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The First-order Logic of Hyperproperties Joint work with Bernd Finkbeiner (Saarland University) Martin Zimmermann Saarland University September, 13th 2017 Highlights Conference, London, UK Martin Zimmermann Saarland University The

  1. The First-order Logic of Hyperproperties Joint work with Bernd Finkbeiner (Saarland University) Martin Zimmermann Saarland University September, 13th 2017 Highlights Conference, London, UK Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 1/8

  2. LTL vs. First-order Logic Theorem (Kamp ’68, Gabbay et al. ’80) LTL and FO [ < ] are expressively equivalent. Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 2/8

  3. HyperLTL A new logic: ∀ π ∀ π ′ . F on π ↔ on π ′ Extend LTL by trace quantifiers to express security, privacy, and information flow properties Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 3/8

  4. HyperLTL A new logic: ∀ π ∀ π ′ . F on π ↔ on π ′ Extend LTL by trace quantifiers to express security, privacy, and information flow properties Models are sets of traces! Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 3/8

  5. HyperLTL A new logic: ∀ π ∀ π ′ . F on π ↔ on π ′ Extend LTL by trace quantifiers to express security, privacy, and information flow properties Models are sets of traces! Is there a first-order logic that is expressively equivalent to HyperLTL? Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 3/8

  6. An Example Fix AP = { a } and consider the conjunction ϕ of ∀ π. ( ¬ a π ) U ( a π ∧ X G ¬ a π ) Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 4/8

  7. An Example Fix AP = { a } and consider the conjunction ϕ of ∀ π. ( ¬ a π ) U ( a π ∧ X G ¬ a π ) ∃ π. a π Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 4/8

  8. An Example Fix AP = { a } and consider the conjunction ϕ of ∀ π. ( ¬ a π ) U ( a π ∧ X G ¬ a π ) ∃ π. a π { a } ∅ ∅ ∅ ∅ ∅ ∅ ∅ · · · Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 4/8

  9. An Example Fix AP = { a } and consider the conjunction ϕ of ∀ π. ( ¬ a π ) U ( a π ∧ X G ¬ a π ) ∃ π. a π ∀ π. ∃ π ′ . F ( a π ∧ X a π ′ ) { a } ∅ ∅ ∅ ∅ ∅ ∅ ∅ · · · Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 4/8

  10. An Example Fix AP = { a } and consider the conjunction ϕ of ∀ π. ( ¬ a π ) U ( a π ∧ X G ¬ a π ) ∃ π. a π ∀ π. ∃ π ′ . F ( a π ∧ X a π ′ ) { a } ∅ ∅ ∅ ∅ ∅ ∅ ∅ · · · ∅ { a } ∅ ∅ ∅ ∅ ∅ ∅ · · · Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 4/8

  11. An Example Fix AP = { a } and consider the conjunction ϕ of ∀ π. ( ¬ a π ) U ( a π ∧ X G ¬ a π ) ∃ π. a π ∀ π. ∃ π ′ . F ( a π ∧ X a π ′ ) { a } ∅ ∅ ∅ ∅ ∅ ∅ ∅ · · · ∅ { a } ∅ ∅ ∅ ∅ ∅ ∅ · · · ∅ ∅ { a } ∅ ∅ ∅ ∅ ∅ · · · Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 4/8

  12. An Example Fix AP = { a } and consider the conjunction ϕ of ∀ π. ( ¬ a π ) U ( a π ∧ X G ¬ a π ) ∃ π. a π ∀ π. ∃ π ′ . F ( a π ∧ X a π ′ ) { a } ∅ ∅ ∅ ∅ ∅ ∅ ∅ · · · ∅ { a } ∅ ∅ ∅ ∅ ∅ ∅ · · · ∅ ∅ { a } ∅ ∅ ∅ ∅ ∅ · · · ∅ ∅ ∅ { a } ∅ ∅ ∅ ∅ · · · . . . . . . . . . . . . . . . . . . . . . . . . Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 4/8

  13. An Example Fix AP = { a } and consider the conjunction ϕ of ∀ π. ( ¬ a π ) U ( a π ∧ X G ¬ a π ) ∃ π. a π ∀ π. ∃ π ′ . F ( a π ∧ X a π ′ ) { a } ∅ ∅ ∅ ∅ ∅ ∅ ∅ · · · ∅ { a } ∅ ∅ ∅ ∅ ∅ ∅ · · · ∅ ∅ { a } ∅ ∅ ∅ ∅ ∅ · · · ∅ ∅ ∅ { a } ∅ ∅ ∅ ∅ · · · . . . . . . . . . . . . . . . . . . . . . . . . The unique model of ϕ is {∅ n { a } ∅ ω | n ∈ N } . Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 4/8

  14. First-order Logic for Hyperproperties N · · · < Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 5/8

  15. First-order Logic for Hyperproperties N · · · < · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · A · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 5/8

  16. First-order Logic for Hyperproperties N · · · < · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · E A · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 5/8

  17. First-order Logic for Hyperproperties N · · · < · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · E A · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FO [ <, E ] : first-order logic with equality over the signature { <, E } ∪ { P a | a ∈ AP } over structures with universe A × N . Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 5/8

  18. First-order Logic for Hyperproperties N · · · < · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · E A · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FO [ <, E ] : first-order logic with equality over the signature { <, E } ∪ { P a | a ∈ AP } over structures with universe A × N . Proposition For every HyperLTL sentence there is an equivalent FO [ <, E ] sentence. Martin Zimmermann Saarland University The First-order Logic of Hyperproperties 5/8

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