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

T ˙ ⊥ M T Model existence Completeness Markov’s principle

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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 sI,ρ tI,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

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Model semantics Dialogues Conclusions

A constructive proof

T ˙ ⊥ M T Model existence Completeness

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Model semantics Dialogues Conclusions

Kripke models

• K = (I, W, , Pu)

∀ u v. Pu ⊆ Pv

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Model semantics Dialogues Conclusions

Universal Kripke model1

Γ Γ, ϕ Γ, ϕ, τ ... Γ, ϕ, σ ... Γ, ψ ... K = (I, L(F), ⊆, λ Γst. 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, , Pu, ⊥u)

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Model semantics Dialogues Conclusions

Partial History of Dialogue Semantics

1958 Lorenzen describes material dialogues 2 1961 Lorenz formalizes dialogues as games 3 1985 Felscher gives a rigorous completeness proof 4 2007 Sørensen and Urzyczyn give a generic completeness proof 5

• 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) Attacks Da ⊥ ⊳ A⊥ — ϕ → ψ ⊳ A→ | ϕ {ψ} ϕ ∨ ψ ⊳ A∨ {ϕ, ψ} ϕ ∧ ψ ⊳ AL {ϕ} ϕ ∧ ψ ⊳ AR {ψ} ∀ϕ ⊳ At {ϕ[t]} ∃ϕ ⊳ A∃ {ϕ[t] | t : T} ϕ ⊳ a := ϕ ⊳ a | ∅

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Model semantics Dialogues Conclusions

Dialogues

(P(x) → Q(x)) → P(x) → P(x) ∧ Q(x)

“Let’s assume P(x) → Q(x).”

O: 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)

“Yes.”

P: P(x) ∧ Q(x) O: AR

“So Q(x) holds?” “As P(x) → Q(x), Q(x) holds?”

P: A→ P(x)

“Yes.”

O: Q(x)

“Then Q(x) holds.”

P: Q(x)

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Model semantics Dialogues Conclusions

Structure of dialogues

Two player game Opponent makes admissions Proponent makes claim Players take turns, either attack or defend O: P(x) → Q(x) P: P(x) → P(x) ∧ Q(x) O: A→ P(x) O: AR P: A→ P(x) P: P(x) ∧ Q(x) P: Q(x) O: Q(x)

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Model semantics Dialogues Conclusions

Structure of dialogues

Opponent reacts to previous move Proponent may attack any admission Proponent may defend against the last attack Proponent may only admit atomic formulas after the

• pponent has done so

A dialogue is won if the

• pponent can’t react

O: P(x) → Q(x) P: P(x) → P(x) ∧ Q(x) O: A→ P(x) O: AR P: A→ P(x) P: P(x) ∧ Q(x) P: Q(x) O: Q(x)

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Model semantics Dialogues Conclusions

Winning & Validity

O: P(x) → Q(x) O: A→ P(x) (P(x) → Q(x)) → P(x) → ⊥ ∧ Q(x) P: P(x) → ⊥ ∧ Q(x) P: ⊥ ∧ Q(x) O: AR P: A→ P(x) O: Q(x) P: Q(x) O: AL P: ⊥ O: A⊥

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Model semantics Dialogues Conclusions

Formalizing Dialogues

L(F) × A M := PA (a : A) | PD (ϕ : F) p : S → M → P

• : S → M → S → P

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Model semantics Dialogues Conclusions

Proponent moves

Proponent may attack any admission Proponent may defend against the last attack Proponent may only ad- mit atomic formulas after the opponent has done so ϕ ∈ Ao ϕ ⊳ a | ψ justified Ao ψ (Ao, c) p PA a ϕ ∈ Dc justified Ao ϕ (Ao, c) p PD ϕ justified Ao ϕ := ϕ ∈ Fa → ϕ ∈ Ao

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Model semantics Dialogues Conclusions

Opponent moves

Opponent may attack preceding defense Opponent may defend against preceding attack Opponent may counter preceding attack ϕ ⊳ c′ | ψ (Ao, c) ; PD ϕ o (ψ :: Ao, c′) ϕ ∈ Da (Ao, c) ; PA a o (ϕ :: Ao, c) ϕ ⊳ a | ψ ψ ⊳ c′ | τ (Ao, c) ; PA a o (τ :: Ao, c′)

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Model semantics Dialogues Conclusions

Winning & Validity

s p m ∀s′. s ; m o s′ → Win s′ Win s Γ ϕ := ∀ ϕ ⊳ c | ψ. Win (ψ :: Γ, c)

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Model semantics Dialogues Conclusions

Sequent Calculus LJD

⊢: L(F) → (F → P) → P

L

ϕ ∈ Γ ϕ ⊳ a | ψ justified Γ ψ ∀σ ∈ Da. Γ, σ ⊢ ∆ ∀ψ ⊳ a′ | τ. Γ, τ ⊢ Da′ Γ ⊢ ∆

R

ϕ ∈ ∆ justified Γ ϕ ∀ϕ ⊳ a | ψ. Γ, ψ ⊢ Da Γ ⊢ ∆

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Model semantics Dialogues Conclusions

Soundness & Completeness

Theorem

Γ ⊢ {ϕ} → Γ ϕ Γ ϕ → Γ ⊢ {ϕ}

Proof.

Show ∀ Γ, ∆. Γ ⊢ ∆ → ∀c. ∆ ⊆ Dc → Win (Γ, c). Show ∀Ao, c. Win (Ao, c) → Ao ⊢ Dc.

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Model semantics Dialogues Conclusions

Intuitionistic results (∀,→,⊥-fragment)

Kripke

• E. Kripke

LJT ND LJD D-Dialogues E-Dialogues MP Formalized Future work

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Model semantics Dialogues Conclusions

Classical results

ND Tarski

• E. Tarski
• Min. ND
• Min. Tarski

MP MP? Formalized Future work

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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.

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