Toward Uniform Random Generation in 1-safe Petri Nets Yi-Ting Chen - - PowerPoint PPT Presentation

Toward Uniform Random Generation in 1-safe Petri Nets

Yi-Ting Chen (LIP6 / Sorbonne Université)

Advisor: Jean Mairesse, Samy Abbes

2019 Apr 23 - LIPN

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Motivation

• Complexity and scale in software systems are increasing.
• The crucial factor is related to concurrency.

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Motivation

• Complexity and scale in software systems are increasing.
• The crucial factor is related to concurrency.
• Difficulty : "Combinatorial explosion problems"

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Motivation

• Complexity and scale in software systems are increasing.
• The crucial factor is related to concurrency.
• Difficulty : "Combinatorial explosion problems"
• Approach : Statistical model checking

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Motivation

• Complexity and scale in software systems are increasing.
• The crucial factor is related to concurrency.
• Difficulty : "Combinatorial explosion problems"
• Approach : Statistical model checking

→ probabilistic framework in a trace monoid

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Motivation

• Complexity and scale in software systems are increasing.
• The crucial factor is related to concurrency.
• Difficulty : "Combinatorial explosion problems"
• Approach : Statistical model checking

→ probabilistic framework in a trace monoid

• Goal : Random generation for concurrent systems

→ 1-safe Petri nets

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - 1-safe Petri nets

a b c

M0

M0 M1 c c a b

Reachability graph

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - 1-safe Petri nets

a b c

M1

M0 M1 c c a b

Reachability graph

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - 1-safe Petri nets

a b c

M0

M0 M1 c c a b

Reachability graph

• Concurrency :
• Casuality :
• Conflit :

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - 1-safe Petri nets

a b c

M0

M0 M1 c c a b

Reachability graph

• Concurrency : a, c
• Casuality :
• Conflit :

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - 1-safe Petri nets

a b c

M0

M0 M1 c c a b

Reachability graph

• Concurrency : a, c
• Casuality : a, b
• Conflit :

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - 1-safe Petri nets

a b c

M0

M0 M1 c c a b

Reachability graph

• Concurrency : a, c
• Casuality : a, b
• Conflit :

b, c

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - 1-safe Petri nets

a b c

M0

M0 M1 c c a b

Reachability graph

• From M0, abacb is a valid firing sequence.

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - 1-safe Petri nets

a b c

M0

M0 M1 c c a b

Reachability graph

• From M0, abacb is a valid firing sequence.
• We lost the feature of concurrency by viewing the firing

sequences as the sequential executions. ex : abacb = abcab

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - trace monoids

Trace monoid M

• Alphabet : Σ = {a, b, c}
• Independent relation :

I = {(a, c)} Heap of pieces

• Pieces:

a b c

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - trace monoids

Trace monoid M

• Alphabet : Σ = {a, b, c}
• Independent relation :

I = {(a, c)} Heap of pieces

• Pieces:

a b c

• Example of heap :

a b a c b abacb = abcab

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - trace monoids

Trace monoid M

• Alphabet : Σ = {a, b, c}
• Independent relation :

I = {(a, c)}

• Canonical normal form :

abacb = a · b · ac · b Heap of pieces

• Pieces:

a b c

• Example of heap :

a b a c b abacb = abcab

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Concurrent models - trace monoids

Trace monoid M

• Alphabet : Σ = {a, b, c}
• Independent relation :

I = {(a, c)}

• Canonical normal form :

abacb = a · b · ac · b

a c b ac

Heap of pieces

• Pieces:

a b c

• Example of heap :

a b a c b abacb = abcab

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Framework- random sampling from a Markov chain

• Take account of "concurrency" and "states"

a c b ac

traces in a trace monoid

M0 M1 c c a b

words in an automaton

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Framework- random sampling from a Markov chain

• Take account of "concurrency" and "states"

a c b ac

traces in a trace monoid

M0 M1 c c a b

words in an automaton

M0, a M1, b M1, c M0, ac M0, c

traces in an automaton

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Framework- random sampling from a Markov chain

• Take account of "concurrency" and "states"
• The executions of 1-safe Petri nets are understood up to

traces.

a c b ac

traces in a trace monoid

M0 M1 c c a b

words in an automaton

M0, a M1, b M1, c M0, ac M0, c

traces in an automaton

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniform measure on trace monoids

Trace monoid : M = a, b, c | a · c = c · a

• set of cliques C : ε, a, b, c, ac
• Möbius polynomial µ(x) =

c∈C

(−1)|c|x|c| = 1 − 3x + x2

• Möbius inversion formula : G(x) =

u∈M

x|u| =

1 µ(x)

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniform measure on trace monoids

Trace monoid : M = a, b, c | a · c = c · a

• set of cliques C : ε, a, b, c, ac
• Möbius polynomial µ(x) =

c∈C

(−1)|c|x|c| = 1 − 3x + x2

• Möbius inversion formula : G(x) =

u∈M

x|u| =

1 µ(x)

Theorem (Abbes, Mairesse 2015)

There exists a unique uniform measure ν on ∂M, satisfying: ∀u ∈ M, ν(↑ u) = p|u| p0 : the root of smallest modulus of µ(x).

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniform measure on trace monoids

Theorem (Abbes, Mairesse 2015)

Let ν be the uniform measure on ∂M. Then the canonical normal decomposition of a trace is a realization of the Markov chain with initial probability measure h which is the Möbius transform of ν.

a c b ac

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniform measure on trace monoids

Theorem (Abbes, Mairesse 2015)

Let ν be the uniform measure on ∂M. Then the canonical normal decomposition of a trace is a realization of the Markov chain with initial probability measure h which is the Möbius transform of ν.

• # paths with length k in the

automaton = # traces with height k in a trace monoid

a c b ac

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniform measure for actions on trace monoids

Theorem (Abbes 2015)

Let X × M → X be an irreducible partial action. Then there exists a uniform Markov measure, satisfying : ∀α ∈ X ∀x ∈ Mα, να(↑ x) = p|x|

0 Γ(α, α · x).

a b c

• Gα(x) =
• u∈Mα

x|u|

• Γ(α, β) = lim

x→p0 Gβ(x) Gα(x)

• µα,β(x) =
• γ∈Cα,β

(−1)|γ|x|γ|

• Möbius matrix:

µ(x) = (µα,β)(x)

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Properties of Γ function

• Define Γ(α, β) = lim

x→p0 Gβ(x) Gα(x)

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Properties of Γ function

• Define Γ(α, β) = lim

x→p0 Gβ(x) Gα(x)

• cocycle relation: Γ(α, γ) = Γ(α, β)Γ(β, γ)

Γ(α, α) = 1

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Properties of Γ function

• Define Γ(α, β) = lim

x→p0 Gβ(x) Gα(x)

• cocycle relation: Γ(α, γ) = Γ(α, β)Γ(β, γ)

Γ(α, α) = 1

• Fix a state α0,

(Γ(α0, β))β ∈ ker µ(p0)

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Calculation of Γ function

a b c

M0 → M0 : ε, c, M0 → M1 : a, ac, M1 → M0 : b, M1 → M1 : ε, c.

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Calculation of Γ function

a b c

M0 → M0 : ε, c, M0 → M1 : a, ac, M1 → M0 : b, M1 → M1 : ε, c. µ(x) = M0 M1 1 − x −x + x2 −x 1 − x

• .

p0 =

√ 5+1 2

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Calculation of Γ function

a b c

M0 → M0 : ε, c, M0 → M1 : a, ac, M1 → M0 : b, M1 → M1 : ε, c. µ(x) = M0 M1 1 − x −x + x2 −x 1 − x

• .

p0 =

√ 5+1 2

Let

• Γ(M0, M0)

Γ(M0, M1) T =

• 1

λ T ∈ ker µ(p0)

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Calculation of Γ function

a b c

M0 → M0 : ε, c, M0 → M1 : a, ac, M1 → M0 : b, M1 → M1 : ε, c. µ(x) = M0 M1 1 − x −x + x2 −x 1 − x

• .

p0 =

√ 5+1 2

Let

• Γ(M0, M0)

Γ(M0, M1) T =

• 1

λ T ∈ ker µ(p0) −p0 + (1 − p0) · λ = 0

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Calculation of Γ function

a b c

M0 → M0 : ε, c, M0 → M1 : a, ac, M1 → M0 : b, M1 → M1 : ε, c. µ(x) = M0 M1 1 − x −x + x2 −x 1 − x

• .

p0 =

√ 5+1 2

Let

• Γ(M0, M0)

Γ(M0, M1) T =

• 1

λ T ∈ ker µ(p0) −p0 + (1 − p0) · λ = 0 = ⇒ λ = p0 1 − p0 = 1 p0 = √ 5 − 1 2 .

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on trace monoids

New proof from the linear algebra point of view

• Construct the expanded automaton of cliques

M = a, b, c | a · c = c · a

a b c ac1 ac2

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on trace monoids

New proof from the linear algebra point of view

• Construct the expanded automaton of cliques

M = a, b, c | a · c = c · a

a b c ac1 ac2

(a, 1) (b, 1) (c, 1) (ac, 1) (ac, 2)       1 1 1 1 1 1 1 1 1 1 1 1 1      

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on trace monoids

New proof from the linear algebra point of view

• Construct the expanded automaton of cliques

M = a, b, c | a · c = c · a

a b c ac1 ac2

(a, 1) (b, 1) (c, 1) (ac, 1) (ac, 2)       1 1 1 1 1 1 1 1 1 1 1 1 1      

• Find the Perron eigenvector of the incidence matrix of

this automaton

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on trace monoids

• Find the Perron eigenvector of the incidence matrix of

this automaton (a, 1) (b, 1) (c, 1) (ac, 1) (ac, 2)       1 1 1 1 1 1 1 1 1 1 1 1 1       v(c,i) = 1 pi−1h(c).

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on trace monoids

• Find the Perron eigenvector of the incidence matrix of

this automaton (a, 1) (b, 1) (c, 1) (ac, 1) (ac, 2)       1 1 1 1 1 1 1 1 1 1 1 1 1       v(c,i) = 1 pi−1h(c). v =       h(a) h(b) h(c) h(ac)

1 ph(ac)

     

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on trace monoids

• Find the Perron eigenvector of the incidence matrix of

this automaton (a, 1) (b, 1) (c, 1) (ac, 1) (ac, 2)       1 1 1 1 1 1 1 1 1 1 1 1 1       v(c,i) = 1 pi−1h(c). v =       h(a) h(b) h(c) h(ac)

1 ph(ac)

     

• Since the incidence matrix is irreducible and aperiodic
• Apply Perron-FrobeniusTheorem

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on trace monoids

• Find the Perron eigenvector of the incidence matrix of

this automaton (a, 1) (b, 1) (c, 1) (ac, 1) (ac, 2)       1 1 1 1 1 1 1 1 1 1 1 1 1       v(c,i) = 1 pi−1h(c). v =       h(a) h(b) h(c) h(ac)

1 ph(ac)

     

• Since the incidence matrix is irreducible and aperiodic
• Apply Perron-FrobeniusTheorem
• Get the uniqueness of the uniform measure

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on actions

Example for 1-safe petri net

a b c

M0, a M1, b M1, c M0, ac M0, c

νM0(C1 = c) = νM0(↑ c) − νM0(↑ (ac))

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on actions

Example for 1-safe petri net

a b c

M0, a M1, b M1, c M0, ac M0, c

νM0(C1 = c) = νM0(↑ c) − νM0(↑ (ac)) = p0 · Γ(M0, M0) − p2

0 · Γ(M0, M1)

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on actions

Example for 1-safe petri net

a b c

M0, a M1, b M1, c M0, ac M0, c

νM0(C1 = c) = νM0(↑ c) − νM0(↑ (ac)) = p0 · Γ(M0, M0) − p2

0 · Γ(M0, M1)

= p0 · 1 − p2

0 · 1

p0 = 0

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on actions

Example for 1-safe petri net

a b c

M0, a M1, b M1, c M0, ac M0, c

• The automaton is NOT strongly connected

→ can not apply Perron-Frobenius Theorem

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Uniqueness of uniform measure on actions

Example for 1-safe petri net

a b c

M0, a M1, b M1, c M0, ac M0, c

• The automaton is NOT strongly connected

→ can not apply Perron-Frobenius Theorem

• Some state never go through under νM0

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Ongoing directions

• A systematic way to calculate Γ function

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Ongoing directions

• A systematic way to calculate Γ function
• Complete the proof of the uniqueness of uniform measure
• n actions in a trace monoid

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Ongoing directions

• A systematic way to calculate Γ function
• Complete the proof of the uniqueness of uniform measure
• n actions in a trace monoid
• Random generation for actions on a trace monoid

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Ongoing directions

• A systematic way to calculate Γ function
• Complete the proof of the uniqueness of uniform measure
• n actions in a trace monoid
• Random generation for actions on a trace monoid
• For the purpose of model checking (application of this

theory)

Introduction Uniform measure Uniform measure for actions Uniqueness of uniform measure Ongoing directions

Thank you!