Rosalie Iemho Utrecht University Workshop on Admissible ... · (Negri)Fix a labelled sequent...

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?

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, whenadded, preserve cut-elimination.

(Ciabattoni, Galatos, Terui) Fix a sequent calculus and determine which axiomsor structural rules, when added, preserve cut-elimination.

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

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Proof systems

Related work:

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Proof systems

Related work:

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Proof systems

Related work:

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Proof systems

Related work:

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Proof systems

Related work:

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Proof systems

Related work:

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Uniform interpolation

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

Dfn A propositional (modal) logic has uniform interpolation if the interpolantdepends 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) =∧{ψ(q̄) | ` ϕ(p, q̄)→ ψ(q̄)}

∀pϕ(p, q̄) =∨{ψ(q̄) | ` ψ(q̄)→ ϕ(p, q̄)}

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` ψ → ϕ ⇔ ` ψ → ∀pϕ ` ϕ→ ψ ⇔ ` ∃pϕ→ ψ.

∃pϕ(p, q̄) =∧{ψ(q̄) | ` ϕ(p, q̄)→ ψ(q̄)}

∀pϕ(p, q̄) =∨{ψ(q̄) | ` ψ(q̄)→ ϕ(p, q̄)}

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` ψ → ϕ ⇔ ` ψ → ∀pϕ ` ϕ→ ψ ⇔ ` ∃pϕ→ ψ.

∃pϕ(p, q̄) =∧{ψ(q̄) | ` ϕ(p, q̄)→ ψ(q̄)}

∀pϕ(p, q̄) =∨{ψ(q̄) | ` ψ(q̄)→ ϕ(p, q̄)}

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` ψ → ϕ ⇔ ` ψ → ∀pϕ ` ϕ→ ψ ⇔ ` ∃pϕ→ ψ.

∃pϕ(p, q̄) =∧{ψ(q̄) | ` ϕ(p, q̄)→ ψ(q̄)}

∀pϕ(p, q̄) =∨{ψ(q̄) | ` ψ(q̄)→ ϕ(p, q̄)}

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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, thenit has uniform interpolation.

Therefore no modal or intermediate logic without uniform interpolation hassuch an such a calculus.

Modularity: The possibility to determine whether the addition of a new rulewill preserve uniform interpolation.

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Modularity

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Modularity

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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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Focussed rules

(Γ⇒ ∆) · (Π⇒ Σ) ≡ (Γ,Π⇒ ∆,Σ).

S · S1 . . . S · Sn

S · S0

Γ⇒ A,∆ Γ⇒ B,∆

Γ⇒ A ∧ B,∆

Γ,B → C ⇒ A→ B Γ,C ⇒ D

Γ, (A→ B)→ C ⇒ D

Γ, p ⇒ p,∆ Γ,⊥ ⇒ ∆ Γ⇒ >,∆ . . .

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Focussed rules

(Γ⇒ ∆) · (Π⇒ Σ) ≡ (Γ,Π⇒ ∆,Σ).

S · S1 . . . S · Sn

S · S0

Γ⇒ A,∆ Γ⇒ B,∆

Γ⇒ A ∧ B,∆

Γ,B → C ⇒ A→ B Γ,C ⇒ D

Γ, (A→ B)→ C ⇒ D

Γ, p ⇒ p,∆ Γ,⊥ ⇒ ∆ Γ⇒ >,∆ . . .

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Focussed rules

(Γ⇒ ∆) · (Π⇒ Σ) ≡ (Γ,Π⇒ ∆,Σ).

S · S1 . . . S · Sn

S · S0

Γ⇒ A,∆ Γ⇒ B,∆

Γ⇒ A ∧ B,∆

Γ,B → C ⇒ A→ B Γ,C ⇒ D

Γ, (A→ B)→ C ⇒ D

Γ, p ⇒ p,∆ Γ,⊥ ⇒ ∆ Γ⇒ >,∆ . . .

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Focussed rules

(Γ⇒ ∆) · (Π⇒ Σ) ≡ (Γ,Π⇒ ∆,Σ).

S · S1 . . . S · Sn

S · S0

Γ⇒ A,∆ Γ⇒ B,∆

Γ⇒ A ∧ B,∆

Γ,B → C ⇒ A→ B Γ,C ⇒ D

Γ, (A→ B)→ C ⇒ D

Γ, p ⇒ p,∆ Γ,⊥ ⇒ ∆ Γ⇒ >,∆ . . .

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Propositional logic

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

Thm Every terminating calculus that consists of focussed axioms and rules hasuniform interpolation.

Cor Classical propositional logic has uniform interpolation.

Cor Intuitionistic propositional logic has uniform interpolation.

Cor Except the seven intermediate logics that have interpolation, nointermediate logic has a terminating sequent calculus that consists of focussedrules and axioms.

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Propositional logic

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Propositional logic

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Propositional logic

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Propositional logic

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Propositional logic

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

Thm Every logic with a terminating calculus that consists of focussed axiomsand rules has uniform interpolation.

Proof idea:Define interpolation for rules. For every instance

S1 . . . Sn

of a rule, define the formula ∀RpS0 in terms of ∀pSi (i > 0). For example,

∀RpS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn.

Then inductively define

∀pS ≡∨{∀RpS | 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. a

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Propositional logic

S1 . . . Sn

∀RpS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn.

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Propositional logic

S1 . . . Sn

∀RpS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn.

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Propositional logic

S1 . . . Sn

∀RpS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn.

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Propositional logic

S1 . . . Sn

∀RpS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn.

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Propositional logic

S1 . . . Sn

∀RpS0 ≡ ∀pS1 ∧ . . . ∧ ∀pSn.

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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 ⇒ S s ;

(∀r) for all p: ` S l · (⇒ ∀pS r ) if S l · S r is derivable and S l 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 S l = (ψ ⇒ ) and S r = (⇒ ϕ). Hence ` ψ → ϕ ⇔ ` ψ → ∀pϕ, if ψdoes not contain p.

A logic satisfies the interpolant properties if it satisfies:

(1) {Sj · (∀pSj ⇒ ) | 1 ≤ j ≤ n} ` S0 · (∀RpS0 ⇒ ).

(2) {S lj · (⇒ ∀pS r

j ) | 1 ≤ j ≤ n} ` S l0 · (⇒ ∀

(3) If S r0 is no conclusion of R there exists . . .

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

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Propositional logic

(1) {Sj · (∀pSj ⇒ ) | 1 ≤ j ≤ n} ` S0 · (∀RpS0 ⇒ ).

(2) {S lj · (⇒ ∀pS r

j ) | 1 ≤ j ≤ n} ` S l0 · (⇒ ∀

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Propositional logic

(1) {Sj · (∀pSj ⇒ ) | 1 ≤ j ≤ n} ` S0 · (∀RpS0 ⇒ ).

(2) {S lj · (⇒ ∀pS r

j ) | 1 ≤ j ≤ n} ` S l0 · (⇒ ∀

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Propositional logic

(1) {Sj · (∀pSj ⇒ ) | 1 ≤ j ≤ n} ` S0 · (∀RpS0 ⇒ ).

(2) {S lj · (⇒ ∀pS r

j ) | 1 ≤ j ≤ n} ` S l0 · (⇒ ∀

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Propositional logic

(1) {Sj · (∀pSj ⇒ ) | 1 ≤ j ≤ n} ` S0 · (∀RpS0 ⇒ ).

(2) {S lj · (⇒ ∀pS r

j ) | 1 ≤ j ≤ n} ` S l0 · (⇒ ∀

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Propositional logic

(1) {Sj · (∀pSj ⇒ ) | 1 ≤ j ≤ n} ` S0 · (∀RpS0 ⇒ ).

(2) {S lj · (⇒ ∀pS r

j ) | 1 ≤ j ≤ n} ` S l0 · (⇒ ∀

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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

Π,2Γ⇒ 2p,ΣRK

2Γ, p ⇒ 2∆

Π,2Γ,2p ⇒ 2∆,Σ

p ⇒ ∆

Π,2p ⇒ 2∆,ΣROK

Cor A modal logic with a balanced terminating calculus that consists offocussed axioms and focussed (modal) rules and contains RK or ROK, hasuniform interpolation.

Cor Any normal modal logic with a balanced terminating calculus that consistsof focussed (modal) axioms and rules, has uniform interpolation. (Ex: K)

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

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Modal logic

2S1 · S0

S2 · 2S1 · 2S0

Γ⇒ p

Π,2Γ⇒ 2p,ΣRK

2Γ, p ⇒ 2∆

Π,2Γ,2p ⇒ 2∆,Σ

p ⇒ ∆

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Modal logic

2S1 · S0

S2 · 2S1 · 2S0

Γ⇒ p

Π,2Γ⇒ 2p,ΣRK

2Γ, p ⇒ 2∆

Π,2Γ,2p ⇒ 2∆,Σ

p ⇒ ∆

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Modal logic

2S1 · S0

S2 · 2S1 · 2S0

Γ⇒ p

Π,2Γ⇒ 2p,ΣRK

2Γ, p ⇒ 2∆

Π,2Γ,2p ⇒ 2∆,Σ

p ⇒ ∆

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Modal logic

2S1 · S0

S2 · 2S1 · 2S0

Γ⇒ p

Π,2Γ⇒ 2p,ΣRK

2Γ, p ⇒ 2∆

Π,2Γ,2p ⇒ 2∆,Σ

p ⇒ ∆

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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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Rosalie Iemho Utrecht University Workshop on Admissible ... · (Negri)Fix a labelled sequent...

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