1. On the origins of Bisimulation & Coinduction
Speaker: Filippo Bonchi
(work by Davide Sangiorgi)

2. Plan of the talk Bisimulation & Coinduction In Computer Science
In Set Theory
In Philosophical Logic
My work on Bisimulation

3. Plan of the talk Bisimulation & Coinduction In Computer Science
In Set Theory
In Philosophical Logic
My work on Bisimulation

4. Sequentiality and Concurrency
Sequential programs can be seen as function
X:= X:=1 ; X:=X+1

5. Sequentiality and Concurrency
Sequential programs can be seen as function
X:= X:=1 ; X:=X+1
Put them in a parallel context - |X:=2
X:=2 | X:= X:=1 ; X:=X+1 | X:=2

6. Sequentiality and Concurrency
Sequential programs can be seen as function
X:= X:=1 ; X:=X+1
Put them in a parallel context - |X:=2
X:=2 | X:= X:=1 ; X:=X+1 | X:=2
Thus Concurrent Programs are not functions
(non terminating process could be usefull -
Non deterministic behaviour)

7. Sequentiality and Concurrency
What is a process? When are two process equivalent? Thus Concurrent Programs are not functions (non terminating process could be usefull - Non deterministic behaviour)

8. Labeled Transition SystemX P A A A Q R Y B C B C W Z S T

9. Trace Equivalence X
tr(X)={ab, ac}=tr(P) P A A A Q R Y B C B C W Z S T

10. Bisimulation X P A A A Q R Y B C B C W Z S T

11. Bisimulation X P A A A Q R Y B C B C W Z S T

12. Bisimulation X P A A A Q R Y B C B C W Z S T

13. Bisimulation X P A A A A Y Z Q R B C B C W S T

14. Bisimulation X P A A A A Y Z Q R B C B C W S T

15. Bisimulation X P A A A A Y Z Q R B C B C W S T

16. Bisimulation X P A A A A Y Z Q R B C B C W S T

17. Bisimulation Proof Method
In order to prove that s ~ t, we have to build a bisimulation R such that s R t
(another proof method through the minimal canonical representative)
It is pratically interesting mainly for:
The locality of checks on the states
The lack of hierarchy on the pairs of bisimulation

18. A (wrong) Inductive Definition
This definition requires a hierachy, since the checks on (s1, s2) must follow those on (s1’, s2’).
The definition is ill-founded if the states space is infinite or includes loops.

19. A (wrong) Inductive Definition
The definition is ill-founded if the states space is infinite or includes loops. a a S1 T1 S2 a
S1=S2 if T1=S2
T1=S2 if S1=S2

20. Stratification of Bisimilarity
Let W be the set of states of an LTS:

21. A0

22. A1 A0 a

23. … An
An-1 … A2 A1 A0 a a a a a a

24. a Aw … An
An-1 … A2 A1 A0 a a a a a a

25. a Aw T a a a a a a … An
An-1 … A2 A1 A0 a a a a a a

26. a Aw T a a S a a a a a a a a a a a … An
An-1 … A2 A1 A0 a a a a a a

27. Coinduction
A set A is defined coinductively if It is the greatest solution of a set of inequation of a certain form The coinduction proof principle just say that: Any set that is solution of these inequations is contained in A

28. Bisimulation as Coinduction

29. Plan of the talk Bisimulation & Coinduction In Computer Science
In Set Theory
In Philosophical Logic
My work on Bisimulation

30. From Automata to Bisimulation
Algebraic Theory of Automata (60s): Notion of Simulation on Mealy Automata
Late 60s: Works on program correctness (Floyd ‘67- Manna ‘69 – Landin ‘69)
Early 70s, Milner uses the notion of simulation to prove equivalence of sequential (possibly not terminating) programs: locality is central in the proof technique
Milner 1980: CCS & Bisimulation

31. … and Coinduction (David Park 1980)
Bisimilarity was defined by Milner as
Park was one of the top-expert in fixed point theory (and was sleeping in Milner’s house)
Bisimulation as the greatest fixed point
Park: Mimicry
Milner: Bisimilarity

32. Plan of the talk Bisimulation & Coinduction In Computer Science
In Set Theory
In Philosophical Logic
My work on Bisimulation

33. “Two sets are equal iff they have exactly the same elements”
Set Theory
Foundational theory of modern mathematics introduced by Cantor in the late XIX century
Paradoxes: Burali-Forti (1897), Russell (1901)
Axioms System by Zermelo (1908):
Foundation Axiom:
“The membership relation does not give rise to infinite descending chains”
Extensionality Axiom:
“Two sets are equal iff they have exactly the same elements”

34. “Two sets are equal iff they have exactly the same elements”
Set Theory
Foundational theory of modern mathematics introduced by Cantor in the late XIX century
Paradoxes: Burali-Forti (1897), Russell (1901)
Axioms System by Zermelo (1908):
Foundation Axiom:
“The membership relation does not give rise to infinite descending chains”
Extensionality Axiom:
“Two sets are equal iff they have exactly the same elements”

35. Non-Well-Founded sets (Dimitry Mirimanoff 1917)
“not all the circular definition are dangerous, and it is a task for the logician to isolate the good one”

36. Equivalence for Non-Well-founded Sets
Are A and B equal?
If we apply the extensionality axiom we get:
“A and B are equal iff A and B are equal”

37. Anti Foundation Axiom (Marco Forti, Furio Honsell 1983)
This formulation is due to Peter Aczel 1988 A decoration for a graph maps nodes to sets s.t. the set assigned to a node is equal to the set of the children of the node

38. Anti Foundation Axiom (Marco Forti, Furio Honsell 1983)
This formulation is due to Peter Aczel 1988 A decoration for a graph maps nodes to sets s.t. the set assigned to a node is equal to the set of the children of the node

39. Anti Foundation Axiom (Marco Forti, Furio Honsell 1983)
This formulation is due to Peter Aczel 1988 A decoration for a graph maps nodes to sets s.t. the set assigned to a node is equal to the set of the children of the node

40. Anti Foundation Axiom (Marco Forti, Furio Honsell 1983)
This formulation is due to Peter Aczel 1988 A decoration for a graph maps nodes to sets s.t. the set assigned to a node is equal to the set of the children of the node

41. Anti Foundation Axiom (Marco Forti, Furio Honsell 1983)
This formulation is due to Peter Aczel 1988 A decoration for a graph maps nodes to sets s.t. the set assigned to a node is equal to the set of the children of the node

42. Anti Foundation Axiom (Marco Forti, Furio Honsell 1983)
This formulation is due to Peter Aczel 1988 A decoration for a graph maps nodes to sets s.t. the set assigned to a node is equal to the set of the children of the node

43. Anti Foundation Axiom (Marco Forti, Furio Honsell 1983)
This formulation is due to Peter Aczel 1988 A decoration for a graph maps nodes to sets s.t. the set assigned to a node is equal to the set of the children of the node Anti-Foundation-Axiom: “every graph has a unique decoration”

44. Anti Foundation Axiom (Marco Forti, Furio Honsell 1983)
This formulation is due to Peter Aczel 1988 A decoration for a graph maps nodes to sets s.t. the set assigned to a node is equal to the set of the children of the node Anti-Foundation-Axiom: “every graph has a unique decoration” are all the same set

45. Plan of the talk Bisimulation & Coinduction In Computer Science
In Set Theory
In Philosophical Logic
My work on Bisimulation

46. Modal Logics Logic of necessity and possibility
Temporal logic (“it always be truth that” “it was the case that”)
Epistemic Logic (“it certenlay true that” “it may be true that”)
Deontic Logic (“it is obligatory that” “it is permitted that”) …

47. Basic Modal Logic x y z w
{p1,p2}
{ }
{p1}
{p2,p3}

48. Semantics of Modal Logicx y
{ }
{p,q}

49. Homomorphisms
When is the truth of a formula preserved when the model changes?
Homomorphisms are maps that preserve structure N M u x y H
{ }
{ }
{p,q}

50. P-Morphisms Krister Segeberg ‘71

51. P-Morphisms Krister Segeberg ‘71
P-Morphisms are maps that preserve and reflect structure Modal Formula are invariant under p-morphism x y z P u v
{ }
{p,q}
{ }
{p,q}
{ }

52. P-Relations (Bisimulations) Johan van Benthem ‘76

53. P-Relations (Bisimulations) Johan van Benthem ‘76
Which First Order Logic formulas can be translated in Modal Logic? xTx A First Order Logic Formula is equivalent to a modal formula iff it is bisimulation invariant u v u
{p}
{p}
{p}