Download Uniform Interpolation and Proof Systems

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
or 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
or 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, q̄) =
V
{ψ(q̄) | ` ϕ(p, q̄) → ψ(q̄)}
∀pϕ(p, q̄) =
W
{ψ(q̄) | ` ψ(q̄) → ϕ(p, q̄)}
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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
...
S0
Sn
R
R
of a rule, define the formula ∀p S0 in terms of ∀p Si (i > 0). For example,
R
∀p S0 ≡ ∀p S1 ∧ . . . ∧ ∀p Sn .
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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
...
S0
Sn
R
R
of a rule, define the formula ∀p S0 in terms of ∀p Si (i > 0). For example,
R
∀p S0 ≡ ∀p S1 ∧ . . . ∧ ∀p Sn .
Then inductively define
_ R
∀p S ≡ {∀p S | 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
...
S0
Sn
R
R
of a rule, define the formula ∀p S0 in terms of ∀p Si (i > 0). For example,
R
∀p S0 ≡ ∀p S1 ∧ . . . ∧ ∀p Sn .
Then inductively define
_ R
∀p S ≡ {∀p S | R is an instance of a rule with conclusion S}
For free sequents S define ∀p S 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
...
S0
Sn
R
R
of a rule, define the formula ∀p S0 in terms of ∀p Si (i > 0). For example,
R
∀p S0 ≡ ∀p S1 ∧ . . . ∧ ∀p Sn .
Then inductively define
_ R
∀p S ≡ {∀p S | R is an instance of a rule with conclusion S}
For free sequents S define ∀p S separately.
Prove with induction on the order that for all sequents S a uniform interpolant
∀p S exists.
a
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Propositional logic
Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀p ϕ = ∀p ( ⇒ ϕ).
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Propositional logic
Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀p ϕ = ∀p ( ⇒ ϕ).
A logic has uniform interpolation if it satisfies the interpolant properties:
(∀l) for all p: ` S a , ∀p S ⇒ S s ;
(∀r) for all p: ` S l · ( ⇒ ∀p S r ) if S l · S r is derivable and S l does not contain p.
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Propositional logic
Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀p ϕ = ∀p ( ⇒ ϕ).
A logic has uniform interpolation if it satisfies the interpolant properties:
(∀l) for all p: ` S a , ∀p S ⇒ S s ;
(∀r) for all p: ` S l · ( ⇒ ∀p S r ) if S l · S r is derivable and S l does not contain p.
From (∀l) obtain ` ∀p ϕ → ϕ.
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Propositional logic
Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀p ϕ = ∀p ( ⇒ ϕ).
A logic has uniform interpolation if it satisfies the interpolant properties:
(∀l) for all p: ` S a , ∀p S ⇒ S s ;
(∀r) for all p: ` S l · ( ⇒ ∀p S r ) if S l · S r is derivable and S l does not contain p.
From (∀l) obtain ` ∀p ϕ → ϕ.
From (∀r) obtain that ` ψ → ϕ implies ` ψ → ∀p ϕ, if ψ does not contain p,
by taking S l = (ψ ⇒ ) and S r = ( ⇒ ϕ). Hence ` ψ → ϕ ⇔ ` ψ → ∀p ϕ, if ψ
does not contain p.
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Propositional logic
Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀p ϕ = ∀p ( ⇒ ϕ).
A logic has uniform interpolation if it satisfies the interpolant properties:
(∀l) for all p: ` S a , ∀p S ⇒ S s ;
(∀r) for all p: ` S l · ( ⇒ ∀p S r ) if S l · S r is derivable and S l does not contain p.
From (∀l) obtain ` ∀p ϕ → ϕ.
From (∀r) obtain that ` ψ → ϕ implies ` ψ → ∀p ϕ, if ψ does not contain p,
by taking S l = (ψ ⇒ ) and S r = ( ⇒ ϕ). Hence ` ψ → ϕ ⇔ ` ψ → ∀p ϕ, if ψ
does not contain p.
A logic satisfies the interpolant properties if it satisfies:
R
(1) {Sj · (∀p Sj ⇒ ) | 1 ≤ j ≤ n} ` S0 · (∀p S0 ⇒ ).
R
(2) {Sjl · ( ⇒ ∀p Sjr ) | 1 ≤ j ≤ n} ` S0l · ( ⇒ ∀p S0r ).
(3) If S0r is no conclusion of R there exists . . .
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Propositional logic
Dfn (Γ ⇒ ∆)a = Γ and (Γ ⇒ ∆)s = ∆ and ∀p ϕ = ∀p ( ⇒ ϕ).
A logic has uniform interpolation if it satisfies the interpolant properties:
(∀l) for all p: ` S a , ∀p S ⇒ S s ;
(∀r) for all p: ` S l · ( ⇒ ∀p S r ) if S l · S r is derivable and S l does not contain p.
From (∀l) obtain ` ∀p ϕ → ϕ.
From (∀r) obtain that ` ψ → ϕ implies ` ψ → ∀p ϕ, if ψ does not contain p,
by taking S l = (ψ ⇒ ) and S r = ( ⇒ ϕ). Hence ` ψ → ϕ ⇔ ` ψ → ∀p ϕ, if ψ
does not contain p.
A logic satisfies the interpolant properties if it satisfies:
R
(1) {Sj · (∀p Sj ⇒ ) | 1 ≤ j ≤ n} ` S0 · (∀p S0 ⇒ ).
R
(2) {Sjl · ( ⇒ ∀p Sjr ) | 1 ≤ j ≤ n} ` S0l · ( ⇒ ∀p S0r ).
(3) If S0r 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.
a
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Modal logic
Dfn A focussed modal rule is of the form
2S1 · S0
S2 · 2S1 · 2S0
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
2S1 · S0
S2 · 2S1 · 2S0
where S1 and S2 consist of multisets and S0 of multisets and exactly one atom.
Ex The following are focussed modal rules.
Γ⇒p
RK
Π, 2Γ ⇒ 2p, Σ
2Γ, p ⇒ 2∆
Π, 2Γ, 2p ⇒ 2∆, Σ
p⇒∆
ROK
Π, 2p ⇒ 2∆, Σ
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Modal logic
Dfn A focussed modal rule is of the form
2S1 · S0
S2 · 2S1 · 2S0
where S1 and S2 consist of multisets and S0 of multisets and exactly one atom.
Ex The following are focussed modal rules.
Γ⇒p
RK
Π, 2Γ ⇒ 2p, Σ
2Γ, p ⇒ 2∆
Π, 2Γ, 2p ⇒ 2∆, Σ
p⇒∆
ROK
Π, 2p ⇒ 2∆, Σ
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
2S1 · S0
S2 · 2S1 · 2S0
where S1 and S2 consist of multisets and S0 of multisets and exactly one atom.
Ex The following are focussed modal rules.
Γ⇒p
RK
Π, 2Γ ⇒ 2p, Σ
2Γ, p ⇒ 2∆
Π, 2Γ, 2p ⇒ 2∆, Σ
p⇒∆
ROK
Π, 2p ⇒ 2∆, Σ
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
of 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
2S1 · S0
S2 · 2S1 · 2S0
where S1 and S2 consist of multisets and S0 of multisets and exactly one atom.
Ex The following are focussed modal rules.
Γ⇒p
RK
Π, 2Γ ⇒ 2p, Σ
2Γ, p ⇒ 2∆
Π, 2Γ, 2p ⇒ 2∆, Σ
p⇒∆
ROK
Π, 2p ⇒ 2∆, Σ
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
of focussed (modal) axioms and rules, has uniform interpolation. (Ex: K )
Cor The logics K 4 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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