The H , S , P , P + operators H V = {� B, β � ∈ Set Γ | ∃ V ∋ � C, γ � � � � B, β �} S V = {� B, β � ∈ Set Γ | ∃� B, β � � � � C, γ � ∈ V } P V = {� B, β � ∈ Set Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i �} = P + V = {� B, β � ∈ Set Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i � , I � = ∅} = Horn Covarieties for Coalgebras – p.7/26
The variety theorems Let Γ be polynomial and V ⊆ Set Γ . Theorem (Birkhoff variety theorem). Sat ( EqTh V ) = HSP V Horn Covarieties for Coalgebras – p.8/26
The variety theorems Let Γ be polynomial and V ⊆ Set Γ . Theorem (Birkhoff variety theorem). Sat ( EqTh V ) = HSP V Theorem (Quasivariety theorem). Sat ( ImpEqTh V ) = SP V Horn Covarieties for Coalgebras – p.8/26
The variety theorems Let Γ be polynomial and V ⊆ Set Γ . Theorem (Birkhoff variety theorem). Sat ( EqTh V ) = HSP V Theorem (Quasivariety theorem). Sat ( ImpEqTh V ) = SP V Theorem (Horn variety theorem). Sat ( HornEqTh V ) = SP + V Horn Covarieties for Coalgebras – p.8/26
Outline I. Infinitary Horn varieties II. Dual theorems for E Γ III. Examples of Horn covarieties IV. Cofree for H Σ + V coalgebras V. Behavioral Horn covarieties VI. Optimistic promissary note Horn Covarieties for Coalgebras – p.9/26
Outline I. Infinitary Horn varieties II. Dual theorems for E Γ III. Examples of Horn covarieties IV. Cofree for H Σ + V coalgebras V. Behavioral Horn covarieties VI. Optimistic promissary note Horn Covarieties for Coalgebras – p.9/26
Closure operators for coalgebras Recall the algebra operators. H V = {� B, β � ∈ E Γ | ∃ V ∋ � C, γ � � � � B, β �} S V = {� B, β � ∈ E Γ | ∃� B, β � � � � C, γ � ∈ V } P V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i �} = P + V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i � , I � = ∅} = Horn Covarieties for Coalgebras – p.10/26
� � � Closure operators for coalgebras Each algebra operator yields a coalgebra operator. ◦ H V = {� B, β � ∈ E Γ | ∃ V ∋ � C, γ � � B, β � } S V = {� B, β � ∈ E Γ | ∃� B, β � � � � C, γ � ∈ V } P V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i �} = P + V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i � , I � = ∅} = Horn Covarieties for Coalgebras – p.10/26
� � � � � Closure operators for coalgebras Each algebra operator yields a coalgebra operator. ◦ H V = {� B, β � ∈ E Γ | ∃ V ∋ � C, γ � � B, β � } ◦ S V = {� B, β � ∈ E Γ | ∃� B, β � � C, γ � ∈ V } P V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i �} = P + V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i � , I � = ∅} = Horn Covarieties for Coalgebras – p.10/26
� � � � � Closure operators for coalgebras Each algebra operator yields a coalgebra operator. ◦ H V = {� B, β � ∈ E Γ | ∃ V ∋ � C, γ � � B, β � } ◦ S V = {� B, β � ∈ E Γ | ∃� B, β � � C, γ � ∈ V } ◦ P V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i �} = P + V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i � , I � = ∅} = Horn Covarieties for Coalgebras – p.10/26
� � � � � Closure operators for coalgebras Each algebra operator yields a coalgebra operator. ◦ H V = {� B, β � ∈ E Γ | ∃ V ∋ � C, γ � � B, β � } ◦ S V = {� B, β � ∈ E Γ | ∃� B, β � � C, γ � ∈ V } ◦ P V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i �} = ◦ P + V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i � , I � = ∅} = Horn Covarieties for Coalgebras – p.10/26
� � � � � Closure operators for coalgebras Each algebra operator yields a coalgebra operator. S V = {� B, β � ∈ E Γ | ∃ V ∋ � C, γ � � B, β � } H V = {� B, β � ∈ E Γ | ∃� B, β � � C, γ � ∈ V } Σ V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i �} = Σ + V = {� B, β � ∈ E Γ | ∃{� A i , α i �} i ∈ I ⊆ V . � � B, β � ∼ � A i , α i � , I � = ∅} = Horn Covarieties for Coalgebras – p.10/26
Dualizing equations Consider again equations in Set Γ . We consider the mapping { S � � UFX × UFX } → { FX � � � Q, ν �} , and dualize the notion of sets of equations by dualizing quotients of free algebras. Horn Covarieties for Coalgebras – p.11/26
Dualizing equations Consider again equations in Set Γ . We consider the mapping EqTh X → Quot ( FX ) , and dualize the notion of sets of equations by dualizing quotients of free algebras. Horn Covarieties for Coalgebras – p.11/26
Dualizing equations Consider again equations in Set Γ . We consider the mapping EqTh X → Quot ( FX ) , and dualize the notion of sets of equations by dualizing quotients of free algebras. Return to E Γ . Let E , Γ be good (co-good?) and let H be the right adjoint to U : E Γ � E , with counit ε : UH � 1 . Horn Covarieties for Coalgebras – p.11/26
� � � Dualizing equations Return to E Γ . Let E , Γ be good and let H be the right adjoint to U : E Γ � E , with counit ε : UH � 1 . Reminder: Let C ∈ E and � A, α � ∈ E Γ . For any C -coloring p : A � C of A , there exists a unique p : � A, α � homomorphism � � HC making the diagram below commute. UHC U � p ε C A C p Horn Covarieties for Coalgebras – p.11/26
Dualizing equations Return to E Γ . Let E , Γ be good and let H be the right adjoint to U : E Γ � E , with counit ε : UH � 1 . A coequation over C is a regular subobject ϕ ≤ UHC . Horn Covarieties for Coalgebras – p.11/26
� � � � � � � Dualizing equations Return to E Γ . Let E , Γ be good and let H be the right adjoint to U : E Γ � E , with counit ε : UH � 1 . A coequation over C is a regular subobject ϕ ≤ UHC . We write � A, α � | = C ϕ iff for every coloring p : A � C of A , the adjoint transpose U � p factors through ϕ . A � � � p U � p � � � ϕ C UHC ε C Horn Covarieties for Coalgebras – p.11/26
� � � � � � � � Dualizing equations Return to E Γ . Let E , Γ be good and let H be the right adjoint to U : E Γ � E , with counit ε : UH � 1 . A coequation over C is a regular subobject ϕ ≤ UHC . We write � A, α � | = C ϕ iff for every coloring p : A � C of p ) ≤ ϕ . A , Im ( U � A � � � p U � p � � � ϕ C UHC ε C In other words, Hom ( A, C ) ∼ = Hom ( � A, α � , HC ) ∼ = Hom ( � A, α � , ϕ ) . Horn Covarieties for Coalgebras – p.11/26
� � � � � � � Dualizing equations A coequation over C is a regular subobject ϕ ≤ UHC . We write � A, α � | = C ϕ iff for every coloring p : A � C of p ) ≤ ϕ . A , Im ( U � A � � � p U � p � � � ϕ C UHC ε C � A, α � | = C ϕ just in case, however we paint the elements of A , they “look like” elements of ϕ . Horn Covarieties for Coalgebras – p.11/26
Dualizing equations A coequation over C is a regular subobject ϕ ≤ UHC . We write � A, α � | = C ϕ iff for every coloring p : A � C of p ) ≤ ϕ . A , Im ( U � We view coequations ϕ as predicates over UHC . � A, α � | = C ϕ iff, for every p : A � C , we have p ) ⊢ ϕ. Im ( U � Horn Covarieties for Coalgebras – p.11/26
Conditional coequations Let ϕ, ψ ≤ UHC . We write � A, α � | = ϕ ⇒ ψ just in case, for every p : A � C p ) ≤ ϕ , we have Im ( � p ) ≤ ψ . such that Im ( � Horn Covarieties for Coalgebras – p.12/26
� � � � � � � � � � � � Conditional coequations Let ϕ, ψ ≤ UHC . We write � A, α � | = ϕ ⇒ ψ just in case, for every p : A � C p ) ≤ ϕ , we have Im ( � p ) ≤ ψ . such that Im ( � C C � � � � � � p p � � � � � � � � � � � � � � � � � � � � � � ⇒ � � A UHC A UHC U � U � p p � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ϕ ψ Horn Covarieties for Coalgebras – p.12/26
� � � � � Conditional coequations Let ϕ, ψ ≤ UHC . We write � A, α � | = ϕ ⇒ ψ just in case, for every p : A � C p ) ≤ ϕ , we have Im ( � p ) ≤ ψ . such that Im ( � � A, α � | = ϕ ⇒ ψ just in case every homomorphism � A, α � ϕ factors through ψ , i.e., ϕ ) ∼ Hom ( � A, α � , = Hom ( � A, α � , ψ ) . Horn Covarieties for Coalgebras – p.12/26
Dualizing negations Let ϕ ≤ UHC . We write � A, α � | = ϕ just in case for every p : A � C , it is p ) ≤ ϕ . not the case Im ( � Horn Covarieties for Coalgebras – p.13/26
� � � Dualizing negations Let ϕ ≤ UHC . We write � A, α � | = ϕ just in case for every p : A � C , it is p ) ≤ ϕ . not the case Im ( � Equivalently, there is no homomorphism � A, α � ϕ , i.e., Hom ( � A, α � , ϕ ) = ∅ . Horn Covarieties for Coalgebras – p.13/26
� � � Dualizing negations Let ϕ ≤ UHC . We write � A, α � | = ϕ just in case for every p : A � C , it is p ) ≤ ϕ . not the case Im ( � Equivalently, there is no homomorphism � A, α � ϕ , i.e., Hom ( � A, α � , ϕ ) = ∅ . No matter how we paint A , there is some element a ∈ A that doesn’t land in ϕ . Horn Covarieties for Coalgebras – p.13/26
Dualizing negations Let ϕ ≤ UHC . We write � A, α � | = ϕ just in case for every p : A � C , it is p ) ≤ ϕ . not the case Im ( � No matter how we paint A , there is some element a ∈ A that doesn’t land in ϕ . Note: This does not mean that � A, α � | = ¬ ϕ ! “Something in A does not land in ϕ ,” is not the same as, “Everything in A does not land in ϕ .” Horn Covarieties for Coalgebras – p.13/26
A few more things... Let V ⊆ E Γ . CoeqTh ( V ) = { ϕ | ∃ reg. inj. C . ϕ ≤ UHC, V | = C ϕ } Horn Covarieties for Coalgebras – p.14/26
A few more things... Let V ⊆ E Γ . CoeqTh ( V ) = { ϕ | ∃ reg. inj. C . ϕ ≤ UHC, V | = C ϕ } ImpCoeqTh ( V ) = { ϕ ⇒ ψ | ∃ reg. inj. C . ϕ, ψ ≤ UHC, V | = C ϕ ⇒ ψ } Horn Covarieties for Coalgebras – p.14/26
A few more things... Let V ⊆ E Γ . CoeqTh ( V ) = { ϕ | ∃ reg. inj. C . ϕ ≤ UHC, V | = C ϕ } ImpCoeqTh ( V ) = { ϕ ⇒ ψ | ∃ reg. inj. C . ϕ, ψ ≤ UHC, V | = C ϕ ⇒ ψ } HornCoeqTh ( V ) = ImpCoeqTh ( V ) ∪ { ϕ | ∃ reg. inj. C . ϕ ≤ UHC, V | = C ϕ } Horn Covarieties for Coalgebras – p.14/26
A few more things... Let V ⊆ E Γ . CoeqTh ( V ) = { ϕ | ∃ reg. inj. C . ϕ ≤ UHC, V | = C ϕ } ImpCoeqTh ( V ) = { ϕ ⇒ ψ | ∃ reg. inj. C . ϕ, ψ ≤ UHC, V | = C ϕ ⇒ ψ } HornCoeqTh ( V ) = ImpCoeqTh ( V ) ∪ { ϕ | ∃ reg. inj. C . ϕ ≤ UHC, V | = C ϕ } Let S ⊆ HornCoeqTh . Define Sat ( S ) = {� A, α � ∈ E Γ | � A, α � | = S} . Horn Covarieties for Coalgebras – p.14/26
The covariety theorems Let E , Γ be good and V ⊆ E . Theorem (Birkhoff covariety theorem). Sat ( CoeqTh V ) = SH Σ V Horn Covarieties for Coalgebras – p.15/26
The covariety theorems Let E , Γ be good and V ⊆ E . Theorem (Birkhoff covariety theorem). Sat ( CoeqTh V ) = SH Σ V Theorem (Quasi-covariety theorem). Sat ( ImpCoeqTh V ) = H Σ V Horn Covarieties for Coalgebras – p.15/26
The covariety theorems Let E , Γ be good and V ⊆ E . Theorem (Birkhoff covariety theorem). Sat ( CoeqTh V ) = SH Σ V Theorem (Quasi-covariety theorem). Sat ( ImpCoeqTh V ) = H Σ V Theorem (Horn covariety theorem). Sat ( HornCoeqTh V ) = H Σ + V Horn Covarieties for Coalgebras – p.15/26
Outline I. Infinitary Horn varieties II. Dual theorems for E Γ III. Examples of Horn covarieties IV. Cofree for H Σ + V coalgebras V. Behavioral Horn covarieties VI. Optimistic promissary note Horn Covarieties for Coalgebras – p.16/26
Outline I. Infinitary Horn varieties II. Dual theorems for E Γ III. Examples of Horn covarieties IV. Cofree for H Σ + V coalgebras V. Behavioral Horn covarieties VI. Optimistic promissary note Horn Covarieties for Coalgebras – p.16/26
Some simple examples Fix a set Z and consider Γ: Set � Set where Γ X = Z × X . Regard a Γ -coalgebra � A, α � as a set of streams over Z and let h α : A � Z t α : A � A denote the evident head and tail operations. Horn Covarieties for Coalgebras – p.17/26
Some simple examples Fix a set Z and consider Γ: Set � Set where Γ X = Z × X . The following are Horn covarieties. • {� A, α � ∈ Set Γ | ∃ a ∈ A . t α ( a ) = a } . Horn Covarieties for Coalgebras – p.17/26
Some simple examples Fix a set Z and consider Γ: Set � Set where Γ X = Z × X . The following are Horn covarieties. • {� A, α � ∈ Set Γ | ∃ a ∈ A . t α ( a ) = a } . . t n • {� A, α � ∈ Set Γ | A � = ∅ and ∀ a ∈ A ∃ n ∈ α ( a ) = t n +1 ( a ) } . α Horn Covarieties for Coalgebras – p.17/26
Some simple examples Fix a set Z and consider Γ: Set � Set where Γ X = Z × X . The following are Horn covarieties. • {� A, α � ∈ Set Γ | ∃ a ∈ A . t α ( a ) = a } . . t n • {� A, α � ∈ Set Γ | A � = ∅ and ∀ a ∈ A ∃ n ∈ α ( a ) = t n +1 ( a ) } . α • {� A, α � ∈ Set Γ | A � = ∅ and ∀ a ∈ A ∃ n ∈ ∀ m > n . h α ◦ t n α ( a ) = h α ◦ t m α ( a ) } . Horn Covarieties for Coalgebras – p.17/26
Deterministic automata and languages Fix an alphabet I . Let Γ: Set � Set be the functor X �→ 2 × X I . Horn Covarieties for Coalgebras – p.18/26
Deterministic automata and languages Fix an alphabet I . Let Γ: Set � Set be the functor X �→ 2 × X I . A Γ -coalgebra � A, α � is an automaton accepting input from I and outputting either 0 or 1 , where out α ( a ) = π 1 ◦ α ( a ) trans α ( a ) = π 2 ◦ α ( a ) Horn Covarieties for Coalgebras – p.18/26
Deterministic automata and languages Let σ ∈ I <ω and define eval α : A × I <ω � A by eval α ( a, ()) = a, eval α ( a, σ ∗ i ) = trans α ( eval α ( a, σ ))( i ) . eval α ( a, σ ) is the final state of the calculation beginning in a with input σ . Horn Covarieties for Coalgebras – p.18/26
Deterministic automata and languages Define � P ( I <ω ) acc α : A by acc α ( a ) = { σ ∈ I <ω | out α ◦ eval α ( a, σ ) = 1 } . acc α ( a ) is the set of all words accepted by state a . Horn Covarieties for Coalgebras – p.18/26
Deterministic automata and languages Fix a “language” L ⊆ I <ω and define V L = {� A, α � ∈ Set Γ | ∃ a ∈ A . acc α ( a ) = L} . Horn Covarieties for Coalgebras – p.18/26
Deterministic automata and languages Fix a “language” L ⊆ I <ω and define V L = {� A, α � ∈ Set Γ | ∃ a ∈ A . acc α ( a ) = L} . V L is a Horn covariety. Horn Covarieties for Coalgebras – p.18/26
Deterministic automata and languages Fix a “language” L ⊆ I <ω and define V L = {� A, α � ∈ Set Γ | ∃ a ∈ A . acc α ( a ) = L} . V L is a Horn covariety. Explicitly: the class of all automata which have an initial state accepting exactly L is closed under codomains of epis and non-empty coproducts. Furthermore, V L is definable by a Horn coequation. Horn Covarieties for Coalgebras – p.18/26
� Deterministic automata and languages Fix a “language” L ⊆ I <ω and define V L = {� A, α � ∈ Set Γ | ∃ a ∈ A . acc α ( a ) = L} . V L is a Horn covariety. Indeed, let ϕ ≤ UH 1 be the set { c ∈ UH 1 | acc H 1 ( c ) � = L} . Then � A, α � ∈ V L just in case Hom ( � A, α � , ϕ ) = ∅ . Horn Covarieties for Coalgebras – p.18/26
More automata Fix L ⊆ I <ω . 1. Deterministic automata which have an accepting state for L . Horn Covarieties for Coalgebras – p.19/26
More automata Fix L ⊆ I <ω . 1. Deterministic automata which have an accepting state for L . 2. Non-deterministic automata which have a non-empty, deterministic sub-automaton. Horn Covarieties for Coalgebras – p.19/26
More automata Fix L ⊆ I <ω . 1. Deterministic automata which have an accepting state for L . 2. Non-deterministic automata which have a non-empty, deterministic sub-automaton. 3. Non-deterministic automata which have a deterministic sub-automata in (1). Horn Covarieties for Coalgebras – p.19/26
More automata Fix L ⊆ I <ω . 1. Deterministic automata which have an accepting state for L . 2. Non-deterministic automata which have a non-empty, deterministic sub-automaton. 3. Non-deterministic automata which have a deterministic sub-automata in (1). 4. Etc. and so on. Horn Covarieties for Coalgebras – p.19/26
Outline I. Infinitary Horn varieties II. Dual theorems for E Γ III. Examples of Horn covarieties IV. Cofree for H Σ + V coalgebras V. Behavioral Horn covarieties VI. Optimistic promissary note Horn Covarieties for Coalgebras – p.20/26
Outline I. Infinitary Horn varieties II. Dual theorems for E Γ III. Examples of Horn covarieties IV. Cofree for H Σ + V coalgebras V. Behavioral Horn covarieties VI. Optimistic promissary note Horn Covarieties for Coalgebras – p.20/26
Cofree for H Σ + V coalgebras Let V ⊆ E Γ , C ∈ E . Define Θ C = { f : � A, α � � HC | � A, α � ∈ V } , � { Im f | f ∈ Θ C } . ∆ C = Horn Covarieties for Coalgebras – p.21/26
Cofree for H Σ + V coalgebras Let V ⊆ E Γ , C ∈ E . Define Θ C = { f : � A, α � � HC | � A, α � ∈ V } , � { Im f | f ∈ Θ C } . ∆ C = If Θ C � = ∅ , then ∆ C is cofree for H Σ + V over C , i.e., • ∆ C ∈ H Σ + V ; Horn Covarieties for Coalgebras – p.21/26
� � Cofree for H Σ + V coalgebras If Θ C � = ∅ , then ∆ C is cofree for H Σ + V over C , i.e., • ∆ C ∈ H Σ + V ; • If � B, β � ∈ H Σ + V , then for every p : B � C , there is a p : � B, β � � ∆ C such that the unique homomorphism � diagram below commutes. B � p p � UHC ε C � C U ∆ C � � Horn Covarieties for Coalgebras – p.21/26
Cofree for H Σ + V coalgebras If Θ C � = ∅ , then ∆ C is cofree for H Σ + V over C . If E = Set (or any category in which each C � = 0 has a global element) and V � = 0 , then every C � = 0 has a cofree for H Σ + V coalgebra. Horn Covarieties for Coalgebras – p.21/26
� � � � Cofree for H Σ + V coalgebras If Θ C � = ∅ , then ∆ C is cofree for H Σ + V over C . If E = Set (or any category in which each C � = 0 has a global element) and V � = 0 , then every C � = 0 has a cofree for H Σ + V coalgebra. In technical terms, we have damn near an adjunction . Indeed, it arises as the composition of an adjunction and damn near a regular mono-coreflection . H � E Γ � V E U Horn Covarieties for Coalgebras – p.21/26
Outline I. Infinitary Horn varieties II. Dual theorems for E Γ III. Examples of Horn covarieties IV. Cofree for H Σ + V coalgebras V. Behavioral Horn covarieties VI. Optimistic promissary note Horn Covarieties for Coalgebras – p.22/26
Outline I. Infinitary Horn varieties II. Dual theorems for E Γ III. Examples of Horn covarieties IV. Cofree for H Σ + V coalgebras V. Behavioral Horn covarieties VI. Optimistic promissary note Horn Covarieties for Coalgebras – p.22/26
Behavioral classes Consider the following operators. R V = {� B, β � ∈ E Γ | ∃� B, β � � � � A, α � ∈ V } Horn Covarieties for Coalgebras – p.23/26
� � Behavioral classes Consider the following operators. R V = {� B, β � ∈ E Γ | ∃� B, β � � � � A, α � ∈ V } B V = {� B, β � ∈ E Γ | ∃ bisimulation � � A ∈ V } B R Horn Covarieties for Coalgebras – p.23/26
� � � � Behavioral classes Consider the following operators. R V = {� B, β � ∈ E Γ | ∃� B, β � � � � A, α � ∈ V } B V = {� B, β � ∈ E Γ | ∃ bisimulation � � A ∈ V } B R Q V = {� B, β � ∈ E Γ | � � � A, α � ∈ V } ∃ � B, β � C Horn Covarieties for Coalgebras – p.23/26
� � � � Behavioral classes Consider the following operators. R V = {� B, β � ∈ E Γ | ∃� B, β � � � � A, α � ∈ V } B V = {� B, β � ∈ E Γ | ∃ bisimulation � � A ∈ V } B R Q V = {� B, β � ∈ E Γ | � � � A, α � ∈ V } ∃ � B, β � C RH V = BB V = QQ V . Horn Covarieties for Coalgebras – p.23/26
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