Similarity quotients as final coalgebras Paul Blain Levy University - PowerPoint PPT Presentation

Similarity quotients as final coalgebras Paul Blain Levy University of Birmingham February 3, 2010 Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 1 / 32

Outline Examples 1 General Theory 2 Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 2 / 32

Examples We study the following examples: 1 bisimilarity 2 bisimilarity and similarity together 3 similarity 4 upper similarity 5 intersection of lower and upper similarity 6 2-nested similarity Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 3 / 32

Examples We study the following examples: 1 bisimilarity 2 bisimilarity and similarity together 3 similarity 4 upper similarity 5 intersection of lower and upper similarity 6 2-nested similarity In each case we see how to use a final coalgebra how to construct a final coalgebra. Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 3 / 32

Bisimilarity: Using A Final Coalgebra Fix a countable set Act of labels. Let F : X �→ P � ℵ 0 (Act × X ) on Set . A countably branching Act-labelled transition system is an F -coalgebra. Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 4 / 32

Bisimilarity: Using A Final Coalgebra Fix a countable set Act of labels. Let F : X �→ P � ℵ 0 (Act × X ) on Set . A countably branching Act-labelled transition system is an F -coalgebra. Let A be a final F -coalgebra, and σ B the anamorphism from B . Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 4 / 32

Bisimilarity: Using A Final Coalgebra Fix a countable set Act of labels. Let F : X �→ P � ℵ 0 (Act × X ) on Set . A countably branching Act-labelled transition system is an F -coalgebra. Let A be a final F -coalgebra, and σ B the anamorphism from B . Theorem: characterizing bisimilarity b ∈ B is bisimilar to c ∈ C iff σ B b = σ C c . Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 4 / 32

Bisimilarity: Using A Final Coalgebra Fix a countable set Act of labels. Let F : X �→ P � ℵ 0 (Act × X ) on Set . A countably branching Act-labelled transition system is an F -coalgebra. Let A be a final F -coalgebra, and σ B the anamorphism from B . Theorem: characterizing bisimilarity b ∈ B is bisimilar to c ∈ C iff σ B b = σ C c . Theorem: no junk Every element of A is of the form σ B b . Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 4 / 32

Bisimilarity: Constructing A Final Coalgebra Let F : X �→ P � ℵ 0 (Act × X ) on Set . Suppose A is a transition system that is big enough i.e. every b ∈ B is bisimilar to some a ∈ A . Then A modulo bisimilarity (with behaviour map chosen to make � A / � a homomorphism) is a final F -coalgebra. A Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 5 / 32

Bisimilarity: Constructing A Final Coalgebra Let F : X �→ P � ℵ 0 (Act × X ) on Set . Suppose A is a transition system that is big enough i.e. every b ∈ B is bisimilar to some a ∈ A . Then A modulo bisimilarity (with behaviour map chosen to make � A / � a homomorphism) is a final F -coalgebra. A Example of a big enough transition system def A = the disjoint union of all transition systems on initial segments of N . It’s big enough because every ( B , b ) has countably many successors. Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 5 / 32

Bisimilarity: Constructing A Final Coalgebra Let F : X �→ P � ℵ 0 (Act × X ) on Set . Suppose A is a transition system that is big enough i.e. every b ∈ B is bisimilar to some a ∈ A . Then A modulo bisimilarity (with behaviour map chosen to make � A / � a homomorphism) is a final F -coalgebra. A Example of a big enough transition system def A = the disjoint union of all transition systems on initial segments of N . It’s big enough because every ( B , b ) has countably many successors. If A isn’t big enough, then A / � is still subfinal, i.e. parallel morphisms to it are equal. Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 5 / 32

Bisimilarity and Similarity: Using A Final Coalgebra Let G be the endofunctor on Preord mapping ( X , � ) to ( P � ℵ 0 (Act × X ) , Sim( � )) def where U Sim( R ) V ⇔ ∀ x ∈ U . ∃ y ∈ V . u R v . Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 6 / 32

Bisimilarity and Similarity: Using A Final Coalgebra Let G be the endofunctor on Preord mapping ( X , � ) to ( P � ℵ 0 (Act × X ) , Sim( � )) def where U Sim( R ) V ⇔ ∀ x ∈ U . ∃ y ∈ V . u R v . Let A be a final G -coalgebra. Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 6 / 32

Bisimilarity and Similarity: Using A Final Coalgebra Let G be the endofunctor on Preord mapping ( X , � ) to ( P � ℵ 0 (Act × X ) , Sim( � )) def where U Sim( R ) V ⇔ ∀ x ∈ U . ∃ y ∈ V . u R v . Let A be a final G -coalgebra. Any transition system B gives a G -coalgebra ∆ B , using the discrete order. Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 6 / 32

Bisimilarity and Similarity: Using A Final Coalgebra Let G be the endofunctor on Preord mapping ( X , � ) to ( P � ℵ 0 (Act × X ) , Sim( � )) def where U Sim( R ) V ⇔ ∀ x ∈ U . ∃ y ∈ V . u R v . Let A be a final G -coalgebra. Any transition system B gives a G -coalgebra ∆ B , using the discrete order. Theorem: characterizing bisimilarity and similarity b ∈ B is bisimilar to c ∈ C iff σ ∆ B b = σ ∆ C c . b ∈ B is similar to c ∈ C iff σ ∆ B b � σ ∆ C c Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 6 / 32

Bisimilarity and Similarity: Using A Final Coalgebra Let G be the endofunctor on Preord mapping ( X , � ) to ( P � ℵ 0 (Act × X ) , Sim( � )) def where U Sim( R ) V ⇔ ∀ x ∈ U . ∃ y ∈ V . u R v . Let A be a final G -coalgebra. Any transition system B gives a G -coalgebra ∆ B , using the discrete order. Theorem: characterizing bisimilarity and similarity b ∈ B is bisimilar to c ∈ C iff σ ∆ B b = σ ∆ C c . b ∈ B is similar to c ∈ C iff σ ∆ B b � σ ∆ C c Theorem: no junk Every element of A is of the form σ ∆ B b . Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 6 / 32

Bisimilarity and similarity: constructing a final coalgebra (Hughes-Jacobs) Let G be the endofunctor on Preord mapping ( X , � ) to ( P � ℵ 0 (Act × X ) , Sim( � )) Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 7 / 32

Bisimilarity and similarity: constructing a final coalgebra (Hughes-Jacobs) Let G be the endofunctor on Preord mapping ( X , � ) to ( P � ℵ 0 (Act × X ) , Sim( � )) Suppose A is a transition system that is big enough i.e. every b ∈ B is bisimilar to some a ∈ A . Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 7 / 32

Bisimilarity and similarity: constructing a final coalgebra (Hughes-Jacobs) Let G be the endofunctor on Preord mapping ( X , � ) to ( P � ℵ 0 (Act × X ) , Sim( � )) Suppose A is a transition system that is big enough i.e. every b ∈ B is bisimilar to some a ∈ A . Then A modulo bisimilarity, preordered by similarity, is a final G -coalgebra. Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 7 / 32

Quotienting by a preorder If A is a set with an equivalence relation ∼ then A / ∼ is a set consisting of the equivalence classes def [ a ] ∼ = { x ∈ A | x ∼ a } Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 8 / 32

Quotienting by a preorder If A is a set with an equivalence relation ∼ then A / ∼ is a set consisting of the equivalence classes def [ a ] ∼ = { x ∈ A | x ∼ a } More generally if A is a set with a preorder � then A / � is a poset consisting of the principal lower sets def [ a ] � = { x ∈ A | x � a } ordered by inclusion. Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 8 / 32

� � Quotienting by a preorder If A is a set with an equivalence relation ∼ then A / ∼ is a set consisting of the equivalence classes def [ a ] ∼ = { x ∈ A | x ∼ a } More generally if A is a set with a preorder � then A / � is a poset consisting of the principal lower sets def [ a ] � = { x ∈ A | x � a } ordered by inclusion. So Poset is a full reflective subcategory of Preord . Q Poset � � Preord ⊥ Paul Blain Levy (University of Birmingham) Similarity quotients as final coalgebras February 3, 2010 8 / 32