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Focused emulation of modal proof systems Sonia Marin with Dale Miller and Marco Volpe Inria, LIX, Ecole Polytechnique Theory and Logic Group seminar TU Wien October 12, 2016 The quest Modal logics: program verification, artificial

  1. Focused emulation of modal proof systems Sonia Marin with Dale Miller and Marco Volpe Inria, LIX, ´ Ecole Polytechnique Theory and Logic Group seminar TU Wien October 12, 2016

  2. The quest Modal logics: program verification, artificial intelligence, distributed systems . . .

  3. The quest Modal logics: program verification, artificial intelligence, distributed systems . . . What if... one wants to have automated proof search for modal logics?

  4. The quest Modal logics: program verification, artificial intelligence, distributed systems . . . What if... one wants to have automated proof search for modal logics? Their proof theory: tableaux, sequents, hypersequents, nested sequents, labeled sequents . . .

  5. The quest We want to provide a general framework for: 1. comparing formalisms; 2. proof checking; 3. proof reconstruction and sharing.

  6. The quest We want to provide a general framework for: 1. comparing formalisms; 2. proof checking; 3. proof reconstruction and sharing. The ProofCert approach: proof in LKF a proof in S proof in LMF ∗

  7. The quest We want to provide a general framework for: 1. comparing formalisms; 2. proof checking; 3. proof reconstruction and sharing. The ProofCert approach: proof in LKF a proof in S proof in LMF ∗ ◮ LMF ∗ : focused labeled framework for propositional modal logic ◮ LKF a : focused framework for classical first-order logic

  8. Modal logic Formulas: A ::= P | A ∧ A | A ∨ A Logic K : Propositional Logic

  9. Modal logic Formulas: A ::= P | A ∧ A | A ∨ A | ✷ A | ✸ A A Logic K : Propositional Logic + ✷ ( A → B ) → ( ✷ A → ✷ B ) + nec − − − ✷ A

  10. Modal logic Formulas: A ::= P | A ∧ A | A ∨ A | ✷ A | ✸ A A Logic K : Propositional Logic + ✷ ( A → B ) → ( ✷ A → ✷ B ) + nec − − − ✷ A Kripke semantics: Relational structures

  11. Modal logic Formulas: A ::= P | A ∧ A | A ∨ A | ✷ A | ✸ A A Logic K : Propositional Logic + ✷ ( A → B ) → ( ✷ A → ✷ B ) + nec − − − ✷ A Kripke semantics: Relational structures W : set of worlds; R : binary relation on W ; V : valuation at each world.

  12. Modal logic Formulas: A ::= P | A ∧ A | A ∨ A | ✷ A | ✸ A A Logic K : Propositional Logic + ✷ ( A → B ) → ( ✷ A → ✷ B ) + nec − − − ✷ A Kripke semantics: Relational structures W : set of worlds; R : binary relation on W ; V : valuation at each world. M , x | = ✷ A iff for all y . xRy implies M , y | = A M , x | = ✸ A iff there exists y . xRy and M , y | = A .

  13. Modal logic Formulas: A ::= P | A ∧ A | A ∨ A | ✷ A | ✸ A A Logic K : Propositional Logic + ✷ ( A → B ) → ( ✷ A → ✷ B ) + nec − − − ✷ A Kripke semantics: Relational structures W : set of worlds; R : binary relation on W ; V : valuation at each world. Sequent system OS : ⊢ Γ , A ⊢ Γ , B ⊢ Γ , A , B ⊢ Γ , A ∧ − ∨ − id − ✷ K − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − ⊢ Γ , P , ¬ P ⊢ Γ , A ∧ B ⊢ Γ , A ∨ B ⊢ ✸ Γ , ✷ A , ∆

  14. Labeled proof systems Labeled deduction: encode semantical information in the syntax

  15. Labeled proof systems Labeled deduction: encode semantical information in the syntax Two classes of formulas : 1. Labeled logical formulas x : A 2. Relational formulas xRy

  16. Labeled proof systems Labeled deduction: encode semantical information in the syntax Two classes of formulas : 1. Labeled logical formulas x : A 2. Relational formulas xRy ◮ each label x refers to a world in the semantics ◮ an atomic relational symbol R refers to the accessibility relation

  17. A labeled proof system for modal logics (G3K) id − − − − − − − − − − − − − − − − − − − − P , Γ ⊢ ∆ , P B , Γ ⊢ ∆ Γ ⊢ ∆ , Γ ⊢ ∆ , A , A B L ∧ − R ∧ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − A ∧ B , Γ ⊢ ∆ Γ ⊢ ∆ , A ∧ B A , Γ ⊢ ∆ B , Γ ⊢ ∆ Γ ⊢ ∆ , A , B L ∨ − R ∨ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − A ∨ B , Γ ⊢ ∆ Γ ⊢ ∆ , A ∨ B [S. Negri, Proof analysis in modal logic , J. Philos. Logic 2005]

  18. A labeled proof system for modal logics (G3K) id − − − − − − − − − − − − − − − − − − − − x : P , Γ ⊢ ∆ , x : P x : A , x : B , Γ ⊢ ∆ Γ ⊢ ∆ , x : A Γ ⊢ ∆ , x : B L ∧ − R ∧ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − x : A ∧ B , Γ ⊢ ∆ Γ ⊢ ∆ , x : A ∧ B x : A , Γ ⊢ ∆ x : B , Γ ⊢ ∆ Γ ⊢ ∆ , x : A , x : B L ∨ − R ∨ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − x : A ∨ B , Γ ⊢ ∆ Γ ⊢ ∆ , x : A ∨ B [S. Negri, Proof analysis in modal logic , J. Philos. Logic 2005]

  19. A labeled proof system for modal logics (G3K) id − − − − − − − − − − − − − − − − − − − − x : P , Γ ⊢ ∆ , x : P x : A , x : B , Γ ⊢ ∆ Γ ⊢ ∆ , x : A Γ ⊢ ∆ , x : B L ∧ − R ∧ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − x : A ∧ B , Γ ⊢ ∆ Γ ⊢ ∆ , x : A ∧ B x : A , Γ ⊢ ∆ x : B , Γ ⊢ ∆ Γ ⊢ ∆ , x : A , x : B L ∨ − R ∨ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − x : A ∨ B , Γ ⊢ ∆ Γ ⊢ ∆ , x : A ∨ B y : A , x : ✷ A , xRy , Γ ⊢ ∆ xRy , Γ ⊢ ∆ , y : A L ✷ − R ✷ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − x : ✷ A , xRy , Γ ⊢ ∆ Γ ⊢ ∆ , x : ✷ A xRy , y : A , Γ ⊢ ∆ xRy , Γ ⊢ ∆ , x : ✸ A , y : A L ✸ − R ✸ − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − x : ✸ A , Γ ⊢ ∆ xRy , Γ ⊢ ∆ , x : ✸ A In R ✷ , y does not occur in the conclusion. [S. Negri, Proof analysis in modal logic , J. Philos. Logic 2005]

  20. Hocus-Focus Focusing is a way to control non-determinism in proof search and ... ◮ Better organize the structure of derivations. ◮ Emphasis on: non-invertible vs. invertible rules. ◮ Propositional connectives have: ◮ a positive version; ◮ a negative version. ⊢ Θ , B i ⊢ Θ , B 1 , B 2 + − − − ∨ ∨ − − − − − − − − − − − − − − − − − − − − − − − − − − ⊢ Θ , B 1 ∨ B 2 ⊢ Θ , B 1 ∨ B 2 ◮ Polarization of a formula does not affect its provability .

  21. What is a bipole? store ⊢ Θ ⇑ Γ release ⊢ Θ ⇓ A decide

  22. What is a bipole? store (a positive formula to possibly focus on later) − , ∧ − , ∀ ⊢ Θ ⇑ Γ ∨ release + , ∧ + , ∃ ⊢ Θ ⇓ A ∨ decide (on a positive formula to focus on)

  23. What is a bipole? store (a positive formula to possibly focus on later) ⊢ Θ ⇑ Γ NEGATIVE PHASE (invertible) release (change of phase) ⊢ Θ ⇓ A POSITIVE PHASE (non-invertible) decide (on a positive formula to focus on)

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