Soundness and Completeness of Intuitionistic Dialogues Second - PowerPoint PPT Presentation

Soundness and Completeness of Intuitionistic Dialogues Second Bachelor Seminar Talk Dominik Wehr Advisors: Dominik Kirst, Yannick Forster https://www.ps.uni-saarland.de/~wehr/bachelor.php Saarland University 20th March 2019

Model semantics Dialogues Conclusions Recap Model existence T � ˙ M � T ⊥ Markov’s principle Completeness 2 / 22

Model semantics Dialogues Conclusions A constructive proof Definition (Tarski Semantics) Given ρ : N → D , we extend classical I to ρ � I ϕ : P : ρ � I ˙ ⊥ = Q ρ � I P s t = P I s I ,ρ t I ,ρ ρ � I ϕ → ˙ ψ = ρ � I ϕ → ρ � I ψ ρ � I ˙ ∀ x.ϕ = ∀ d : D . ρ [ x �→ d ] � I ϕ A � ϕ := ∀ I ρ. ρ � I A → ρ � I ϕ 3 / 22

Model semantics Dialogues Conclusions A constructive proof Model existence T � ˙ M � T ⊥ Completeness 4 / 22

Model semantics Dialogues Conclusions Kripke models • • • • • • • • • • • K = ( I , W , � , P u ) ∀ u � v. P u ⊆ P v 5 / 22

Model semantics Dialogues Conclusions Universal Kripke model 1 ... Γ , ϕ, τ Γ , ϕ ... Γ , ϕ, σ Γ ... Γ , ψ K = ( I , L ( F ) , ⊆ , λ Γ st. Γ ⊢ P s t ) MP → ρ � Γ ϕ → Γ ⊢ ϕ [ ρ ] 1Herbelin and Lee. “Forcing-based cut-elimination for gentzen-style intuitionistic sequent calculus” 6 / 22

Model semantics Dialogues Conclusions A constructive proof • • • • • • • • • • • K = ( I , W , � , P u , ⊥ u ) 7 / 22

Model semantics Dialogues Conclusions Partial History of Dialogue Semantics Lorenzen describes material dialogues 2 1958 Lorenz formalizes dialogues as games 3 1961 Felscher gives a rigorous completeness proof 4 1985 Sørensen and Urzyczyn give a generic completeness proof 5 2007 2Lorenzen. “Logik und Agon” 3Lorenz. “Arithmetik und Logik als Spiele” 4Felscher. “Dialogues, strategies, and intuitionistic provability” 5Sørensen and Urzyczyn. “Sequent calculus, dialogues, and cut elimination” 8 / 22

Model semantics Dialogues Conclusions Attacks & Defenses ⊳ : F → A → O ( F ) → P D · : A → ( F → P ) D a Attacks ⊥ ⊳ A ⊥ — ϕ → ψ ⊳ A → | � ϕ � { ψ } ϕ ∨ ψ ⊳ A ∨ { ϕ, ψ } ϕ ∧ ψ ⊳ A L { ϕ } ϕ ∧ ψ ⊳ A R { ψ } ∀ ϕ ⊳ A t { ϕ [ t ] } ∃ ϕ ⊳ A ∃ { ϕ [ t ] | t : T } ϕ ⊳ a := ϕ ⊳ a | ∅ 9 / 22

Model semantics Dialogues Conclusions Dialogues ( P ( x ) → Q ( x )) → P ( x ) → P ( x ) ∧ Q ( x ) O: P ( x ) → Q ( x ) “Let’s assume P ( x ) → Q ( x ) .” “Then P ( x ) → P ( x ) ∧ Q ( x ) .” P: P ( x ) → P ( x ) ∧ Q ( x ) “Assuming P ( x ) , P ( x ) ∧ Q ( x ) follows?” O: A → P ( x ) P: P ( x ) ∧ Q ( x ) “Yes.” O: A R “So Q ( x ) holds?” “As P ( x ) → Q ( x ) , Q ( x ) holds?” P: A → P ( x ) O: Q ( x ) “Yes.” P: Q ( x ) “Then Q ( x ) holds.” 10 / 22

Model semantics Dialogues Conclusions Structure of dialogues Two player game O: P ( x ) → Q ( x ) Opponent makes admissions P: P ( x ) → P ( x ) ∧ Q ( x ) Proponent makes claim Players take turns, either O: A → P ( x ) attack or defend P: P ( x ) ∧ Q ( x ) O: A R P: A → P ( x ) O: Q ( x ) P: Q ( x ) 11 / 22

Model semantics Dialogues Conclusions Structure of dialogues O: P ( x ) → Q ( x ) Opponent reacts to previous move P: P ( x ) → P ( x ) ∧ Q ( x ) Proponent may attack any O: A → P ( x ) admission Proponent may defend against P: P ( x ) ∧ Q ( x ) the last attack O: A R Proponent may only admit atomic formulas after the P: A → P ( x ) opponent has done so O: Q ( x ) A dialogue is won if the opponent can’t react P: Q ( x ) 12 / 22

Model semantics Dialogues Conclusions Winning & Validity ( P ( x ) → Q ( x )) → P ( x ) → ⊥ ∧ Q ( x ) O: P ( x ) → Q ( x ) P: P ( x ) → ⊥ ∧ Q ( x ) O: A → P ( x ) P: ⊥ ∧ Q ( x ) O: A R O: A L P: A → P ( x ) P: ⊥ O: Q ( x ) O: A ⊥ P: Q ( x ) 13 / 22

Model semantics Dialogues Conclusions Formalizing Dialogues L ( F ) × A M := PA ( a : A ) | PD ( ϕ : F ) � p : S → M → P � o : S → M → S → P 14 / 22

Model semantics Dialogues Conclusions Proponent moves ϕ ∈ A o ϕ ⊳ a | ψ justified A o ψ Proponent may attack any admission ( A o , c ) � p PA a ϕ ∈ D c justified A o ϕ Proponent may defend against the last attack ( A o , c ) � p PD ϕ Proponent may only ad- justified A o ϕ := ϕ ∈ F a → ϕ ∈ A o mit atomic formulas after the opponent has done so 15 / 22

Model semantics Dialogues Conclusions Opponent moves ϕ ⊳ c ′ | ψ Opponent may attack ( A o , c ) ; PD ϕ � o ( ψ :: A o , c ′ ) preceding defense ϕ ∈ D a Opponent may defend ( A o , c ) ; PA a � o ( ϕ :: A o , c ) against preceding attack ψ ⊳ c ′ | τ ϕ ⊳ a | � ψ � Opponent may counter ( A o , c ) ; PA a � o ( τ :: A o , c ′ ) preceding attack 16 / 22

Model semantics Dialogues Conclusions Winning & Validity ∀ s ′ . s ; m � o s ′ → Win s ′ s � p m Win s Γ � ϕ := ∀ ϕ ⊳ c | ψ. Win ( ψ :: Γ , c ) 17 / 22

Model semantics Dialogues Conclusions Sequent Calculus LJD ⊢ : L ( F ) → ( F → P ) → P ϕ ∈ Γ ϕ ⊳ a | ψ ∀ ψ ⊳ a ′ | τ. Γ , τ ⊢ D a ′ justified Γ ψ ∀ σ ∈ D a . Γ , σ ⊢ ∆ L Γ ⊢ ∆ ϕ ∈ ∆ justified Γ ϕ ∀ ϕ ⊳ a | ψ. Γ , ψ ⊢ D a R Γ ⊢ ∆ 18 / 22

Model semantics Dialogues Conclusions Soundness & Completeness Theorem Γ ⊢ { ϕ } → Γ � ϕ Γ � ϕ → Γ ⊢ { ϕ } Proof. Show ∀ Γ , ∆ . Γ ⊢ ∆ → ∀ c. ∆ ⊆ D c → Win (Γ , c ) . Show ∀ A o , c. Win ( A o , c ) → A o ⊢ D c . 19 / 22

Model semantics Dialogues Conclusions Intuitionistic results ( ∀ , → , ⊥ -fragment) MP D-Dialogues LJT Kripke E. Kripke LJD E-Dialogues ND Formalized Future work 20 / 22

Model semantics Dialogues Conclusions Classical results MP Tarski ND E. Tarski Min. ND Min. Tarski MP? Formalized Future work 21 / 22

Model semantics Dialogues Conclusions References Hugo Herbelin and Gyesik Lee. “Forcing-based cut-elimination for gentzen-style intuitionistic sequent calculus”. In: International Workshop on Logic, Language, Information, and Computation (2009), pp. 209–217. Walter Felscher. “Dialogues, strategies, and intuitionistic provability”. In: Annals of pure and applied logic 28.3 (1985), pp. 217–254. Morten Sørensen and Pavel Urzyczyn. “Sequent calculus, dialogues, and cut elimination”. In: Reflections on Type Theory, λ -Calculus, and the Mind (2007), pp. 253–261. Dominik Wehr. “Soundness and Completeness of Intuitionistic Dialogues”. In: (2019). url : https://www.ps.uni- saarland.de/~wehr/pdf/memo-dialogues.pdf . 22 / 22