Uniform Interpolation Part 3: Case Studies George Metcalfe - PowerPoint PPT Presentation

Normal Modal Logics The basic modal logic K can be defined by extending any axiomatization of classical propositional logic with the axiom schema � ( α → β ) → ( � α → � β ) ( K ) and the necessitation rule : from α , infer � α . A normal modal logic is any axiomatic extension of K; in particular, K4 = K + � α → �� α KT = K + � α → α S4 = K4 + � α → α GL = K4 + � ( � α → α ) → � α S5 = S4 + ♦ α → �♦ α. George Metcalfe (University of Bern) Uniform Interpolation August 2018 5 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C if for any formula α , ⊢ L α ⇐ ⇒ M | = α for every model M based on a frame in C . George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C if for any formula α , ⊢ L α ⇐ ⇒ M | = α for every model M based on a frame in C . The following normal modal logics are complete with respect to the given class of frames: Logic Frames K all frames George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C if for any formula α , ⊢ L α ⇐ ⇒ M | = α for every model M based on a frame in C . The following normal modal logics are complete with respect to the given class of frames: Logic Frames K all frames K4 transitive frames George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C if for any formula α , ⊢ L α ⇐ ⇒ M | = α for every model M based on a frame in C . The following normal modal logics are complete with respect to the given class of frames: Logic Frames K all frames K4 transitive frames KT reflexive frames George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C if for any formula α , ⊢ L α ⇐ ⇒ M | = α for every model M based on a frame in C . The following normal modal logics are complete with respect to the given class of frames: Logic Frames K all frames K4 transitive frames KT reflexive frames S4 preorders George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C if for any formula α , ⊢ L α ⇐ ⇒ M | = α for every model M based on a frame in C . The following normal modal logics are complete with respect to the given class of frames: Logic Frames K all frames K4 transitive frames KT reflexive frames S4 preorders GL transitive and conversely well-founded frames George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C if for any formula α , ⊢ L α ⇐ ⇒ M | = α for every model M based on a frame in C . The following normal modal logics are complete with respect to the given class of frames: Logic Frames K all frames K4 transitive frames KT reflexive frames S4 preorders GL transitive and conversely well-founded frames S5 equivalence relations George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Completeness A normal modal logic L is said to be complete with respect to a class of frames C if for any formula α , ⊢ L α ⇐ ⇒ M | = α for every model M based on a frame in C . The following normal modal logics are complete with respect to the given class of frames: Logic Frames K all frames K4 transitive frames KT reflexive frames S4 preorders GL transitive and conversely well-founded frames S5 equivalence relations Moreover, all these normal modal logics have the finite model property . George Metcalfe (University of Bern) Uniform Interpolation August 2018 6 / 27

Modal Algebras A modal algebra consists of a Boolean algebra supplemented with a unary operation � satisfying � ( x ∧ y ) ≈ � x ∧ � y � ⊤ ≈ ⊤ . and George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 27

Modal Algebras A modal algebra consists of a Boolean algebra supplemented with a unary operation � satisfying � ( x ∧ y ) ≈ � x ∧ � y � ⊤ ≈ ⊤ . and We let K denote the variety of all modal algebras. George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 27

Modal Algebras A modal algebra consists of a Boolean algebra supplemented with a unary operation � satisfying � ( x ∧ y ) ≈ � x ∧ � y � ⊤ ≈ ⊤ . and We let K denote the variety of all modal algebras. In particular, each Kripke frame � W , R � yields a complex modal algebra �P ( W ) , ∩ , ∪ , c , ∅ , W , � � where � A := { w ∈ W | Rwv for all v ∈ A } . George Metcalfe (University of Bern) Uniform Interpolation August 2018 7 / 27

Equivalence Theorem For any normal modal logic L , let V L = { A ∈ K | ⊢ L α = ⇒ A | = α ≈ ⊤} . George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 27

Equivalence Theorem For any normal modal logic L , let V L = { A ∈ K | ⊢ L α = ⇒ A | = α ≈ ⊤} . Then V L is an equivalent algebraic semantics for L George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 27

Equivalence Theorem For any normal modal logic L , let V L = { A ∈ K | ⊢ L α = ⇒ A | = α ≈ ⊤} . Then V L is an equivalent algebraic semantics for L with transformers τ ( α ) = α ≈ ⊤ and ρ ( α ≈ β ) = ( α → β ) ∧ ( β → α ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 27

Equivalence Theorem For any normal modal logic L , let V L = { A ∈ K | ⊢ L α = ⇒ A | = α ≈ ⊤} . Then V L is an equivalent algebraic semantics for L with transformers τ ( α ) = α ≈ ⊤ and ρ ( α ≈ β ) = ( α → β ) ∧ ( β → α ) . That is, for any set of formulas T ∪ { α, β } and set of equations Σ , (a) T ⊢ L α ⇐ ⇒ τ [ T ] | = V L τ ( α ) ; (b) Σ | = V L α ≈ β ⇐ ⇒ ρ [ T ] ⊢ L ρ ( α ≈ β ) ; (c) α ⊢ L ρ ( τ ( α )) and ρ ( τ ( α )) ⊢ L α ; (d) α ≈ β | = V L τ ( ρ ( α ≈ β )) and τ ( ρ ( α ≈ β )) | = V L α ≈ β . George Metcalfe (University of Bern) Uniform Interpolation August 2018 8 / 27

Interpolation in Modal Logic A normal modal logic L admits deductive interpolation George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 27

Interpolation in Modal Logic A normal modal logic L admits deductive interpolation α ( x , y ) ⊢ L β ( y , z ) = ⇒ α ⊢ L γ and γ ⊢ L β for some γ ( y ) George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 27

Interpolation in Modal Logic A normal modal logic L admits deductive interpolation α ( x , y ) ⊢ L β ( y , z ) = ⇒ α ⊢ L γ and γ ⊢ L β for some γ ( y ) if and only if V L admits the amalgamation property . George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 27

Interpolation in Modal Logic A normal modal logic L admits deductive interpolation α ( x , y ) ⊢ L β ( y , z ) = ⇒ α ⊢ L γ and γ ⊢ L β for some γ ( y ) if and only if V L admits the amalgamation property . For example, K, K4, S4, GL, and somewhere between 43 and 49 axiomatic extensions of S4 admit deductive interpolation, but not S5. George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 27

Interpolation in Modal Logic A normal modal logic L admits deductive interpolation α ( x , y ) ⊢ L β ( y , z ) = ⇒ α ⊢ L γ and γ ⊢ L β for some γ ( y ) if and only if V L admits the amalgamation property . For example, K, K4, S4, GL, and somewhere between 43 and 49 axiomatic extensions of S4 admit deductive interpolation, but not S5. Note, however, that L admits Craig interpolation ⊢ L α ( x , y ) → β ( y , z ) = ⇒ ⊢ L α → γ and ⊢ L γ → β for some γ ( y ) George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 27

Interpolation in Modal Logic A normal modal logic L admits deductive interpolation α ( x , y ) ⊢ L β ( y , z ) = ⇒ α ⊢ L γ and γ ⊢ L β for some γ ( y ) if and only if V L admits the amalgamation property . For example, K, K4, S4, GL, and somewhere between 43 and 49 axiomatic extensions of S4 admit deductive interpolation, but not S5. Note, however, that L admits Craig interpolation ⊢ L α ( x , y ) → β ( y , z ) = ⇒ ⊢ L α → γ and ⊢ L γ → β for some γ ( y ) if and only if V L admits the super amalgamation property . George Metcalfe (University of Bern) Uniform Interpolation August 2018 9 / 27

Uniform Interpolation in Modal Logic Theorem (Ghilardi 1995, Visser 1996, Bílková 2007) K has uniform interpolation. George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 27

Uniform Interpolation in Modal Logic Theorem (Ghilardi 1995, Visser 1996, Bílková 2007) K has uniform interpolation. Theorem (Kowalski and Metcalfe 2017) K does not have uniform interpolation. George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 27

Uniform Interpolation in Modal Logic Theorem (Ghilardi 1995, Visser 1996, Bílková 2007) K has uniform Craig interpolation Theorem (Kowalski and Metcalfe 2017) K does not have uniform deductive interpolation. George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 27

Uniform Interpolation in Modal Logic Theorem (Ghilardi 1995, Visser 1996, Bílková 2007) K has uniform Craig interpolation; that is, for any formula α ( x , y ) , there exist formulas α L ( y ) and α R ( y ) such that ⇒ ⊢ K α R ( y ) → β ( y , z ) ⊢ K α ( x , y ) → β ( y , z ) ⇐ ⇒ ⊢ K β ( y , z ) → α L ( y ) . ⊢ K β ( y , z ) → α ( x , y ) ⇐ Theorem (Kowalski and Metcalfe 2017) K does not have uniform deductive interpolation. George Metcalfe (University of Bern) Uniform Interpolation August 2018 10 / 27

Recall. . . V has deductive interpolation if for any set of equations Σ( x , y ) , there exists a set of equations ∆( y ) such that Σ( x , y ) | ⇐ ⇒ ∆( y ) | = V ε ( y , z ) = V ε ( y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 27

Recall. . . V has right uniform deductive interpolation if for any finite set of equations Σ( x , y ) , there exists a finite set of equations ∆( y ) such that Σ( x , y ) | ⇐ ⇒ ∆( y ) | = V ε ( y , z ) = V ε ( y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 27

Recall. . . V has right uniform deductive interpolation if for any finite set of equations Σ( x , y ) , there exists a finite set of equations ∆( y ) such that Σ( x , y ) | ⇐ ⇒ ∆( y ) | = V ε ( y , z ) = V ε ( y , z ) . Equivalently, V has deductive interpolation and for any finite set of equations Σ( x , y ) , there exists a finite set of equations ∆( y ) such that Σ( x , y ) | = V ε ( y ) ⇐ ⇒ ∆( y ) | = V ε ( y ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 11 / 27

Recall also. . . Theorem (Kowalski and Metcalfe 2017) The following are equivalent: (1) For any finite set of equations Σ( x , y ) , there exists a finite set of equations ∆( y ) such that Σ( x , y ) | = V ε ( y ) ⇐ ⇒ ∆( y ) | = V ε ( y ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 27

Recall also. . . Theorem (Kowalski and Metcalfe 2017) The following are equivalent: (1) For any finite set of equations Σ( x , y ) , there exists a finite set of equations ∆( y ) such that Σ( x , y ) | = V ε ( y ) ⇐ ⇒ ∆( y ) | = V ε ( y ) . (2) For finite x , y , the compact lifting of F ( y ) ֒ → F ( x , y ) has a right ⇒ Θ ∩ F ( y ) 2 ∈ KCon F ( y ) . adjoint; that is, Θ ∈ KCon F ( x , y ) = George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 27

Recall also. . . Theorem (Kowalski and Metcalfe 2017) The following are equivalent: (1) For any finite set of equations Σ( x , y ) , there exists a finite set of equations ∆( y ) such that Σ( x , y ) | = V ε ( y ) ⇐ ⇒ ∆( y ) | = V ε ( y ) . (2) For finite x , y , the compact lifting of F ( y ) ֒ → F ( x , y ) has a right ⇒ Θ ∩ F ( y ) 2 ∈ KCon F ( y ) . adjoint; that is, Θ ∈ KCon F ( x , y ) = (3) V is coherent , i.e., all finitely generated subalgebras of finitely presented members of V are finitely presented. George Metcalfe (University of Bern) Uniform Interpolation August 2018 12 / 27

A Failure of Coherence Theorem (Kowalski and Metcalfe 2017) The variety of modal algebras is not coherent; so it does not admit uniform deductive interpolation and its theory does not have a model completion. T. Kowalski and G. Metcalfe. Uniform interpolation and coherence. Submitted (2017). T. Kowalski and G. Metcalfe. Coherence in modal logic. Proceedings of AiML 2018 , College Publications (2018). George Metcalfe (University of Bern) Uniform Interpolation August 2018 13 / 27

Proof Let ⊡ α := � α ∧ α , George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and Claim. Σ | = K ε ( y , z ) ⇐ ⇒ ∆ | = K ε ( y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and Claim. Σ | = K ε ( y , z ) ⇐ ⇒ ∆ | = K ε ( y , z ) . It follows that if K were coherent, then ∆ ′ | = K ∆ for some finite ∆ ′ ⊆ ∆ , George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and Claim. Σ | = K ε ( y , z ) ⇐ ⇒ ∆ | = K ε ( y , z ) . It follows that if K were coherent, then ∆ ′ | = K ∆ for some finite ∆ ′ ⊆ ∆ , = ⊡ n z ≈ ⊡ n + 1 z for some n ∈ N , a contradiction. and from this that K | George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and Claim. Σ | = K ε ( y , z ) ⇐ ⇒ ∆ | = K ε ( y , z ) . It follows that if K were coherent, then ∆ ′ | = K ∆ for some finite ∆ ′ ⊆ ∆ , = ⊡ n z ≈ ⊡ n + 1 z for some n ∈ N , a contradiction. and from this that K | Proof of claim. ( ⇐ ) Just observe that Σ | = K ∆ . George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and Claim. Σ | = K ε ( y , z ) ⇐ ⇒ ∆ | = K ε ( y , z ) . It follows that if K were coherent, then ∆ ′ | = K ∆ for some finite ∆ ′ ⊆ ∆ , = ⊡ n z ≈ ⊡ n + 1 z for some n ∈ N , a contradiction. and from this that K | Proof of claim. ( ⇐ ) Just observe that Σ | = K ∆ . ( ⇒ ) Suppose that ∆ �| = K ε ( y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and Claim. Σ | = K ε ( y , z ) ⇐ ⇒ ∆ | = K ε ( y , z ) . It follows that if K were coherent, then ∆ ′ | = K ∆ for some finite ∆ ′ ⊆ ∆ , = ⊡ n z ≈ ⊡ n + 1 z for some n ∈ N , a contradiction. and from this that K | Proof of claim. ( ⇐ ) Just observe that Σ | = K ∆ . ( ⇒ ) Suppose that ∆ �| = K ε ( y , z ) . Then there is a complete modal algebra A and homomorphism e : Tm ( y , z ) → A such that ∆ ⊆ ker ( e ) and ε �∈ ker ( e ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and Claim. Σ | = K ε ( y , z ) ⇐ ⇒ ∆ | = K ε ( y , z ) . It follows that if K were coherent, then ∆ ′ | = K ∆ for some finite ∆ ′ ⊆ ∆ , = ⊡ n z ≈ ⊡ n + 1 z for some n ∈ N , a contradiction. and from this that K | Proof of claim. ( ⇐ ) Just observe that Σ | = K ∆ . ( ⇒ ) Suppose that ∆ �| = K ε ( y , z ) . Then there is a complete modal algebra A and homomorphism e : Tm ( y , z ) → A such that ∆ ⊆ ker ( e ) and ε �∈ ker ( e ) . Extend e with � ⊡ k e ( z ) . e ( x ) = k ∈ N George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

Proof Let ⊡ α := � α ∧ α , and define ∆ = { y ≤ ⊡ k z | k ∈ N } . Σ = { y ≤ x , x ≤ z , x ≈ ⊡ x } and Claim. Σ | = K ε ( y , z ) ⇐ ⇒ ∆ | = K ε ( y , z ) . It follows that if K were coherent, then ∆ ′ | = K ∆ for some finite ∆ ′ ⊆ ∆ , = ⊡ n z ≈ ⊡ n + 1 z for some n ∈ N , a contradiction. and from this that K | Proof of claim. ( ⇐ ) Just observe that Σ | = K ∆ . ( ⇒ ) Suppose that ∆ �| = K ε ( y , z ) . Then there is a complete modal algebra A and homomorphism e : Tm ( y , z ) → A such that ∆ ⊆ ker ( e ) and ε �∈ ker ( e ) . Extend e with � ⊡ k e ( z ) . e ( x ) = k ∈ N Then also Σ ⊆ ker ( e ) , and hence Σ �| = K ε ( y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 14 / 27

An Obvious Question Can we generalize this proof to other varieties? George Metcalfe (University of Bern) Uniform Interpolation August 2018 15 / 27

A General Criterion Theorem (Kowalski and Metcalfe 2017) Let V be a coherent variety of algebras with a meet-semilattice reduct George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 27

A General Criterion Theorem (Kowalski and Metcalfe 2017) Let V be a coherent variety of algebras with a meet-semilattice reduct and let t ( x , ¯ u ) be a term satisfying V | = t ( x , ¯ u ) ≤ x and V | = x ≤ y ⇒ t ( x , ¯ u ) ≤ t ( y , ¯ u ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 27

A General Criterion Theorem (Kowalski and Metcalfe 2017) Let V be a coherent variety of algebras with a meet-semilattice reduct and let t ( x , ¯ u ) be a term satisfying V | = t ( x , ¯ u ) ≤ x and V | = x ≤ y ⇒ t ( x , ¯ u ) ≤ t ( y , ¯ u ) . Suppose also that for any finitely generated A ∈ V and a , ¯ b ∈ A , there exists B ∈ V containing A as a subalgebra George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 27

A General Criterion Theorem (Kowalski and Metcalfe 2017) Let V be a coherent variety of algebras with a meet-semilattice reduct and let t ( x , ¯ u ) be a term satisfying V | = t ( x , ¯ u ) ≤ x and V | = x ≤ y ⇒ t ( x , ¯ u ) ≤ t ( y , ¯ u ) . Suppose also that for any finitely generated A ∈ V and a , ¯ b ∈ A , there exists B ∈ V containing A as a subalgebra and satisfying � t k ( a , ¯ � t k ( a , ¯ b ) , ¯ b ) = t ( b ) . k ∈ N k ∈ N George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 27

A General Criterion Theorem (Kowalski and Metcalfe 2017) Let V be a coherent variety of algebras with a meet-semilattice reduct and let t ( x , ¯ u ) be a term satisfying V | = t ( x , ¯ u ) ≤ x and V | = x ≤ y ⇒ t ( x , ¯ u ) ≤ t ( y , ¯ u ) . Suppose also that for any finitely generated A ∈ V and a , ¯ b ∈ A , there exists B ∈ V containing A as a subalgebra and satisfying � t k ( a , ¯ � t k ( a , ¯ b ) , ¯ b ) = t ( b ) . k ∈ N k ∈ N = t n ( x , ¯ u ) ≈ t n + 1 ( x , ¯ Then V | u ) for some n ∈ N . George Metcalfe (University of Bern) Uniform Interpolation August 2018 16 / 27

Strong Kripke Completeness A normal modal logic L is called strongly Kripke complete George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 27

Strong Kripke Completeness A normal modal logic L is called strongly Kripke complete if for any set of formulas T ∪ { α } , for any Kripke model M based on a frame for L, T ⊢ L α ⇐ ⇒ M | = β for all β ∈ T = ⇒ M | = α. George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 27

Strong Kripke Completeness A normal modal logic L is called strongly Kripke complete if for any set of formulas T ∪ { α } , for any Kripke model M based on a frame for L, T ⊢ L α ⇐ ⇒ M | = β for all β ∈ T = ⇒ M | = α. E.g., K, KT, K4, S4, and S5 are strongly Kripke complete, but not GL. George Metcalfe (University of Bern) Uniform Interpolation August 2018 17 / 27

Coherence and Weak Transitivity Applying our general criterion with t ( x ) = ⊡ x , using strong Kripke completeness to establish the fixpoint condition, we obtain: George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 27

Coherence and Weak Transitivity Applying our general criterion with t ( x ) = ⊡ x , using strong Kripke completeness to establish the fixpoint condition, we obtain: Theorem Any coherent strongly Kripke-complete variety of modal algebras is weakly transitive : George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 27

Coherence and Weak Transitivity Applying our general criterion with t ( x ) = ⊡ x , using strong Kripke completeness to establish the fixpoint condition, we obtain: Theorem Any coherent strongly Kripke-complete variety of modal algebras is weakly transitive : that is, it satisfies ⊡ n + 1 x ≈ ⊡ n x for some n ∈ N George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 27

Coherence and Weak Transitivity Applying our general criterion with t ( x ) = ⊡ x , using strong Kripke completeness to establish the fixpoint condition, we obtain: Theorem Any coherent strongly Kripke-complete variety of modal algebras is weakly transitive : that is, it satisfies ⊡ n + 1 x ≈ ⊡ n x for some n ∈ N (equivalently, it admits equationally definable principal congruences). George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 27

Coherence and Weak Transitivity Applying our general criterion with t ( x ) = ⊡ x , using strong Kripke completeness to establish the fixpoint condition, we obtain: Theorem Any coherent strongly Kripke-complete variety of modal algebras is weakly transitive : that is, it satisfies ⊡ n + 1 x ≈ ⊡ n x for some n ∈ N (equivalently, it admits equationally definable principal congruences). Hence a large family of varieties of modal algebras for non-weakly-transitive modal logics, including K and KT, are not coherent, do not admit uniform deductive interpolation, and their theories do not have a model completion. George Metcalfe (University of Bern) Uniform Interpolation August 2018 18 / 27

Weakly Transitive Modal Logics (1) We can also show that weakly transitive logics such as K4 and S4 are not coherent George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 27

Weakly Transitive Modal Logics (1) We can also show that weakly transitive logics such as K4 and S4 are not coherent using the ternary term t ( x , y , z ) = ♦ ( y ∧ ♦ ( z ∧ x )) ∧ x . George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 27

Weakly Transitive Modal Logics (1) We can also show that weakly transitive logics such as K4 and S4 are not coherent using the ternary term t ( x , y , z ) = ♦ ( y ∧ ♦ ( z ∧ x )) ∧ x . For any normal modal logic L, V L | = t ( x , y , z ) ≤ x and V L | = u ≤ v ⇒ t ( u , y , z ) ≤ t ( v , y , z ) . George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 27

Weakly Transitive Modal Logics (1) We can also show that weakly transitive logics such as K4 and S4 are not coherent using the ternary term t ( x , y , z ) = ♦ ( y ∧ ♦ ( z ∧ x )) ∧ x . For any normal modal logic L, V L | = t ( x , y , z ) ≤ x and V L | = u ≤ v ⇒ t ( u , y , z ) ≤ t ( v , y , z ) . Lemma Suppose that L admits finite chains : George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 27

Weakly Transitive Modal Logics (1) We can also show that weakly transitive logics such as K4 and S4 are not coherent using the ternary term t ( x , y , z ) = ♦ ( y ∧ ♦ ( z ∧ x )) ∧ x . For any normal modal logic L, V L | = t ( x , y , z ) ≤ x and V L | = u ≤ v ⇒ t ( u , y , z ) ≤ t ( v , y , z ) . Lemma Suppose that L admits finite chains : that is, for each n ∈ N there exists a frame � W , R � for L such that | W | = n and the reflexive closure of R is a total order. George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 27

Weakly Transitive Modal Logics (1) We can also show that weakly transitive logics such as K4 and S4 are not coherent using the ternary term t ( x , y , z ) = ♦ ( y ∧ ♦ ( z ∧ x )) ∧ x . For any normal modal logic L, V L | = t ( x , y , z ) ≤ x and V L | = u ≤ v ⇒ t ( u , y , z ) ≤ t ( v , y , z ) . Lemma Suppose that L admits finite chains : that is, for each n ∈ N there exists a frame � W , R � for L such that | W | = n and the reflexive closure of R is a = t n ( x , y , z ) ≈ t n + 1 ( x , y , z ) for all n ∈ N . total order. Then V L �| George Metcalfe (University of Bern) Uniform Interpolation August 2018 19 / 27

Weakly Transitive Modal Logics (2) Theorem Let L be a normal modal logic admitting finite chains George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 27

Weakly Transitive Modal Logics (2) Theorem Let L be a normal modal logic admitting finite chains such that V L is canonical : that is, closed under taking canonical extensions. George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 27

Weakly Transitive Modal Logics (2) Theorem Let L be a normal modal logic admitting finite chains such that V L is canonical : that is, closed under taking canonical extensions. Then (a) V L is not coherent; (b) V L does not admit uniform deductive interpolation; (c) the first-order theory of V L does not have a model completion. George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 27

Weakly Transitive Modal Logics (2) Theorem Let L be a normal modal logic admitting finite chains such that V L is canonical : that is, closed under taking canonical extensions. Then (a) V L is not coherent; (b) V L does not admit uniform deductive interpolation; (c) the first-order theory of V L does not have a model completion. In particular, V K4 and V S4 are not coherent and do not admit uniform deductive interpolation, and their theories do not have a model completion. George Metcalfe (University of Bern) Uniform Interpolation August 2018 20 / 27

Remarks Note that GL admits finite chains but is not canonical. George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 27

Remarks Note that GL admits finite chains but is not canonical. In fact, V GL is coherent and admits uniform deductive interpolation (Shavrukov 1993); George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 27

Remarks Note that GL admits finite chains but is not canonical. In fact, V GL is coherent and admits uniform deductive interpolation (Shavrukov 1993); also, its theory has a model completion (Ghilardi and Zawadowski 2002). George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 27

Remarks Note that GL admits finite chains but is not canonical. In fact, V GL is coherent and admits uniform deductive interpolation (Shavrukov 1993); also, its theory has a model completion (Ghilardi and Zawadowski 2002). Ghilardi and Zawadowski (2002) have also proved that no logic extending K4 that has the finite model property and admits all finite reflexive chains and the two-element cluster is coherent. S. Ghilardi and M. Zawadowski. Sheaves, Games and Model Completions , Kluwer (2002). George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 27

Remarks Note that GL admits finite chains but is not canonical. In fact, V GL is coherent and admits uniform deductive interpolation (Shavrukov 1993); also, its theory has a model completion (Ghilardi and Zawadowski 2002). Ghilardi and Zawadowski (2002) have also proved that no logic extending K4 that has the finite model property and admits all finite reflexive chains and the two-element cluster is coherent. S. Ghilardi and M. Zawadowski. Sheaves, Games and Model Completions , Kluwer (2002). However, their methods are more complicated and less general than ours; George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 27

Remarks Note that GL admits finite chains but is not canonical. In fact, V GL is coherent and admits uniform deductive interpolation (Shavrukov 1993); also, its theory has a model completion (Ghilardi and Zawadowski 2002). Ghilardi and Zawadowski (2002) have also proved that no logic extending K4 that has the finite model property and admits all finite reflexive chains and the two-element cluster is coherent. S. Ghilardi and M. Zawadowski. Sheaves, Games and Model Completions , Kluwer (2002). However, their methods are more complicated and less general than ours; they also yield similar but incomparable results. George Metcalfe (University of Bern) Uniform Interpolation August 2018 21 / 27

Coherence in Algebra Any locally finite variety (e.g., Boolean algebras, Sugihara monoids, etc.) is coherent George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 27

Coherence in Algebra Any locally finite variety (e.g., Boolean algebras, Sugihara monoids, etc.) is coherent — also the varieties of Heyting algebras, abelian groups, abelian ℓ -groups, and MV-algebras. George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 27

Coherence in Algebra Any locally finite variety (e.g., Boolean algebras, Sugihara monoids, etc.) is coherent — also the varieties of Heyting algebras, abelian groups, abelian ℓ -groups, and MV-algebras. The variety of groups is not coherent, however, since every finitely generated recursively presented group embeds into some finitely presented group (Higman 1961). George Metcalfe (University of Bern) Uniform Interpolation August 2018 22 / 27

Lattices Theorem (Schmidt 1981) The variety LAT of lattices is not coherent, does not admit uniform deductive interpolation, and its theory does not have a model completion. George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 27

Lattices Theorem (Schmidt 1981) The variety LAT of lattices is not coherent, does not admit uniform deductive interpolation, and its theory does not have a model completion. We obtain an easy proof of this result using our criterion with the term t ( x , u 1 , u 2 , u 3 ) = ( u 1 ∧ ( u 2 ∨ ( u 3 ∧ x ))) ∧ x . George Metcalfe (University of Bern) Uniform Interpolation August 2018 23 / 27