Focused emulation of modal proof systems Sonia Marin with Dale - PowerPoint PPT Presentation

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 intelligence, distributed systems . . .

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

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 . . .

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

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 ∗

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

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

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

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

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.

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 .

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 , ∆

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

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

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

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]

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]

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]

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 .

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

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)

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)

Focused emulation of modal proof systems Sonia Marin with Dale - PowerPoint PPT Presentation

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

Proof systems for modal logics Emil Je r abek jerabek@math.cas.cz Institute of Mathematics

A Formalization of a Henkin-style Completeness Proof for Propositional Modal Logic in Lean Bruno

Getting to the Box of it: proof-theory for meta-programming and modal type theory Murdoch J.

Practical Proof Systems for SAT and QBF Marijn J.H. Heule Dagstuhl Seminar on SAT and

Decomposing labelled proof theory for intuitionistic modal logic Sonia Marin , Marianela Morales,

Quantitative Quantitative Quantitative Quantitative Modal Modal Transition Transition

W HAT IS EHD? Introduction EHD without cross-flow Modal Dielectric fluid Non-modal EHD with

VILLAS concept Virtually Interconnected Laboratories for LArge systems Simulation/emulation

Intuitionistic and Modal Logic Dick de Jongh Institute for Logic, Language and Computation

MAPS UMTS for IuCS, IuH Interfaces Emulator (IuCS Emulation over IP and ATM; and IuH Emulation

KeYmaera Improving the Proof Experience Corwin de Boor Cyber-Physical Systems

Emulation in ns Presented by Alefiya Hussain What is Emulation Ability to introduce the

Just some space Axiomatizing a Reflexive Real-Valued Modal Logic Laura Janina Schn uriger

Proof Systems that Take Advice Olaf Beyersdorff Institut fr Theoretische Informatik

Modal logic for concurrent processes Lu s Soares Barbosa Interaction & Concurrency

Uniform Interpolation and Proof Systems Rosalie Iemhoff Utrecht University Workshop on

Proofs, disproofs, and their duals Heinrich Wansing Advances in Modal Logic 2010 1 / 49 Syntax

Polluted Resolution and other Combined Proof Search Methods for Propositional Modal Logics

Proof Systems and Proof Complexity Marijn J.H. Heule http://www.cs.cmu.edu/~mheule/15816-f19/

Label-free Modular Systems for Classical and Intuitionistic Modal Logics Sonia Marin Lutz

Multiple outputs operational modal identification of time-varying systems Mathieu BERTHA

A cut-free proof system for pseudo-transitive modal logics Sonia Marin With Lutz Straburger

Emulation Outline Emulation Interpretation basic, threaded, directed threaded

Modal dal Logic ic Submitted to Prof . Lubomr Popelnsk, Masaryk University Prepared by

Focused emulation of modal proof systems Sonia Marin with Dale - PowerPoint PPT Presentation

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

Proof systems for modal logics Emil Je r abek jerabek@math.cas.cz Institute of Mathematics

A Formalization of a Henkin-style Completeness Proof for Propositional Modal Logic in Lean Bruno

Getting to the Box of it: proof-theory for meta-programming and modal type theory Murdoch J.

Practical Proof Systems for SAT and QBF Marijn J.H. Heule Dagstuhl Seminar on SAT and

Decomposing labelled proof theory for intuitionistic modal logic Sonia Marin , Marianela Morales,

Quantitative Quantitative Quantitative Quantitative Modal Modal Transition Transition

W HAT IS EHD? Introduction EHD without cross-flow Modal Dielectric fluid Non-modal EHD with

VILLAS concept Virtually Interconnected Laboratories for LArge systems Simulation/emulation

Intuitionistic and Modal Logic Dick de Jongh Institute for Logic, Language and Computation

MAPS UMTS for IuCS, IuH Interfaces Emulator (IuCS Emulation over IP and ATM; and IuH Emulation

KeYmaera Improving the Proof Experience Corwin de Boor Cyber-Physical Systems

Emulation in ns Presented by Alefiya Hussain What is Emulation Ability to introduce the

Just some space Axiomatizing a Reflexive Real-Valued Modal Logic Laura Janina Schn uriger

Proof Systems that Take Advice Olaf Beyersdorff Institut fr Theoretische Informatik

Modal logic for concurrent processes Lu s Soares Barbosa Interaction &amp; Concurrency

Uniform Interpolation and Proof Systems Rosalie Iemhoff Utrecht University Workshop on

Proofs, disproofs, and their duals Heinrich Wansing Advances in Modal Logic 2010 1 / 49 Syntax

Polluted Resolution and other Combined Proof Search Methods for Propositional Modal Logics

Proof Systems and Proof Complexity Marijn J.H. Heule http://www.cs.cmu.edu/~mheule/15816-f19/

Label-free Modular Systems for Classical and Intuitionistic Modal Logics Sonia Marin Lutz

Multiple outputs operational modal identification of time-varying systems Mathieu BERTHA

A cut-free proof system for pseudo-transitive modal logics Sonia Marin With Lutz Straburger

Emulation Outline Emulation Interpretation basic, threaded, directed threaded

Modal dal Logic ic Submitted to Prof . Lubomr Popelnsk, Masaryk University Prepared by

Modal logic for concurrent processes Lu s Soares Barbosa Interaction & Concurrency