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A Kleene algebra with tests for union bound reasoning about probabilistic programs

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Title: A Kleene algebra with tests for union bound reasoning about probabilistic programs
Authors: Gomes, Leandro, Baillot, Patrick, Gaboardi, Marco
Contributors: Gomes, Leandro, Leandro Gomes and Patrick Baillot and Marco Gaboardi
Publisher Information: Array, 2025.
Publication Year: 2025
Subject Terms: union bound, Kleene algebras with tests Hoare logic equational reasoning probabilistic programs union bound formal verification, Theory of computation → Hoare logic, Hoare logic, equational reasoning, [INFO] Computer Science [cs], Theory of computation → Logic and verification, Theory of computation → Algebraic semantics, Kleene algebras with tests, probabilistic programs, ddc:004, formal verification, Theory of computation → Pre- and post-conditions
Description: Kleene Algebra with Tests (KAT) provides a framework for algebraic equational reasoning about imperative programs. The recent variant Guarded KAT (GKAT) allows to reason on non-probabilistic properties of probabilistic programs. Here we introduce an extension of this framework called approximate GKAT (aGKAT), which equips GKAT with a partially ordered monoid (real numbers) enabling to express satisfaction of (deterministic) properties except with a probability up to a certain bound. This allows to represent in equational reasoning ` a la KAT’ proofs of probabilistic programs based on the union bound, a technique from basic probability theory. We show how a propositional variant of approximate Hoare Logic (aHL), a program logic for union bound, can be soundly encoded in our system aGKAT. We then illustrate the use of aGKAT with an example of accuracy analysis from the field of differential privacy.
Document Type: Conference object
Article
File Description: application/pdf
Language: English
DOI: 10.4230/lipics.csl.2025.35
Access URL: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.35
Rights: CC BY
Accession Number: edsair.dedup.wf.002..a48c47eb52a13a33ecec27824b4a671f
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  Data: A Kleene algebra with tests for union bound reasoning about probabilistic programs
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  Data: <searchLink fieldCode="AR" term="%22Gomes%2C+Leandro%22">Gomes, Leandro</searchLink><br /><searchLink fieldCode="AR" term="%22Baillot%2C+Patrick%22">Baillot, Patrick</searchLink><br /><searchLink fieldCode="AR" term="%22Gaboardi%2C+Marco%22">Gaboardi, Marco</searchLink>
– Name: Author
  Label: Contributors
  Group: Au
  Data: Gomes, Leandro<br />Leandro Gomes and Patrick Baillot and Marco Gaboardi
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  Data: Array, 2025.
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  Data: 2025
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  Data: <searchLink fieldCode="DE" term="%22union+bound%22">union bound</searchLink><br /><searchLink fieldCode="DE" term="%22Kleene+algebras+with+tests+Hoare+logic+equational+reasoning+probabilistic+programs+union+bound+formal+verification%22">Kleene algebras with tests Hoare logic equational reasoning probabilistic programs union bound formal verification</searchLink><br /><searchLink fieldCode="DE" term="%22Theory+of+computation+→+Hoare+logic%22">Theory of computation → Hoare logic</searchLink><br /><searchLink fieldCode="DE" term="%22Hoare+logic%22">Hoare logic</searchLink><br /><searchLink fieldCode="DE" term="%22equational+reasoning%22">equational reasoning</searchLink><br /><searchLink fieldCode="DE" term="%22[INFO]+Computer+Science+[cs]%22">[INFO] Computer Science [cs]</searchLink><br /><searchLink fieldCode="DE" term="%22Theory+of+computation+→+Logic+and+verification%22">Theory of computation → Logic and verification</searchLink><br /><searchLink fieldCode="DE" term="%22Theory+of+computation+→+Algebraic+semantics%22">Theory of computation → Algebraic semantics</searchLink><br /><searchLink fieldCode="DE" term="%22Kleene+algebras+with+tests%22">Kleene algebras with tests</searchLink><br /><searchLink fieldCode="DE" term="%22probabilistic+programs%22">probabilistic programs</searchLink><br /><searchLink fieldCode="DE" term="%22ddc%3A004%22">ddc:004</searchLink><br /><searchLink fieldCode="DE" term="%22formal+verification%22">formal verification</searchLink><br /><searchLink fieldCode="DE" term="%22Theory+of+computation+→+Pre-+and+post-conditions%22">Theory of computation → Pre- and post-conditions</searchLink>
– Name: Abstract
  Label: Description
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  Data: Kleene Algebra with Tests (KAT) provides a framework for algebraic equational reasoning about imperative programs. The recent variant Guarded KAT (GKAT) allows to reason on non-probabilistic properties of probabilistic programs. Here we introduce an extension of this framework called approximate GKAT (aGKAT), which equips GKAT with a partially ordered monoid (real numbers) enabling to express satisfaction of (deterministic) properties except with a probability up to a certain bound. This allows to represent in equational reasoning ` a la KAT’ proofs of probabilistic programs based on the union bound, a technique from basic probability theory. We show how a propositional variant of approximate Hoare Logic (aHL), a program logic for union bound, can be soundly encoded in our system aGKAT. We then illustrate the use of aGKAT with an example of accuracy analysis from the field of differential privacy.
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  Data: 10.4230/lipics.csl.2025.35
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RecordInfo BibRecord:
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        Value: 10.4230/lipics.csl.2025.35
    Languages:
      – Text: English
    Subjects:
      – SubjectFull: union bound
        Type: general
      – SubjectFull: Kleene algebras with tests Hoare logic equational reasoning probabilistic programs union bound formal verification
        Type: general
      – SubjectFull: Theory of computation → Hoare logic
        Type: general
      – SubjectFull: Hoare logic
        Type: general
      – SubjectFull: equational reasoning
        Type: general
      – SubjectFull: [INFO] Computer Science [cs]
        Type: general
      – SubjectFull: Theory of computation → Logic and verification
        Type: general
      – SubjectFull: Theory of computation → Algebraic semantics
        Type: general
      – SubjectFull: Kleene algebras with tests
        Type: general
      – SubjectFull: probabilistic programs
        Type: general
      – SubjectFull: ddc:004
        Type: general
      – SubjectFull: formal verification
        Type: general
      – SubjectFull: Theory of computation → Pre- and post-conditions
        Type: general
    Titles:
      – TitleFull: A Kleene algebra with tests for union bound reasoning about probabilistic programs
        Type: main
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            NameFull: Gomes, Leandro
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            NameFull: Gaboardi, Marco
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            NameFull: Gomes, Leandro
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              M: 01
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              Y: 2025
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