Statistical model checking for parameterized model t Delahaye 1 - - PowerPoint PPT Presentation

Statistical model checking for parameterized model

Benoˆ ıt Delahaye1 Paulin Fournier1 Didier Lime2

Universit´ e de Nantes - LS2N, UMR 6004 - Nantes, France ´ Ecole Centrale de Nantes - LS2N, UMR 6004 - Nantes, France

SynCoP 2018

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 1 / 23

Introduction

Motivation

Model-checking of pMC

◮ Suffers from the same problems as standard MC: state space explosion ◮ Produces large rational functions as the probability of satisfying a

property

◮ Existing cannot handle large number of parameters

In some cases (large models, large number of parameters), it could be beneficial to use approximation techniques such as Statistical Model Checking, but

◮ SMC is limited to models without non-determinism ◮ Attempts to extend SMC to models with non-det have limitations

◮ No precision / confidence intervals ◮ Cannot be easily adapted to parameters (dependent transition

probabilities)

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 2 / 23

Introduction

Contribution

We propose a parametric version of SMC for parameterized models (pMC).

◮ Computes an approximation of the probability of satisfying a property

◮ as a parametric function ◮ polynomial ◮ with parametric confidence intervals

◮ Allows to compute the value of the probability for all parameter

values (with varying precision)

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 3 / 23

Introduction

Outline

Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 4 / 23

Parametric Markov Chains and Properties

Outline

Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 5 / 23

Parametric Markov Chains and Properties Background - Properties

Markov Chains

Definition (Markov chain)

A Markov chain (MC, for short) is a tuple M = (S, s0, P) where S is a denumerable set of states, s0 ∈ S is the initial state and P : S × S → [0, 1] is the transition probability function.

◮ Finite run: ρ = s0s1 . . . sn s.t. P(si, si+1) > 0 ◮ Γ(l): set of all runs of length l in M ◮ Probability of a finite run: PM(ρ) = Πn i=1P(si−1, si)

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 6 / 23

Parametric Markov Chains and Properties Background - Properties

Properties

In the context of SMC, we only consider properties on bounded runs. Let r : Γ(l) → R be a reward function. Reachability PM(♦≤ls). ρ | = ♦≤ls, if there exists i ≤ l such that si = s. Safety PM(=lE). ρ | = =lE, if for all i ≤ l, si ∈ E. Expected reward El

M(r). El M(r) = ρ∈Γ(l) PM(ρ)r(ρ) is the expected

value of r on the runs of length l.

Remark

For any property ϕ ⊆ Γ(l), PM(ϕ) = El

M(✶ϕ)

⇒ we focus on properties of the form El

M(r).

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 7 / 23

Parametric Markov Chains and Properties Parametric Markov Chains

Outline

Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 8 / 23

Parametric Markov Chains and Properties Parametric Markov Chains

Parametric Markov Chains (pMCs)

Definition (Parametric Markov chain)

A Parametric Markov chain is a tuple M = (S, s0, P, X) such that S is a finite set of states, s0 ∈ S is the initial state, X is a finite set of parameters, and P : S × S → Poly(X) is a parametric transition probability function. If v ∈ RX is a valuation of the parameters,

◮ Pv: transition probabilities under v : Pv(s, s′) = P(s, s′)(v) ◮ v is valid if (S, s0, Pv) is a MC ◮ Mv = (S, s0, Pv) ◮ Runs and probabilities are similar to MC, but parametric

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 9 / 23

Parametric Markov Chains and Properties Parametric Markov Chains

Example 1

1 2 3 4 0.5 0.5 q r p 1 r q p 1

pMC M1

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 10 / 23

Parametric Markov Chains and Properties Parametric Markov Chains

Example 1

1 2 3 4 0.5 0.5 q r p 1 r q p 1

pMC M1

1 2 3 4 0.5 0.5 0.5 0.5 1 0.5 0.5 1

MC Mv

1 for parameter valuation v

such that v(p) = v(q) = 0.5 and v(r) = 0

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 10 / 23

Monte Carlo and pMCs

Outline

Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 11 / 23

Monte Carlo and pMCs

Monte Carlo for MCs

1 2 3 4 0.5 0.5 0.5 0.5 1 0.5 0.5 1

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

Monte Carlo and pMCs

Monte Carlo for MCs

1 2 3 4 0.5 0.5 0.5 0.5 1 0.5 0.5 1

◮ Run n simulations ρi of length l

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

Monte Carlo and pMCs

Monte Carlo for MCs

1 2 3 4 0.5 0.5 0.5 0.5 1 0.5 0.5 1

ρ1 = 0 · 1 · 1 · 1 · 1 · 1 ρ2 = 0 · 1 · 0 · 3 · 4 · 4 ρ3 = 0 · 3 · 2 · 2 · 2 · 2 ρ4 = 0 · 1 · 0 · 1 · 0 · 3 ρ5 = 0 · 3 · 4 · 4 · 4 · 1 ρ6 = 0 · 3 · 2 · 2 · 2 · 2 ρ7 = 0 · 1 · 0 · 3 · 2 · 2 ρ8 = 0 · 1 · 0 · 3 · 4 · 4 ◮ Run n simulations ρi of length l

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

Monte Carlo and pMCs

Monte Carlo for MCs

1 2 3 4 0.5 0.5 0.5 0.5 1 0.5 0.5 1

ρ1 = 0 · 1 · 1 · 1 · 1 · 1 ρ2 = 0 · 1 · 0 · 3 · 4 · 4 ρ3 = 0 · 3 · 2 · 2 · 2 · 2 ρ4 = 0 · 1 · 0 · 1 · 0 · 3 ρ5 = 0 · 3 · 4 · 4 · 4 · 1 ρ6 = 0 · 3 · 2 · 2 · 2 · 2 ρ7 = 0 · 1 · 0 · 3 · 2 · 2 ρ8 = 0 · 1 · 0 · 3 · 4 · 4 ◮ Run n simulations ρi of length l ◮ r(ρi) = 1 if ρi reaches 4

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

Monte Carlo and pMCs

Monte Carlo for MCs

1 2 3 4 0.5 0.5 0.5 0.5 1 0.5 0.5 1

ρ1 = 0 · 1 · 1 · 1 · 1 · 1 r(ρ1) = 0 ρ2 = 0 · 1 · 0 · 3 · 4 · 4 r(ρ2) = 1 ρ3 = 0 · 3 · 2 · 2 · 2 · 2 r(ρ3) = 0 ρ4 = 0 · 1 · 0 · 1 · 0 · 3 r(ρ4) = 0 ρ5 = 0 · 3 · 4 · 4 · 4 · 1 r(ρ5) = 1 ρ6 = 0 · 3 · 2 · 2 · 2 · 2 r(ρ6) = 0 ρ7 = 0 · 1 · 0 · 3 · 2 · 2 r(ρ7) = 0 ρ8 = 0 · 1 · 0 · 3 · 4 · 4 r(ρ8) = 1 ◮ Run n simulations ρi of length l ◮ r(ρi) = 1 if ρi reaches 4

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

Monte Carlo and pMCs

Monte Carlo for MCs

1 2 3 4 0.5 0.5 0.5 0.5 1 0.5 0.5 1

ρ1 = 0 · 1 · 1 · 1 · 1 · 1 r(ρ1) = 0 ρ2 = 0 · 1 · 0 · 3 · 4 · 4 r(ρ2) = 1 ρ3 = 0 · 3 · 2 · 2 · 2 · 2 r(ρ3) = 0 ρ4 = 0 · 1 · 0 · 1 · 0 · 3 r(ρ4) = 0 ρ5 = 0 · 3 · 4 · 4 · 4 · 1 r(ρ5) = 1 ρ6 = 0 · 3 · 2 · 2 · 2 · 2 r(ρ6) = 0 ρ7 = 0 · 1 · 0 · 3 · 2 · 2 r(ρ7) = 0 ρ8 = 0 · 1 · 0 · 3 · 4 · 4 r(ρ8) = 1 ◮ Run n simulations ρi of length l ◮ r(ρi) = 1 if ρi reaches 4 ◮ El M(r) ∼ r(ρi) n

⇒ Here, E5

M(r) ∼ 0.375 (exact: 0.3125)

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 12 / 23

Monte Carlo and pMCs

Intuition for pMCs

1 2 3 4 0.5 0.5 q r p 1 r q p 1

◮ How to run simulations?

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

Monte Carlo and pMCs

Intuition for pMCs

1 2 3 4 0.5 0.5 0.33 — q 0.33 — r 0.33 — p 1 0.33 — r 0.33 — q 0.33 — p 1

◮ How to run simulations? Use a normalization function f (uniform?)

→ Mf

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

Monte Carlo and pMCs

Intuition for pMCs

1 2 3 4 0.5 0.5 0.33 — q 0.33 — r 0.33 — p 1 0.33 — r 0.33 — q 0.33 — p 1

ρ1 = 0 · 1 · 1 · 2 · 2 · 2 ρ2 = 0 · 1 · 0 · 3 · 3 · 4 ρ3 = 0 · 3 · 2 · 2 · 2 · 2 ρ4 = 0 · 1 · 0 · 1 · 1 · 0 ρ5 = 0 · 3 · 4 · 4 · 4 · 4 ρ6 = 0 · 3 · 3 · 3 · 4 · 4 ρ7 = 0 · 1 · 0 · 3 · 2 · 2 ρ8 = 0 · 1 · 2 · 2 · 2 · 2 ◮ How to run simulations? Use a normalization function f (uniform?)

→ Mf

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

Monte Carlo and pMCs

Intuition for pMCs

1 2 3 4 0.5 0.5 0.33 — q 0.33 — r 0.33 — p 1 0.33 — r 0.33 — q 0.33 — p 1

ρ1 = 0 · 1 · 1 · 2 · 2 · 2 ρ2 = 0 · 1 · 0 · 3 · 3 · 4 ρ3 = 0 · 3 · 2 · 2 · 2 · 2 ρ4 = 0 · 1 · 0 · 1 · 1 · 0 ρ5 = 0 · 3 · 4 · 4 · 4 · 4 ρ6 = 0 · 3 · 3 · 3 · 4 · 4 ρ7 = 0 · 1 · 0 · 3 · 2 · 2 ρ8 = 0 · 1 · 2 · 2 · 2 · 2 ◮ How to run simulations? Use a normalization function f (uniform?)

→ Mf

◮ R(ρi) = PM(ρ) if ρi reaches 4, 0 otherwise

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

Monte Carlo and pMCs

Intuition for pMCs

1 2 3 4 0.5 0.5 0.33 — q 0.33 — r 0.33 — p 1 0.33 — r 0.33 — q 0.33 — p 1

ρ1 = 0 · 1 · 1 · 2 · 2 · 2 R(ρ1) = 0 ρ2 = 0 · 1 · 0 · 3 · 3 · 4 R(ρ2) = 0.25pqr ρ3 = 0 · 3 · 2 · 2 · 2 · 2 R(ρ3) = 0 ρ4 = 0 · 1 · 0 · 1 · 1 · 0 R(ρ4) = 0 ρ5 = 0 · 3 · 4 · 4 · 4 · 4 R(ρ5) = 0.5q ρ6 = 0 · 3 · 3 · 3 · 4 · 4 R(ρ6) = 0.5qr2 ρ7 = 0 · 1 · 0 · 3 · 2 · 2 R(ρ7) = 0 ρ8 = 0 · 1 · 2 · 2 · 2 · 2 R(ρ8) = 0 ◮ How to run simulations? Use a normalization function f (uniform?)

→ Mf

◮ R(ρi) = PM(ρ) if ρi reaches 4, 0 otherwise

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

Monte Carlo and pMCs

Intuition for pMCs

1 2 3 4 0.5 0.5 0.33 — q 0.33 — r 0.33 — p 1 0.33 — r 0.33 — q 0.33 — p 1

ρ1 = 0 · 1 · 1 · 2 · 2 · 2 R(ρ1) = 0 ρ2 = 0 · 1 · 0 · 3 · 3 · 4 R(ρ2) = 0.25pqr ρ3 = 0 · 3 · 2 · 2 · 2 · 2 R(ρ3) = 0 ρ4 = 0 · 1 · 0 · 1 · 1 · 0 R(ρ4) = 0 ρ5 = 0 · 3 · 4 · 4 · 4 · 4 R(ρ5) = 0.5q ρ6 = 0 · 3 · 3 · 3 · 4 · 4 R(ρ6) = 0.5qr2 ρ7 = 0 · 1 · 0 · 3 · 2 · 2 R(ρ7) = 0 ρ8 = 0 · 1 · 2 · 2 · 2 · 2 R(ρ8) = 0 ◮ How to run simulations? Use a normalization function f (uniform?)

→ Mf

◮ R(ρi) = PM(ρ) if ρi reaches 4, 0 otherwise ◮ El M(R) ??

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 13 / 23

Monte Carlo and pMCs

Results

Let r′(ρ) =

R(ρ) PMf (ρ)r(ρ). Due to the central limit theorem: ◮ El M(r′)(v) = El Mv (r) for all valid parameter valuation v ◮ Condition: Under v, M must conserve the same structure as under f

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 14 / 23

Monte Carlo and pMCs

Results

Let r′(ρ) =

R(ρ) PMf (ρ)r(ρ). Due to the central limit theorem: ◮ El M(r′)(v) = El Mv (r) for all valid parameter valuation v ◮ Condition: Under v, M must conserve the same structure as under f ◮ Moreover, the confidence interval can be expressed as a polynomial

function of the parameters. For n large enough and an error rate of 5%, its size is 3.92 σ/√n where

• σ2 =

1 (n − 1)

n

• i=1

(r′(ρi))2 − (n

i=1 r′(ρi))2

n(n − 1) .

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 14 / 23

Monte Carlo and pMCs

Results

Let r′(ρ) =

R(ρ) PMf (ρ)r(ρ). Due to the central limit theorem: ◮ El M(r′)(v) = El Mv (r) for all valid parameter valuation v ◮ Condition: Under v, M must conserve the same structure as under f ◮ Moreover, the confidence interval can be expressed as a polynomial

function of the parameters. For n large enough and an error rate of 5%, its size is 3.92 σ/√n where

• σ2 =

1 (n − 1)

n

• i=1

(r′(ρi))2 − (n

i=1 r′(ρi))2

n(n − 1) .

◮ Here, E5 M(r′) ∼ 3.48pqr + 0.38q + 3.47qr2.

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 14 / 23

Monte Carlo and pMCs

Results

Let r′(ρ) =

R(ρ) PMf (ρ)r(ρ). Due to the central limit theorem: ◮ El M(r′)(v) = El Mv (r) for all valid parameter valuation v ◮ Condition: Under v, M must conserve the same structure as under f ◮ Moreover, the confidence interval can be expressed as a polynomial

function of the parameters. For n large enough and an error rate of 5%, its size is 3.92 σ/√n where

• σ2 =

1 (n − 1)

n

• i=1

(r′(ρi))2 − (n

i=1 r′(ρi))2

n(n − 1) .

◮ Here, E5 M(r′) ∼ 3.48pqr + 0.38q + 3.47qr2. ◮ For v(p) = v(q) = 0.25, v(r) = 0.5: E5 M(r′)(v) ∼ 0.42 (exact:

0.261..)

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 14 / 23

Monte Carlo and pMCs

Application to Example 1

For 1000 runs, we get

E5

M(r′) ∼0.189 ∗ p ∗ q2 + 0.405 ∗ p ∗ q ∗ r + 0.252 ∗ p ∗ q + 0.729 ∗ q ∗ r3

+ 0.567 ∗ q ∗ r2 + 0.54 ∗ q ∗ r + 0.492 ∗ q

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 15 / 23

Monte Carlo and pMCs

Application to Example 1

For 1000 runs, we get

E5

M(r′) ∼0.189 ∗ p ∗ q2 + 0.405 ∗ p ∗ q ∗ r + 0.252 ∗ p ∗ q + 0.729 ∗ q ∗ r3

+ 0.567 ∗ q ∗ r2 + 0.54 ∗ q ∗ r + 0.492 ∗ q ∼ 0.2801

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 15 / 23

Monte Carlo and pMCs

Application to Example 1

For 1000 runs, we get

E5

M(r′) ∼0.189 ∗ p ∗ q2 + 0.405 ∗ p ∗ q ∗ r + 0.252 ∗ p ∗ q + 0.729 ∗ q ∗ r3

+ 0.567 ∗ q ∗ r2 + 0.54 ∗ q ∗ r + 0.492 ∗ q ∼ 0.2801

and the size of the CI is

(0.062 ∗ (5.108 ∗ p2 ∗ q4 + 10.946 ∗ p2 ∗ q2 ∗ r2 + 2.27 ∗ p2 ∗ q2 + 59.108 ∗ q2 ∗ r6 + 15.324 ∗ q2 ∗ r4 + 4.865 ∗ q2 ∗ r2 + 1.478 ∗ q2 − 1.0 ∗ (0.189 ∗ p ∗ q2 + 0.405 ∗ p ∗ q ∗ r + 0.252 ∗ p ∗ q + 0.729 ∗ q ∗ r3 + 0.567 ∗ q ∗ r2 + 0.54 ∗ q ∗ r + 0.492 ∗ q)2))

1 2

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 15 / 23

Monte Carlo and pMCs

Application to Example 1

For 1000 runs, we get

E5

M(r′) ∼0.189 ∗ p ∗ q2 + 0.405 ∗ p ∗ q ∗ r + 0.252 ∗ p ∗ q + 0.729 ∗ q ∗ r3

+ 0.567 ∗ q ∗ r2 + 0.54 ∗ q ∗ r + 0.492 ∗ q ∼ 0.2801

and the size of the CI is

(0.062 ∗ (5.108 ∗ p2 ∗ q4 + 10.946 ∗ p2 ∗ q2 ∗ r2 + 2.27 ∗ p2 ∗ q2 + 59.108 ∗ q2 ∗ r6 + 15.324 ∗ q2 ∗ r4 + 4.865 ∗ q2 ∗ r2 + 1.478 ∗ q2 − 1.0 ∗ (0.189 ∗ p ∗ q2 + 0.405 ∗ p ∗ q ∗ r + 0.252 ∗ p ∗ q + 0.729 ∗ q ∗ r3 + 0.567 ∗ q ∗ r2 + 0.54 ∗ q ∗ r + 0.492 ∗ q)2))

1 2 ∼ 0.0296

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 15 / 23

Implementation

Outline

Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 16 / 23

Implementation

Implementation

◮ In Python (not optimized) ◮ Input: text format similar to prism ◮ Output: Polynomial, graph...

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 17 / 23

Implementation

Exemple 1

(a) With 10 000 simulations and uniform

normalization

(b) Theoretical probability Figure: Results on pMC M1 (E100

M (r ′))

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 18 / 23

Implementation

Exemple 2

Figure: Results on the Crowds protocol model for our implementation (left) and from the PARAM website (right).

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 19 / 23

Discussion

Outline

Introduction Parametric Markov Chains and Properties Background - Properties Parametric Markov Chains Monte Carlo and pMCs Implementation Discussion

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 20 / 23

Discussion

Improvements

◮ Choice of normalization function

Depending on this choice, the size of the confidence interval may vary drastically

◮ Structure of the pMC

If the structure is different under valuation and normalization, the estimated number of runs may change

◮ Complement property

Only for probability (not expected value): Evaluating the property and its negation in parallel may improve precision

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 21 / 23

Discussion

Impact of normalization function

init fail win p 1 − p

Impact of the choice of the normalization function f on the size of confidence intervals.

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 22 / 23

Discussion

◮ Summary:

◮ Parametric Monte Carlo procedure for pMC ◮ Polynomial funtions instead of rational; ◮ Polynomial parametric confidence interval; ◮ Prototype implementation Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 23 / 23

Discussion

◮ Summary:

◮ Parametric Monte Carlo procedure for pMC ◮ Polynomial funtions instead of rational; ◮ Polynomial parametric confidence interval; ◮ Prototype implementation

◮ Future work:

◮ Test and implement improvements ◮ Comparison/Integration(?) to existing tools ◮ Extension to parametric timed systems (Non trivial) Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 23 / 23

Discussion

◮ Summary:

◮ Parametric Monte Carlo procedure for pMC ◮ Polynomial funtions instead of rational; ◮ Polynomial parametric confidence interval; ◮ Prototype implementation

◮ Future work:

◮ Test and implement improvements ◮ Comparison/Integration(?) to existing tools ◮ Extension to parametric timed systems (Non trivial)

Thank you for your Attention

Benoˆ ıt Delahaye (Univ Nantes/LS2N) SMC of parametric MC 2018-04-15 23 / 23