Uniform Interpolation Part II: An Algebraic Framework 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
Yesterday Theorem (Pitts 1992) For any formula α ( x , y ) of intuitionistic propositional logic IL , 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 2 / 30
Remarks from Yesterday 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 3 / 30
Remarks from Yesterday Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018). There are exactly seven consistent intermediate logics that admit Craig interpolation (Maksimova 1977), George Metcalfe (University of Bern) Uniform Interpolation August 2018 3 / 30
Remarks from Yesterday Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018). There are exactly seven consistent 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 3 / 30
Remarks from Yesterday Other proofs of Pitts’ theorem have been given using bisimulations (Ghilardi 1995, Visser 1996) and duality (van Gool and Reggio 2018). There are exactly seven consistent 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 3 / 30
Today What does (uniform) interpolation mean algebraically ? George Metcalfe (University of Bern) Uniform Interpolation August 2018 4 / 30
Equations and Equational Classes Let Tm ( x ) denote the term algebra of an algebraic language L with at least one constant over a set of variables x . George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30
Equations and Equational Classes Let Tm ( x ) denote the term algebra of an algebraic language L with at least one constant over a set of variables x . An L - equation is an ordered pair � s , t � of L -terms, also written s ≈ t . George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30
Equations and Equational Classes Let Tm ( x ) denote the term algebra of an algebraic language L with at least one constant over a set of variables x . An L - equation is an ordered pair � s , t � of L -terms, also written s ≈ t . We let V be any variety defined by L -equations, e.g., Boolean algebras, Heyting algebras, MV-algebras, modal algebras, bounded lattices, groups. . . George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 30
Equational Consequence For any set of L -equations Σ ∪ { ε } with variables in x , we write Σ | = V ε George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30
Equational Consequence For any set of L -equations Σ ∪ { ε } with variables in x , we write Σ | = V ε if for any A ∈ V and homomorphism e : Tm ( x ) → A , George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30
Equational Consequence For any set of L -equations Σ ∪ { ε } with variables in x , we write Σ | = V ε if for any A ∈ V and homomorphism e : Tm ( x ) → A , Σ ⊆ ker ( e ) = ⇒ ε ∈ ker ( e ) , where ker ( e ) := {� s , t � | e ( s ) = e ( t ) } is the kernel of e . George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30
Equational Consequence For any set of L -equations Σ ∪ { ε } with variables in x , we write Σ | = V ε if for any A ∈ V and homomorphism e : Tm ( x ) → A , Σ ⊆ ker ( e ) = ⇒ ε ∈ ker ( e ) , where ker ( e ) := {� s , t � | e ( s ) = e ( t ) } is the kernel of e . We also write Σ | = V ∆ to denote that Σ | = V ε for all ε ∈ ∆ . George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 30
Deductive Interpolation V admits deductive interpolation if whenever Σ( x , y ) | = V ε ( y , z ) , there exists a set of equations ∆( y ) such that Σ( x , y ) | = V ∆( y ) and ∆( y ) | = V ε ( y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 30
Deductive Interpolation V admits deductive interpolation if whenever Σ( x , y ) | = V ε ( y , z ) , there exists a set of equations ∆( y ) such that Σ( x , y ) | = V ∆( y ) and ∆( y ) | = V ε ( y , z ) . Equivalently, V admits deductive interpolation if for any set of equations Σ( x , y ) , there exists a set of equations ∆( y ) such that Σ( x , y ) | = V ε ( y , z ) ⇐ ⇒ ∆( y ) | = V ε ( y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 30
Deductive Interpolation V admits deductive interpolation if whenever Σ( x , y ) | = V ε ( y , z ) , there exists a set of equations ∆( y ) such that Σ( x , y ) | = V ∆( y ) and ∆( y ) | = V ε ( y , z ) . Equivalently, V admits deductive interpolation if for any set of equations Σ( x , y ) , there exists a set of equations ∆( y ) such that Σ( x , y ) | = V ε ( y , z ) ⇐ ⇒ ∆( y ) | = V ε ( y , z ) . (Just define ∆( y ) := { ε ( y ) | Σ( x , y ) | = V ε ( y ) } .) George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 30
Congruences A congruence Θ on an algebra A is an equivalence relation satisfying � a 1 , b 1 � , . . . , � a n , b n � ∈ Θ = ⇒ � ⋆ ( a 1 , . . . , a n ) , ⋆ ( b 1 , . . . , b n ) � ∈ Θ for every n -ary operation ⋆ of A . George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30
Congruences A congruence Θ on an algebra A is an equivalence relation satisfying � a 1 , b 1 � , . . . , � a n , b n � ∈ Θ = ⇒ � ⋆ ( a 1 , . . . , a n ) , ⋆ ( b 1 , . . . , b n ) � ∈ Θ for every n -ary operation ⋆ of A . The congruences of A always form a complete lattice Con A . George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 30
Free Algebras The free algebra of V over a set of variables x can be defined as F ( x ) = Tm ( x ) / Θ V where s Θ V t : ⇐ ⇒ V | = s ≈ t . George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 30
Free Algebras The free algebra of V over a set of variables x can be defined as F ( x ) = Tm ( x ) / Θ V where s Θ V t : ⇐ ⇒ V | = s ≈ t . We write t to denote both a term t in Tm ( x ) and [ t ] in F ( x ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 30
Equational Consequence Again Lemma For any set of equations Σ ∪ { s ≈ t } with variables in x , Σ | = V s ≈ t ⇐ ⇒ � s , t � ∈ Cg F ( x ) (Σ) , where Cg F ( x ) (Σ) is the congruence on F ( x ) generated by Σ . George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 30
Lifting Inclusions The inclusion map i : F ( y ) → F ( x , y ) George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30
Lifting Inclusions The inclusion map i : F ( y ) → F ( x , y ) “lifts” to the maps i ∗ : Con F ( y ) → Con F ( x , y ); Θ �→ Cg F ( x , y ) ( i [Θ]) George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30
Lifting Inclusions The inclusion map i : F ( y ) → F ( x , y ) “lifts” to the maps i ∗ : Con F ( y ) → Con F ( x , y ); Θ �→ Cg F ( x , y ) ( i [Θ]) i − 1 : Con F ( x , y ) → Con F ( y ); Ψ �→ i − 1 [Ψ] = Ψ ∩ F ( y ) 2 . George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30
Lifting Inclusions The inclusion map i : F ( y ) → F ( x , y ) “lifts” to the maps i ∗ : Con F ( y ) → Con F ( x , y ); Θ �→ Cg F ( x , y ) ( i [Θ]) i − 1 : Con F ( x , y ) → Con F ( y ); Ψ �→ i − 1 [Ψ] = Ψ ∩ F ( y ) 2 . Note that the pair � i ∗ , i − 1 � is an adjunction , i.e., i ∗ (Θ) ⊆ Ψ Θ ⊆ i − 1 (Ψ) . ⇐ ⇒ George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 30
Deductive Interpolation Again The following are equivalent: (1) V admits deductive interpolation , i.e., for any set of equations Σ( x , y ) , there exists a set of equations ∆( y ) such that Σ( x , y ) | = V ε ( y , z ) ⇐ ⇒ ∆( y ) | = V ε ( y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 30
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