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

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 S 1 . . . S n where S , S 1 , . . . , S n are sequents S 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 S 1 . . . S n where S , S 1 , . . . , S n are sequents S 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 ( id ) Γ , α ⇒ α George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Sequent Calculus GIL for Intuitionistic Logic Identity Axioms ( id ) Γ , α ⇒ α 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 ( id ) Γ , α ⇒ α Left Operation Rules Right Operation Rules ( ⊥⇒ ) ( ⇒⊤ ) Γ , ⊥ ⇒ δ Γ ⇒ ⊤ Γ , α, β ⇒ δ Γ ⇒ α Γ ⇒ β ( ∧⇒ ) ( ⇒∧ ) Γ , α ∧ β ⇒ δ Γ ⇒ α ∧ β George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 30

A Sequent Calculus GIL for Intuitionistic Logic Identity Axioms ( id ) Γ , α ⇒ α Left Operation Rules Right Operation Rules ( ⊥⇒ ) ( ⇒⊤ ) Γ , ⊥ ⇒ δ Γ ⇒ ⊤ Γ , α, β ⇒ δ Γ ⇒ α Γ ⇒ β ( ∧⇒ ) ( ⇒∧ ) Γ , α ∧ β ⇒ δ Γ ⇒ α ∧ β Γ , α ⇒ δ Γ , β ⇒ δ Γ ⇒ β Γ ⇒ α ( ∨⇒ ) ( ⇒∨ ) l ( ⇒∨ ) r 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 ( id ) Γ , α ⇒ α Left Operation Rules Right Operation Rules ( ⊥⇒ ) ( ⇒⊤ ) Γ , ⊥ ⇒ δ Γ ⇒ ⊤ Γ , α, β ⇒ δ Γ ⇒ α Γ ⇒ β ( ∧⇒ ) ( ⇒∧ ) Γ , α ∧ β ⇒ δ Γ ⇒ α ∧ β Γ , α ⇒ δ Γ , β ⇒ δ Γ ⇒ β Γ ⇒ α ( ∨⇒ ) ( ⇒∨ ) l ( ⇒∨ ) r Γ , α ∨ β ⇒ δ Γ ⇒ α ∨ β Γ ⇒ α ∨ β Γ , α → β ⇒ α Γ , β ⇒ δ Γ , α ⇒ β ( →⇒ ) ( ⇒→ ) Γ , α → β ⇒ δ 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 Π , α ⇒ δ ( id ) (cut) Γ , α ⇒ α Γ , Π ⇒ δ Left Operation Rules Right Operation Rules ( ⊥⇒ ) ( ⇒⊤ ) Γ , ⊥ ⇒ δ Γ ⇒ ⊤ Γ , α, β ⇒ δ Γ ⇒ α Γ ⇒ β ( ∧⇒ ) ( ⇒∧ ) Γ , α ∧ β ⇒ δ Γ ⇒ α ∧ β Γ , α ⇒ δ Γ , β ⇒ δ Γ ⇒ β Γ ⇒ α ( ∨⇒ ) ( ⇒∨ ) l ( ⇒∨ ) r Γ , α ∨ β ⇒ δ Γ ⇒ α ∨ β Γ ⇒ α ∨ β Γ , α → β ⇒ α Γ , β ⇒ δ Γ , α ⇒ β ( →⇒ ) ( ⇒→ ) Γ , α → β ⇒ δ 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 ( id ) α → β, α ⇒ α ( →⇒ ) 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 ( id ) ( ⇒∨ ) l α → β, α ⇒ α β, α ⇒ β ∨ γ ( →⇒ ) 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 ( id ) β, α ⇒ β ( id ) ( ⇒∨ ) l α → β, α ⇒ α β, α ⇒ β ∨ γ ( →⇒ ) 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 ( id ) β, α ⇒ β ( id ) ( ⇒∨ ) l α → β, α ⇒ α β, α ⇒ β ∨ γ ( →⇒ ) ( ⇒∨ ) r α → β, α ⇒ β ∨ γ α → β, γ ⇒ 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 ( id ) β, α ⇒ β ( id ) ( ⇒∨ ) l ( id ) α → β, α ⇒ α β, α ⇒ β ∨ γ α → β, γ ⇒ γ ( →⇒ ) ( ⇒∨ ) r α → β, α ⇒ β ∨ γ α → β, γ ⇒ 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

Soundness and Completeness Theorem For any formulas α 1 , . . . , α n , β : ⊢ GIL α 1 , . . . , α n ⇒ β ⇐ ⇒ { α 1 , . . . , α n } ⊢ IL β. George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 30

Soundness and Completeness Theorem For any formulas α 1 , . . . , α n , β : ⊢ GIL α 1 , . . . , α n ⇒ β ⇐ ⇒ { α 1 , . . . , α n } ⊢ 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 α 1 , . . . , α n , β : ⊢ GIL α 1 , . . . , α n ⇒ β ⇐ ⇒ { α 1 , . . . , α n } ⊢ 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.. . . . . . Π , δ ⇒ δ ( id ) Γ ⇒ 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.. . . . . . . . Π , δ ⇒ δ ( id ) ⇒ = . Γ ⇒ δ (cut) Γ , Π ⇒ δ 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.. . . . . . . . Π , δ ⇒ δ ( id ) ⇒ = . Γ ⇒ δ (cut) Γ , Π ⇒ δ Γ , Π ⇒ δ . . . . . . . . . Π , α ⇒ δ Π , β ⇒ δ Γ ⇒ α Γ ⇒ α ∨ β ( ⇒∨ ) l ( ∨⇒ ) Π , α ∨ β ⇒ 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.. . . . . . . . Π , δ ⇒ δ ( id ) ⇒ = . Γ ⇒ δ (cut) Γ , Π ⇒ δ Γ , Π ⇒ δ . . . . . . . . . . . . . . . Π , α ⇒ δ Π , β ⇒ δ = ⇒ Γ ⇒ α Γ ⇒ α ∨ β ( ⇒∨ ) l ( ∨⇒ ) Γ ⇒ α Π , α ⇒ δ Π , α ∨ β ⇒ δ (cut) (cut) Γ , Π ⇒ δ 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.. . . . . . . . Π , δ ⇒ δ ( id ) ⇒ = . Γ ⇒ δ (cut) Γ , Π ⇒ δ Γ , Π ⇒ δ . . . . . . . . . . . . . . . Π , α ⇒ δ Π , β ⇒ δ = ⇒ Γ ⇒ α Γ ⇒ α ∨ β ( ⇒∨ ) l ( ∨⇒ ) Γ ⇒ α Π , α ⇒ δ Π , α ∨ β ⇒ δ (cut) (cut) Γ , Π ⇒ δ Γ , Π ⇒ δ 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 ) , there exists a formula γ ( y ) such that ⊢ GIL Σ , Π ⇒ δ = ⇒ ⊢ 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 ) , there exists a formula γ ( y ) such that ⊢ GIL Σ , Π ⇒ δ = ⇒ ⊢ 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 ) , there exists a formula γ ( y ) such that ⊢ GIL Σ , Π ⇒ δ = ⇒ ⊢ 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 ) , there exists a formula γ ( y ) such that ⊢ GIL Σ , Π ⇒ δ = ⇒ ⊢ 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 γ 1 ( y ) , γ 2 ( 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 γ 1 ( y ) , γ 2 ( y ) such that the following sequents are GIL-derivable: Σ ′ , α → β, γ 1 ⇒ α ; Π ⇒ γ 1 ; 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 γ 1 ( y ) , γ 2 ( y ) such that the following sequents are GIL-derivable: Σ ′ , α → β, γ 1 ⇒ α ; Σ ′ , β ⇒ γ 2 ; Π ⇒ γ 1 ; and Π , γ 2 ⇒ 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 γ 1 ( y ) , γ 2 ( y ) such that the following sequents are GIL-derivable: Σ ′ , α → β, γ 1 ⇒ α ; Σ ′ , β ⇒ γ 2 ; Π ⇒ γ 1 ; and Π , γ 2 ⇒ δ. We obtain an interpolant γ 1 → γ 2 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 γ 1 ( y ) , γ 2 ( y ) such that the following sequents are GIL-derivable: Σ ′ , α → β, γ 1 ⇒ α ; Σ ′ , β ⇒ γ 2 ; Π ⇒ γ 1 ; and Π , γ 2 ⇒ δ. We obtain an interpolant γ 1 → γ 2 with derivations ending with . . . . . . Σ ′ , α → β, γ 1 ⇒ α Σ ′ , β, γ 1 ⇒ γ 2 ( →⇒ ) Σ ′ , α → β, γ 1 ⇒ γ 2 ( ⇒→ ) Σ ′ , α → β ⇒ γ 1 → γ 2 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 γ 1 ( y ) , γ 2 ( y ) such that the following sequents are GIL-derivable: Σ ′ , α → β, γ 1 ⇒ α ; Σ ′ , β ⇒ γ 2 ; Π ⇒ γ 1 ; and Π , γ 2 ⇒ δ. We obtain an interpolant γ 1 → γ 2 with derivations ending with . . . . . . . . Σ ′ , α → β, γ 1 ⇒ α Σ ′ , β, γ 1 ⇒ γ 2 . . . . ( →⇒ ) Σ ′ , α → β, γ 1 ⇒ γ 2 Π , γ 1 → γ 2 ⇒ γ 1 Π , γ 2 ⇒ δ ( ⇒→ ) ( →⇒ ) Σ ′ , α → β ⇒ γ 1 → γ 2 Π , γ 1 → γ 2 ⇒ 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 B 1 A 1 2 B 2 Corollary (Day 1972) HA admits the amalgamation property ; that is, for any A , B 1 , B 2 ∈ HA , and embeddings σ 1 : A → B 1 , σ 2 : A → B 2 , George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

An Algebraic Consequence B 1 C A 1 2 B 2 Corollary (Day 1972) HA admits the amalgamation property ; that is, for any A , B 1 , B 2 ∈ HA , and embeddings σ 1 : A → B 1 , σ 2 : A → B 2 , there exist C ∈ HA George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

An Algebraic Consequence B 1 C A 1 ! 1 2 ! 2 B 2 Corollary (Day 1972) HA admits the amalgamation property ; that is, for any A , B 1 , B 2 ∈ HA , and embeddings σ 1 : A → B 1 , σ 2 : A → B 2 , there exist C ∈ HA and embeddings τ 1 : B 1 → C and τ 2 : B 2 → C George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 30

An Algebraic Consequence B 1 C A 1 ! 1 2 ! 2 B 2 Corollary (Day 1972) HA admits the amalgamation property ; that is, for any A , B 1 , B 2 ∈ HA , and embeddings σ 1 : A → B 1 , σ 2 : A → B 2 , there exist C ∈ HA and embeddings τ 1 : B 1 → C and τ 2 : B 2 → C such that τ 1 σ 1 = τ 2 σ 2 . 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 α L ( y ) and α R ( 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 α L ( y ) and α R ( y ) such that for any formula β ( y , z ) , α R ( y ) ⊢ IL β ( y , z ) α ( x , 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 α L ( y ) and α R ( y ) such that for any formula β ( y , z ) , α R ( y ) ⊢ IL β ( y , z ) α ( x , y ) ⊢ IL β ( y , z ) ⇐ ⇒ β ( y , z ) ⊢ IL α L ( y ) . β ( y , z ) ⊢ IL α ( x , 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 , x , β ⇒ δ Γ ⇒ δ Γ , ⊥ → 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 , α 1 → ( α 2 → β ) ⇒ δ Γ , x , β ⇒ δ Γ ⇒ δ Γ , ⊥ → β ⇒ δ Γ , x , x → β ⇒ δ Γ , ( α 1 ∧ α 2 ) → 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 , α 1 → ( α 2 → β ) ⇒ δ Γ , x , β ⇒ δ Γ ⇒ δ Γ , ⊥ → β ⇒ δ Γ , x , x → β ⇒ δ Γ , ( α 1 ∧ α 2 ) → 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 , α 1 → ( α 2 → β ) ⇒ δ Γ , x , β ⇒ δ Γ ⇒ δ Γ , ⊥ → β ⇒ δ Γ , x , x → β ⇒ δ Γ , ( α 1 ∧ α 2 ) → β ⇒ δ Γ , β ⇒ δ Γ , α 1 → β, α 2 → β ⇒ δ Γ , ⊤ → β ⇒ δ Γ , ( α 1 ∨ α 2 ) → 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 , α 1 → ( α 2 → β ) ⇒ δ Γ , x , β ⇒ δ Γ ⇒ δ Γ , ⊥ → β ⇒ δ Γ , x , x → β ⇒ δ Γ , ( α 1 ∧ α 2 ) → β ⇒ δ Γ , β ⇒ δ Γ , α 1 → β, α 2 → β ⇒ δ Γ , α 2 → β ⇒ α 1 → α 2 Γ , β ⇒ δ Γ , ⊤ → β ⇒ δ Γ , ( α 1 ∨ α 2 ) → β ⇒ δ Γ , ( α 1 → α 2 ) → 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 , α 1 → ( α 2 → β ) ⇒ δ Γ , x , β ⇒ δ Γ ⇒ δ Γ , ⊥ → β ⇒ δ Γ , x , x → β ⇒ δ Γ , ( α 1 ∧ α 2 ) → β ⇒ δ Γ , β ⇒ δ Γ , α 1 → β, α 2 → β ⇒ δ Γ , α 2 → β ⇒ α 1 → α 2 Γ , β ⇒ δ Γ , ⊤ → β ⇒ δ Γ , ( α 1 ∨ α 2 ) → β ⇒ δ Γ , ( α 1 → α 2 ) → 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