Aiming low is harder: induction for lower bounds in probabilistic program verification

We present a new inductive rule for verifying lower bounds on expected values of random variables after execution of probabilistic loops as well as on their expected runtimes. Our rule is simple in the sense that loop body semantics need to be applied only finitely often in order to verify that the candidates are indeed lower bounds. In particular, it is not necessary to find the limit of a sequence as in many previous rules.

[1]  Annabelle McIver,et al.  Abstraction and refinement in probabilistic systems , 2005, PERV.

[2]  Bart Jacobs,et al.  Healthiness from Duality , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[3]  Ugo Dal Lago,et al.  On the Termination Problem for Probabilistic Higher-Order Recursive Programs , 2018, 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[4]  Joost-Pieter Katoen,et al.  Weakest Precondition Reasoning for Expected Runtimes of Randomized Algorithms , 2018, J. ACM.

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

[6]  Joost-Pieter Katoen,et al.  A weakest pre-expectation semantics for mixed-sign expectations , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[7]  Krishnendu Chatterjee,et al.  Termination of Nondeterministic Probabilistic Programs , 2019, VMCAI.

[8]  Moshe Y. Vardi.  Automatic verification of probabilistic concurrent finite state programs , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[9]  Lijun Zhang,et al.  Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation , 2015, CAV.

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

[11]  Van Chan Ngo,et al.  Bounded expectations: resource analysis for probabilistic programs , 2017, PLDI.

[12]  Ali Esmaili,et al.  Probability and Random Processes , 2005, Technometrics.

[13]  Gilles Barthe,et al.  Synthesizing Probabilistic Invariants via Doob's Decomposition , 2016, CAV.

[14]  Annabelle McIver,et al.  Conditioning in Probabilistic Programming , 2015, MFPS.

[15]  Bican Xia,et al.  Finding Polynomial Loop Invariants for Probabilistic Programs , 2017, ATVA.

[16]  Christine Paulin-Mohring,et al.  Proofs of randomized algorithms in Coq , 2006, Sci. Comput. Program..

[17]  Annabelle McIver,et al.  Linear-Invariant Generation for Probabilistic Programs: - Automated Support for Proof-Based Methods , 2010, SAS.

[18]  David Cock.  pGCL for Isabelle , 2014, Arch. Formal Proofs.

[19]  Klaus Keimel.  Healthiness Conditions for Predicate Transformers , 2015, MFPS.

[20]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

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

[22]  Andrei Baranga,et al.  The contraction principle as a particular case of Kleene's fixed point theorem , 1991, Discret. Math..

[23]  Krishnendu Chatterjee,et al.  New Approaches for Almost-Sure Termination of Probabilistic Programs , 2018, APLAS.

[24]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[25]  Benjamin Lucien Kaminski.  Advanced weakest precondition calculi for probabilistic programs , 2019 .

[26]  Joost-Pieter Katoen,et al.  Aiming Low Is Harder - Inductive Proof Rules for Lower Bounds on Weakest Preexpectations in Probabilistic Program Verification , 2019, ArXiv.

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

[28]  Ralph-Johan Back,et al.  Refinement Calculus: A Systematic Introduction , 1998 .

[29]  Joost-Pieter Katoen,et al.  Quantitative separation logic: a logic for reasoning about probabilistic pointer programs , 2018, Proc. ACM Program. Lang..

[30]  Annabelle McIver,et al.  An Expectation-Transformer Model for Probabilistic Temporal Logic , 1999, Log. J. IGPL.

[31]  Joost-Pieter Katoen,et al.  Reasoning about Recursive Probabilistic Programs* , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

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

[33]  Joost-Pieter Katoen,et al.  How long, O Bayesian network, will I sample thee? A program analysis perspective on expected sampling times , 2018, ESOP.

[34]  Krishnendu Chatterjee,et al.  Lexicographic ranking supermartingales: an efficient approach to termination of probabilistic programs , 2017, Proc. ACM Program. Lang..

[35]  Joost-Pieter Katoen,et al.  On the hardness of analyzing probabilistic programs , 2018, Acta Informatica.

[36]  Jürgen Giesl,et al.  Lower Runtime Bounds for Integer Programs , 2016, IJCAR.

[37]  Thomas A. Henzinger,et al.  Probabilistic programming , 2014, FOSE.

[38]  H. Bauer,et al.  Probability Theory and Elements of Measure Theory , 1982 .

[39]  Claire Jones,et al.  Probabilistic non-determinism , 1990 .

[40]  Annabelle McIver,et al.  Partial correctness for probabilistic demonic programs , 2001, Theor. Comput. Sci..

[41]  Eric C. R. Hehner,et al.  A probability perspective , 2011, Formal Aspects of Computing.

[42]  Edsger W. Dijkstra,et al.  Guarded commands, nondeterminacy and formal derivation of programs , 1975, Commun. ACM.

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

[44]  Sriram Sankaranarayanan,et al.  Expectation Invariants for Probabilistic Program Loops as Fixed Points , 2014, SAS.

[45]  Krishnendu Chatterjee,et al.  Algorithmic analysis of qualitative and quantitative termination problems for affine probabilistic programs , 2015, POPL.

[46]  G. Grimmett,et al.  Probability and random processes , 2002 .

[47]  Krishnendu Chatterjee,et al.  Cost analysis of nondeterministic probabilistic programs , 2019, PLDI.

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

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

[50]  G. Pólya,et al.  Eine Wahrscheinlichkeitsaufgabe in der Kundenwerbung , 1930 .

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

[52]  Liz Sonenberg,et al.  Fixed Point Theorems and Semantics: A Folk Tale , 1982, Inf. Process. Lett..

[53]  David Monniaux.  Abstract interpretation of programs as Markov decision processes , 2005, Sci. Comput. Program..

[54]  Piotr Pokarowski,et al.  The Tarski–Kantorovitch prinicple and the theory of iterated function systems , 2000, Bulletin of the Australian Mathematical Society.

[55]  Nils Jansen,et al.  Bounded Model Checking for Probabilistic Programs , 2016, ATVA.

[56]  Annabelle McIver,et al.  A new proof rule for almost-sure termination , 2017, Proc. ACM Program. Lang..