Uniform Interpolation Part 1: Intuitionistic Logic George Metcalfe - - PowerPoint PPT Presentation

Uniform Interpolation

Part 1: Intuitionistic Logic George Metcalfe

Mathematical Institute University of Bern BLAST 2018, University of Denver, 6-10 August 2018

George Metcalfe (University of Bern) Uniform Interpolation August 2018 1 / 30

The Craig Interpolation Theorem

Theorem (Craig 1957)

If ϕ and ψ are sentences of first-order logic such that ϕ ⊢ ψ, ϕ ⊢ ψ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 2 / 30

The Craig Interpolation Theorem

Theorem (Craig 1957)

If ϕ and ψ are sentences of first-order logic such that ϕ ⊢ ψ, ϕ ⊢ ψ language of ϕ language of ψ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 2 / 30

The Craig Interpolation Theorem

Theorem (Craig 1957)

If ϕ and ψ are sentences of first-order logic such that ϕ ⊢ ψ, then there exists a sentence χ with Rel(χ) ⊆ Rel(ϕ) ∩ Rel(ψ) such that ϕ ⊢ χ and χ ⊢ ψ. ϕ ⊢ χ ⊢ ψ language of ϕ language of ψ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 2 / 30

Origins

“Although I was aware of the mathematical interest of questions related to elimination problems in logic, my main aim, initially unfocused, was to try to use methods and results from logic to clarify or illuminate a topic that seems central to empiricist programs: In epistemology, the relationship between the external world and sense data; in philosophy of science, that between theoretical constructs and observed data.” William Craig (2008).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 3 / 30

Craig Interpolation in Classical Logic

Theorem

For any formulas α(x, y), β(y, z) of classical propositional logic satisfying α ⊢CL β,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30

Craig Interpolation in Classical Logic

Theorem

For any formulas α(x, y), β(y, z) of classical propositional logic satisfying α ⊢CL β, there exists a formula γ(y) such that α ⊢CL γ and γ ⊢CL β.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30

Craig Interpolation in Classical Logic

Theorem

For any formulas α(x, y), β(y, z) of classical propositional logic satisfying α ⊢CL β, there exists a formula γ(y) such that α ⊢CL γ and γ ⊢CL β. For example. . . α = ¬(x → y) β = y → ¬z γ = 001 101 100 000 011 111 110 010

George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30

Craig Interpolation in Classical Logic

Theorem

For any formulas α(x, y), β(y, z) of classical propositional logic satisfying α ⊢CL β, there exists a formula γ(y) such that α ⊢CL γ and γ ⊢CL β. For example. . . α = ¬(x → y) β = y → ¬z γ = 001 101 100 000 011 111 110 010 α

George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30

Craig Interpolation in Classical Logic

Theorem

For any formulas α(x, y), β(y, z) of classical propositional logic satisfying α ⊢CL β, there exists a formula γ(y) such that α ⊢CL γ and γ ⊢CL β. For example. . . α = ¬(x → y) β = y → ¬z γ = 001 101 100 000 011 111 110 010 α β

George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30

Craig Interpolation in Classical Logic

Theorem

For any formulas α(x, y), β(y, z) of classical propositional logic satisfying α ⊢CL β, there exists a formula γ(y) such that α ⊢CL γ and γ ⊢CL β. For example. . . α = ¬(x → y) β = y → ¬z γ = ¬y 001 101 100 000 011 111 110 010 α β γ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30

Craig Interpolation in Classical Logic

Theorem

For any formulas α(x, y), β(y, z) of classical propositional logic satisfying α ⊢CL β, there exists a formula γ(y) such that α ⊢CL γ and γ ⊢CL β. For example. . . α = ¬(x → y) β = y → ¬z γ = ¬y In fact, for any formula δ(y, z), α ⊢CL δ ⇐ ⇒ γ ⊢CL δ. 001 101 100 000 011 111 110 010 α β γ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30

Uniform Interpolation in Classical Logic

Theorem

For any formula α(x, y) of classical propositional logic, there exist formulas αL(y) and αR(y)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30

Uniform Interpolation in Classical Logic

Theorem

For any formula α(x, y) of classical propositional logic, there exist formulas αL(y) and αR(y) such that for any formula β(y, z), β(y, z) ⊢CL α(x, y) ⇐ ⇒ β(y, z) ⊢CL αL(y)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30

Uniform Interpolation in Classical Logic

Theorem

For any formula α(x, y) of classical propositional logic, there exist formulas αL(y) and αR(y) such that for any formula β(y, z), β(y, z) ⊢CL α(x, y) ⇐ ⇒ β(y, z) ⊢CL αL(y) α(x, y) ⊢CL β(y, z) ⇐ ⇒ αR(y) ⊢CL β(y, z).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30

Uniform Interpolation in Classical Logic

Theorem

For any formula α(x, y) of classical propositional logic, there exist formulas αL(y) and αR(y) such that for any formula β(y, z), β(y, z) ⊢CL α(x, y) ⇐ ⇒ β(y, z) ⊢CL αL(y) α(x, y) ⊢CL β(y, z) ⇐ ⇒ αR(y) ⊢CL β(y, z).

Proof.

Given any formula α(x, y), we just define αL(y) = {α(¯ a, y) | ¯ a ⊆ {0, 1}}

George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30

Uniform Interpolation in Classical Logic

Theorem

For any formula α(x, y) of classical propositional logic, there exist formulas αL(y) and αR(y) such that for any formula β(y, z), β(y, z) ⊢CL α(x, y) ⇐ ⇒ β(y, z) ⊢CL αL(y) α(x, y) ⊢CL β(y, z) ⇐ ⇒ αR(y) ⊢CL β(y, z).

Proof.

Given any formula α(x, y), we just define αL(y) = {α(¯ a, y) | ¯ a ⊆ {0, 1}} αR(y) = {α(¯ a, y) | ¯ a ⊆ {0, 1}}.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30

This Tutorial

What does (uniform) interpolation mean in logic and algebra?

George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30

Today

What does (uniform) interpolation mean for intuitionistic logic?

George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 30

Intuitionistic Logic

Intuitionistic logic was introduced by Heyting in the 1930s to formalize certain principles used in Brouwer’s constructive mathematics.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30

Intuitionistic Logic

Intuitionistic logic was introduced by Heyting in the 1930s to formalize certain principles used in Brouwer’s constructive mathematics. The BHK-interpretation presents the validity of formulas in intuitionistic logic in terms of the construction of proofs,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30

Intuitionistic Logic

Intuitionistic logic was introduced by Heyting in the 1930s to formalize certain principles used in Brouwer’s constructive mathematics. The BHK-interpretation presents the validity of formulas in intuitionistic logic in terms of the construction of proofs, e.g., “A proof of α ∨ β is given via a proof of α or a proof of β.”

George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30

Intuitionistic Logic

Intuitionistic logic was introduced by Heyting in the 1930s to formalize certain principles used in Brouwer’s constructive mathematics. The BHK-interpretation presents the validity of formulas in intuitionistic logic in terms of the construction of proofs, e.g., “A proof of α ∨ β is given via a proof of α or a proof of β.” Intuitionistic logic may be presented syntactically via axiom systems omitting the law of excluded middle, natural deduction, sequent calculi, etc.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30

Intuitionistic Logic

Intuitionistic logic was introduced by Heyting in the 1930s to formalize certain principles used in Brouwer’s constructive mathematics. The BHK-interpretation presents the validity of formulas in intuitionistic logic in terms of the construction of proofs, e.g., “A proof of α ∨ β is given via a proof of α or a proof of β.” Intuitionistic logic may be presented syntactically via axiom systems omitting the law of excluded middle, natural deduction, sequent calculi, etc.

• r semantically via

Kripke models, Heyting algebras, topological semantics, etc.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30

An Axiom System for Intuitionistic Logic

Let us fix a language with connectives ∧, ∨, →, ⊥, ⊤ and write T ⊢IL α if a formula α is derivable from a set of formulas T using the axiom schema

• 1. α → (β → α)
• 2. (α → (β → γ)) → ((α → β) → (α → γ))
• 3. (α ∧ β) → α
• 4. (α ∧ β) → β
• 5. α → (β → (α ∧ β))
• 6. α → (α ∨ β)
• 7. β → (α ∨ β)
• 8. (α → γ) → ((β → γ) → ((α ∨ β) → γ))
• 9. ⊥ → α
• 10. α → ⊤

together with the modus ponens rule: from α and α → β, infer β.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 30

Heyting Algebras

A Heyting algebra is an algebraic structure A, ∧, ∨, →, ⊥, ⊤ such that

George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30

Heyting Algebras

A Heyting algebra is an algebraic structure A, ∧, ∨, →, ⊥, ⊤ such that (i) A, ∧, ∨, ⊥, ⊤ is a bounded lattice;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30

Heyting Algebras

A Heyting algebra is an algebraic structure A, ∧, ∨, →, ⊥, ⊤ such that (i) A, ∧, ∨, ⊥, ⊤ is a bounded lattice; (ii) a ≤ b → c ⇐ ⇒ a ∧ b ≤ c for all a, b, c ∈ A.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30

Heyting Algebras

A Heyting algebra is an algebraic structure A, ∧, ∨, →, ⊥, ⊤ such that (i) A, ∧, ∨, ⊥, ⊤ is a bounded lattice; (ii) a ≤ b → c ⇐ ⇒ a ∧ b ≤ c for all a, b, c ∈ A. The class HA of Heyting algebras forms a variety.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30

Heyting Algebras

A Heyting algebra is an algebraic structure A, ∧, ∨, →, ⊥, ⊤ such that (i) A, ∧, ∨, ⊥, ⊤ is a bounded lattice; (ii) a ≤ b → c ⇐ ⇒ a ∧ b ≤ c for all a, b, c ∈ A. The class HA of Heyting algebras forms a variety. Examples:

• 1. any Boolean algebra;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30

Heyting Algebras

A Heyting algebra is an algebraic structure A, ∧, ∨, →, ⊥, ⊤ such that (i) A, ∧, ∨, ⊥, ⊤ is a bounded lattice; (ii) a ≤ b → c ⇐ ⇒ a ∧ b ≤ c for all a, b, c ∈ A. The class HA of Heyting algebras forms a variety. Examples:

• 1. any Boolean algebra;
• 2. letting U be the set of upsets of a poset X, ≤,

U, ∩, ∪, →, ∅, X where Y → Z = {a ∈ X | a ≤ b ∈ Y ⇒ b ∈ Z};

George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30

Heyting Algebras

A Heyting algebra is an algebraic structure A, ∧, ∨, →, ⊥, ⊤ such that (i) A, ∧, ∨, ⊥, ⊤ is a bounded lattice; (ii) a ≤ b → c ⇐ ⇒ a ∧ b ≤ c for all a, b, c ∈ A. The class HA of Heyting algebras forms a variety. Examples:

• 1. any Boolean algebra;
• 2. letting U be the set of upsets of a poset X, ≤,

U, ∩, ∪, →, ∅, X where Y → Z = {a ∈ X | a ≤ b ∈ Y ⇒ b ∈ Z};

• 3. letting O be the set of open subsets of R with the usual topology,

O, ∩, ∪, →, ∅, R where Y → Z = int(Y c ∪ Z).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30

Equational Consequence

For any set of equations Σ ∪ {α ≈ β} over the language of Heyting algebras with variables in x, we write Σ | =HA α ≈ β

George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30

Equational Consequence

For any set of equations Σ ∪ {α ≈ β} over the language of Heyting algebras with variables in x, we write Σ | =HA α ≈ β if for any homomorphism e from the term algebra over x to some A ∈ HA, e(γ) = e(δ) for all γ ≈ δ ∈ Σ = ⇒ e(α) = e(β).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30

Equivalence

Theorem

HA is an equivalent algebraic semantics for IL

George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30

Equivalence

Theorem

HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ ⊤ and ρ(α ≈ β) = (α → β) ∧ (β → α).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30

Equivalence

Theorem

HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ ⊤ and ρ(α ≈ β) = (α → β) ∧ (β → α). (a) For any set of formulas T ∪ {α}, T ⊢IL α ⇐ ⇒ τ[T] | =HA τ(α).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30

Equivalence

Theorem

HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ ⊤ and ρ(α ≈ β) = (α → β) ∧ (β → α). (a) For any set of formulas T ∪ {α}, T ⊢IL α ⇐ ⇒ τ[T] | =HA τ(α). (b) For any set of equations Σ ∪ {α ≈ β}, Σ | =HA α ≈ β ⇐ ⇒ ρ[T] ⊢IL ρ(α ≈ β).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30

Equivalence

Theorem

HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ ⊤ and ρ(α ≈ β) = (α → β) ∧ (β → α). (a) For any set of formulas T ∪ {α}, T ⊢IL α ⇐ ⇒ τ[T] | =HA τ(α). (b) For any set of equations Σ ∪ {α ≈ β}, Σ | =HA α ≈ β ⇐ ⇒ ρ[T] ⊢IL ρ(α ≈ β). (c) α ⊢IL ρ(τ(α)) and ρ(τ(α)) ⊢IL α.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30

Equivalence

Theorem

HA is an equivalent algebraic semantics for IL with transformers τ(α) = α ≈ ⊤ and ρ(α ≈ β) = (α → β) ∧ (β → α). (a) For any set of formulas T ∪ {α}, T ⊢IL α ⇐ ⇒ τ[T] | =HA τ(α). (b) For any set of equations Σ ∪ {α ≈ β}, Σ | =HA α ≈ β ⇐ ⇒ ρ[T] ⊢IL ρ(α ≈ β). (c) α ⊢IL ρ(τ(α)) and ρ(τ(α)) ⊢IL α. (d) α ≈ β | =HA τ(ρ(α ≈ β)) and τ(ρ(α ≈ β)) | =HA α ≈ β.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30

Sequents

A sequent is an ordered pair consisting of a finite multiset of formulas Γ and a formula α, written Γ ⇒ α.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 30

Sequents

A sequent is an ordered pair consisting of a finite multiset of formulas Γ and a formula α, written Γ ⇒ α. We typically write Γ, Π for the multiset sum of Γ and Π, and omit brackets.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 30

Sequents

A sequent is an ordered pair consisting of a finite multiset of formulas Γ and a formula α, written Γ ⇒ α. We typically write Γ, Π for the multiset sum of Γ and Π, and omit brackets. A sequent calculus GL consists of a set of rules with instances S1 . . . Sn S where S, S1, . . . , Sn are sequents

George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 30

Sequents

A sequent is an ordered pair consisting of a finite multiset of formulas Γ and a formula α, written Γ ⇒ α. We typically write Γ, Π for the multiset sum of Γ and Π, and omit brackets. A sequent calculus GL consists of a set of rules with instances S1 . . . Sn S where S, S1, . . . , Sn are sequents A GL-derivation of a sequent S is a finite tree of sequents with root S built using the rules of GL; if such a derivation exists, we write ⊢GL S.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 30

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms Γ, α ⇒ α

(id)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms Γ, α ⇒ α

(id)

Left Operation Rules Right Operation Rules Γ, ⊥ ⇒ δ

(⊥⇒)

Γ ⇒ ⊤

(⇒⊤)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms Γ, α ⇒ α

(id)

Left Operation Rules Right Operation Rules Γ, ⊥ ⇒ δ

(⊥⇒)

Γ ⇒ ⊤

(⇒⊤)

Γ, α, β ⇒ δ Γ, α ∧ β ⇒ δ

(∧⇒)

Γ ⇒ α Γ ⇒ β Γ ⇒ α ∧ β

(⇒∧)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms Γ, α ⇒ α

(id)

Left Operation Rules Right Operation Rules Γ, ⊥ ⇒ δ

(⊥⇒)

Γ ⇒ ⊤

(⇒⊤)

Γ, α, β ⇒ δ Γ, α ∧ β ⇒ δ

(∧⇒)

Γ ⇒ α Γ ⇒ β Γ ⇒ α ∧ β

(⇒∧)

Γ, α ⇒ δ Γ, β ⇒ δ Γ, α ∨ β ⇒ δ

(∨⇒)

Γ ⇒ α Γ ⇒ α ∨ β

(⇒∨)l

Γ ⇒ β Γ ⇒ α ∨ β

(⇒∨)r

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms Γ, α ⇒ α

(id)

Left Operation Rules Right Operation Rules Γ, ⊥ ⇒ δ

(⊥⇒)

Γ ⇒ ⊤

(⇒⊤)

Γ, α, β ⇒ δ Γ, α ∧ β ⇒ δ

(∧⇒)

Γ ⇒ α Γ ⇒ β Γ ⇒ α ∧ β

(⇒∧)

Γ, α ⇒ δ Γ, β ⇒ δ Γ, α ∨ β ⇒ δ

(∨⇒)

Γ ⇒ α Γ ⇒ α ∨ β

(⇒∨)l

Γ ⇒ β Γ ⇒ α ∨ β

(⇒∨)r

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

Γ, α ⇒ β Γ ⇒ α → β

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Sequent Calculus GIL for Intuitionistic Logic

Identity Axioms Cut Rule Γ, α ⇒ α

(id)

Γ ⇒ α Π, α ⇒ δ Γ, Π ⇒ δ

(cut)

Left Operation Rules Right Operation Rules Γ, ⊥ ⇒ δ

(⊥⇒)

Γ ⇒ ⊤

(⇒⊤)

Γ, α, β ⇒ δ Γ, α ∧ β ⇒ δ

(∧⇒)

Γ ⇒ α Γ ⇒ β Γ ⇒ α ∧ β

(⇒∧)

Γ, α ⇒ δ Γ, β ⇒ δ Γ, α ∨ β ⇒ δ

(∨⇒)

Γ ⇒ α Γ ⇒ α ∨ β

(⇒∨)l

Γ ⇒ β Γ ⇒ α ∨ β

(⇒∨)r

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

Γ, α ⇒ β Γ ⇒ α → β

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

An Example Derivation

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

An Example Derivation

(α → β) ∧ (α ∨ γ) ⇒ β ∨ γ

(∧⇒)

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

An Example Derivation

α → β, α ∨ γ ⇒ β ∨ γ

(∨⇒)

(α → β) ∧ (α ∨ γ) ⇒ β ∨ γ

(∧⇒)

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

An Example Derivation

α → β, α ⇒ β ∨ γ

(→⇒)

α → β, α ∨ γ ⇒ β ∨ γ

(∨⇒)

(α → β) ∧ (α ∨ γ) ⇒ β ∨ γ

(∧⇒)

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

An Example Derivation

α → β, α ⇒ α

(id)

α → β, α ⇒ β ∨ γ

(→⇒)

α → β, α ∨ γ ⇒ β ∨ γ

(∨⇒)

(α → β) ∧ (α ∨ γ) ⇒ β ∨ γ

(∧⇒)

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

An Example Derivation

α → β, α ⇒ α

(id)

β, α ⇒ β ∨ γ

(⇒∨)l

α → β, α ⇒ β ∨ γ

(→⇒)

α → β, α ∨ γ ⇒ β ∨ γ

(∨⇒)

(α → β) ∧ (α ∨ γ) ⇒ β ∨ γ

(∧⇒)

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

An Example Derivation

α → β, α ⇒ α

(id)

β, α ⇒ β

(id)

β, α ⇒ β ∨ γ

(⇒∨)l

α → β, α ⇒ β ∨ γ

(→⇒)

α → β, α ∨ γ ⇒ β ∨ γ

(∨⇒)

(α → β) ∧ (α ∨ γ) ⇒ β ∨ γ

(∧⇒)

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

An Example Derivation

α → β, α ⇒ α

(id)

β, α ⇒ β

(id)

β, α ⇒ β ∨ γ

(⇒∨)l

α → β, α ⇒ β ∨ γ

(→⇒)

α → β, γ ⇒ β ∨ γ

(⇒∨)r

α → β, α ∨ γ ⇒ β ∨ γ

(∨⇒)

(α → β) ∧ (α ∨ γ) ⇒ β ∨ γ

(∧⇒)

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

An Example Derivation

α → β, α ⇒ α

(id)

β, α ⇒ β

(id)

β, α ⇒ β ∨ γ

(⇒∨)l

α → β, α ⇒ β ∨ γ

(→⇒)

α → β, γ ⇒ γ

(id)

α → β, γ ⇒ β ∨ γ

(⇒∨)r

α → β, α ∨ γ ⇒ β ∨ γ

(∨⇒)

(α → β) ∧ (α ∨ γ) ⇒ β ∨ γ

(∧⇒)

⇒ ((α → β) ∧ (α ∨ γ)) → (β ∨ γ)

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 30

Soundness and Completeness

Theorem

For any formulas α1, . . . , αn, β: ⊢GIL α1, . . . , αn ⇒ β ⇐ ⇒ {α1, . . . , αn} ⊢IL β.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

Soundness and Completeness

Theorem

For any formulas α1, . . . , αn, β: ⊢GIL α1, . . . , αn ⇒ β ⇐ ⇒ {α1, . . . , αn} ⊢IL β.

Proof.

(⇒) It suffices to check that the rules of GIL preserve derivability in IL, e.g., Γ ∪ {α} ⊢IL δ and Γ ∪ {β} ⊢IL δ = ⇒ Γ ∪ {α ∨ β} ⊢IL δ.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

Soundness and Completeness

Theorem

For any formulas α1, . . . , αn, β: ⊢GIL α1, . . . , αn ⇒ β ⇐ ⇒ {α1, . . . , αn} ⊢IL β.

Proof.

(⇒) It suffices to check that the rules of GIL preserve derivability in IL, e.g., Γ ∪ {α} ⊢IL δ and Γ ∪ {β} ⊢IL δ = ⇒ Γ ∪ {α ∨ β} ⊢IL δ. (⇐) It suffices to check that the axioms of IL are GIL-derivable and that (using the cut rule!) modus ponens preserves GIL-derivability.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

Cut Elimination

Theorem (Gentzen 1935)

Any GIL-derivable sequent is cut-free GIL-derivable.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 30

Cut Elimination

Theorem (Gentzen 1935)

Any GIL-derivable sequent is cut-free GIL-derivable. Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g.. . .

George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 30

Cut Elimination

Theorem (Gentzen 1935)

Any GIL-derivable sequent is cut-free GIL-derivable. Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g.. . .

. . . Γ ⇒ δ Π, δ ⇒ δ (id) Γ, Π ⇒ δ (cut)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 30

Cut Elimination

Theorem (Gentzen 1935)

Any GIL-derivable sequent is cut-free GIL-derivable. Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g.. . .

. . . Γ ⇒ δ Π, δ ⇒ δ (id) Γ, Π ⇒ δ (cut)

= ⇒

. . . Γ, Π ⇒ δ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 30

Cut Elimination

Theorem (Gentzen 1935)

Any GIL-derivable sequent is cut-free GIL-derivable. Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g.. . .

. . . Γ ⇒ δ Π, δ ⇒ δ (id) Γ, Π ⇒ δ (cut)

= ⇒

. . . Γ, Π ⇒ δ . . . Γ ⇒ α Γ ⇒ α ∨ β (⇒∨)l . . . Π, α ⇒ δ . . . Π, β ⇒ δ Π, α ∨ β ⇒ δ (∨⇒) Γ, Π ⇒ δ (cut)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 30

Cut Elimination

Theorem (Gentzen 1935)

Any GIL-derivable sequent is cut-free GIL-derivable. Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g.. . .

. . . Γ ⇒ δ Π, δ ⇒ δ (id) Γ, Π ⇒ δ (cut)

= ⇒

. . . Γ, Π ⇒ δ . . . Γ ⇒ α Γ ⇒ α ∨ β (⇒∨)l . . . Π, α ⇒ δ . . . Π, β ⇒ δ Π, α ∨ β ⇒ δ (∨⇒) Γ, Π ⇒ δ (cut)

= ⇒

. . . Γ ⇒ α . . . Π, α ⇒ δ Γ, Π ⇒ δ (cut)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 30

Cut Elimination

Theorem (Gentzen 1935)

Any GIL-derivable sequent is cut-free GIL-derivable. Proof idea. We push uppermost cuts upwards in GIL-derivations until they reach axioms and disappear, e.g.. . .

. . . Γ ⇒ δ Π, δ ⇒ δ (id) Γ, Π ⇒ δ (cut)

= ⇒

. . . Γ, Π ⇒ δ . . . Γ ⇒ α Γ ⇒ α ∨ β (⇒∨)l . . . Π, α ⇒ δ . . . Π, β ⇒ δ Π, α ∨ β ⇒ δ (∨⇒) Γ, Π ⇒ δ (cut)

= ⇒

. . . Γ ⇒ α . . . Π, α ⇒ δ Γ, Π ⇒ δ (cut)

Corollary (Gentzen 1935)

Intuitionistic propositional logic is decidable.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 30

Craig Interpolation for Intuitionistic Logic

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α ⊢IL β, then there exists a formula γ(y) such that α ⊢IL γ and γ ⊢IL β.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 30

Craig Interpolation for Intuitionistic Logic

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α ⊢IL β, then there exists a formula γ(y) such that α ⊢IL γ and γ ⊢IL β. Proof Idea. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z),

George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 30

Craig Interpolation for Intuitionistic Logic

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α ⊢IL β, then there exists a formula γ(y) such that α ⊢IL γ and γ ⊢IL β. Proof Idea. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z), ⊢GIL Σ, Π ⇒ δ = ⇒ there exists a formula γ(y) such that ⊢GIL Σ ⇒ γ and ⊢GIL Π, γ ⇒ δ,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 30

Craig Interpolation for Intuitionistic Logic

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α ⊢IL β, then there exists a formula γ(y) such that α ⊢IL γ and γ ⊢IL β. Proof Idea. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z), ⊢GIL Σ, Π ⇒ δ = ⇒ there exists a formula γ(y) such that ⊢GIL Σ ⇒ γ and ⊢GIL Π, γ ⇒ δ, by induction on the height of a cut-free GIL-derivation of Σ, Π ⇒ δ.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 30

Craig Interpolation for Intuitionistic Logic

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α ⊢IL β, then there exists a formula γ(y) such that α ⊢IL γ and γ ⊢IL β. Proof Idea. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z), ⊢GIL Σ, Π ⇒ δ = ⇒ there exists a formula γ(y) such that ⊢GIL Σ ⇒ γ and ⊢GIL Π, γ ⇒ δ, by induction on the height of a cut-free GIL-derivation of Σ, Π ⇒ δ. Base case. E.g., if δ ∈ Σ, let γ = δ;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 30

Craig Interpolation for Intuitionistic Logic

Theorem (Schütte 1962)

If α(x, y) and β(y, z) are formulas such that α ⊢IL β, then there exists a formula γ(y) such that α ⊢IL γ and γ ⊢IL β. Proof Idea. We prove that for any sequent Σ(x, y), Π(y, z) ⇒ δ(y, z), ⊢GIL Σ, Π ⇒ δ = ⇒ there exists a formula γ(y) such that ⊢GIL Σ ⇒ γ and ⊢GIL Π, γ ⇒ δ, by induction on the height of a cut-free GIL-derivation of Σ, Π ⇒ δ. Base case. E.g., if δ ∈ Σ, let γ = δ; if δ ∈ Π, let γ = ⊤.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 30

Craig Interpolation for Intuitionistic Logic

Inductive step. E.g., if Σ is Σ′, α → β and the derivation ends with

. . . Σ′, α → β, Π ⇒ α . . . Σ′, β, Π ⇒ δ Σ′, α → β, Π ⇒ δ

(→⇒)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 30

Craig Interpolation for Intuitionistic Logic

Inductive step. E.g., if Σ is Σ′, α → β and the derivation ends with

. . . Σ′, α → β, Π ⇒ α . . . Σ′, β, Π ⇒ δ Σ′, α → β, Π ⇒ δ

(→⇒)

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable:

George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 30

Craig Interpolation for Intuitionistic Logic

Inductive step. E.g., if Σ is Σ′, α → β and the derivation ends with

. . . Σ′, α → β, Π ⇒ α . . . Σ′, β, Π ⇒ δ Σ′, α → β, Π ⇒ δ

(→⇒)

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable: Π ⇒ γ1; Σ′, α → β, γ1 ⇒ α;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 30

Craig Interpolation for Intuitionistic Logic

Inductive step. E.g., if Σ is Σ′, α → β and the derivation ends with

. . . Σ′, α → β, Π ⇒ α . . . Σ′, β, Π ⇒ δ Σ′, α → β, Π ⇒ δ

(→⇒)

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable: Π ⇒ γ1; Σ′, α → β, γ1 ⇒ α; Σ′, β ⇒ γ2; and Π, γ2 ⇒ δ.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 30

Craig Interpolation for Intuitionistic Logic

Inductive step. E.g., if Σ is Σ′, α → β and the derivation ends with

. . . Σ′, α → β, Π ⇒ α . . . Σ′, β, Π ⇒ δ Σ′, α → β, Π ⇒ δ

(→⇒)

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable: Π ⇒ γ1; Σ′, α → β, γ1 ⇒ α; Σ′, β ⇒ γ2; and Π, γ2 ⇒ δ. We obtain an interpolant γ1 → γ2 with derivations ending with

George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 30

Craig Interpolation for Intuitionistic Logic

Inductive step. E.g., if Σ is Σ′, α → β and the derivation ends with

. . . Σ′, α → β, Π ⇒ α . . . Σ′, β, Π ⇒ δ Σ′, α → β, Π ⇒ δ

(→⇒)

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable: Π ⇒ γ1; Σ′, α → β, γ1 ⇒ α; Σ′, β ⇒ γ2; and Π, γ2 ⇒ δ. We obtain an interpolant γ1 → γ2 with derivations ending with

. . . Σ′, α → β, γ1 ⇒ α . . . Σ′, β, γ1 ⇒ γ2 Σ′, α → β, γ1 ⇒ γ2

(→⇒)

Σ′, α → β ⇒ γ1 → γ2

(⇒→)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 30

Craig Interpolation for Intuitionistic Logic

Inductive step. E.g., if Σ is Σ′, α → β and the derivation ends with

. . . Σ′, α → β, Π ⇒ α . . . Σ′, β, Π ⇒ δ Σ′, α → β, Π ⇒ δ

(→⇒)

then by the induction hypothesis twice, there exist formulas γ1(y), γ2(y) such that the following sequents are GIL-derivable: Π ⇒ γ1; Σ′, α → β, γ1 ⇒ α; Σ′, β ⇒ γ2; and Π, γ2 ⇒ δ. We obtain an interpolant γ1 → γ2 with derivations ending with

. . . Σ′, α → β, γ1 ⇒ α . . . Σ′, β, γ1 ⇒ γ2 Σ′, α → β, γ1 ⇒ γ2

(→⇒)

Σ′, α → β ⇒ γ1 → γ2

(⇒→)

. . . Π, γ1 → γ2 ⇒ γ1 . . . Π, γ2 ⇒ δ Π, γ1 → γ2 ⇒ δ

(→⇒)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 30

An Algebraic Consequence

Corollary (Day 1972)

HA admits the amalgamation property;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

An Algebraic Consequence

B1

A

1 2

B2

Corollary (Day 1972)

HA admits the amalgamation property; that is, for any A, B1, B2 ∈ HA, and embeddings σ1 : A → B1, σ2 : A → B2,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

An Algebraic Consequence

B1

C A

1 2

B2

Corollary (Day 1972)

HA admits the amalgamation property; that is, for any A, B1, B2 ∈ HA, and embeddings σ1 : A → B1, σ2 : A → B2, there exist C ∈ HA

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

An Algebraic Consequence

B1

C A

1 !1 2 !2

B2

Corollary (Day 1972)

HA admits the amalgamation property; that is, for any A, B1, B2 ∈ HA, and embeddings σ1 : A → B1, σ2 : A → B2, there exist C ∈ HA and embeddings τ1 : B1 → C and τ2 : B2 → C

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

An Algebraic Consequence

B1

C A

1 !1 2 !2

B2

Corollary (Day 1972)

HA admits the amalgamation property; that is, for any A, B1, B2 ∈ HA, and embeddings σ1 : A → B1, σ2 : A → B2, there exist C ∈ HA and embeddings τ1 : B1 → C and τ2 : B2 → C such that τ1σ1 = τ2σ2.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

Uniform Interpolation in Intuitionistic Logic

Theorem (Pitts 1992)

For any formula α(x, y) of intuitionistic propositional logic, there exist formulas αL(y) and αR(y)

A.M. Pitts. On an interpretation of second-order quantification in first-order intuitionistic propositional logic. Journal of Symbolic Logic 57 (1992), 33–52.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 30

Uniform Interpolation in Intuitionistic Logic

Theorem (Pitts 1992)

For any formula α(x, y) of intuitionistic propositional logic, there exist formulas αL(y) and αR(y) such that for any formula β(y, z), α(x, y) ⊢IL β(y, z) ⇐ ⇒ αR(y) ⊢IL β(y, z)

A.M. Pitts. On an interpretation of second-order quantification in first-order intuitionistic propositional logic. Journal of Symbolic Logic 57 (1992), 33–52.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 30

Uniform Interpolation in Intuitionistic Logic

Theorem (Pitts 1992)

For any formula α(x, y) of intuitionistic propositional logic, there exist formulas αL(y) and αR(y) such that for any formula β(y, z), α(x, y) ⊢IL β(y, z) ⇐ ⇒ αR(y) ⊢IL β(y, z) β(y, z) ⊢IL α(x, y) ⇐ ⇒ β(y, z) ⊢IL αL(y).

A.M. Pitts. On an interpretation of second-order quantification in first-order intuitionistic propositional logic. Journal of Symbolic Logic 57 (1992), 33–52.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic by removing the cut rule from GIL and replacing the implication left rule

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic by removing the cut rule from GIL and replacing the implication left rule

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

with the decomposition rules

Γ ⇒ δ Γ, ⊥ → β ⇒ δ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic by removing the cut rule from GIL and replacing the implication left rule

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

with the decomposition rules

Γ ⇒ δ Γ, ⊥ → β ⇒ δ Γ, x, β ⇒ δ Γ, x, x → β ⇒ δ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic by removing the cut rule from GIL and replacing the implication left rule

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

with the decomposition rules

Γ ⇒ δ Γ, ⊥ → β ⇒ δ Γ, x, β ⇒ δ Γ, x, x → β ⇒ δ Γ, α1 → (α2 → β) ⇒ δ Γ, (α1 ∧ α2) → β ⇒ δ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic by removing the cut rule from GIL and replacing the implication left rule

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

with the decomposition rules

Γ ⇒ δ Γ, ⊥ → β ⇒ δ Γ, x, β ⇒ δ Γ, x, x → β ⇒ δ Γ, α1 → (α2 → β) ⇒ δ Γ, (α1 ∧ α2) → β ⇒ δ Γ, β ⇒ δ Γ, ⊤ → β ⇒ δ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic by removing the cut rule from GIL and replacing the implication left rule

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

with the decomposition rules

Γ ⇒ δ Γ, ⊥ → β ⇒ δ Γ, x, β ⇒ δ Γ, x, x → β ⇒ δ Γ, α1 → (α2 → β) ⇒ δ Γ, (α1 ∧ α2) → β ⇒ δ Γ, β ⇒ δ Γ, ⊤ → β ⇒ δ Γ, α1 → β, α2 → β ⇒ δ Γ, (α1 ∨ α2) → β ⇒ δ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic by removing the cut rule from GIL and replacing the implication left rule

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

with the decomposition rules

Γ ⇒ δ Γ, ⊥ → β ⇒ δ Γ, x, β ⇒ δ Γ, x, x → β ⇒ δ Γ, α1 → (α2 → β) ⇒ δ Γ, (α1 ∧ α2) → β ⇒ δ Γ, β ⇒ δ Γ, ⊤ → β ⇒ δ Γ, α1 → β, α2 → β ⇒ δ Γ, (α1 ∨ α2) → β ⇒ δ Γ, α2 → β ⇒ α1 → α2 Γ, β ⇒ δ Γ, (α1 → α2) → β ⇒ δ

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

A Terminating Sequent Calculus

We obtain a terminating sequent calculus GIL∗ for intuitionistic logic by removing the cut rule from GIL and replacing the implication left rule

Γ, α → β ⇒ α Γ, β ⇒ δ Γ, α → β ⇒ δ

(→⇒)

with the decomposition rules

Γ ⇒ δ Γ, ⊥ → β ⇒ δ Γ, x, β ⇒ δ Γ, x, x → β ⇒ δ Γ, α1 → (α2 → β) ⇒ δ Γ, (α1 ∧ α2) → β ⇒ δ Γ, β ⇒ δ Γ, ⊤ → β ⇒ δ Γ, α1 → β, α2 → β ⇒ δ Γ, (α1 ∨ α2) → β ⇒ δ Γ, α2 → β ⇒ α1 → α2 Γ, β ⇒ δ Γ, (α1 → α2) → β ⇒ δ

Theorem (Dyckhoff 1992)

A sequent is derivable in GIL∗ if and only if it is derivable in GIL.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 30

Weighing Formulas

The weight wt(α) of a formula α is defined inductively by

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

Weighing Formulas

The weight wt(α) of a formula α is defined inductively by wt(x) = wt(⊥) = wt(⊤) = 1;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

Weighing Formulas

The weight wt(α) of a formula α is defined inductively by wt(x) = wt(⊥) = wt(⊤) = 1; wt(α ∨ β) = wt(α → β) = wt(α) + wt(β) + 1;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

Weighing Formulas

The weight wt(α) of a formula α is defined inductively by wt(x) = wt(⊥) = wt(⊤) = 1; wt(α ∨ β) = wt(α → β) = wt(α) + wt(β) + 1; wt(α ∧ β) = wt(α) + wt(β) + 2,

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

Weighing Formulas

The weight wt(α) of a formula α is defined inductively by wt(x) = wt(⊥) = wt(⊤) = 1; wt(α ∨ β) = wt(α → β) = wt(α) + wt(β) + 1; wt(α ∧ β) = wt(α) + wt(β) + 2, yielding a well-ordering ≺ on formulas α ≺ β :⇐ ⇒ wt(α) < wt(β).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 30

Weighing Sequents

We obtain also a well-ordering on multisets of formulas

George Metcalfe (University of Bern) Uniform Interpolation August 2018 24 / 30

Weighing Sequents

We obtain also a well-ordering on multisets of formulas Γ ≺ Π :⇐ ⇒ Γ = Γ′, ∆ and Π = Π′, ∆ with Π′ = ∅ and each α ∈ Γ′ is ≺-smaller than some β ∈ Π′

George Metcalfe (University of Bern) Uniform Interpolation August 2018 24 / 30

Weighing Sequents

We obtain also a well-ordering on multisets of formulas Γ ≺ Π :⇐ ⇒ Γ = Γ′, ∆ and Π = Π′, ∆ with Π′ = ∅ and each α ∈ Γ′ is ≺-smaller than some β ∈ Π′ and on sequents by defining Γ ⇒ α ≺ Π ⇒ β :⇐ ⇒ Γ, α ≺ Π, β.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 24 / 30

Weighing Sequents

We obtain also a well-ordering on multisets of formulas Γ ≺ Π :⇐ ⇒ Γ = Γ′, ∆ and Π = Π′, ∆ with Π′ = ∅ and each α ∈ Γ′ is ≺-smaller than some β ∈ Π′ and on sequents by defining Γ ⇒ α ≺ Π ⇒ β :⇐ ⇒ Γ, α ≺ Π, β. For each rule of GIL∗, the premises are all ≺-smaller than the conclusion, so the calculus terminates.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 24 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

Lemma

For any sequent Γ ⇒ α and variable x, there exist formulas Ex(Γ) and Ax(Γ; α) such that

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

Lemma

For any sequent Γ ⇒ α and variable x, there exist formulas Ex(Γ) and Ax(Γ; α) such that (i) Var(Ex(Γ)) ⊆ Var(Γ)\{x} and Var(Ax(Γ; α)) ⊆ Var(Γ, α)\{x};

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

Lemma

For any sequent Γ ⇒ α and variable x, there exist formulas Ex(Γ) and Ax(Γ; α) such that (i) Var(Ex(Γ)) ⊆ Var(Γ)\{x} and Var(Ax(Γ; α)) ⊆ Var(Γ, α)\{x}; (ii) ⊢GIL Γ ⇒ Ex(Γ) and ⊢GIL Γ, Ax(Γ; α) ⇒ α;

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

Lemma

For any sequent Γ ⇒ α and variable x, there exist formulas Ex(Γ) and Ax(Γ; α) such that (i) Var(Ex(Γ)) ⊆ Var(Γ)\{x} and Var(Ax(Γ; α)) ⊆ Var(Γ, α)\{x}; (ii) ⊢GIL Γ ⇒ Ex(Γ) and ⊢GIL Γ, Ax(Γ; α) ⇒ α; (iii) whenever ⊢GIL Π, Γ ⇒ α and x ∈ Var(Π),

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

Lemma

For any sequent Γ ⇒ α and variable x, there exist formulas Ex(Γ) and Ax(Γ; α) such that (i) Var(Ex(Γ)) ⊆ Var(Γ)\{x} and Var(Ax(Γ; α)) ⊆ Var(Γ, α)\{x}; (ii) ⊢GIL Γ ⇒ Ex(Γ) and ⊢GIL Γ, Ax(Γ; α) ⇒ α; (iii) whenever ⊢GIL Π, Γ ⇒ α and x ∈ Var(Π), ⊢GIL Π, Ex(Γ) ⇒ α if x ∈ Var(α)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

Lemma

For any sequent Γ ⇒ α and variable x, there exist formulas Ex(Γ) and Ax(Γ; α) such that (i) Var(Ex(Γ)) ⊆ Var(Γ)\{x} and Var(Ax(Γ; α)) ⊆ Var(Γ, α)\{x}; (ii) ⊢GIL Γ ⇒ Ex(Γ) and ⊢GIL Γ, Ax(Γ; α) ⇒ α; (iii) whenever ⊢GIL Π, Γ ⇒ α and x ∈ Var(Π), ⊢GIL Π, Ex(Γ) ⇒ α if x ∈ Var(α) and ⊢GIL Π, Ex(Γ) ⇒ Ax(Γ; α).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

Lemma

For any sequent Γ ⇒ α and variable x, there exist formulas Ex(Γ) and Ax(Γ; α) such that (i) Var(Ex(Γ)) ⊆ Var(Γ)\{x} and Var(Ax(Γ; α)) ⊆ Var(Γ, α)\{x}; (ii) ⊢GIL Γ ⇒ Ex(Γ) and ⊢GIL Γ, Ax(Γ; α) ⇒ α; (iii) whenever ⊢GIL Π, Γ ⇒ α and x ∈ Var(Π), ⊢GIL Π, Ex(Γ) ⇒ α if x ∈ Var(α) and ⊢GIL Π, Ex(Γ) ⇒ Ax(Γ; α). Pitts’ theorem then follows by defining for any formula α(x, y), αL(y) = Ax(∅; α)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

The Key Lemma

Let Var(Γ) denote the variables occurring in a finite multiset of formulas Γ.

Lemma

For any sequent Γ ⇒ α and variable x, there exist formulas Ex(Γ) and Ax(Γ; α) such that (i) Var(Ex(Γ)) ⊆ Var(Γ)\{x} and Var(Ax(Γ; α)) ⊆ Var(Γ, α)\{x}; (ii) ⊢GIL Γ ⇒ Ex(Γ) and ⊢GIL Γ, Ax(Γ; α) ⇒ α; (iii) whenever ⊢GIL Π, Γ ⇒ α and x ∈ Var(Π), ⊢GIL Π, Ex(Γ) ⇒ α if x ∈ Var(α) and ⊢GIL Π, Ex(Γ) ⇒ Ax(Γ; α). Pitts’ theorem then follows by defining for any formula α(x, y), αL(y) = Ax(∅; α) and αR(y) = Az(∅; Ax(∅; α → z) → z).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 25 / 30

Proof Idea

The formulas Ex(Γ) and Ax(Γ; α) are defined simultaneously by induction

• ver the well-ordering ≺

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Proof Idea

The formulas Ex(Γ) and Ax(Γ; α) are defined simultaneously by induction

• ver the well-ordering ≺ via finite sets of formulas Ex(Γ) and Ax(Γ; α):

Ex(Γ) :=

• Ex(Γ)

and Ax(Γ; α) :=

• Ax(Γ; α).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Proof Idea

The formulas Ex(Γ) and Ax(Γ; α) are defined simultaneously by induction

• ver the well-ordering ≺ via finite sets of formulas Ex(Γ) and Ax(Γ; α):

Ex(Γ) :=

• Ex(Γ)

and Ax(Γ; α) :=

• Ax(Γ; α).

The elements of Ex(Γ) and Ax(Γ; α) are given by clauses

Γ matches Ex(Γ) contains

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Proof Idea

The formulas Ex(Γ) and Ax(Γ; α) are defined simultaneously by induction

• ver the well-ordering ≺ via finite sets of formulas Ex(Γ) and Ax(Γ; α):

Ex(Γ) :=

• Ex(Γ)

and Ax(Γ; α) :=

• Ax(Γ; α).

The elements of Ex(Γ) and Ax(Γ; α) are given by clauses

Γ matches Ex(Γ) contains Γ′, y Ex(Γ′) ∧ y

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Proof Idea

The formulas Ex(Γ) and Ax(Γ; α) are defined simultaneously by induction

• ver the well-ordering ≺ via finite sets of formulas Ex(Γ) and Ax(Γ; α):

Ex(Γ) :=

• Ex(Γ)

and Ax(Γ; α) :=

• Ax(Γ; α).

The elements of Ex(Γ) and Ax(Γ; α) are given by clauses

Γ matches Ex(Γ) contains Γ′, y Ex(Γ′) ∧ y Γ′, β1 ∧ β2 Ex(Γ′, β1, β2)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Proof Idea

The formulas Ex(Γ) and Ax(Γ; α) are defined simultaneously by induction

• ver the well-ordering ≺ via finite sets of formulas Ex(Γ) and Ax(Γ; α):

Ex(Γ) :=

• Ex(Γ)

and Ax(Γ; α) :=

• Ax(Γ; α).

The elements of Ex(Γ) and Ax(Γ; α) are given by clauses

Γ matches Ex(Γ) contains Γ′, y Ex(Γ′) ∧ y Γ′, β1 ∧ β2 Ex(Γ′, β1, β2) Γ′, β1 ∨ β2 Ex(Γ′, β1) ∨ Ex(Γ′, β2)

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Proof Idea

The formulas Ex(Γ) and Ax(Γ; α) are defined simultaneously by induction

• ver the well-ordering ≺ via finite sets of formulas Ex(Γ) and Ax(Γ; α):

Ex(Γ) :=

• Ex(Γ)

and Ax(Γ; α) :=

• Ax(Γ; α).

The elements of Ex(Γ) and Ax(Γ; α) are given by clauses

Γ matches Ex(Γ) contains Γ′, y Ex(Γ′) ∧ y Γ′, β1 ∧ β2 Ex(Γ′, β1, β2) Γ′, β1 ∨ β2 Ex(Γ′, β1) ∨ Ex(Γ′, β2) Γ′, (β1 → β2) → β3 (Ex(Γ′, β2 → β3) → Ax(Γ, β2 → β3; δ1 → β2)) → Ex(Γ′, β3) . . . . . .

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Proof Idea

The formulas Ex(Γ) and Ax(Γ; α) are defined simultaneously by induction

• ver the well-ordering ≺ via finite sets of formulas Ex(Γ) and Ax(Γ; α):

Ex(Γ) :=

• Ex(Γ)

and Ax(Γ; α) :=

• Ax(Γ; α).

The elements of Ex(Γ) and Ax(Γ; α) are given by clauses

Γ matches Ex(Γ) contains Γ′, y Ex(Γ′) ∧ y Γ′, β1 ∧ β2 Ex(Γ′, β1, β2) Γ′, β1 ∨ β2 Ex(Γ′, β1) ∨ Ex(Γ′, β2) Γ′, (β1 → β2) → β3 (Ex(Γ′, β2 → β3) → Ax(Γ, β2 → β3; δ1 → β2)) → Ex(Γ′, β3) . . . . . .

The calculus GIL∗ is then used to check that conditions (i)-(iii) are satisfied.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 26 / 30

Interpolation and Coherence

Right uniform interpolation for intuitionistic logic consists of two parts:

George Metcalfe (University of Bern) Uniform Interpolation August 2018 27 / 30

Interpolation and Coherence

Right uniform interpolation for intuitionistic logic consists of two parts: Craig interpolation: for any α(x, y), β(y, z) satisfying α(x, y) ⊢IL β(y, z), there exists γ(y) such that α(x, y) ⊢IL γ(y) and γ(y) ⊢IL β(y, z);

George Metcalfe (University of Bern) Uniform Interpolation August 2018 27 / 30

Interpolation and Coherence

Right uniform interpolation for intuitionistic logic consists of two parts: Craig interpolation: for any α(x, y), β(y, z) satisfying α(x, y) ⊢IL β(y, z), there exists γ(y) such that α(x, y) ⊢IL γ(y) and γ(y) ⊢IL β(y, z); Coherence: for any α(x, y), there exists αR(y) such that α(x, y) ⊢IL β(y) ⇐ ⇒ αR(y) ⊢IL β(y).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 27 / 30

Interpolation and Coherence

Right uniform interpolation for intuitionistic logic consists of two parts: Craig interpolation: for any α(x, y), β(y, z) satisfying α(x, y) ⊢IL β(y, z), there exists γ(y) such that α(x, y) ⊢IL γ(y) and γ(y) ⊢IL β(y, z); Coherence: for any α(x, y), there exists αR(y) such that α(x, y) ⊢IL β(y) ⇐ ⇒ αR(y) ⊢IL β(y). Coherence corresponds to the fact that every finitely generated subalgebra

• f a finitely presented Heyting algebra is finitely presented.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 27 / 30

A Model-Theoretic Consequence

A first-order theory T ∗ is a model completion of a universal theory T if (a) T and T ∗ entail the same universal sentences; (b) T ∗ admits quantifier elimination.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 28 / 30

A Model-Theoretic Consequence

A first-order theory T ∗ is a model completion of a universal theory T if (a) T and T ∗ entail the same universal sentences; (b) T ∗ admits quantifier elimination. Moreover, T ∗ is then the theory of the existentially closed models for T.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 28 / 30

A Model-Theoretic Consequence

A first-order theory T ∗ is a model completion of a universal theory T if (a) T and T ∗ entail the same universal sentences; (b) T ∗ admits quantifier elimination. Moreover, T ∗ is then the theory of the existentially closed models for T.

Theorem (Ghilardi and Zawadowski 1997)

The first-order theory of Heyting algebras admits a model completion.

• S. Ghilardi and M. Zawadowski.

Sheaves, Games and Model Completions, Kluwer (2002).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 28 / 30

Remarks

Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 29 / 30

Remarks

Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018). There are exactly eight intermediate logics that admit Craig interpolation (Maksimova 1977), and all of these also have uniform interpolation (Ghilardi and Zawadowski 2002).

George Metcalfe (University of Bern) Uniform Interpolation August 2018 29 / 30

Remarks

Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018). There are exactly eight intermediate logics that admit Craig interpolation (Maksimova 1977), and all of these also have uniform interpolation (Ghilardi and Zawadowski 2002). Iemhoff has shown recently that any logic admitting a certain Dyckhoff-style decomposition calculus has uniform interpolation.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 29 / 30

Tomorrow

We present a general framework for studying (uniform) interpolation in algebra and logic.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 30 / 30

Tomorrow

We present a general framework for studying (uniform) interpolation in algebra and logic. We relate deductive interpolation to amalgamation and deductive uniform interpolation to coherence.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 30 / 30

Tomorrow

We present a general framework for studying (uniform) interpolation in algebra and logic. We relate deductive interpolation to amalgamation and deductive uniform interpolation to coherence. We give sufficient conditions for the theory of a variety to admit a model completion.

George Metcalfe (University of Bern) Uniform Interpolation August 2018 30 / 30