Uniform Interpolation and Proof Systems Rosalie Iemhoff Utrecht - - PowerPoint PPT Presentation

Uniform Interpolation and Proof Systems

Rosalie Iemhoff Utrecht University Workshop on Admissible Rules and Unification II Les Diablerets, February 2, 2015

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Proof systems An old question: When does a logic have a decent proof system?

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Proof systems An old question: When does a logic have a decent proof system? Another old question: When does a logic have a sequent calculus?

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Proof systems An old question: When does a logic have a decent proof system? Another old question: When does a logic have a sequent calculus? Answers: Many positive instances. Less negative ones.

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Proof systems An old question: When does a logic have a decent proof system? Another old question: When does a logic have a sequent calculus? Answers: Many positive instances. Less negative ones. Related work:

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Proof systems An old question: When does a logic have a decent proof system? Another old question: When does a logic have a sequent calculus? Answers: Many positive instances. Less negative ones. Related work: (Negri) Fix a labelled sequent calculus and determine which axioms, when added, preserve cut-elimination.

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Proof systems An old question: When does a logic have a decent proof system? Another old question: When does a logic have a sequent calculus? Answers: Many positive instances. Less negative ones. Related work: (Negri) Fix a labelled sequent calculus and determine which axioms, when added, preserve cut-elimination. (Ciabattoni, Galatos, Terui) Fix a sequent calculus and determine which axioms

• r structural rules, when added, preserve cut-elimination.

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Proof systems An old question: When does a logic have a decent proof system? Another old question: When does a logic have a sequent calculus? Answers: Many positive instances. Less negative ones. Related work: (Negri) Fix a labelled sequent calculus and determine which axioms, when added, preserve cut-elimination. (Ciabattoni, Galatos, Terui) Fix a sequent calculus and determine which axioms

• r structural rules, when added, preserve cut-elimination.

Aim: Formulate properties that, when violated by a logic, imply that the logic does not have a sequent calculus of a certain form.

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Uniform interpolation Dfn A logic L has interpolation if whenever ⊢ ϕ → ψ there is a χ in the common language L(ϕ) ∩ L(ψ) such that ⊢ ϕ → χ and ⊢ χ → ψ.

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Uniform interpolation Dfn A logic L has interpolation if whenever ⊢ ϕ → ψ there is a χ in the common language L(ϕ) ∩ L(ψ) such that ⊢ ϕ → χ and ⊢ χ → ψ. Dfn A propositional (modal) logic has uniform interpolation if the interpolant depends only on the premiss or the conclusion: For all ϕ there are formulas ∃pϕ and ∀pϕ not containing p such that for all ψ not containing p: ⊢ ψ → ϕ ⇔ ⊢ ψ → ∀pϕ ⊢ ϕ → ψ ⇔ ⊢ ∃pϕ → ψ.

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Uniform interpolation Dfn A logic L has interpolation if whenever ⊢ ϕ → ψ there is a χ in the common language L(ϕ) ∩ L(ψ) such that ⊢ ϕ → χ and ⊢ χ → ψ. Dfn A propositional (modal) logic has uniform interpolation if the interpolant depends only on the premiss or the conclusion: For all ϕ there are formulas ∃pϕ and ∀pϕ not containing p such that for all ψ not containing p: ⊢ ψ → ϕ ⇔ ⊢ ψ → ∀pϕ ⊢ ϕ → ψ ⇔ ⊢ ∃pϕ → ψ. Algebraic view (next talk).

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Uniform interpolation Dfn A logic L has interpolation if whenever ⊢ ϕ → ψ there is a χ in the common language L(ϕ) ∩ L(ψ) such that ⊢ ϕ → χ and ⊢ χ → ψ. Dfn A propositional (modal) logic has uniform interpolation if the interpolant depends only on the premiss or the conclusion: For all ϕ there are formulas ∃pϕ and ∀pϕ not containing p such that for all ψ not containing p: ⊢ ψ → ϕ ⇔ ⊢ ψ → ∀pϕ ⊢ ϕ → ψ ⇔ ⊢ ∃pϕ → ψ. Algebraic view (next talk). Note A locally tabular logic that has interpolation, has uniform interpolation. ∃pϕ(p, ¯ q) = {ψ(¯ q) | ⊢ ϕ(p, ¯ q) → ψ(¯ q)} ∀pϕ(p, ¯ q) = {ψ(¯ q) | ⊢ ψ(¯ q) → ϕ(p, ¯ q)}

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Modal and intermediate logics Thm (Pitts ’92) IPC has uniform interpolation.

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Modal and intermediate logics Thm (Pitts ’92) IPC has uniform interpolation. Thm (Shavrukov ’94) GL has uniform interpolation.

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Modal and intermediate logics Thm (Pitts ’92) IPC has uniform interpolation. Thm (Shavrukov ’94) GL has uniform interpolation. Thm (Ghilardi & Zawadowski ’95) K has uniform interpolation. S4 does not.

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Modal and intermediate logics Thm (Pitts ’92) IPC has uniform interpolation. Thm (Shavrukov ’94) GL has uniform interpolation. Thm (Ghilardi & Zawadowski ’95) K has uniform interpolation. S4 does not. Thm (Bilkova ’06) KT has uniform interpolation. K4 does not.

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Modal and intermediate logics Thm (Pitts ’92) IPC has uniform interpolation. Thm (Shavrukov ’94) GL has uniform interpolation. Thm (Ghilardi & Zawadowski ’95) K has uniform interpolation. S4 does not. Thm (Bilkova ’06) KT has uniform interpolation. K4 does not. Thm (Maxsimova ’77, Ghilardi & Zawadowski ’02) There are exactly seven intermediate logics with (uniform) interpolation: IPC, Sm, GSc, LC, KC, Bd2, CPC.

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Modal and intermediate logics Thm (Pitts ’92) IPC has uniform interpolation. Thm (Shavrukov ’94) GL has uniform interpolation. Thm (Ghilardi & Zawadowski ’95) K has uniform interpolation. S4 does not. Thm (Bilkova ’06) KT has uniform interpolation. K4 does not. Thm (Maxsimova ’77, Ghilardi & Zawadowski ’02) There are exactly seven intermediate logics with (uniform) interpolation: IPC, Sm, GSc, LC, KC, Bd2, CPC. Pitts uses Dyckhoff’s ’92 sequent calculus for IPC.

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Modularity Aim: If a modal or intermediate logic has such an such a sequent calculus, then it has uniform interpolation.

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Modularity Aim: If a modal or intermediate logic has such an such a sequent calculus, then it has uniform interpolation. Therefore no modal or intermediate logic without uniform interpolation has such an such a calculus.

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Modularity Aim: If a modal or intermediate logic has such an such a sequent calculus, then it has uniform interpolation. Therefore no modal or intermediate logic without uniform interpolation has such an such a calculus. Modularity: The possibility to determine whether the addition of a new rule will preserve uniform interpolation.

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Focussed rules Dfn Multiplication of sequents: (Γ ⇒ ∆) · (Π ⇒ Σ) ≡ (Γ, Π ⇒ ∆, Σ).

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Focussed rules Dfn Multiplication of sequents: (Γ ⇒ ∆) · (Π ⇒ Σ) ≡ (Γ, Π ⇒ ∆, Σ). Dfn A rule is focussed if it is of the form S · S1 . . . S · Sn S · S0 where S, Si are sequents and S0 contains exactly one formula.

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Focussed rules Dfn Multiplication of sequents: (Γ ⇒ ∆) · (Π ⇒ Σ) ≡ (Γ, Π ⇒ ∆, Σ). Dfn A rule is focussed if it is of the form S · S1 . . . S · Sn S · S0 where S, Si are sequents and S0 contains exactly one formula. Ex The following rules are focussed. Γ ⇒ A, ∆ Γ ⇒ B, ∆ Γ ⇒ A ∧ B, ∆

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Focussed rules Dfn Multiplication of sequents: (Γ ⇒ ∆) · (Π ⇒ Σ) ≡ (Γ, Π ⇒ ∆, Σ). Dfn A rule is focussed if it is of the form S · S1 . . . S · Sn S · S0 where S, Si are sequents and S0 contains exactly one formula. Ex The following rules are focussed. Γ ⇒ A, ∆ Γ ⇒ B, ∆ Γ ⇒ A ∧ B, ∆ Γ, B → C ⇒ A → B Γ, C ⇒ D Γ, (A → B) → C ⇒ D

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Focussed rules Dfn Multiplication of sequents: (Γ ⇒ ∆) · (Π ⇒ Σ) ≡ (Γ, Π ⇒ ∆, Σ). Dfn A rule is focussed if it is of the form S · S1 . . . S · Sn S · S0 where S, Si are sequents and S0 contains exactly one formula. Ex The following rules are focussed. Γ ⇒ A, ∆ Γ ⇒ B, ∆ Γ ⇒ A ∧ B, ∆ Γ, B → C ⇒ A → B Γ, C ⇒ D Γ, (A → B) → C ⇒ D Dfn An axiom is focussed if it is of the form Γ, p ⇒ p, ∆ Γ, ⊥ ⇒ ∆ Γ ⇒ ⊤, ∆ . . .

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Propositional logic Dfn A calculus is terminating if there exists a well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . .

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Propositional logic Dfn A calculus is terminating if there exists a well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every terminating calculus that consists of focussed axioms and rules has uniform interpolation.

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Propositional logic Dfn A calculus is terminating if there exists a well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every terminating calculus that consists of focussed axioms and rules has uniform interpolation. Cor Classical propositional logic has uniform interpolation.

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Propositional logic Dfn A calculus is terminating if there exists a well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every terminating calculus that consists of focussed axioms and rules has uniform interpolation. Cor Classical propositional logic has uniform interpolation. Cor Intuitionistic propositional logic has uniform interpolation.

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Propositional logic Dfn A calculus is terminating if there exists a well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every terminating calculus that consists of focussed axioms and rules has uniform interpolation. Cor Classical propositional logic has uniform interpolation. Cor Intuitionistic propositional logic has uniform interpolation. Cor Except the seven intermediate logics that have interpolation, no intermediate logic has a terminating sequent calculus that consists of focussed rules and axioms.

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Propositional logic Dfn A calculus is terminating if there exists an well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . .

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Propositional logic Dfn A calculus is terminating if there exists an well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every logic with a terminating calculus that consists of focussed axioms and rules has uniform interpolation.

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Propositional logic Dfn A calculus is terminating if there exists an well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every logic with a terminating calculus that consists of focussed axioms and rules has uniform interpolation. Proof idea: Define interpolation for rules. For every instance S1 . . . Sn S0 R

• f a rule, define the formula ∀

R

pS0 in terms of ∀pSi (i > 0). For example, ∀

R

pS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn.

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Propositional logic Dfn A calculus is terminating if there exists an well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every logic with a terminating calculus that consists of focussed axioms and rules has uniform interpolation. Proof idea: Define interpolation for rules. For every instance S1 . . . Sn S0 R

• f a rule, define the formula ∀

R

pS0 in terms of ∀pSi (i > 0). For example, ∀

R

pS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn. Then inductively define ∀pS ≡

• {∀

R

pS | R is an instance of a rule with conclusion S}

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Propositional logic Dfn A calculus is terminating if there exists an well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every logic with a terminating calculus that consists of focussed axioms and rules has uniform interpolation. Proof idea: Define interpolation for rules. For every instance S1 . . . Sn S0 R

• f a rule, define the formula ∀

R

pS0 in terms of ∀pSi (i > 0). For example, ∀

R

pS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn. Then inductively define ∀pS ≡

• {∀

R

pS | R is an instance of a rule with conclusion S} For free sequents S define ∀pS separately.

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Propositional logic Dfn A calculus is terminating if there exists an well-founded order on sequents such that in every rule the premisses come befor the conclusion, and . . . Thm Every logic with a terminating calculus that consists of focussed axioms and rules has uniform interpolation. Proof idea: Define interpolation for rules. For every instance S1 . . . Sn S0 R

• f a rule, define the formula ∀

R

pS0 in terms of ∀pSi (i > 0). For example, ∀

R

pS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn. Then inductively define ∀pS ≡

• {∀

R

pS | R is an instance of a rule with conclusion S} For free sequents S define ∀pS separately. Prove with induction on the order that for all sequents S a uniform interpolant ∀pS exists. ⊣

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Propositional logic Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀pϕ = ∀p( ⇒ ϕ).

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Propositional logic Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀pϕ = ∀p( ⇒ ϕ). A logic has uniform interpolation if it satisfies the interpolant properties: (∀l) for all p: ⊢ Sa, ∀pS ⇒ Ss; (∀r) for all p: ⊢ Sl · ( ⇒ ∀pSr) if Sl · Sr is derivable and Sl does not contain p.

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Propositional logic Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀pϕ = ∀p( ⇒ ϕ). A logic has uniform interpolation if it satisfies the interpolant properties: (∀l) for all p: ⊢ Sa, ∀pS ⇒ Ss; (∀r) for all p: ⊢ Sl · ( ⇒ ∀pSr) if Sl · Sr is derivable and Sl does not contain p. From (∀l) obtain ⊢ ∀pϕ → ϕ.

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Propositional logic Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀pϕ = ∀p( ⇒ ϕ). A logic has uniform interpolation if it satisfies the interpolant properties: (∀l) for all p: ⊢ Sa, ∀pS ⇒ Ss; (∀r) for all p: ⊢ Sl · ( ⇒ ∀pSr) if Sl · Sr is derivable and Sl does not contain p. From (∀l) obtain ⊢ ∀pϕ → ϕ. From (∀r) obtain that ⊢ ψ → ϕ implies ⊢ ψ → ∀pϕ, if ψ does not contain p, by taking Sl = (ψ ⇒ ) and Sr = ( ⇒ ϕ). Hence ⊢ ψ → ϕ ⇔ ⊢ ψ → ∀pϕ, if ψ does not contain p.

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Propositional logic Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀pϕ = ∀p( ⇒ ϕ). A logic has uniform interpolation if it satisfies the interpolant properties: (∀l) for all p: ⊢ Sa, ∀pS ⇒ Ss; (∀r) for all p: ⊢ Sl · ( ⇒ ∀pSr) if Sl · Sr is derivable and Sl does not contain p. From (∀l) obtain ⊢ ∀pϕ → ϕ. From (∀r) obtain that ⊢ ψ → ϕ implies ⊢ ψ → ∀pϕ, if ψ does not contain p, by taking Sl = (ψ ⇒ ) and Sr = ( ⇒ ϕ). Hence ⊢ ψ → ϕ ⇔ ⊢ ψ → ∀pϕ, if ψ does not contain p. A logic satisfies the interpolant properties if it satisfies: (1) {Sj · (∀pSj ⇒ ) | 1 ≤ j ≤ n} ⊢ S0 · (∀

R

pS0 ⇒ ). (2) {Sl

j · ( ⇒ ∀pSr j ) | 1 ≤ j ≤ n} ⊢ Sl 0 · ( ⇒ ∀ R

pSr

0).

(3) If Sr

0 is no conclusion of R there exists . . .

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Propositional logic Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀pϕ = ∀p( ⇒ ϕ). A logic has uniform interpolation if it satisfies the interpolant properties: (∀l) for all p: ⊢ Sa, ∀pS ⇒ Ss; (∀r) for all p: ⊢ Sl · ( ⇒ ∀pSr) if Sl · Sr is derivable and Sl does not contain p. From (∀l) obtain ⊢ ∀pϕ → ϕ. From (∀r) obtain that ⊢ ψ → ϕ implies ⊢ ψ → ∀pϕ, if ψ does not contain p, by taking Sl = (ψ ⇒ ) and Sr = ( ⇒ ϕ). Hence ⊢ ψ → ϕ ⇔ ⊢ ψ → ∀pϕ, if ψ does not contain p. A logic satisfies the interpolant properties if it satisfies: (1) {Sj · (∀pSj ⇒ ) | 1 ≤ j ≤ n} ⊢ S0 · (∀

R

pS0 ⇒ ). (2) {Sl

j · ( ⇒ ∀pSr j ) | 1 ≤ j ≤ n} ⊢ Sl 0 · ( ⇒ ∀ R

pSr

0).

(3) If Sr

0 is no conclusion of R there exists . . .

A logic satisfies the above properties if it has a terminating calculus that consists of focussed axioms and rules. ⊣

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Modal logic Dfn A focussed modal rule is of the form ✷S1 · S0 S2 · ✷S1 · ✷S0 where S1 and S2 consist of multisets and S0 of multisets and exactly one atom.

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Modal logic Dfn A focussed modal rule is of the form ✷S1 · S0 S2 · ✷S1 · ✷S0 where S1 and S2 consist of multisets and S0 of multisets and exactly one atom. Ex The following are focussed modal rules. Γ ⇒ p Π, ✷Γ ⇒ ✷p, Σ RK ✷Γ, p ⇒ ✷∆ Π, ✷Γ, ✷p ⇒ ✷∆, Σ p ⇒ ∆ Π, ✷p ⇒ ✷∆, Σ ROK

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Modal logic Dfn A focussed modal rule is of the form ✷S1 · S0 S2 · ✷S1 · ✷S0 where S1 and S2 consist of multisets and S0 of multisets and exactly one atom. Ex The following are focussed modal rules. Γ ⇒ p Π, ✷Γ ⇒ ✷p, Σ RK ✷Γ, p ⇒ ✷∆ Π, ✷Γ, ✷p ⇒ ✷∆, Σ p ⇒ ∆ Π, ✷p ⇒ ✷∆, Σ ROK Cor A modal logic with a balanced terminating calculus that consists of focussed axioms and focussed (modal) rules and contains RK or ROK, has uniform interpolation.

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Modal logic Dfn A focussed modal rule is of the form ✷S1 · S0 S2 · ✷S1 · ✷S0 where S1 and S2 consist of multisets and S0 of multisets and exactly one atom. Ex The following are focussed modal rules. Γ ⇒ p Π, ✷Γ ⇒ ✷p, Σ RK ✷Γ, p ⇒ ✷∆ Π, ✷Γ, ✷p ⇒ ✷∆, Σ p ⇒ ∆ Π, ✷p ⇒ ✷∆, Σ ROK Cor A modal logic with a balanced terminating calculus that consists of focussed axioms and focussed (modal) rules and contains RK or ROK, has uniform interpolation. Cor Any normal modal logic with a balanced terminating calculus that consists

• f focussed (modal) axioms and rules, has uniform interpolation. (Ex: K)

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Modal logic Dfn A focussed modal rule is of the form ✷S1 · S0 S2 · ✷S1 · ✷S0 where S1 and S2 consist of multisets and S0 of multisets and exactly one atom. Ex The following are focussed modal rules. Γ ⇒ p Π, ✷Γ ⇒ ✷p, Σ RK ✷Γ, p ⇒ ✷∆ Π, ✷Γ, ✷p ⇒ ✷∆, Σ p ⇒ ∆ Π, ✷p ⇒ ✷∆, Σ ROK Cor A modal logic with a balanced terminating calculus that consists of focussed axioms and focussed (modal) rules and contains RK or ROK, has uniform interpolation. Cor Any normal modal logic with a balanced terminating calculus that consists

• f focussed (modal) axioms and rules, has uniform interpolation. (Ex: K)

Cor The logics K4 and S4 do not have balanced terminating sequent calculi that consist of focussed (modal) axioms and rules.

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Questions

• Extend the method to other modal logics, such as GL and KT.
• Extend the method to hypersequents.
• Use other proof systems than sequent calculi.

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Finis

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