Coalgebra Love and Beauty in Science Ana Sokolova Logic Mentoring - PowerPoint PPT Presentation

Coalgebra Love and Beauty in Science Ana Sokolova Logic Mentoring Workshop 2019

Do you know any coalgebra? Ana Sokolova LMW 2019 22-6-19

Do you know any coalgebra? Yes, you know many coalgebras ! Ana Sokolova LMW 2019 22-6-19

Some coalgebras c X Ñ FX Ana Sokolova LMW 2019 22-6-19

| " k ✏ Some coalgebras NFA X ➝ 2 x ( P X) A x 1 a a x 2 x 3 b ˚ c X Ñ FX Ana Sokolova LMW 2019 22-6-19

k ✏ ✏ ✏ �  | " ✏ Some coalgebras MC NFA X ➝ D X + 1 X ➝ 2 x ( P X) A x 1 1 1 2 2 x 1 a a / x 3 x 2 2 x 2 x 3 1 3 1 3 ✏ x 4 x 5 b ˚ ˚ ˚ c X Ñ FX Ana Sokolova LMW 2019 22-6-19

k  | " k � | " | k | 3 � ✏ ✏ ✏ ✏ Some coalgebras MC NFA X ➝ D X + 1 X ➝ 2 x ( P X) A x 1 1 1 2 2 x 1 a a / x 3 x 2 2 x 2 x 3 1 3 1 3 ✏ PA x 4 x 5 b ˚ X ➝ ( PD X) A ˚ ˚ b x 1 a a c X Ñ FX 2 1 1 1 3 3 " 2 2 " x 2 x 3 x 4 a b b Ana Sokolova LMW 2019 22-6-19

| " 3 � k " k | | ✏ | k  � ✏ ✏ ✏ Some coalgebras MC NFA X ➝ D X + 1 X ➝ 2 x ( P X) A x 1 1 1 2 2 x 1 a a / x 3 x 2 2 x 2 x 3 1 3 1 3 ✏ PA x 4 x 5 b ˚ X ➝ ( PD X) A ˚ ˚ Various b transitions systems / x 1 a a automata are c X Ñ FX coalgebras 2 1 1 1 3 3 " 2 2 " x 2 x 3 x 4 a b b Ana Sokolova LMW 2019 22-6-19

Where do they live ? c X Ñ FX Ana Sokolova LMW 2019 22-6-19

Where do they live ? 3D Organ Model 2D Tissue Model c X Ñ FX Ana Sokolova LMW 2019 22-6-19

Where do they live ? 3D Organ Model 2D Tissue Model Bartocci et al. TCS09, CAV11 c X Ñ FX Ana Sokolova LMW 2019 22-6-19

Where do they live ? 3D Organ Model 2D Tissue Model Bartocci et al. Verification requires clear TCS09, CAV11 semantics c X Ñ FX Ana Sokolova LMW 2019 22-6-19

Where do they live ? 3D Organ Model 2D Tissue Model Bartocci et al. Verification requires clear TCS09, CAV11 semantics c X Ñ FX and suffers from state-space explosion Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A language equivalence Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence An equivalence relation R Ñ X ˆ X is a bisimulation of the NFA p o, n q : X Ñ 2 ˆ p P X q A i ff whenever p x, y q P R , we have o p x q “ o p y q and for all a P A Ñ y 1 ^ p x 1 , y 1 q P R. a a Ñ x 1 D y 1 .y ñ x Bisimilarity, denoted by „ , is the largest bisimulation. bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence bisimulation R bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence bisimulation R bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence bisimulation R a bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence bisimulation R a a bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence bisimulation R a a R bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ „ NFA X ➝ 2 x ( P X) A Two states are equivalent i ff the languages recognised from these two states are the same. language equivalence bisimulation R largest bisimulation a a R bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 An equivalence relation R Ñ X ˆ X is a bisimulation of the MC c : X Ñ D X ` 1 i ff whenever p x, y q P R , then either c p x q “ c p y q “ ˚ or for all R -equivalence classes C we have ÿ ÿ c p x qp z q “ c p y qp z q . z P C z P C Bisimilarity, denoted by „ , is the largest bisimulation. bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 An equivalence relation R Ñ X ˆ X is a bisimulation of the MC c : X Ñ D X ` 1 i ff whenever p x, y q P R , then either c p x q “ c p y q “ ˚ or for all R -equivalence classes C we have ÿ ÿ c p x qp z q “ c p y qp z q . z P C z P C Bisimilarity, denoted by „ , is the largest bisimulation. bisimilarity Why are they both called bisimilarity ? Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 An equivalence relation R Ñ X ˆ X is a bisimulation of the MC c : X Ñ D X ` 1 i ff whenever p x, y q P R , then either c p x q “ c p y q “ ˚ or for all R -equivalence classes C we have ÿ ÿ c p x qp z q “ c p y qp z q . z P C z P C Bisimilarity, denoted by „ , is the largest bisimulation. bisimilarity What do they have in common ? Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimulation R bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimulation R bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimulation R μ bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimulation R μ ৵ bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimulation R μ ৵ ” R bisimilarity Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimulation R μ ৵ ” R bisimilarity lifting of R to distributions Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimulation R μ ৵ ” R bisimilarity assign the same probability lifting of R to distributions to “R-classes” Ana Sokolova LMW 2019 22-6-19

Behavioural Equivalences „ MC „ X ➝ D X + 1 bisimulation R largest bisimulation μ ৵ ” R bisimilarity assign the same probability lifting of R to distributions to “R-classes” Ana Sokolova LMW 2019 22-6-19

Coalgebra Uniform framework for dynamic transition systems, based on category theory. A coalgebra is generic transition system: Ana Sokolova LMW 2019 22-6-19

Coalgebra Uniform framework for dynamic transition systems, based on category theory. A coalgebra is generic transition system: c X Ñ FX Ana Sokolova LMW 2019 22-6-19

Coalgebra Uniform framework for dynamic transition systems, based on category theory. A coalgebra is generic transition system: c X Ñ FX states Ana Sokolova LMW 2019 22-6-19

Coalgebra Uniform framework for dynamic transition systems, based on category theory. A coalgebra is generic transition system: c X Ñ FX states object in the base category C Ana Sokolova LMW 2019 22-6-19

Coalgebra Uniform framework for dynamic transition systems, based on category theory. A coalgebra is generic transition system: c X Ñ FX behaviour states type object in the base category C Ana Sokolova LMW 2019 22-6-19

Coalgebra Uniform framework for dynamic transition systems, based on category theory. A coalgebra is generic transition system: c X Ñ FX behaviour states type functor on the object in the base base category C category C Ana Sokolova LMW 2019 22-6-19

Coalgebra Uniform framework for dynamic transition systems, based on category theory. A coalgebra is generic transition system: c X Ñ FX behaviour form a states type category too functor on the object in the base base category C category C Ana Sokolova LMW 2019 22-6-19