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

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 termination". The rule uses both (non-strict) super martingales and guarantees of progress, together, and it seems to cover significant cases that earlier methods do not. 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 a proof exists. Non-determinism (i.e. demonic choice) is supported; but we do currently restrict to discrete distributions.

[1]  K. Knopp.  Theory and Application of Infinite Series , 1990 .

[2]  Annabelle McIver,et al.  Compositional Specification and Analysis of Cost-Based Properties in Probabilistic Programs , 2005, FM.

[3]  David Blackwell,et al.  On Transient Markov Processes with a Countable Number of States and Stationary Transition Probabilities , 1955 .

[4]  Roberto Segala,et al.  Modeling and verification of randomized distributed real-time systems , 1996 .

[5]  Aleksandar Chakarov.  Deductive Verification of Infinite-State Stochastic Systems using Martingales , 2016 .

[6]  Annabelle McIver,et al.  Probabilistic Termination in B , 2003, ZB.

[7]  Krishnendu Chatterjee,et al.  Stochastic invariants for probabilistic termination , 2016, POPL.

[8]  Sriram Sankaranarayanan,et al.  Probabilistic Program Analysis with Martingales , 2013, CAV.

[9]  S. Hart,et al.  Termination of Probabilistic Concurrent Programs. , 1982 .

[10]  Annabelle McIver,et al.  Probabilistic predicate transformers , 1996, TOPL.

[11]  Annabelle McIver,et al.  Operational versus weakest pre-expectation semantics for the probabilistic guarded command language , 2014, Perform. Evaluation.

[12]  Dexter Kozen,et al.  A probabilistic PDL , 1983, J. Comput. Syst. Sci..

[13]  F. J. Anscombe,et al.  On Markov chains with an enumerable infinity of states , 1952 .

[14]  Annabelle McIver,et al.  Almost-certain eventualities and abstract probabilities in the quantitative temporal logic qTL , 2003, Theor. Comput. Sci..

[15]  Javier Esparza,et al.  Proving Termination of Probabilistic Programs Using Patterns , 2012, CAV.

[16]  Holger Hermanns,et al.  Probabilistic Termination , 2015, POPL.

[17]  Edsger W. Dijkstra,et al.  A Discipline of Programming , 1976 .

[18]  Joost-Pieter Katoen,et al.  Weakest Precondition Reasoning for Expected Run-Times of Probabilistic Programs , 2016, ESOP.

[19]  Micha Sharir,et al.  Termination of Probabilistic Concurrent Program , 1983, TOPL.

[20]  Carroll Morgan,et al.  Proof rules for probabilistic loops , 1996 .

[21]  Annabelle McIver,et al.  Abstraction, Refinement and Proof for Probabilistic Systems , 2004, Monographs in Computer Science.

[22]  David G. Kendall,et al.  On non-dissipative Markoff chains with an enumerable infinity of states , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  C. Jones,et al.  A probabilistic powerdomain of evaluations , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[24]  Pablo Lessa.  Recurrence vs transience: An introduction to random walks , 2016 .

[25]  Annabelle McIver,et al.  Probabilistic Models for the Guarded Command Language , 1997, Sci. Comput. Program..