move with a probability proportional to the cumulative payoff she obtained in the past with that move.  ...  This proves that, π-almost surely: lim t→∞ p t (a * (e n), ω) = lim t→∞ q t (e n, ω) = 1. (c) Induction. For the general induction step, consider any non-terminal node n.  ... 
doi:10.1007/s001820400194 fatcat:oqia6etodrfebkwu7njzwuqdne

Modular Verification for Almost-Sure Termination of Probabilistic Programs [article]

Mingzhang Huang, Hongfei Fu, Krishnendu Chatterjee, Amir Kafshdar Goharshady
2019 arXiv   pre-print
Besides establishing unsoundness of the previous rule, our contributions are as follows: First, we present a sound modular rule for almost-sure termination of probabilistic programs.  ...  In this work, we consider the almost-sure termination problem for probabilistic programs that asks whether a given probabilistic program terminates with probability 1.  ...  Yuxi Fu, director of the support of the BASICS Lab at Shanghai Jiao Tong University, for his support.  ... 
arXiv:1901.06087v3 fatcat:5dy3h7fe2nbcdhmvkawylbt6ee

Runtime Analysis of Probabilistic Programs with Unbounded Recursion [article]

Tomas Brazdil, Stefan Kiefer, Antonin Kucera, Ivana Hutarova Varekova
2012 arXiv   pre-print
We obtain tail bounds for the distribution of termination time for pPDA.  ...  We also study the recurrence time for probabilistic recursive programs that are not supposed to terminate (such as system daemons, network servers, etc.).  ...  Since ∆ is almost surely terminating, the length of c(X 1 ) is at most |Γ | for all X 1 ∈ Γ .  ... 
arXiv:1007.1710v2 fatcat:3w6u72o35bddlbakfecjvzxhoy

Probabilistic Rewrite Strategies. Applications to ELAN [chapter]

Olivier Bournez, Claude Kirchner
2002 Lecture Notes in Computer Science  
Recently rule based languages focussed on the use of rewriting as a modeling tool which results in making specifications executable.  ...  This leads us to interesting new problems, and we address the generalization of the usual concepts in abstract reduction systems to randomized systems.  ...  We will then say that a PARS is locally confluent (LC) (respectively confluent (C) , normalizing (N) , terminating (T) , unique normal form property (UN) , inductive (Ind) , increasing (Inc) , finitely  ... 
doi:10.1007/3-540-45610-4_18 fatcat:jnf4m7qw45dfplstqy4uui75pi

Probabilistic Termination

Luis María Ferrer Fioriti, Holger Hermanns
2015 Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '15  
We propose a framework to prove almost sure termination for probabilistic programs with real valued variables.  ...  We complement this foundational insight by a practical proof methodology, based on sound conditions that enable compositional reasoning and are amenable to a direct implementation using modern theorem  ...  D'Argenio and Jan Krcál for their comments in a preliminary version of this paper, Hassan Hatefi for many insightful discussions, and the many anonymous reviewers for their invaluable comments.  ... 
doi:10.1145/2676726.2677001 dblp:conf/popl/FioritiH15 fatcat:gdc7m4nopbfgthafxixdqjbkna

A new rule for almost-certain termination of probabilistic- and demonic programs [article]

Annabelle McIver, Carroll Morgan
2017 arXiv   pre-print
Extending our own and others' earlier approaches to reasoning about termination of probabilistic programs, we propose and prove a new rule for termination with probability one, also known as "almost-certain  ...  In particular, it suffices for termination of the unbounded symmetric random walk in both one- and two dimensions: for the first, we give a proof; for the second, we use a theorem of Foster to argue that  ...  And thanks particularly to Andreas Lochbihler, who shared with us the probabilistic termination problem that led to it.  ... 
arXiv:1612.01091v2 fatcat:ei6cxa442rd65lhrb63bacikge

An Assertion-Based Program Logic for Probabilistic Programs [article]

Gilles Barthe and Thomas Espitau and Marco Gaboardi and Benjamin Grégoire and Justin Hsu and Pierre-Yves Strub
2018 arXiv   pre-print
Ellora features new proof rules for loops and adversarial code, and supports richer assertions than existing program logics.  ...  We also show that Ellora allows convenient reasoning about complex probabilistic concepts by developing a new program logic for probabilistic independence and distribution law, and then smoothly embedding  ...  We thank the reviewers for their helpful comments. This work benefited from discussions with Dexter Kozen, Annabelle McIver, and Carroll Morgan.  ... 
arXiv:1803.05535v1 fatcat:6rsr2vcc6ve3xjwraql75uhh7q

Language equivalence of probabilistic pushdown automata

Vojtěch Forejt, Petr Jančar, Stefan Kiefer, James Worrell
2014 Information and Computation  
In contrast, for the case of a one-letter input alphabet we show that pPDA language equivalence (and hence multiplicity equivalence of context-free grammars) is in PSPACE and at least as hard as the polynomial  ...  Interreducibility also holds for pPDA with one control state.  ...  Stefan Kiefer is supported by a Royal Society University Research Fellowship.  ... 
doi:10.1016/j.ic.2014.04.003 fatcat:vvgom2hdv5ftpn52bue7c3agga

Aiming Low Is Harder - Inductive Proof Rules for Lower Bounds on Weakest Preexpectations in Probabilistic Program Verification [article]

Marcel Hark, Benjamin Lucien Kaminski, Jürgen Giesl, Joost-Pieter Katoen
2019 arXiv   pre-print
We present a new inductive proof rule for reasoning about lower bounds on weakest preexpectations, i.e., expected values of random variables after execution of a probabilistic loop.  ...  Our rule is simple in the sense that the semantics of the loop needs to be applied to a candidate lower bound only a finite number of times in order to verify that the candidate is indeed a lower bound  ...  Furthermore, we would like to thank Florian Frohn and Christoph Matheja for many fruitful discussions on examples and counterexamples.  ... 
arXiv:1904.01117v1 fatcat:ca6l3zcndvegzprdi6hiuf4ksm
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