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Jan Rutten - Concrete coalgebra: an introduction by examples - Lecture 1

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This is part one of Jan Rutten's lectures on Coalgebra in Tehran, Iran.

Abstract: Since the early nineties, coalgebra has become an active area of research in which one tries to understand all kinds of infinite data types, automata, transition systems and dynamical systems from a unifying perspective. The focus of coalgebra is on observable behaviour and one uses coinduction as a central methodology, both for behavioural specifications and to prove behavioural equivalences. These days, one uses coalgebraic techniques in a wide variety of areas, ranging from automata theory to software engineering to economy.

In a series of three lectures, we will discuss the following subjects:
(i) The coalgebraic method
(ii) A coinductive calculus of streams
(iii) Algebra meets coalgebra: automata and formal languages

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Jan Rutten - Concrete coalgebra: an introduction by examples - Lecture 1

  1. 1. Lecture one: The Method of Coalgebra – in some detail – Jan Rutten CWI Amsterdam & Radboud University Nijmegen IPM, Tehran - 13 January 2016
  2. 2. Acknowledgements My esteemed co-authors - Marcello Bonsangue (Leiden, CWI) - Helle Hansen (Delft) - Alexandra Silva (City College London) - Milad Niqui - Clemens Kupke (Glasgow) - Prakash Panangaden (McGill, Montreal) - Filippo Bonchi (ENS, Lyon) - Joost Winter (Warsaw), Jurriaan Rot (ENS, Lyon) - and many others. Samson Abramsky (Oxford) Damien Pous (ENS, Lyon)
  3. 3. Acknowledgements My esteemed co-authors - Marcello Bonsangue (Leiden, CWI) - Helle Hansen (Delft) - Alexandra Silva (City College London) - Milad Niqui - Clemens Kupke (Glasgow) - Prakash Panangaden (McGill, Montreal) - Filippo Bonchi (ENS, Lyon) - Joost Winter (Warsaw), Jurriaan Rot (ENS, Lyon) - and many others. Samson Abramsky (Oxford) Damien Pous (ENS, Lyon)
  4. 4. Overview of todays lectures Lecture one: The method of coalgebra Lecture two: A coinductive calculus of streams Lecture three: Automata and the algebra-coalgebra duality Lecture four: Coalgebraic up-to techniques
  5. 5. Overview of Lecture one 1. Category theory (where coalgebra comes from) 2. Duality (where coalgebra comes from) 3. How coalgebra works (the method in slogans) 4. Duality: induction and coinduction 5. What coalgebra studies (its subject) 6. Discussion
  6. 6. 1. Category theory (where coalgebra comes from)
  7. 7. Why categories? From Samson Abramsky’s tutorial: Categories, why and how? (Dagstuhl, January 2015)
  8. 8. Why categories? For logicians: gives a syntax-independent view of the fundamental structures of logic, opens up new kinds of models and interpretations. For philosophers: a fresh approach to structuralist foundations of mathematics and science; an alternative to the traditional focus on set theory. For computer scientists: gives a precise handle on abstraction, representation-independence, genericity and more. Gives the fundamental mathematical structures underpinning programming concepts.
  9. 9. Why categories? For mathematicians: organizes your previous mathematical experience in a new and powerful way, reveals new connections and structure, allows you to “think bigger thoughts”. For physicists: new ways of formulating physical theories in a structural form. Recent applications to Quantum Information and Computation. For economists and game theorists: new tools, bringing complex phenomena into the scope of formalisation.
  10. 10. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  11. 11. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  12. 12. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  13. 13. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  14. 14. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  15. 15. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  16. 16. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  17. 17. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  18. 18. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  19. 19. Category Theory in 10 Slogans 1. Always ask: what are the types? 2. Think in terms of arrows rather than elements. 3. Ask what mathematical structures do, not what they are. 4. Categories as mathematical contexts. 5. Categories as mathematical structures. 6. Make definitions and constructions as general as possible. 7. Functoriality! 8. Naturality! 9. Universality! 10. Adjoints are everywhere.
  20. 20. Categories: basic definitions A category C consists of - Objects A, B, C, . . . - Morphisms/arrows: for each pair of objects A, B, a set of morphisms C(A, B) with domain A and codomain B - Composition of morphisms: g ◦ f: A g ◦ f TT f GG B g GG C - Axioms: h ◦ (g ◦ f) = (h ◦ g) ◦ f f ◦ id = f = id ◦ f
  21. 21. Categories: basic definitions A category C consists of - Objects A, B, C, . . . - Morphisms/arrows: for each pair of objects A, B, a set of morphisms C(A, B) with domain A and codomain B - Composition of morphisms: g ◦ f: A g ◦ f TT f GG B g GG C - Axioms: h ◦ (g ◦ f) = (h ◦ g) ◦ f f ◦ id = f = id ◦ f
  22. 22. Categories: basic definitions A category C consists of - Objects A, B, C, . . . - Morphisms/arrows: for each pair of objects A, B, a set of morphisms C(A, B) with domain A and codomain B - Composition of morphisms: g ◦ f: A g ◦ f TT f GG B g GG C - Axioms: h ◦ (g ◦ f) = (h ◦ g) ◦ f f ◦ id = f = id ◦ f
  23. 23. Categories: basic definitions A category C consists of - Objects A, B, C, . . . - Morphisms/arrows: for each pair of objects A, B, a set of morphisms C(A, B) with domain A and codomain B - Composition of morphisms: g ◦ f: A g ◦ f TT f GG B g GG C - Axioms: h ◦ (g ◦ f) = (h ◦ g) ◦ f f ◦ id = f = id ◦ f
  24. 24. Categories: basic definitions A category C consists of - Objects A, B, C, . . . - Morphisms/arrows: for each pair of objects A, B, a set of morphisms C(A, B) with domain A and codomain B - Composition of morphisms: g ◦ f: A g ◦ f TT f GG B g GG C - Axioms: h ◦ (g ◦ f) = (h ◦ g) ◦ f f ◦ id = f = id ◦ f
  25. 25. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  26. 26. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  27. 27. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  28. 28. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  29. 29. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  30. 30. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  31. 31. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  32. 32. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  33. 33. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  34. 34. Categories: examples • Any kind of mathematical structure, together with structure preserving functions, forms a category. E.g. - sets and functions - groups and group homomorphisms - monoids and monoid homomorphisms - vector spaces over a field k, and linear maps - topological spaces and continuous functions - partially ordered sets and monotone functions • Monoids are one-object categories • algebras, and algebra homomorphisms • coalgebras, and coalgebra homomorphisms
  35. 35. Arrows rather than elements A function f : X → Y (between sets) is: - injective if ∀x, y ∈ X, f(x) = f(y) ⇒ x = y - surjective if ∀y ∈ Y, ∃x ∈ X, f(x) = y - monic if ∀g, h, f ◦ g = f ◦ h ⇒ g = h - epic if ∀g, h, g ◦ f = h ◦ f ⇒ g = h Proposition • m is injective iff m is monic. • e is surjective iff e is epic.
  36. 36. Arrows rather than elements A function f : X → Y (between sets) is: - injective if ∀x, y ∈ X, f(x) = f(y) ⇒ x = y - surjective if ∀y ∈ Y, ∃x ∈ X, f(x) = y - monic if ∀g, h, f ◦ g = f ◦ h ⇒ g = h - epic if ∀g, h, g ◦ f = h ◦ f ⇒ g = h Proposition • m is injective iff m is monic. • e is surjective iff e is epic.
  37. 37. Arrows rather than elements A function f : X → Y (between sets) is: - injective if ∀x, y ∈ X, f(x) = f(y) ⇒ x = y - surjective if ∀y ∈ Y, ∃x ∈ X, f(x) = y - monic if ∀g, h, f ◦ g = f ◦ h ⇒ g = h - epic if ∀g, h, g ◦ f = h ◦ f ⇒ g = h Proposition • m is injective iff m is monic. • e is surjective iff e is epic.
  38. 38. Arrows rather than elements A function f : X → Y (between sets) is: - injective if ∀x, y ∈ X, f(x) = f(y) ⇒ x = y - surjective if ∀y ∈ Y, ∃x ∈ X, f(x) = y - monic if ∀g, h, f ◦ g = f ◦ h ⇒ g = h - epic if ∀g, h, g ◦ f = h ◦ f ⇒ g = h Proposition • m is injective iff m is monic. • e is surjective iff e is epic.
  39. 39. Arrows rather than elements A function f : X → Y (between sets) is: - injective if ∀x, y ∈ X, f(x) = f(y) ⇒ x = y - surjective if ∀y ∈ Y, ∃x ∈ X, f(x) = y - monic if ∀g, h, f ◦ g = f ◦ h ⇒ g = h - epic if ∀g, h, g ◦ f = h ◦ f ⇒ g = h Proposition • m is injective iff m is monic. • e is surjective iff e is epic.
  40. 40. Arrows rather than elements A function f : X → Y (between sets) is: - injective if ∀x, y ∈ X, f(x) = f(y) ⇒ x = y - surjective if ∀y ∈ Y, ∃x ∈ X, f(x) = y - monic if ∀g, h, f ◦ g = f ◦ h ⇒ g = h - epic if ∀g, h, g ◦ f = h ◦ f ⇒ g = h Proposition • m is injective iff m is monic. • e is surjective iff e is epic.
  41. 41. Arrows rather than elements A function f : X → Y (between sets) is: - injective if ∀x, y ∈ X, f(x) = f(y) ⇒ x = y - surjective if ∀y ∈ Y, ∃x ∈ X, f(x) = y - monic if ∀g, h, f ◦ g = f ◦ h ⇒ g = h - epic if ∀g, h, g ◦ f = h ◦ f ⇒ g = h Proposition • m is injective iff m is monic. • e is surjective iff e is epic.
  42. 42. Arrows rather than elements A function f : X → Y (between sets) is: - injective if ∀x, y ∈ X, f(x) = f(y) ⇒ x = y - surjective if ∀y ∈ Y, ∃x ∈ X, f(x) = y - monic if ∀g, h, f ◦ g = f ◦ h ⇒ g = h - epic if ∀g, h, g ◦ f = h ◦ f ⇒ g = h Proposition • m is injective iff m is monic. • e is surjective iff e is epic.
  43. 43. Arrows rather than elements Defining the Cartesian product . . . - with elements: A × B = { a, b | a ∈ A, b ∈ B } where a, b = {{a, b}, b} - with arrows (expressing a universal property): ∀C f ÐÐ g 11 f, g A A × Bπ1 oo π2 GG B
  44. 44. Arrows rather than elements Defining the Cartesian product . . . - with elements: A × B = { a, b | a ∈ A, b ∈ B } where a, b = {{a, b}, b} - with arrows (expressing a universal property): ∀C f ÐÐ g 11 f, g A A × Bπ1 oo π2 GG B
  45. 45. Arrows rather than elements Defining the Cartesian product . . . - with elements: A × B = { a, b | a ∈ A, b ∈ B } where a, b = {{a, b}, b} - with arrows (expressing a universal property): ∀C f ÐÐ g 11 f, g A A × Bπ1 oo π2 GG B
  46. 46. 2. Duality (where coalgebra comes from) An additional slogan for categories: duality is omnipresent - epi - mono - product - sum - initial object - final object - algebra - coalgebra
  47. 47. 2. Duality (where coalgebra comes from) An additional slogan for categories: duality is omnipresent - epi - mono - product - sum - initial object - final object - algebra - coalgebra
  48. 48. Duality: monos and epis - f is monic: ∀g, h, f ◦ g = f ◦ h ⇒ g = h • g 99 h UU • f GG • - f is epic: ∀g, h, g ◦ f = h ◦ f ⇒ g = h • • h gg g ww • foo Proposition: f is monic in C iff f is epic in Cop.
  49. 49. Duality: monos and epis - f is monic: ∀g, h, f ◦ g = f ◦ h ⇒ g = h • g 99 h UU • f GG • - f is epic: ∀g, h, g ◦ f = h ◦ f ⇒ g = h • • h gg g ww • foo Proposition: f is monic in C iff f is epic in Cop.
  50. 50. Duality: monos and epis - f is monic: ∀g, h, f ◦ g = f ◦ h ⇒ g = h • g 99 h UU • f GG • - f is epic: ∀g, h, g ◦ f = h ◦ f ⇒ g = h • • h gg g ww • foo Proposition: f is monic in C iff f is epic in Cop.
  51. 51. Duality: monos and epis - f is monic: ∀g, h, f ◦ g = f ◦ h ⇒ g = h • g 99 h UU • f GG • - f is epic: ∀g, h, g ◦ f = h ◦ f ⇒ g = h • • h gg g ww • foo Proposition: f is monic in C iff f is epic in Cop.
  52. 52. Duality: products and coproducts The product of A and B: ∀C f ÐÐ g 11 f, g A A × Bπ1 oo π2 GG B The coproduct of A and B: ∀C A κ1 GG f PP A + B [f, g] yy Bκ2 oo gll Proposition: O is product in C iff O is coproduct in Cop.
  53. 53. Duality: products and coproducts The product of A and B: ∀C f ÐÐ g 11 f, g A A × Bπ1 oo π2 GG B The coproduct of A and B: ∀C A κ1 GG f PP A + B [f, g] yy Bκ2 oo gll Proposition: O is product in C iff O is coproduct in Cop.
  54. 54. Duality: products and coproducts The product of A and B: ∀C f ÐÐ g 11 f, g A A × Bπ1 oo π2 GG B The coproduct of A and B: ∀C A κ1 GG f PP A + B [f, g] yy Bκ2 oo gll Proposition: O is product in C iff O is coproduct in Cop.
  55. 55. Duality: products and coproducts The product of A and B: ∀C f ÐÐ g 11 f, g A A × Bπ1 oo π2 GG B The coproduct of A and B: ∀C A κ1 GG f PP A + B [f, g] yy Bκ2 oo gll Proposition: O is product in C iff O is coproduct in Cop.
  56. 56. Duality: initial and final objects An object A in a category C is . . . - initial if for any object B there exists a unique arrow A ! GG B - final if for any object B there exists a unique arrow B ! GG A Proposition: A is initial in C iff A is final in Cop. Proposition: Initial and final objects are unique up-to isomorphism.
  57. 57. Duality: initial and final objects An object A in a category C is . . . - initial if for any object B there exists a unique arrow A ! GG B - final if for any object B there exists a unique arrow B ! GG A Proposition: A is initial in C iff A is final in Cop. Proposition: Initial and final objects are unique up-to isomorphism.
  58. 58. Duality: initial and final objects An object A in a category C is . . . - initial if for any object B there exists a unique arrow A ! GG B - final if for any object B there exists a unique arrow B ! GG A Proposition: A is initial in C iff A is final in Cop. Proposition: Initial and final objects are unique up-to isomorphism.
  59. 59. Duality: initial and final objects An object A in a category C is . . . - initial if for any object B there exists a unique arrow A ! GG B - final if for any object B there exists a unique arrow B ! GG A Proposition: A is initial in C iff A is final in Cop. Proposition: Initial and final objects are unique up-to isomorphism.
  60. 60. Duality: initial and final objects An object A in a category C is . . . - initial if for any object B there exists a unique arrow A ! GG B - final if for any object B there exists a unique arrow B ! GG A Proposition: A is initial in C iff A is final in Cop. Proposition: Initial and final objects are unique up-to isomorphism.
  61. 61. Duality: initial and final objects An object A in a category C is . . . - initial if for any object B there exists a unique arrow A ! GG B - final if for any object B there exists a unique arrow B ! GG A Proposition: A is initial in C iff A is final in Cop. Proposition: Initial and final objects are unique up-to isomorphism.
  62. 62. Where coalgebra comes from By duality. From algebra! Classically, algebras are sets with operations. Ex. (N, 0, succ), with 0 ∈ N and succ : N → N. Equivalently, 1 + N [zero, succ] N where 1 = {∗} and zero(∗) = 0.
  63. 63. Where coalgebra comes from By duality. From algebra! Classically, algebras are sets with operations. Ex. (N, 0, succ), with 0 ∈ N and succ : N → N. Equivalently, 1 + N [zero, succ] N where 1 = {∗} and zero(∗) = 0.
  64. 64. Where coalgebra comes from By duality. From algebra! Classically, algebras are sets with operations. Ex. (N, 0, succ), with 0 ∈ N and succ : N → N. Equivalently, 1 + N [zero, succ] N where 1 = {∗} and zero(∗) = 0.
  65. 65. Algebra Classically, algebras are sets with operations. Ex. Prog × Prog α Prog with α(P1, P2) = P1; P2.
  66. 66. Algebra, categorically F(X) α X where F is the type of the algebra.
  67. 67. Coalgebra, dually X α F(X) where F is the type of the coalgebra.
  68. 68. Example: streams Streams are our favourite example of a coalgebra: Nω head, tail N × Nω where head(σ) = σ(0) tail(σ) = (σ(1), σ(2), σ(3), . . .) for any stream σ = (σ(0), σ(1), σ(2), . . .) ∈ Nω.
  69. 69. 3. How coalgebra works (its method in slogans) • be precise about types • ask what a system does rather than what it is • functoriality • interaction through homomorphisms • aim for universality Note that all these slogans are part of the categorical approach.
  70. 70. 3. How coalgebra works (its method in slogans) • be precise about types • ask what a system does rather than what it is • functoriality • interaction through homomorphisms • aim for universality Note that all these slogans are part of the categorical approach.
  71. 71. Starting point: the system’s type A coalgebra of type F is a pair (X, α) with α : X → F(X) For instance, non-deterministic transition systems: X α X α X α X α Pf (A×X) Pf (X)A P(X)A 2 × P(X)A Formally, the type F of a coalgebra/system is a functor.
  72. 72. The importance of knowing the system’s type The type F of a coalgebra/system α : X → F(X) determines - a canonical notion of system equivalence: bisimulation - a canonical notion of minimization - a canonical interpretation: final coalgebra semantics - (a canonical logic)
  73. 73. Doing versus being Doing Being Behaviour Construction Systems as black boxes (with internal states) Behavioural specification
  74. 74. Doing versus being Doing Being Behaviour Construction Systems as black boxes (with internal states) Behavioural specification
  75. 75. Doing versus being Doing Being Behaviour Construction Systems as black boxes (with internal states) Behavioural specification
  76. 76. Doing versus being Doing Being Behaviour Construction Systems as black boxes (with internal states) Behavioural specification
  77. 77. Doing versus being Doing Being Behaviour Construction Systems as black boxes (with internal states) Behavioural specification
  78. 78. Example: the shuffle product of streams Being: (σ ⊗ τ) (n) = n k=0 n k · σ(k) · τ(n − k) Doing: σ ⊗ τ σ(0) · τ(0) GG (σ ⊗ τ) + (σ ⊗ τ )
  79. 79. Example: the shuffle product of streams Being: (σ ⊗ τ) (n) = n k=0 n k · σ(k) · τ(n − k) Doing: σ ⊗ τ σ(0) · τ(0) GG (σ ⊗ τ) + (σ ⊗ τ )
  80. 80. Example: the shuffle product of streams Being: (σ ⊗ τ) (n) = n k=0 n k · σ(k) · τ(n − k) Doing: σ ⊗ τ σ(0) · τ(0) GG (σ ⊗ τ) + (σ ⊗ τ )
  81. 81. Example: the Hamming numbers Being: The increasing stream h of all natural numbers that are divisible by only 2, 3, or 5: h = (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, . . . ) h(n) = ? Doing: h 1 GG (2 · h) (3 · h) (5 · h)
  82. 82. Example: the Hamming numbers Being: The increasing stream h of all natural numbers that are divisible by only 2, 3, or 5: h = (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, . . . ) h(n) = ? Doing: h 1 GG (2 · h) (3 · h) (5 · h)
  83. 83. Example: the Hamming numbers Being: The increasing stream h of all natural numbers that are divisible by only 2, 3, or 5: h = (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, . . . ) h(n) = ? Doing: h 1 GG (2 · h) (3 · h) (5 · h)
  84. 84. Example: the Hamming numbers Being: The increasing stream h of all natural numbers that are divisible by only 2, 3, or 5: h = (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, . . . ) h(n) = ? Doing: h 1 GG (2 · h) (3 · h) (5 · h)
  85. 85. Homomorphisms X α h GG Y β F(X) F(h) GG F(Y) . . . are for systems/coalgebras what functions are for sets. . . . are behaviour preserving functions.
  86. 86. Functoriality X α h GG Y β F(X) F(h) GG F(Y) Note that for the definition of homomorphism, the type F needs to be a functor: F acts on sets: F(X), F(Y) and on functions: F(h)
  87. 87. Example of a homomorphism X h GG Y O × X id × h GG O × Y x0 a GG x1 b GG x2 a GG x3 b ww h GG y0 a VV y1 b xx Minimization through (canonical) homomorphism.
  88. 88. Universality Always aim at universal/canonical formulations. For instance: final coalgebras In final coalgebras: Being = Doing ⇒ coinduction (to be discussed shortly) ⇒ semantics
  89. 89. Universality Always aim at universal/canonical formulations. For instance: final coalgebras In final coalgebras: Being = Doing ⇒ coinduction (to be discussed shortly) ⇒ semantics
  90. 90. Universality Always aim at universal/canonical formulations. For instance: final coalgebras In final coalgebras: Being = Doing ⇒ coinduction (to be discussed shortly) ⇒ semantics
  91. 91. Universality Always aim at universal/canonical formulations. For instance: final coalgebras In final coalgebras: Being = Doing ⇒ coinduction (to be discussed shortly) ⇒ semantics
  92. 92. Universality Always aim at universal/canonical formulations. For instance: final coalgebras In final coalgebras: Being = Doing ⇒ coinduction (to be discussed shortly) ⇒ semantics
  93. 93. Universality Always aim at universal/canonical formulations. For instance: final coalgebras In final coalgebras: Being = Doing ⇒ coinduction (to be discussed shortly) ⇒ semantics
  94. 94. Semantics by finality: streams The final homomorphism into the set of streams: X ∃! h GG Oω O × X id × h GG O × Oω maps any system X to its minimization: e.g., x0 a GG x1 b GG x2 a GG x3 b ww h GG (ab)ω a RR (ba)ω btt x0, x2 h GG (ab)ω x1, x3 h GG (ba)ω
  95. 95. 4. Duality: induction and coinduction - initial algebra - final coalgebra - congruence - bisimulation - induction - coinduction - least fixed point - greatest fixed point
  96. 96. Initial algebra The natural numbers are an example of an initial algebra: 1 + N [zero, succ] GG 1 + S β∀ N ∃ ! GG S Note: any two homomorphisms from N to S are equal. Note: id : N → N is a homomorphism. Note: [zero, succ] : 1 + N ∼= N.
  97. 97. Initial algebra The natural numbers are an example of an initial algebra: 1 + N [zero, succ] GG 1 + S β∀ N ∃ ! GG S Note: any two homomorphisms from N to S are equal. Note: id : N → N is a homomorphism. Note: [zero, succ] : 1 + N ∼= N.
  98. 98. Initial algebra The natural numbers are an example of an initial algebra: 1 + N [zero, succ] GG 1 + S β∀ N ∃ ! GG S Note: any two homomorphisms from N to S are equal. Note: id : N → N is a homomorphism. Note: [zero, succ] : 1 + N ∼= N.
  99. 99. Initial algebra The natural numbers are an example of an initial algebra: 1 + N [zero, succ] GG 1 + S β∀ N ∃ ! GG S Note: any two homomorphisms from N to S are equal. Note: id : N → N is a homomorphism. Note: [zero, succ] : 1 + N ∼= N.
  100. 100. Final coalgebra Streams are an example of a final coalgebra: S β∀ ∃ ! GG Nω head, tail N × S GG N × Nω (Note: instead of N, we could have taken any set.) Note: any two homomorphisms from S to Nω are equal. Note: id : Nω → Nω is a homomorphism. Note: head, tail : Nω ∼= N × Nω.
  101. 101. Final coalgebra Streams are an example of a final coalgebra: S β∀ ∃ ! GG Nω head, tail N × S GG N × Nω (Note: instead of N, we could have taken any set.) Note: any two homomorphisms from S to Nω are equal. Note: id : Nω → Nω is a homomorphism. Note: head, tail : Nω ∼= N × Nω.
  102. 102. Final coalgebra Streams are an example of a final coalgebra: S β∀ ∃ ! GG Nω head, tail N × S GG N × Nω (Note: instead of N, we could have taken any set.) Note: any two homomorphisms from S to Nω are equal. Note: id : Nω → Nω is a homomorphism. Note: head, tail : Nω ∼= N × Nω.
  103. 103. Final coalgebra Streams are an example of a final coalgebra: S β∀ ∃ ! GG Nω head, tail N × S GG N × Nω (Note: instead of N, we could have taken any set.) Note: any two homomorphisms from S to Nω are equal. Note: id : Nω → Nω is a homomorphism. Note: head, tail : Nω ∼= N × Nω.
  104. 104. Algebra and induction Induction = definition and proof principle for algebras. Ex. mathematical induction: for all P ⊆ N, ( P(0) and (∀n : P(n) ⇒ P(succ(n))) ) ⇒ ∀n : P(n) (Other examples: transfinite, well-founded, tree, structural, etc.) We show that induction is a property of initial algebras.
  105. 105. Algebra and induction Induction = definition and proof principle for algebras. Ex. mathematical induction: for all P ⊆ N, ( P(0) and (∀n : P(n) ⇒ P(succ(n))) ) ⇒ ∀n : P(n) (Other examples: transfinite, well-founded, tree, structural, etc.) We show that induction is a property of initial algebras.
  106. 106. Algebra and induction Induction = definition and proof principle for algebras. Ex. mathematical induction: for all P ⊆ N, ( P(0) and (∀n : P(n) ⇒ P(succ(n))) ) ⇒ ∀n : P(n) (Other examples: transfinite, well-founded, tree, structural, etc.) We show that induction is a property of initial algebras.
  107. 107. Algebra and induction Induction = definition and proof principle for algebras. Ex. mathematical induction: for all P ⊆ N, ( P(0) and (∀n : P(n) ⇒ P(succ(n))) ) ⇒ ∀n : P(n) (Other examples: transfinite, well-founded, tree, structural, etc.) We show that induction is a property of initial algebras.
  108. 108. Algebras and congruences (ex. natural numbers) We call R ⊆ N × N a congruence if (i) (0, 0) ∈ R and (ii) (n, m) ∈ R ⇒ (succ(n), succ(m)) ∈ R (Note: R is not required to be an equivalence relation.) Equivalently, R ⊆ N × N is a congruence if 1 + N [zero, succ] 1 + R γ∃ oo GG 1 + N [zero, succ] N Rπ1 oo π2 GG N for some function γ : 1 + R → R.
  109. 109. Algebras and congruences (ex. natural numbers) We call R ⊆ N × N a congruence if (i) (0, 0) ∈ R and (ii) (n, m) ∈ R ⇒ (succ(n), succ(m)) ∈ R (Note: R is not required to be an equivalence relation.) Equivalently, R ⊆ N × N is a congruence if 1 + N [zero, succ] 1 + R γ∃ oo GG 1 + N [zero, succ] N Rπ1 oo π2 GG N for some function γ : 1 + R → R.
  110. 110. Initial algebras and congruences Theorem: induction proof principle Every congruence R ⊆ N × N contains the diagonal: ∆ ⊆ R where ∆ = {(n, n) | n ∈ N}. Proof: Because (N, [zero, succ]) is an initial algebra, 1 + N [zero, succ] CC 1 + R γ∃ oo GG 1 + N [zero, succ] ss N ! AA Rπ1 oo π2 GG N ! uu we have π1◦! = id = π2◦!, which implies !(n) = (n, n), all n ∈ N.
  111. 111. Initial algebras and congruences Theorem: induction proof principle Every congruence R ⊆ N × N contains the diagonal: ∆ ⊆ R where ∆ = {(n, n) | n ∈ N}. Proof: Because (N, [zero, succ]) is an initial algebra, 1 + N [zero, succ] CC 1 + R γ∃ oo GG 1 + N [zero, succ] ss N ! AA Rπ1 oo π2 GG N ! uu we have π1◦! = id = π2◦!, which implies !(n) = (n, n), all n ∈ N.
  112. 112. Initial algebras and induction Theorem: The following are equivalent: 1. For every congruence relation R ⊆ N × N, ∆ ⊆ R 2. For every predicate P ⊆ N, ( P(0) and (∀n : P(n) ⇒ P(succ(n))) ) ⇒ ∀n : P(n) Proof: Exercise. In other words: two equivalent formulations of induction!
  113. 113. Initial algebras and induction Theorem: The following are equivalent: 1. For every congruence relation R ⊆ N × N, ∆ ⊆ R 2. For every predicate P ⊆ N, ( P(0) and (∀n : P(n) ⇒ P(succ(n))) ) ⇒ ∀n : P(n) Proof: Exercise. In other words: two equivalent formulations of induction!
  114. 114. Initial algebras and induction Theorem: The following are equivalent: 1. For every congruence relation R ⊆ N × N, ∆ ⊆ R 2. For every predicate P ⊆ N, ( P(0) and (∀n : P(n) ⇒ P(succ(n))) ) ⇒ ∀n : P(n) Proof: Exercise. In other words: two equivalent formulations of induction!
  115. 115. Coalgebra and coinduction Coinduction = definition and proof principle for coalgebras. Coinduction is dual to induction, in a very precise way. Categorically, coinduction is a property of final coalgebras. Algorithmically, coinduction generalises Robin Milner’s bisimulation proof method.
  116. 116. Coalgebra and coinduction Coinduction = definition and proof principle for coalgebras. Coinduction is dual to induction, in a very precise way. Categorically, coinduction is a property of final coalgebras. Algorithmically, coinduction generalises Robin Milner’s bisimulation proof method.
  117. 117. Coalgebra and coinduction Coinduction = definition and proof principle for coalgebras. Coinduction is dual to induction, in a very precise way. Categorically, coinduction is a property of final coalgebras. Algorithmically, coinduction generalises Robin Milner’s bisimulation proof method.
  118. 118. Coalgebra and coinduction Coinduction = definition and proof principle for coalgebras. Coinduction is dual to induction, in a very precise way. Categorically, coinduction is a property of final coalgebras. Algorithmically, coinduction generalises Robin Milner’s bisimulation proof method.
  119. 119. Coalgebras and bisimulations (ex. streams) We call R ⊆ Nω × Nω a bisimulation if, for all (σ, τ) ∈ R, (i) head(σ) = head(τ) and (ii) (tail(σ), tail(τ)) ∈ R Equivalently, R ⊆ Nω × Nω is a bisimulation if Nω head, tail R γ∃ π1oo π2 GG Nω head, tail N × Nω N × Roo GG N × Nω for some function γ : R → N × R.
  120. 120. Coalgebras and bisimulations (ex. streams) We call R ⊆ Nω × Nω a bisimulation if, for all (σ, τ) ∈ R, (i) head(σ) = head(τ) and (ii) (tail(σ), tail(τ)) ∈ R Equivalently, R ⊆ Nω × Nω is a bisimulation if Nω head, tail R γ∃ π1oo π2 GG Nω head, tail N × Nω N × Roo GG N × Nω for some function γ : R → N × R.
  121. 121. Final coalgebras and bisimulations Theorem: coinduction proof principle Every bisimulation R ⊆ Nω × Nω is contained in the diagonal: R ⊆ ∆ where ∆ = {(σ, σ) | σ ∈ Nω}. Proof: Because (Nω, head, tail ) is a final coalgebra, Nω head, tail R γ∃ π1oo π2 GG Nω head, tail N × Nω N × Roo GG N × Nω we have π1 = π2, which implies σ = τ, for all (σ, τ) ∈ Nω.
  122. 122. Final coalgebras and bisimulations Theorem: coinduction proof principle Every bisimulation R ⊆ Nω × Nω is contained in the diagonal: R ⊆ ∆ where ∆ = {(σ, σ) | σ ∈ Nω}. Proof: Because (Nω, head, tail ) is a final coalgebra, Nω head, tail R γ∃ π1oo π2 GG Nω head, tail N × Nω N × Roo GG N × Nω we have π1 = π2, which implies σ = τ, for all (σ, τ) ∈ Nω.
  123. 123. Final coalgebras and coinduction The following are equivalent: 1. For every bisimulation relation R ⊆ Nω × Nω, R ⊆ ∆ 2. ?? In other words: no obvious equivalent formulation of coinduction!
  124. 124. Final coalgebras and coinduction The following are equivalent: 1. For every bisimulation relation R ⊆ Nω × Nω, R ⊆ ∆ 2. ?? In other words: no obvious equivalent formulation of coinduction!
  125. 125. Final coalgebras and coinduction The following are equivalent: 1. For every bisimulation relation R ⊆ Nω × Nω, R ⊆ ∆ 2. ?? In other words: no obvious equivalent formulation of coinduction!
  126. 126. Congruences and bisimulations: dual? R ⊆ N × N is a congruence if 1 + N [zero, succ] 1 + R γ∃ oo GG 1 + N [zero, succ] N Rπ1 oo π2 GG N R ⊆ Nω × Nω is a bisimulation if Nω head, tail R γ∃ π1oo π2 GG Nω head, tail N × Nω N × Roo GG N × Nω
  127. 127. Congruences and bisimulations: dual? R ⊆ S × T is an F-congruence if F(S) α F(R) γ∃ oo GG F(T) β S Rπ1 oo π2 GG T R ⊆ S × T is an F-bisimulation if S α R γ∃ π1oo π2 GG T β F(S) F(R)oo GG F(T)
  128. 128. Induction and coinduction: dual? For every congruence relation R ⊆ N × N, ∆ ⊆ R For every bisimulation relation R ⊆ Nω × Nω, R ⊆ ∆
  129. 129. Induction and coinduction: dual? For every congruence relation R on an initial algebra: ∆ ⊆ R For every bisimulation relation R on a final coalgebra: R ⊆ ∆
  130. 130. An aside: fixed points Let (P, ≤) be a preorder and f : P → P a monotone map. Classically, least fixed point induction is: ∀p ∈ P : f(p) ≤ p ⇒ µf ≤ p Classically, greatest fixed point coinduction is: ∀p ∈ P : p ≤ f(p) ⇒ p ≤ νf
  131. 131. An aside: fixed points Let (P, ≤) be a preorder and f : P → P a monotone map. Classically, least fixed point induction is: ∀p ∈ P : f(p) ≤ p ⇒ µf ≤ p Classically, greatest fixed point coinduction is: ∀p ∈ P : p ≤ f(p) ⇒ p ≤ νf
  132. 132. An aside: fixed points Let (P, ≤) be a preorder and f : P → P a monotone map. Classically, least fixed point induction is: ∀p ∈ P : f(p) ≤ p ⇒ µf ≤ p Classically, greatest fixed point coinduction is: ∀p ∈ P : p ≤ f(p) ⇒ p ≤ νf
  133. 133. An aside: fixed points Any preorder (P, ≤) is a category, with arrows: p → q ≡ p ≤ q Any monotone map is a functor: p → q → f(p) → f(q) Lfp induction and gfp coinduction become: f(µf) GG f(p) µf GG p p GG νf f(p) GG f(νf)
  134. 134. An aside: fixed points Any preorder (P, ≤) is a category, with arrows: p → q ≡ p ≤ q Any monotone map is a functor: p → q → f(p) → f(q) Lfp induction and gfp coinduction become: f(µf) GG f(p) µf GG p p GG νf f(p) GG f(νf)
  135. 135. An aside: fixed points Any preorder (P, ≤) is a category, with arrows: p → q ≡ p ≤ q Any monotone map is a functor: p → q → f(p) → f(q) Lfp induction and gfp coinduction become: f(µf) GG f(p) µf GG p p GG νf f(p) GG f(νf)
  136. 136. Fixed point (co)induction = initiality and finality f(µf) GG f(p) µf GG p p GG νf f(p) GG f(νf) F(A) GG F(S) A ∃ ! GG S S ∃ ! GG Z F(S) GG F(Z)
  137. 137. Fixed point (co)induction = initiality and finality f(µf) GG f(p) µf GG p p GG νf f(p) GG f(νf) F(A) GG F(S) A ∃ ! GG S S ∃ ! GG Z F(S) GG F(Z)
  138. 138. 5. What coalgebra studies • the behaviour of – often infinite, circular – systems (their equivalence, minimization, synthesis) • rather: the universal principles underlying this behaviour • these days applied in many different scientific disciplines
  139. 139. 5. What coalgebra studies • the behaviour of – often infinite, circular – systems (their equivalence, minimization, synthesis) • rather: the universal principles underlying this behaviour • these days applied in many different scientific disciplines
  140. 140. 5. What coalgebra studies • the behaviour of – often infinite, circular – systems (their equivalence, minimization, synthesis) • rather: the universal principles underlying this behaviour • these days applied in many different scientific disciplines
  141. 141. 5. What coalgebra studies • the behaviour of – often infinite, circular – systems (their equivalence, minimization, synthesis) • rather: the universal principles underlying this behaviour • these days applied in many different scientific disciplines
  142. 142. Example: dynamical systems A dynamical system is: set of states X and a transition function t : X → X Notation for transitions: x → y ≡ t(x) = y Examples: x GG y GG z poo r GG s q dd
  143. 143. Example: systems with output A system with output: o, t : X → O × X Notation: x a GG y ≡ o(x) = a and t(x) = y. x0 a GG x1 b GG x2 a GG x3 b ww y0 a VV y1 b xx
  144. 144. Example: infinite data types For instance, streams of natural numbers: Nω = {σ | σ : N → N } The behaviour of streams: (σ(0), σ(1), σ(2), . . .) σ(0) GG (σ(1), σ(2), σ(3), . . .) where we call σ(0): the initial value (= head) σ = (σ(1), σ(2), σ(3), . . .): the derivative (= tail)
  145. 145. Example: streams (1, 1, 1, . . .) 1 GG (1, 1, 1, . . .) 1 GG (1, 1, 1, . . .) 1 GG · · · (1, 1, 1, . . .) 1
  146. 146. Example: streams (1, 1, 1, . . .) 1 GG (1, 1, 1, . . .) 1 GG (1, 1, 1, . . .) 1 GG · · · (1, 1, 1, . . .) 1
  147. 147. Example: streams (1, 2, 3, . . .) 1 GG (2, 3, 4, . . .) 2 GG (3, 4, 5, . . .) 3 GG · · · (1, 2, 3, . . .) 1 GG (2, 3, 4, . . .) = (1, 2, 3, . . .) + (1, 1, 1, . . .) (1, 2, 3, . . .) 1 GG 1 (1, 1, 1, . . .) 1
  148. 148. Example: streams (1, 2, 3, . . .) 1 GG (2, 3, 4, . . .) 2 GG (3, 4, 5, . . .) 3 GG · · · (1, 2, 3, . . .) 1 GG (2, 3, 4, . . .) = (1, 2, 3, . . .) + (1, 1, 1, . . .) (1, 2, 3, . . .) 1 GG 1 (1, 1, 1, . . .) 1
  149. 149. Example: streams (1, 2, 3, . . .) 1 GG (2, 3, 4, . . .) 2 GG (3, 4, 5, . . .) 3 GG · · · (1, 2, 3, . . .) 1 GG (2, 3, 4, . . .) = (1, 2, 3, . . .) + (1, 1, 1, . . .) (1, 2, 3, . . .) 1 GG 1 (1, 1, 1, . . .) 1
  150. 150. Example: non-well-founded sets Historically important: Peter Aczel’s book. x = {x} y = {y} xDD y rr x = {y} y = {z} z = {x, y} x GG y GG z yy ––
  151. 151. Example: non-well-founded sets Historically important: Peter Aczel’s book. x = {x} y = {y} xDD y rr x = {y} y = {z} z = {x, y} x GG y GG z yy ––
  152. 152. Example: non-well-founded sets Historically important: Peter Aczel’s book. x = {x} y = {y} xDD y rr x = {y} y = {z} z = {x, y} x GG y GG z yy ––
  153. 153. Example: automata A deterministic automaton GG x b a GG z b zz a }} y b yy a tt • initial state: x • final states: y and z • L(x) = {a, b}∗ a
  154. 154. Example: automata A deterministic automaton GG x b a GG z b zz a }} y b yy a tt • initial state: x • final states: y and z • L(x) = {a, b}∗ a
  155. 155. Example: automata A deterministic automaton GG x b a GG z b zz a }} y b yy a tt • initial state: x • final states: y and z • L(x) = {a, b}∗ a
  156. 156. All these examples: circular behaviour GG x b a GG z b zz a }} y b yy a tt y0 a VV y1 b xx (1, 2, 3, . . .) 1 GG 1 (1, 1, 1, . . .) 1
  157. 157. Where coalgebra is used • logic, set theory • automata • control theory • data types • dynamical systems • games • economy • ecology
  158. 158. 6. Discussion • New way of thinking – give it time • Extensive example: streams (Lecture two) • Algebra and coalgebra (Lecture three and four) - bisimulation up-to - cf. CALCO • Algorithms, tools (Lecture four) - Cf. Hacking nondeterminism with induction and coinduction Bonchi and Pous, Comm. ACM Vol. 58(2), 2015

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