[PPT] - A Formalized Hierarchy of Probabilistic System Types Proof Pearl PowerPoint Presentation

SLIDE 1

A Formalized Hierarchy of Probabilistic System Types

Proof Pearl Johannes H¨

lzl1, Andreas Lochbihler 2, and Dmitriy Traytel1,2

1Institut f¨

ur Informatik TU M¨ unchen, Germany

2Institute of Information Security

ETH Zurich, Switzerland

ITP 2015

SLIDE 2

Zoo of Probabilistic System Types

Det. automaton

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 3

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 4

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 5

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 6

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 7

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system Generative system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 8

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system Generative system Stratified system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 9

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system Generative system Stratified system Alternating system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 10

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system Generative system Stratified system Alternating system Segala system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 11

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system Generative system Stratified system Alternating system Segala system Simple Segala system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 12

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system Generative system Stratified system Alternating system Segala system Simple Segala system Bundle system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 13

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system Generative system Stratified system Alternating system Segala system Simple Segala system Bundle system Pnueli-Zuck system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 14

Zoo of Probabilistic System Types

Det. automaton

Non-det. automaton Labeled Markov chain Labeled Markov decision process Reactive system Generative system Stratified system Alternating system Segala system Simple Segala system Bundle system Pnueli-Zuck system Most general system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 2 / 19

SLIDE 15

Hierarchy of Probabilistic System Types

Ana Sokolva – Coalgebraic Analysis of Probabilistic Systems (2005):

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 3 / 19

SLIDE 16

Hierarchy of Probabilistic Systems Types

How to . . .

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 17

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types?

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 18

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types? . . . compare systems of same type?

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 19

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types? . . . compare systems of same type? . . . compare different system types?

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 20

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types? . . . compare systems of same type? . . . compare different system types? Coalgebras

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 21

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types? . . . compare systems of same type? . . . compare different system types? Coalgebras Bisimulation

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 22

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types? . . . compare systems of same type? . . . compare different system types? Coalgebras Bisimulation Embedding respecting bisimulation

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 23

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types? . . . compare systems of same type? . . . compare different system types? Coalgebras Bisimulation Embedding respecting bisimulation

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 24

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types? . . . compare systems of same type? . . . compare different system types? Coalgebras Bisimulation Embedding respecting bisimulation . . . formalize it in Isabelle/HOL?

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 25

Hierarchy of Probabilistic Systems Types

How to . . . . . . model system types? . . . compare systems of same type? . . . compare different system types? Coalgebras Bisimulation Embedding respecting bisimulation . . . formalize it in Isabelle/HOL? codatatype + Probability Mass Func. + Eisbach

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 4 / 19

SLIDE 26

Coalgebras

◮ Functor F describes the system type

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

SLIDE 27

Coalgebras

◮ Functor F describes the system type ◮ Examples:

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

SLIDE 28

Coalgebras

◮ Functor F describes the system type ◮ Examples:

Deterministic System α × (β ⇒ )

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

SLIDE 29

Coalgebras

◮ Functor F describes the system type ◮ Examples:

Deterministic System α × (β ⇒ ) Non-Deterministic System α × (β ⇒ set)

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

SLIDE 30

Coalgebras

◮ Functor F describes the system type ◮ Examples:

Deterministic System α × (β ⇒ ) Non-Deterministic System α × (β ⇒ set)

◮ System (σ, s) of type F:

σ type of states, s :: σ ⇒ σ F transition system

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

SLIDE 31

Coalgebras

◮ Functor F describes the system type ◮ Examples:

Deterministic System α × (β ⇒ ) Non-Deterministic System α × (β ⇒ set)

◮ System (σ, s) of type F:

σ type of states, s :: σ ⇒ σ F transition system

◮ (σ, s) is a F-coalgebra

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 5 / 19

SLIDE 32

Types of Transition System

Property

Functor

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19

SLIDE 33

Types of Transition System

p

q r Property

◮ Probability p

Functor pmf

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19

SLIDE 34

Types of Transition System

α

p q r Property

◮ Probability p ◮ Label α

Functor pmf α × ( pmf )

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19

SLIDE 35

Types of Transition System

α

p q r Property

◮ Probability p ◮ Label α ◮ Non-determinism

Functor pmf α × ( pmf ) α × ( pmf set)

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19

SLIDE 36

Types of Transition System

α

β1 β2 p q r Property

◮ Probability p ◮ Label α ◮ Non-determinism ◮ Reactive β

Functor pmf α × ( pmf ) α × ( pmf set) α × (β ⇒ pmf)

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19

SLIDE 37

Types of Transition System

α

β1 β2 p, δ1 q, δ2 r, δ3 Property

◮ Probability p ◮ Label α ◮ Non-determinism ◮ Reactive β ◮ Generative δ

Functor pmf α × ( pmf ) α × ( pmf set) α × (β ⇒ pmf) α × (β ⇒ (δ × ) pmf)

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 6 / 19

SLIDE 38

Bisimulation

F = α × ( pmf ) ⋄ ⋄ ⋄ . . . △ △ △ 0.5 0.5 0.5 0.5 0.5 0.5 ⋄ △ 0.5 0.5

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 7 / 19

SLIDE 39

Bisimulation

F = α × ( pmf ) ⋄ ⋄ ⋄ . . . △ △ △ 0.5 0.5 0.5 0.5 0.5 0.5 ⋄ △ 0.5 0.5

◮ Expresses observational equality

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 7 / 19

SLIDE 40

Bisimulation

F = α × ( pmf ) ⋄ ⋄ ⋄ . . . △ △ △ 0.5 0.5 0.5 0.5 0.5 0.5 ⋄ △ 0.5 0.5

◮ Expresses observational equality ◮ Bisimulation relation R :: (σ × τ) set for a system type F:

∀(x, y) ∈ R. (s x, t y) ∈ relF R

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 7 / 19

SLIDE 41

Bisimulation

F = α × ( pmf ) ⋄ ⋄ ⋄ . . . △ △ △ 0.5 0.5 0.5 0.5 0.5 0.5 ⋄ △ 0.5 0.5

◮ Expresses observational equality ◮ Bisimulation relation R :: (σ × τ) set for a system type F:

∀(x, y) ∈ R. (s x, t y) ∈ relF R

◮ State x in system s is bisimilar to state y in system t iff

∃ bisimulation relation R with (x, y) ∈ R

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 7 / 19

SLIDE 42

Coalgebras in Isabelle/HOL

Idea: Analyse transition systems modulo bisimulation!

Equality :⇐ ⇒ Bisimulation

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 8 / 19

SLIDE 43

Coalgebras in Isabelle/HOL

Idea: Analyse transition systems modulo bisimulation!

Equality :⇐ ⇒ Bisimulation

How to model all F-coalgebras as type?

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 8 / 19

SLIDE 44

Coalgebras in Isabelle/HOL

Idea: Analyse transition systems modulo bisimulation!

Equality :⇐ ⇒ Bisimulation

How to model all F-coalgebras as type? codatatype τF = C (τF F)

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 8 / 19

SLIDE 45

Coalgebras in Isabelle/HOL

Idea: Analyse transition systems modulo bisimulation!

Equality :⇐ ⇒ Bisimulation

How to model all F-coalgebras as type? codatatype τF = C (τF F) Example (Labeled Markov Chains where F = α × pmf ): codatatype α mc = MC (α × α mc pmf )

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 8 / 19

SLIDE 46

Bounded Natural Functors (BNFs)

Traytel, Popescu & Blanchette: Foundational, compositional (co)datatypes for HOL

Codatatype only allows nesting through BNFs:

◮ Examples: products, sums, functions, lists

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 9 / 19

SLIDE 47

Bounded Natural Functors (BNFs)

Traytel, Popescu & Blanchette: Foundational, compositional (co)datatypes for HOL

Codatatype only allows nesting through BNFs:

◮ Examples: products, sums, functions, lists ◮ Has map and set functions:

mapF :: (σ ⇒ τ) ⇒ σ F ⇒ τ F setF :: σ F ⇒ σ set

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 9 / 19

SLIDE 48

Bounded Natural Functors (BNFs)

Traytel, Popescu & Blanchette: Foundational, compositional (co)datatypes for HOL

Codatatype only allows nesting through BNFs:

◮ Examples: products, sums, functions, lists ◮ Has map and set functions:

mapF :: (σ ⇒ τ) ⇒ σ F ⇒ τ F setF :: σ F ⇒ σ set

◮ Bound κ on the elements |setF x| ≤ κ

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 9 / 19

SLIDE 49

Bounded Natural Functors (BNFs)

Traytel, Popescu & Blanchette: Foundational, compositional (co)datatypes for HOL

Codatatype only allows nesting through BNFs:

◮ Examples: products, sums, functions, lists ◮ Has map and set functions:

mapF :: (σ ⇒ τ) ⇒ σ F ⇒ τ F setF :: σ F ⇒ σ set

◮ Bound κ on the elements |setF x| ≤ κ ◮ Has relator lifting relations R component wise:

relF :: (σ × τ) set ⇒ (σ F × τ F) set relF R := {(mapF π1 z, mapF π2 z) | z :: (σ × τ) F. setF z ⊆ R}

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 9 / 19

SLIDE 50

Bounded Natural Functors (BNFs)

Traytel, Popescu & Blanchette: Foundational, compositional (co)datatypes for HOL

Codatatype only allows nesting through BNFs:

◮ Examples: products, sums, functions, lists ◮ Has map and set functions:

mapF :: (σ ⇒ τ) ⇒ σ F ⇒ τ F setF :: σ F ⇒ σ set

◮ Bound κ on the elements |setF x| ≤ κ ◮ Has relator lifting relations R component wise:

relF :: (σ × τ) set ⇒ (σ F × τ F) set relF R := {(mapF π1 z, mapF π2 z) | z :: (σ × τ) F. setF z ⊆ R} Example: R = {(a, i), (b, j), (c, k), (d, l)} a b c d i j k l

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 9 / 19

SLIDE 51

Bounded Natural Functors (BNFs)

Traytel, Popescu & Blanchette: Foundational, compositional (co)datatypes for HOL

Codatatype only allows nesting through BNFs:

◮ Examples: products, sums, functions, lists ◮ Has map and set functions:

mapF :: (σ ⇒ τ) ⇒ σ F ⇒ τ F setF :: σ F ⇒ σ set

◮ Bound κ on the elements |setF x| ≤ κ ◮ Has relator lifting relations R component wise:

relF :: (σ × τ) set ⇒ (σ F × τ F) set relF R := {(mapF π1 z, mapF π2 z) | z :: (σ × τ) F. setF z ⊆ R} Example: R = {(a, i), (b, j), (c, k), (d, l)} a b c d (a, i) (b, j) (c, k) (d, l) i j k l

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 9 / 19

SLIDE 52

Bounded Natural Functors (BNFs)

Traytel, Popescu & Blanchette: Foundational, compositional (co)datatypes for HOL

Codatatype only allows nesting through BNFs:

◮ Examples: products, sums, functions, lists ◮ Has map and set functions:

mapF :: (σ ⇒ τ) ⇒ σ F ⇒ τ F setF :: σ F ⇒ σ set

◮ Bound κ on the elements |setF x| ≤ κ ◮ Has relator lifting relations R component wise:

relF :: (σ × τ) set ⇒ (σ F × τ F) set relF R := {(mapF π1 z, mapF π2 z) | z :: (σ × τ) F. setF z ⊆ R} relF R ◦ relF Q ⊆ relF (R ◦ Q) Example: R = {(a, i), (b, j), (c, k), (d, l)} a b c d (a, i) (b, j) (c, k) (d, l) i j k l

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 9 / 19

SLIDE 53

Bounded Natural Functors (BNFs)

Traytel, Popescu & Blanchette: Foundational, compositional (co)datatypes for HOL

Codatatype only allows nesting through BNFs:

◮ Examples: products, sums, functions, lists ◮ Has map and set functions:

mapF :: (σ ⇒ τ) ⇒ σ F ⇒ τ F setF :: σ F ⇒ σ set

◮ Bound κ on the elements |setF x| ≤ κ ◮ Has relator lifting relations R component wise:

relF :: (σ × τ) set ⇒ (σ F × τ F) set relF R := {(mapF π1 z, mapF π2 z) | z :: (σ × τ) F. setF z ⊆ R} relF R ◦ relF Q ⊆ relF (R ◦ Q) Example: R = {(a, i), (b, j), (c, k), (d, l)} a b c d (a, i) (b, j) (c, k) (d, l) i j k l

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 9 / 19

SLIDE 54

BNF: Bounded Sets To model non-determinism we need sets!

◮ Problem: Bound of |X :: σ set| depends on σ

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 10 / 19

SLIDE 55

BNF: Bounded Sets To model non-determinism we need sets!

◮ Problem: Bound of |X :: σ set| depends on σ ◮ Introduce type σ setκ of sets bounded by κ: |X :: σ setκ| ≤ κ

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 10 / 19

SLIDE 56

BNF: Bounded Sets To model non-determinism we need sets!

◮ Problem: Bound of |X :: σ set| depends on σ ◮ Introduce type σ setκ of sets bounded by κ: |X :: σ setκ| ≤ κ

Map is direct image

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 10 / 19

SLIDE 57

BNF: Bounded Sets To model non-determinism we need sets!

◮ Problem: Bound of |X :: σ set| depends on σ ◮ Introduce type σ setκ of sets bounded by κ: |X :: σ setκ| ≤ κ

Map is direct image Set is identity

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 10 / 19

SLIDE 58

BNF: Bounded Sets To model non-determinism we need sets!

◮ Problem: Bound of |X :: σ set| depends on σ ◮ Introduce type σ setκ of sets bounded by κ: |X :: σ setκ| ≤ κ

Map is direct image Set is identity Bound is κ, needs to be infinite! – wrong claim in previous publications

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 10 / 19

SLIDE 59

BNF: Bounded Sets To model non-determinism we need sets!

◮ Problem: Bound of |X :: σ set| depends on σ ◮ Introduce type σ setκ of sets bounded by κ: |X :: σ setκ| ≤ κ

Map is direct image Set is identity Bound is κ, needs to be infinite! – wrong claim in previous publications Relator (∀a ∈ A. ∃b ∈ B. (a, b) ∈ R) ∧ (∀b ∈ B. ∃a ∈ A. (a, b) ∈ R) A · · · B · ·

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 10 / 19

SLIDE 60

BNF: Probability Mass Function Model probabilistic transitions!

µ :: σ pmf ≈ µ :: σ ⇒ [0, 1],

x µ x = 1

≈ µ :: σ measure, µ U = 1, discrete Similar to Audebaud & Paulin-Mohring (2009)

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 11 / 19

SLIDE 61

BNF: Probability Mass Function Model probabilistic transitions!

µ :: σ pmf ≈ µ :: σ ⇒ [0, 1],

x µ x = 1

≈ µ :: σ measure, µ U = 1, discrete Similar to Audebaud & Paulin-Mohring (2009) Map is λx.

f y=x

y Set is {x | µ x = 0} Bound is ℵ0

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 11 / 19

SLIDE 62

BNF: Probability Mass Function Model probabilistic transitions!

µ :: σ pmf ≈ µ :: σ ⇒ [0, 1],

x µ x = 1

≈ µ :: σ measure, µ U = 1, discrete Similar to Audebaud & Paulin-Mohring (2009) Map is λx.

f y=x

y Set is {x | µ x = 0} Bound is ℵ0 Relator a b c x R = {(a, e), (b, e), (b, d), (c, d)} relpmf R x y ? e d y

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 11 / 19

SLIDE 63

BNF: Probability Mass Function Model probabilistic transitions!

µ :: σ pmf ≈ µ :: σ ⇒ [0, 1],

x µ x = 1

≈ µ :: σ measure, µ U = 1, discrete Similar to Audebaud & Paulin-Mohring (2009) Map is λx.

f y=x

y Set is {x | µ x = 0} Bound is ℵ0 Relator a b c x R = {(a, e), (b, e), (b, d), (c, d)} relpmf R x y ? z e d y

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 11 / 19

SLIDE 64

BNF: Probability Mass Function Model probabilistic transitions!

µ :: σ pmf ≈ µ :: σ ⇒ [0, 1],

x µ x = 1

≈ µ :: σ measure, µ U = 1, discrete Similar to Audebaud & Paulin-Mohring (2009) Map is λx.

f y=x

y Set is {x | µ x = 0} Bound is ℵ0 Relator a b c x e d y R′ = {(a, e), (b, e), (c, d)}

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 11 / 19

SLIDE 65

BNF: Probability Mass Function Model probabilistic transitions!

µ :: σ pmf ≈ µ :: σ ⇒ [0, 1],

x µ x = 1

≈ µ :: σ measure, µ U = 1, discrete Similar to Audebaud & Paulin-Mohring (2009) Map is λx.

f y=x

y Set is {x | µ x = 0} Bound is ℵ0 Relator a b c x e d y R′ = {(a, e), (b, e), (c, d)} ¬relpmf R′ x y

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 11 / 19

SLIDE 66

Proving the Relator Property

relpmf R ◦ relpmf Q ⊆ relpmf (R ◦ Q) lines

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 12 / 19

SLIDE 67

Proving the Relator Property

relpmf R ◦ relpmf Q ⊆ relpmf (R ◦ Q) lines 577 a a Elementary, based on A. Sokolova’s thesis

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 12 / 19

SLIDE 68

Proving the Relator Property

relpmf R ◦ relpmf Q ⊆ relpmf (R ◦ Q) lines 577 a 406 b a Elementary, based on A. Sokolova’s thesis b Using mappmf

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 12 / 19

SLIDE 69

Proving the Relator Property

relpmf R ◦ relpmf Q ⊆ relpmf (R ◦ Q) lines 577 a 406 b 101 c a Elementary, based on A. Sokolova’s thesis b Using mappmf c Based on Jonsson et al.

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 12 / 19

SLIDE 70

Proving the Relator Property

relpmf R ◦ relpmf Q ⊆ relpmf (R ◦ Q) lines 577 a 406 b 101 c 46 d a Elementary, based on A. Sokolova’s thesis b Using mappmf c Based on Jonsson et al. d Using mappmf, bindpmf, and condpmf

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 12 / 19

SLIDE 71

Proving the Relator Property

relpmf R ◦ relpmf Q ⊆ relpmf (R ◦ Q) lines 577 a 406 b 101 c 46 d 337 e a Elementary, based on A. Sokolova’s thesis b Using mappmf c Based on Jonsson et al. d Using mappmf, bindpmf, and condpmf e Zanella et al. in Coq

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 12 / 19

SLIDE 72

System Types

Name Functor Codatatype Markov chain σ pmf MC Labeled MC α × σ pmf α LMC Labeled MDP α × σ pmf setκ

1

α LMDPκ

Det. automaton

α ⇒ σ option α DLTS Non-det. automaton (α × σ) setκ α LTSκ Reactive system α ⇒ σ pmf option α React Generative system (α × σ) pmf option α Gen Stratified system σ pmf + (α × σ) option α Str Alternating system σ pmf + (α × σ) setκ α Altκ Simple Segala system (α × σ pmf) setκ α SSegκ Segala system (α × σ) pmf setκ α Segκ Bundle system (α × σ) setκ pmf α Bunκ Pnueli-Zuck system (α × σ) setκ1 pmf setκ2 α PZκ1, κ2 Most general system (α × σ + σ) setκ1 pmf setκ2 α MGκ1, κ2

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 13 / 19

SLIDE 73

Hierarchy of Probabilistic System Types

Ana Sokolva – Coalgebraic Analysis of Probabilistic Systems (2005):

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 14 / 19

SLIDE 74

Proving the Hierarchy in Isabelle/HOL G is at least as expressive as F, iff ∃G of F :: σ F ⇒ σ G preserving and reflecting bisimilarity Lift to G of F :: τF ⇒ τG Theorem: G of F is injective Proof: by coinduction (proof method in Eisbach)

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 15 / 19

SLIDE 75

Hierarchy

α MGκ1, κ2 α option PZκ1, κ2 α option Bunκ α option Segκ α Altκ α PZκ1, κ2 α Bunκ α Segκ α Gen α SSegκ α option SSegκ α Str α LMDPκ α LMC α React α LTSκ MC α DLTS

κ≤κ1 κ≤κ2 κ≤κ1 κ≤κ2 α set≤κ α set≤κ

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 16 / 19

SLIDE 76

Hierarchy of Probabilistic System Types

Ana Sokolva – Coalgebraic Analysis of Probabilistic Systems (2005):

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 17 / 19

SLIDE 77

Problem with Vardi Systems

◮ Varκ = (α × ) pmf + (α × ) setκ

with: returnpmf (a, s) = {(a, s)}

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 18 / 19

SLIDE 78

Problem with Vardi Systems

◮ Varκ = (α × ) pmf + (α × ) setκ

with: returnpmf (a, s) = {(a, s)}

◮ Couldn’t prove that this is a BNF!

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 18 / 19

SLIDE 79

Problem with Vardi Systems

◮ Varκ = (α × ) pmf + (α × ) setκ

with: returnpmf (a, s) = {(a, s)}

◮ Couldn’t prove that this is a BNF! ◮ Bisimulation not transitive: relVar R ◦ relVar Q ⊆ relVar (R ◦ Q)

△

△

relset R (x, y) ∈ R ↔ y = △

0.5

△ 0.5 △ 1 relpmf Q = (x, y) ∈ Q ↔ x = △

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 18 / 19

SLIDE 80

Problem with Vardi Systems

◮ Varκ = (α × ) pmf + (α × ) setκ

with: returnpmf (a, s) = {(a, s)}

◮ Couldn’t prove that this is a BNF! ◮ Bisimulation not transitive: relVar R ◦ relVar Q ⊆ relVar (R ◦ Q)

△

△

relset R (x, y) ∈ R ↔ y = △

0.5

△ 0.5 △ 1 relpmf Q = (x, y) ∈ Q ↔ x = △

◮ Approach not possible

That is even a flaw in the original proof!

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 18 / 19

SLIDE 81

Conclusion

◮ Formalized hierarchy of probabilistic systems types

α MGκ1, κ2 α option PZκ1, κ2 α option Bunκ α option Segκ α Altκ α PZκ1, κ2 α Bunκ α Segκ α Gen α SSegκ α option SSegκ α Str α LMDPκ α LMC α React α LTSκ MC α DLTS

κ≤κ1 κ≤κ2 κ≤κ1 κ≤κ2 α set≤κ α set≤κ

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 19 / 19

SLIDE 82

Conclusion

◮ Formalized hierarchy of probabilistic systems types

α MGκ1, κ2 α option PZκ1, κ2 α option Bunκ α option Segκ α Altκ α PZκ1, κ2 α Bunκ α Segκ α Gen α SSegκ α option SSegκ α Str α LMDPκ α LMC α React α LTSκ MC α DLTS

κ≤κ1 κ≤κ2 κ≤κ1 κ≤κ2 α set≤κ α set≤κ

◮ Found two flaws:

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 19 / 19

SLIDE 83

Conclusion

◮ Formalized hierarchy of probabilistic systems types

α MGκ1, κ2 α option PZκ1, κ2 α option Bunκ α option Segκ α Altκ α PZκ1, κ2 α Bunκ α Segκ α Gen α SSegκ α option SSegκ α Str α LMDPκ α LMC α React α LTSκ MC α DLTS

κ≤κ1 κ≤κ2 κ≤κ1 κ≤κ2 α set≤κ α set≤κ

◮ Found two flaws:

◮ Bisimulation on Vardi systems is no equiv. relation H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 19 / 19

SLIDE 84

Conclusion

◮ Formalized hierarchy of probabilistic systems types

α MGκ1, κ2 α option PZκ1, κ2 α option Bunκ α option Segκ α Altκ α PZκ1, κ2 α Bunκ α Segκ α Gen α SSegκ α option SSegκ α Str α LMDPκ α LMC α React α LTSκ MC α DLTS

κ≤κ1 κ≤κ2 κ≤κ1 κ≤κ2 α set≤κ α set≤κ

◮ Found two flaws:

◮ Bisimulation on Vardi systems is no equiv. relation ◮ Bounded sets are BNFs for κ ≥ ℵ0 H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 19 / 19

SLIDE 85

Conclusion

◮ Formalized hierarchy of probabilistic systems types

α MGκ1, κ2 α option PZκ1, κ2 α option Bunκ α option Segκ α Altκ α PZκ1, κ2 α Bunκ α Segκ α Gen α SSegκ α option SSegκ α Str α LMDPκ α LMC α React α LTSκ MC α DLTS

κ≤κ1 κ≤κ2 κ≤κ1 κ≤κ2 α set≤κ α set≤κ

◮ Found two flaws:

◮ Bisimulation on Vardi systems is no equiv. relation ◮ Bounded sets are BNFs for κ ≥ ℵ0

codatatype + PMF + Eisbach = Hierarchy

H¨

lzl, Lochbihler & Traytel

A Formalized Hierarchy of Probabilistic System Types ITP 2015 19 / 19