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

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

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

Sequentiality and Concurrency Sequential programs can be seen as function X:=2 X:=1 ; X:=X+1

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

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

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)

Labeled Transition System X P A A A Q R Y B C B C W Z S T

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

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

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

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

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

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

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

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

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

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.

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

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

A0

A1 A0 a

… An An-1 … A2 A1 A0 a a a a a a

a Aw … An An-1 … A2 A1 A0 a a a a a a

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

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

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

Bisimulation as Coinduction

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

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

… 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

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

“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”

“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”

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”

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”

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 (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 (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 (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 (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 (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 (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”

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

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

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”) …

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

Semantics of Modal Logic x y { } {p,q}

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}

P-Morphisms Krister Segeberg ‘71

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} { }

P-Relations (Bisimulations) Johan van Benthem ‘76

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}