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Step-Indexed Logical Relations for Countable Nondeterminism and Probabilistic Choice

Authors:

Alejandro Aguirre,

Lars BirkedalAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 7, Issue POPL

Article No.: 2, Pages 33 - 60

https://doi.org/10.1145/3571195

Published: 11 January 2023 Publication History

Abstract

Developing denotational models for higher-order languages that combine probabilistic and nondeterministic choice is known to be very challenging. In this paper, we propose an alternative approach based on operational techniques. We study a higher-order language combining parametric polymorphism, recursive types, discrete probabilistic choice and countable nondeterminism. We define probabilistic generalizations of may- and must-termination as the optimal and pessimal probabilities of termination. Then we define step-indexed logical relations and show that they are sound and complete with respect to the induced contextual preorders. For may-equivalence we use step-indexing over the natural numbers whereas for must-equivalence we index over the countable ordinals. We then show than the probabilities of may- and must-termination coincide with the maximal and minimal probabilities of termination under all schedulers. Finally we derive the equational theory induced by contextual equivalence and show that it validates the distributive combination of the algebraic theories for probabilistic and nondeterministic choice.

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Cited By

Stassen PMøgelberg RZwart MAguirre ABirkedal L(2025)Modelling Recursion and Probabilistic Choice in Guarded Type TheoryProceedings of the ACM on Programming Languages10.1145/37048849:POPL(1417-1445)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704884
Haselwarter PLi KAguirre AGregersen STassarotti JBirkedal L(2025)Approximate Relational Reasoning for Higher-Order Probabilistic ProgramsProceedings of the ACM on Programming Languages10.1145/37048779:POPL(1196-1226)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704877
Zilberstein NKozen DSilva ATassarotti J(2025)A Demonic Outcome Logic for Randomized NondeterminismProceedings of the ACM on Programming Languages10.1145/37048559:POPL(539-568)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704855
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Index Terms

Step-Indexed Logical Relations for Countable Nondeterminism and Probabilistic Choice
1. Theory of computation
  1. Models of computation
    1. Probabilistic computation
  2. Semantics and reasoning
    1. Program reasoning
      1. Program verification
    2. Program semantics
      1. Operational semantics

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cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 7, Issue POPL

January 2023

2196 pages

EISSN:2475-1421

DOI:10.1145/3554308

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Published: 11 January 2023

Published in PACMPL Volume 7, Issue POPL

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Cited By

Stassen PMøgelberg RZwart MAguirre ABirkedal L(2025)Modelling Recursion and Probabilistic Choice in Guarded Type TheoryProceedings of the ACM on Programming Languages10.1145/37048849:POPL(1417-1445)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704884
Haselwarter PLi KAguirre AGregersen STassarotti JBirkedal L(2025)Approximate Relational Reasoning for Higher-Order Probabilistic ProgramsProceedings of the ACM on Programming Languages10.1145/37048779:POPL(1196-1226)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704877
Zilberstein NKozen DSilva ATassarotti J(2025)A Demonic Outcome Logic for Randomized NondeterminismProceedings of the ACM on Programming Languages10.1145/37048559:POPL(539-568)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704855
Bittner PSchultheiß AMoosherr BYoung JTeixeira LWalkingshaw EAtaei PThüm T(2024)On the Expressive Power of Languages for Static VariabilityProceedings of the ACM on Programming Languages10.1145/36897478:OOPSLA2(1018-1050)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689747
Rajani VBarthe GGarg D(2024)A Modal Type Theory of Expected Cost in Higher-Order Probabilistic ProgramsProceedings of the ACM on Programming Languages10.1145/36897258:OOPSLA2(389-414)Online publication date: 8-Oct-2024
https://dl.acm.org/doi/10.1145/3689725
Gregersen SAguirre AHaselwarter PTassarotti JBirkedal L(2024)Almost-Sure Termination by Guarded RefinementProceedings of the ACM on Programming Languages10.1145/36746328:ICFP(203-233)Online publication date: 15-Aug-2024
https://dl.acm.org/doi/10.1145/3674632
Lago UGalal ZGiusti G(2024)On Computational Indistinguishability and Logical RelationsProgramming Languages and Systems10.1007/978-981-97-8943-6_12(241-263)Online publication date: 23-Oct-2024
https://dl.acm.org/doi/10.1007/978-981-97-8943-6_12

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