Cut Elimination for Modal Logics without Implicit Contractions - - PowerPoint PPT Presentation

Cut Elimination for Modal Logics without Implicit Contractions

Paolo Maffezioli

Ruhr-Universit¨ at Bochum

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Contraction in LK

◮ Contraction rules are not admissible in Gentzen’s LK

A, A, Γ ⇒ ∆ A, Γ ⇒ ∆

C

Γ ⇒ ∆, A, A Γ ⇒ ∆, A

C

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Contraction in LK

◮ Contraction rules are not admissible in Gentzen’s LK

A, A, Γ ⇒ ∆ A, Γ ⇒ ∆

C

Γ ⇒ ∆, A, A Γ ⇒ ∆, A

C ◮ ⇒ ∃x(Px ⊃ ∀yPy) has no contraction-free proof in LK

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Contraction in LK

◮ Contraction rules are not admissible in Gentzen’s LK

A, A, Γ ⇒ ∆ A, Γ ⇒ ∆

C

Γ ⇒ ∆, A, A Γ ⇒ ∆, A

C ◮ ⇒ ∃x(Px ⊃ ∀yPy) has no contraction-free proof in LK ◮ LK without contraction rules is incomplete

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Contraction in LK

◮ Contraction rules are not admissible in Gentzen’s LK

A, A, Γ ⇒ ∆ A, Γ ⇒ ∆

C

Γ ⇒ ∆, A, A Γ ⇒ ∆, A

C ◮ ⇒ ∃x(Px ⊃ ∀yPy) has no contraction-free proof in LK ◮ LK without contraction rules is incomplete ◮ LK without contraction rules is decidable

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Contraction in LK

◮ Contraction rules are not admissible in Gentzen’s LK

A, A, Γ ⇒ ∆ A, Γ ⇒ ∆

C

Γ ⇒ ∆, A, A Γ ⇒ ∆, A

C ◮ ⇒ ∃x(Px ⊃ ∀yPy) has no contraction-free proof in LK ◮ LK without contraction rules is incomplete ◮ LK without contraction rules is decidable

Ketonen and Weyhrauch. A decidable fragment of predicate

• calculus. Theoretical Computer Science, 32:297-307, 1984.

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Contraction in G3c

◮ Variants of LK with implicit contraction, i.e. calculi where

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Contraction in G3c

◮ Variants of LK with implicit contraction, i.e. calculi where ◮ logical rules are such that contraction rules become admissible

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Contraction in G3c

◮ Variants of LK with implicit contraction, i.e. calculi where ◮ logical rules are such that contraction rules become admissible ◮ Final systematization: G3c by Troesltra and Schwichtenberg

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Contraction in G3c

◮ Variants of LK with implicit contraction, i.e. calculi where ◮ logical rules are such that contraction rules become admissible ◮ Final systematization: G3c by Troesltra and Schwichtenberg

Troesltra and Schwichtenberg. Basic Proof Theory. Cambridge University Press, 2000.

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Contraction in G3c

◮ What does make contraction admissible in G3c?

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Contraction in G3c

◮ What does make contraction admissible in G3c? ◮ Invertibility of the logical rules. How to get there?

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Contraction in G3c

◮ What does make contraction admissible in G3c? ◮ Invertibility of the logical rules. How to get there? ◮ Replace Gentzen’s A ⇒ A with P, Γ ⇒ ∆, P ◮ (A arbitrary, P atomic, Γ, ∆ multisets)

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Contraction in G3c

◮ What does make contraction admissible in G3c? ◮ Invertibility of the logical rules. How to get there? ◮ Replace Gentzen’s A ⇒ A with P, Γ ⇒ ∆, P ◮ (A arbitrary, P atomic, Γ, ∆ multisets) ◮ All two-premise rules additive (context-sharing)

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Contraction in G3c

◮ What does make contraction admissible in G3c? ◮ Invertibility of the logical rules. How to get there? ◮ Replace Gentzen’s A ⇒ A with P, Γ ⇒ ∆, P ◮ (A arbitrary, P atomic, Γ, ∆ multisets) ◮ All two-premise rules additive (context-sharing) ◮ All one-premise rules multiplicative (context-independent)

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Contraction in G3c

◮ E.g. replace Gentzen’s multiplicative rule for ∧

Γ ⇒ ∆, A Π ⇒ Σ, B Γ, Π ⇒ Σ, ∆, A ∧ B

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Contraction in G3c

◮ E.g. replace Gentzen’s multiplicative rule for ∧

Γ ⇒ ∆, A Π ⇒ Σ, B Γ, Π ⇒ Σ, ∆, A ∧ B

◮ with its additive companion

Γ ⇒ ∆, A Γ ⇒ ∆, B Γ ⇒ ∆, A ∧ B

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Contraction in G3c

◮ E.g. replace Gentzen’s multiplicative rule for ∧

Γ ⇒ ∆, A Π ⇒ Σ, B Γ, Π ⇒ Σ, ∆, A ∧ B

◮ with its additive companion

Γ ⇒ ∆, A Γ ⇒ ∆, B Γ ⇒ ∆, A ∧ B

◮ in the additive rule Γ ⇒ ∆ occurs twice in the premises

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Contraction in G3c

◮ E.g. replace Gentzen’s multiplicative rule for ∧

Γ ⇒ ∆, A Π ⇒ Σ, B Γ, Π ⇒ Σ, ∆, A ∧ B

◮ with its additive companion

Γ ⇒ ∆, A Γ ⇒ ∆, B Γ ⇒ ∆, A ∧ B

◮ in the additive rule Γ ⇒ ∆ occurs twice in the premises ◮ but only once in the conclusion, a contraction.

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Contraction in G3c

◮ Similarly, replace Gentzen’s the additive rule for ∧

A, Γ ⇒ ∆ A ∧ B, Γ ⇒ ∆

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Contraction in G3c

◮ Similarly, replace Gentzen’s the additive rule for ∧

A, Γ ⇒ ∆ A ∧ B, Γ ⇒ ∆

◮ with its multiplicative companion

A, B, Γ ⇒ ∆ A ∧ B, Γ ⇒ ∆

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Contraction in G3c

◮ Replace Gentzen’s additive quantifier rules

A(t/x), Γ ⇒ ∆ ∀xA, Γ ⇒ ∆ Γ ⇒ ∆, A(t/x) Γ ⇒ ∆, ∃xA

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Contraction in G3c

◮ Replace Gentzen’s additive quantifier rules

A(t/x), Γ ⇒ ∆ ∀xA, Γ ⇒ ∆ Γ ⇒ ∆, A(t/x) Γ ⇒ ∆, ∃xA

◮ with their multiplicative (cumulative) companions

A(t/x), ∀xA, Γ ⇒ ∆ ∀xA, Γ ⇒ ∆ Γ ⇒ ∆, ∃xA, A(t/x) Γ ⇒ ∆, ∃xA

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Contraction in G3c

◮ Replace Gentzen’s additive quantifier rules

A(t/x), Γ ⇒ ∆ ∀xA, Γ ⇒ ∆ Γ ⇒ ∆, A(t/x) Γ ⇒ ∆, ∃xA

◮ with their multiplicative (cumulative) companions

A(t/x), ∀xA, Γ ⇒ ∆ ∀xA, Γ ⇒ ∆ Γ ⇒ ∆, ∃xA, A(t/x) Γ ⇒ ∆, ∃xA

◮ cumulative = ∀xA and ∃xA are repeated into the premise

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G3c

P, Γ ⇒ ∆, P ⊥, Γ ⇒ ∆ A, B, Γ ⇒ ∆ A ∧ B, Γ ⇒ ∆

L∧

Γ ⇒ ∆, A Γ ⇒ ∆, B Γ ⇒ ∆, A ∧ B

R∧

A, Γ ⇒ ∆ B, Γ ⇒ ∆ A ∨ B, Γ ⇒ ∆

L∨

Γ ⇒ ∆, A, B Γ ⇒ ∆, A ∨ B

R∨

Γ ⇒ ∆, A B, Γ ⇒ ∆ A ⊃ B, Γ ⇒ ∆

L⊃

A, Γ ⇒ ∆, B Γ ⇒ ∆, A ⊃ B

R⊃

A(t/x), ∀xA, Γ ⇒ ∆ ∀xA, Γ ⇒ ∆

L∀

Γ ⇒ ∆, A(y/x) Γ ⇒ ∆, ∀xA

R∀∗

A(y/x), Γ ⇒ ∆ ∃xA, Γ ⇒ ∆

L∃∗

Γ ⇒ ∆, ∃xA, A(t/x) Γ ⇒ ∆, ∃xA

R∃

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Invertibility in G3c

◮ In G3c all rules are height-preserving invertible, i.e.

the derivability of the conclusion of each of the rules implies the derivability of any of the premisses [. . . ] In addition, in all cases, the derivation of the premiss has a height that does not exceed the height of the derivation of the conclusion.

• A. Dragalin. Mathematical Intuitionism. Introduction to Proof

Theory, volume 67 of Translations of Mathematical Monographs. American Mathematical Society, 1988.

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Contraction in G3c

◮ Contraction admissibility (Dragalin) via invertibility

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Contraction in G3c

◮ Contraction admissibility (Dragalin) via invertibility ◮ Idea: apply invertibility so as to permute up contraction

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Contraction in G3c

◮ Contraction admissibility (Dragalin) via invertibility ◮ Idea: apply invertibility so as to permute up contraction

A, B, A ∧ B, Γ ⇒ ∆ A ∧ B, A ∧ B, Γ ⇒ ∆

L∧

A ∧ B, Γ ⇒ ∆

C

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Contraction in G3c

◮ Contraction admissibility (Dragalin) via invertibility ◮ Idea: apply invertibility so as to permute up contraction

A, B, A ∧ B, Γ ⇒ ∆ A ∧ B, A ∧ B, Γ ⇒ ∆

L∧

A ∧ B, Γ ⇒ ∆

C

• 10 / 53

Contraction in G3c

◮ Contraction admissibility (Dragalin) via invertibility ◮ Idea: apply invertibility so as to permute up contraction

A, B, A ∧ B, Γ ⇒ ∆ A ∧ B, A ∧ B, Γ ⇒ ∆

L∧

A ∧ B, Γ ⇒ ∆

C

• A, B, A ∧ B, Γ ⇒ ∆

A, B, A, B, Γ ⇒ ∆

Inv

A, B, Γ ⇒ ∆

C

A ∧ B, Γ ⇒ ∆

L∧

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Contraction in G3c

◮ ⇒ ∃x(Px ⊃ ∀yPy) has a contraction-free proof in G3c

Px, Py ⇒ Py, ∀yPy, ∃x(Px ⊃ ∀yPy) Px ⇒ Py, Py ⊃ ∀yPy, ∃x(Px ⊃ ∀yPy) Px ⇒ Py, ∃x(Px ⊃ ∀yPy)

R∃

Px ⇒ ∀yPy, ∃x(Px ⊃ ∀yPy) ⇒ Px ⊃ ∀yPy, ∃x(Px ⊃ ∀yPy) ⇒ ∃x(Px ⊃ ∀yPy)

R∃ ◮ The two applications of R∃ are on the same formula

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Road to cut elimination in G3c

◮ Contraction is admissible but does not disappear altogether

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Road to cut elimination in G3c

◮ Contraction is admissible but does not disappear altogether ◮ it is still used in the proof of cut elimination

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Road to cut elimination in G3c

◮ Contraction is admissible but does not disappear altogether ◮ it is still used in the proof of cut elimination

Γ ⇒ ∆, A Γ ⇒ ∆, B Γ ⇒ ∆, A ∧ B

R∧

A, B, Π ⇒ Σ A ∧ B, Π ⇒ Σ

L∧

Γ, Π ⇒ Σ, ∆

Cut

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Road to cut elimination in G3c

◮ Contraction is admissible but does not disappear altogether ◮ it is still used in the proof of cut elimination

Γ ⇒ ∆, A Γ ⇒ ∆, B Γ ⇒ ∆, A ∧ B

R∧

A, B, Π ⇒ Σ A ∧ B, Π ⇒ Σ

L∧

Γ, Π ⇒ Σ, ∆

Cut

Γ ⇒ ∆, B Γ ⇒ ∆, A A, B, Π ⇒ Σ Γ, B, Π ⇒ ∆, Σ

Cut

Γ, Π ⇒ Σ, ∆

Cut

Γ, Γ, Π ⇒ Σ, ∆, ∆

C

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Road to cut elimination in G3c

◮ From invertibility of logical rules

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Road to cut elimination in G3c

◮ From invertibility of logical rules ◮ to admissibility of contraction

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Road to cut elimination in G3c

◮ From invertibility of logical rules ◮ to admissibility of contraction ◮ to cut elimination

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Labelled systems

◮ Variant of Gentzen sequents for modal logics

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Labelled systems

◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics)

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Labelled systems

◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) ◮ combination of modal and (a fragment of) first-order language

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Labelled systems

◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) ◮ combination of modal and (a fragment of) first-order language ◮ multisets of labelled formulas x : A or relational atoms xRy

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Labelled systems

◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) ◮ combination of modal and (a fragment of) first-order language ◮ multisets of labelled formulas x : A or relational atoms xRy ◮ logical rules for labelled formulas

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Labelled systems

◮ Variant of Gentzen sequents for modal logics ◮ (in fact, for any logic with a Kripke semantics) ◮ combination of modal and (a fragment of) first-order language ◮ multisets of labelled formulas x : A or relational atoms xRy ◮ logical rules for labelled formulas ◮ non-logical rules for relational atoms

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Labelled systems

◮ Similarity between and ∀

xRy, Γ ⇒ ∆, y : A Γ ⇒ ∆, x : A

R∗

Γ ⇒ ∆, A(y/x) Γ ⇒ ∆, ∀xA

R∀∗

y : A, x : , xRy, Γ ⇒ ∆ x : A, xRy, Γ ⇒ ∆

L

A(t/x), ∀xA, Γ ⇒ ∆ ∀xA, Γ ⇒ ∆

L∀

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Labelled systems

◮ Similarity between ♦ and ∃

xRy, Γ ⇒ ∆, x : ♦A, y : A xRy, Γ ⇒ ∆, x : ♦A

R♦

Γ ⇒ ∆, ∃xA, A(y/x) Γ ⇒ ∆, ∃xA

R∃

xRy, y : A, Γ ⇒ ∆ x : ♦A, Γ ⇒ ∆

L♦∗

A(y/x), Γ ⇒ ∆ ∃xA, Γ ⇒ ∆

L∃∗

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G3K

x : P, Γ ⇒ ∆, x : P x : ⊥, Γ ⇒ ∆ x : A, x : B, Γ ⇒ ∆ x : A ∧ B, Γ ⇒ ∆

L∧

Γ ⇒ ∆, x : A Γ ⇒ ∆, x : B Γ ⇒ ∆, x : A ∧ B

R∧

x : A, Γ ⇒ ∆ x : B, Γ ⇒ ∆ x : A ∨ B, Γ ⇒ ∆

L∨

Γ ⇒ ∆, x : A, x : B Γ ⇒ ∆, x : A ∨ B

R∨

Γ ⇒ ∆, x : A x : B, Γ ⇒ ∆ x : A ⊃ B, Γ ⇒ ∆

L⊃

x : A, Γ ⇒ ∆, x : B Γ ⇒ ∆, x : A ⊃ B

R⊃

y : A, x : A, xRy, Γ ⇒ ∆ x : A, xRy, Γ ⇒ ∆

L

xRy, Γ ⇒ ∆, y : A Γ ⇒ ∆, x : A

R∗

xRy, y : A, Γ ⇒ ∆ x : ♦A, Γ ⇒ ∆

L♦∗

xRy, Γ ⇒ ∆, x : ♦A, y : A xRy, Γ ⇒ ∆, x : ♦A

R♦

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Extensions of G3K

◮ Idea: reach stronger logics by adding rules for R

❷ ❶

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Extensions of G3K

◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e.

❷ ❶

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Extensions of G3K

◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. ◮ criteria for a new rule to be ❷good❶ w.r.t cut elimination.

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Extensions of G3K

◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. ◮ criteria for a new rule to be ❷good❶ w.r.t cut elimination. ◮ Example: R is an equivalence relation

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Extensions of G3K

◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. ◮ criteria for a new rule to be ❷good❶ w.r.t cut elimination. ◮ Example: R is an equivalence relation ◮ Reflexivity and Euclideaness of R as axioms

⇒ xRx xRy, xRz ⇒ yRz

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Extensions of G3K

◮ Idea: reach stronger logics by adding rules for R ◮ Problem: cut-free Gentzen system with new rules, i.e. ◮ criteria for a new rule to be ❷good❶ w.r.t cut elimination. ◮ Example: R is an equivalence relation ◮ Reflexivity and Euclideaness of R as axioms

⇒ xRx xRy, xRz ⇒ yRz

◮ No cut-free derivation of the symmetry of R

⇒ xRx xRy, xRx ⇒ yRx xRy ⇒ yRx

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Extensions of G3K

◮ Reflexivity and Euclideaness of R as rules of inference

xRx, Γ ⇒ ∆ Γ ⇒ ∆

RefR

yRz, xRy, xRz, Γ ⇒ ∆ xRy, xRz, Γ ⇒ ∆

EuclR

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Extensions of G3K

◮ Reflexivity and Euclideaness of R as rules of inference

xRx, Γ ⇒ ∆ Γ ⇒ ∆

RefR

yRz, xRy, xRz, Γ ⇒ ∆ xRy, xRz, Γ ⇒ ∆

EuclR ◮ Cut-free derivation of the symmetry of R

yRx ⇒ yRx xRy, xRx ⇒ yRx

EuclR

xRy ⇒ yRx

RefR

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Labelled systems

◮ What conditions should R satisfy so as to yield a cut-free

extension of G3K?

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Labelled systems

◮ What conditions should R satisfy so as to yield a cut-free

extension of G3K?

◮ R can be any first-order condition

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Labelled systems

◮ What conditions should R satisfy so as to yield a cut-free

extension of G3K?

◮ R can be any first-order condition ◮ what modal logics can be captured?

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Labelled systems

◮ What conditions should R satisfy so as to yield a cut-free

extension of G3K?

◮ R can be any first-order condition ◮ what modal logics can be captured? ◮ any modal logic characterized by a first-order frame condition

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Labelled systems

◮ What conditions should R satisfy so as to yield a cut-free

extension of G3K?

◮ R can be any first-order condition ◮ what modal logics can be captured? ◮ any modal logic characterized by a first-order frame condition

Dyckhoff and Negri. Geometrisation of first-order logic. The Bulletin of Symbolic Logic, 21(2):123-163, 2015.

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Extensions of G3K

◮ G3T = G3K + Ref

xRx, Γ ⇒ ∆ Γ ⇒ ∆

Ref ◮ G3S4 = G3T + Trans

xRz, xRy, yRz, Γ ⇒ ∆ xRy, yRz, Γ ⇒ ∆

Trans ◮ G3S5 = G3S4 + Sym

yRx, xRy, Γ ⇒ ∆ xRy, Γ ⇒ ∆

Sym

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Cut elimination in (extensions of) G3K

Theorem (Negri)

In G3KT, G3KS4 and G3KS5

◮ Logical rules are hp-invertible ◮ Contraction is hp-admissible ◮ Cut is admissible

• Negri. Proof analysis in modal logic. Journal of Philosophical

Logic, 34(5-6):507-544, 2005.

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Cut elimination in (extensions of) G3K

◮ Moreover, hyperexponential growth of cut-free derivations

• Schwichtenberg. Proof theory: some applications of cut
• elimination. In J. Barwise, editor, Handbook of Mathematical

Logic, volume 90, pages 867-895. North-Holland, 1977.

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Completeness of (extensions of) G3K

◮ No obvious formula-interpretation of a labelled sequent

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Completeness of (extensions of) G3K

◮ No obvious formula-interpretation of a labelled sequent ◮ In ordinary sequent calculi

Γ ⇒ ∆ iff

• Γ ⊃
• ∆

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Completeness of (extensions of) G3K

◮ No obvious formula-interpretation of a labelled sequent ◮ In ordinary sequent calculi

Γ ⇒ ∆ iff

• Γ ⊃
• ∆

◮ Two forms of completeness for a labelled system S

◮ S is weakly complete if it derives all valid labelled formulas ◮ S is strongly complete if it derives all labelled sequents 24 / 53

Completeness of (extensions of) G3K

Theorem (Negri)

G3KT, G3KS4 and G3KS5 are strongly complete.

• Negri. Kripke completeness revisited. in G. Primiero and S.

Rahman (eds.), Acts of Knowledge - History, Philosophy and Logic, College Publications, 2009

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Completeness of (extensions of) G3K

◮ Weak completeness is proved via derivability of axiom system

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Completeness of (extensions of) G3K

◮ Weak completeness is proved via derivability of axiom system ◮ (and completeness of the axiom system itself)

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Completeness of (extensions of) G3K

◮ Weak completeness is proved via derivability of axiom system ◮ (and completeness of the axiom system itself) ◮ G3K derives all valid labelled formulas

◮ x : A, x : (A ⊃ B) ⇒ x : B (axiom K) ◮ from ⇒ x : A infer ⇒ x : A (necessitation rule) 26 / 53

Completeness of G3K

◮ In the derivation of K

xRy, y : A ⇒ y : B, y : A y : B, y : A ⇒ y : B xRy, y : A, y : A ⊃ B ⇒ y : B xRy, y : A, x : (A ⊃ B) ⇒ y : B

L

xRy, x : A, x : (A ⊃ B) ⇒ y : B

L

x : A, x : (A ⊃ B) ⇒ x : B

◮ the two applications of L are on distinct labelled formulas ◮ no labelled formula get analyzed twice by the same rule

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Doing away with repetitions

◮ Consider L without the principal labelled formula repeated

y : A, xRy, Γ ⇒ ∆ x : A, xRy, Γ ⇒ ∆ L♯

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Doing away with repetitions

◮ Consider L without the principal labelled formula repeated

y : A, xRy, Γ ⇒ ∆ x : A, xRy, Γ ⇒ ∆ L♯

◮ and K is still derivable

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Doing away with repetitions

◮ Consider L without the principal labelled formula repeated

y : A, xRy, Γ ⇒ ∆ x : A, xRy, Γ ⇒ ∆ L♯

◮ and K is still derivable ◮ does this count as a proof that G3K♯ is weakly complete?

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Doing away with repetitions

◮ No, modus ponens still needs to be proved admissible

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Doing away with repetitions

◮ No, modus ponens still needs to be proved admissible ◮ this require cut elimination of G3K♯

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Doing away with repetitions

◮ No, modus ponens still needs to be proved admissible ◮ this require cut elimination of G3K♯ ◮ but cut elimination requires contraction

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Doing away with repetitions

◮ No, modus ponens still needs to be proved admissible ◮ this require cut elimination of G3K♯ ◮ but cut elimination requires contraction ◮ and contraction requires invertibility

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Doing away with repetitions

◮ No, modus ponens still needs to be proved admissible ◮ this require cut elimination of G3K♯ ◮ but cut elimination requires contraction ◮ and contraction requires invertibility ◮ and L♯ is not invertible

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Doing away with repetitions

◮ Is it possible to get rid of the repetition in L

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Doing away with repetitions

◮ Is it possible to get rid of the repetition in L ◮ without losing weak completeness?

30 / 53

Doing away with repetitions

◮ Is it possible to get rid of the repetition in L ◮ without losing weak completeness? ◮ are there modal validities that can be derived using L

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Doing away with repetitions

◮ Is it possible to get rid of the repetition in L ◮ without losing weak completeness? ◮ are there modal validities that can be derived using L ◮ and that are not derivable using L♯?

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Doing away with repetitions

◮ In G3c the trade-off between repetition of quantifiers and

contraction is unavoidable,

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Doing away with repetitions

◮ In G3c the trade-off between repetition of quantifiers and

contraction is unavoidable,

◮ (on pain of decidability of first-order logic)

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Doing away with repetitions

◮ In G3c the trade-off between repetition of quantifiers and

contraction is unavoidable,

◮ (on pain of decidability of first-order logic) ◮ but many modal logics are decidable

31 / 53

Doing away with repetitions

◮ In G3c the trade-off between repetition of quantifiers and

contraction is unavoidable,

◮ (on pain of decidability of first-order logic) ◮ but many modal logics are decidable ◮ there could be a way to avoid cumulative modal rules . . .

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Doing away with repetitions

◮ In G3c the trade-off between repetition of quantifiers and

contraction is unavoidable,

◮ (on pain of decidability of first-order logic) ◮ but many modal logics are decidable ◮ there could be a way to avoid cumulative modal rules . . . ◮ . . . without having contraction rules explicit

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Doing away with repetitions

◮ How far we can get without repetition? Not very far. . .

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Doing away with repetitions

◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L to be derived

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Doing away with repetitions

◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L to be derived ◮ (♦(A ⊃ B) ∧ ♦(A ⊃ ¬B)) ⊃ ♦¬A

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Doing away with repetitions

◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L to be derived ◮ (♦(A ⊃ B) ∧ ♦(A ⊃ ¬B)) ⊃ ♦¬A ◮ Fitting’s example is derivable in G3S5 but not in G3S5♯

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Doing away with repetitions

◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L to be derived ◮ (♦(A ⊃ B) ∧ ♦(A ⊃ ¬B)) ⊃ ♦¬A ◮ Fitting’s example is derivable in G3S5 but not in G3S5♯ ◮ G3S5♯ is weakly incomplete

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Doing away with repetitions

◮ How far we can get without repetition? Not very far. . . ◮ Notable modal validities do need L to be derived ◮ (♦(A ⊃ B) ∧ ♦(A ⊃ ¬B)) ⊃ ♦¬A ◮ Fitting’s example is derivable in G3S5 but not in G3S5♯ ◮ G3S5♯ is weakly incomplete ◮ (and hence also strongly incomplete)

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Doing away with repetitions

◮ (A ∧ ¬A) ⊃ ⊥ is der. in G3K4 but not in G3K4♯

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Doing away with repetitions

◮ (A ∧ ¬A) ⊃ ⊥ is der. in G3K4 but not in G3K4♯ ◮ ♦(A ⊃ A) is der. in G3T but not in G3X♯ (X ∈ {T, S4, S5})

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Doing away with repetitions

◮ (A ∧ ¬A) ⊃ ⊥ is der. in G3K4 but not in G3K4♯ ◮ ♦(A ⊃ A) is der. in G3T but not in G3X♯ (X ∈ {T, S4, S5}) ◮ G3K4♯, G3T♯, G3S4♯, G3S5♯ are all weakly incomplete

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Doing away with repetitions

◮ (A ∧ ¬A) ⊃ ⊥ is der. in G3K4 but not in G3K4♯ ◮ ♦(A ⊃ A) is der. in G3T but not in G3X♯ (X ∈ {T, S4, S5}) ◮ G3K4♯, G3T♯, G3S4♯, G3S5♯ are all weakly incomplete ◮ (and hence also strongly incomplete)

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Is at least G3K♯ weakly complete?

◮ Minari: G3K♯ is weakly complete . . .

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Is at least G3K♯ weakly complete?

◮ Minari: G3K♯ is weakly complete . . . ◮ . . . although it is not is strongly complete

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Is at least G3K♯ weakly complete?

◮ Minari: G3K♯ is weakly complete . . . ◮ . . . although it is not is strongly complete ◮ xRx, x : ♦A ⇒ x : ♦(A ∧ ♦A) is der. in G3K but not in G3K♯

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Is at least G3K♯ weakly complete?

◮ Minari: G3K♯ is weakly complete . . . ◮ . . . although it is not is strongly complete ◮ xRx, x : ♦A ⇒ x : ♦(A ∧ ♦A) is der. in G3K but not in G3K♯ ◮ what is the class of sequents w.r.t which G3K♯ is strongly

complete?

• Minari. Labeled sequent calculi for modal logics and implicit
• contractions. Archive for Mathematical Logic, 52:881-907, 2013.

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Tree-labelled sequents

◮ A sequent is tree-labelled when

◮ it does not contain R-cycles of labels ◮ every label is R-related to at most one label 35 / 53

Tree-labelled sequents

Theorem (Minari)

G3K♯ is strongly complete w.r.t. tree-labelled sequents

Corollary

G3K♯ is weakly complete and cut-free.

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Cut elimination in G3K♯

◮ Minari’s proof of cut elimination is indirect (via completeness)

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Cut elimination in G3K♯

◮ Minari’s proof of cut elimination is indirect (via completeness) ◮ how to actually get rid of cuts is still open problem

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Cut elimination in G3K♯

◮ Ideas

◮ Contraction admissibility without invertibility ◮ Cut admissibility without contraction ◮ . . . 38 / 53

Possible ways out

◮ Cut elimination with

◮ sequents as sets (Negri and von Plato) ◮ non-standard multiset union (Indrzejczak) ◮ additive cut in place of multiplicative cut 39 / 53

Sequents as sets

◮ terms, formulas, sequents are concrete objects ◮ sets are abstracta ◮ expressions are better understood as lists than sets

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Sequents as sets

The idea has been often entertained of dispensing, next to order, also with multiplicity, known as the sequents with sets idea. There will be some difficulties in carrying the idea through on the level of formalization Negri and von Plato. Cut elimination in sequent calculi with implicit contraction, with a conjecture on the origin of Gentzen’s altitude line construction. D. Probst and P. Schuster (eds.), Concepts of Proof in Mathematics, Philosophy, and Computer

• Science. de Gruyter, pp. 269-290, 2016

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Sequents as sets

◮ Rules for first-order intuitionistic and classical sequent calculus ◮ with annotated contexts Γ ◮ Proof of cut elimination

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Non-standard multiset union (Indrzejczak)

◮ In multiset theory there are two unions, additive and ordinary ◮ E.g. if Γ = [A, A, B] and ∆ = [A, C]

◮ additive union: Γ ∪ ∆ = [A, A, A, B, C] ◮ ordinary union: Γ ⊎ ∆ = [A, A, B, C]

• Blizard. Multiset theory. Notre Dame Journal of Formal Logic,

30(1):36-66, 1989

• Casari. La matematica della verit`
• a. Boringhieri:Torino, 2006.

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CPLM (Indrzejczak)

P ⇒ P A, B, Γ ⇒ ∆ A ∧ B, Γ ⇒ ∆ Γ ⇒ ∆, A Π ⇒ Σ, B Γ, Π ⇒ Σ, ∆, A ∧ B A, Γ ⇒ ∆ B, Π ⇒ Σ A ∨ B, Γ, Π ⇒ Σ, ∆ Γ ⇒ ∆, A, B Γ ⇒ ∆, A ∨ B Γ ⇒ ∆, A ¬A, Γ ⇒ ∆ A, Γ ⇒ ∆ Γ ⇒ ∆, ¬A Γ ⇒ ∆ A, Γ ⇒ ∆ Γ ⇒ ∆ Γ ⇒ ∆, A

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CPLM (Indrzejczak)

Theorem (Indrzejczak)

CPLM is cut-free. It seems that in our system contraction is required neither for cut-free proofs of theses nor for proving cut

• elimination. But the last claim needs closer examination.
• Indrzejczak. Contraction contracted. Bulletin of the Section of

Logic, 43(3-4):139-153, 2014.

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CPLM (Indrzejczak)

◮ A reduction step in cut elimination still needs a contraction ◮ such a contraction is not actually needed because . . . ◮ . . . contraction is admissible ◮ Potential problem: contraction is proved admissible via cut

A ⇒ A A, A, Γ ⇒ ∆ A, Γ ⇒ ∆

◮ circularity:

◮ use contraction to prove cut elimination ◮ use cut to prove contraction admissibility 46 / 53

Additive cut

◮ with additive cut, cut elimination does not need contraction

Γ ⇒ ∆, A A, Γ ⇒ ∆ Γ ⇒ ∆

◮ the standard proof does use contraction

We use closure under Contraction and Weakening all the time

◮ but at least for G3K this is not essential

Troesltra and Schwichtenberg. Basic Proof Theory. Cambridge University Press, 2000.

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Additive cut

◮ maybe now additive cut elimination needs invertibility ◮ and, again, L♯ is not invertible

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Mix rule

◮ Or we can live with contraction ◮ and prove cut elimination for G3K♯ `

a la Gentzen

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GK - logical rules

x : P ⇒ x : P x : Ai, Γ ⇒ ∆ x : A1 ∧ A2, Γ ⇒ ∆ Γ ⇒ ∆, x : A Π ⇒ Σ, x : B Γ, Π ⇒ Σ, ∆, x : A ∧ B x : A, Γ ⇒ ∆ x : B, Π ⇒ Σ x : A ∨ B, Γ, Π ⇒ Σ, ∆ Γ ⇒ ∆, x : Ai Γ ⇒ ∆, x : A1 ∨ A2 Γ ⇒ ∆, x : A x : ¬A, Γ ⇒ ∆ x : A, Γ ⇒ ∆ Γ ⇒ ∆, x : ¬A y : A, xRy, Γ ⇒ ∆ x : A, xRy, Γ ⇒ ∆ L♯ xRy, Γ ⇒ ∆, y : A Γ ⇒ ∆, x : A

R

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GK - structural rules

Γ ⇒ ∆ ϕ, Γ ⇒ ∆ Γ ⇒ ∆ Γ ⇒ ∆, ϕ x : A, x : A, Γ ⇒ ∆ x : A, Γ ⇒ ∆ Γ ⇒ ∆, x : A, x : A Γ ⇒ ∆, x : A Γ ⇒ ∆, x : A x : A, Π ⇒ Σ Γ, Π ⇒ Σ, ∆

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GK

◮ Prove cut elimination via admissibility of mix

Γ ⇒ ∆, x : An x : Am, Π ⇒ Σ Γ, Π ⇒ Σ, ∆

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Conclusions

◮ nearly every interesting modal logic needs contraction ◮ either implicitly or explicitly ◮ basic modal logic is contraction-free ◮ but it is had to prove it constructively

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