Modal Logics for Timed Control Patricia Bouyer 1 , Franck Cassez 2 - - PowerPoint PPT Presentation

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Modal Logics for Timed Control Patricia Bouyer 1 , Franck Cassez 2 - - PowerPoint PPT Presentation

Aug 25, 2022 38 likes •983 views

Modal Logics for Timed Control Patricia Bouyer 1 , Franck Cassez 2 and Franois Laroussinie 1 1 LSV, ENS-Cachan 2 IRCCyN, Nantes France CONCUR05 San Francisco, CA Control of Timed Systems Controllability with L Outline of the talk

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SLIDE 1

Modal Logics for Timed Control

Patricia Bouyer1, Franck Cassez2 and François Laroussinie1

1LSV, ENS-Cachan 2IRCCyN, Nantes

France CONCUR’05 San Francisco, CA

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SLIDE 2

Control of Timed Systems Controllability with Lν

Outline of the talk

◮

Control of Timed Systems

◮

Controllability with Lν

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 2 / 30

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SLIDE 3

Control of Timed Systems Controllability with Lν

Outline

◮

Control of Timed Systems

◮

Controllability with Lν

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 3 / 30

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SLIDE 4

Control of Timed Systems Controllability with Lν

Model Checking and Control Problems

S φ

(not bad)

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30

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SLIDE 5

Control of Timed Systems Controllability with Lν

Model Checking and Control Problems

S | = φ

(not bad)

Model Checking Problem

Does S satisfy φ ?

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30

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SLIDE 6

Control of Timed Systems Controllability with Lν

Model Checking and Control Problems

S

c (not bad)

φ

Model Checking Problem

Does S satisfy φ ?

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30

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SLIDE 7

Control of Timed Systems Controllability with Lν

Model Checking and Control Problems

S

X (not bad)

φ

Model Checking Problem

Does S satisfy φ ?

Control Problem

Can S be restricted to satisfy φ ?

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30

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SLIDE 8

Control of Timed Systems Controllability with Lν

Model Checking and Control Problems

S

c (not bad)

φ | = C

  • c

Model Checking Problem

Does S satisfy φ ?

Control Problem

Can S be restricted to satisfy φ ? Is there a Controller C s.t. (S C) | = φ ?

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 4 / 30

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SLIDE 9

Control of Timed Systems Controllability with Lν

Model for Timed Systems: Timed Automata

TA = Finite Automata + clocks

Timed Automata

x := 0 ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 5 / 30

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SLIDE 10

Control of Timed Systems Controllability with Lν

Model for Timed Systems: Timed Automata

TA = Finite Automata + clocks

Timed Automata

x := 0 ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u Semantics = runs = sequences of dense-time and discrete steps ρ : (ℓ0, 0)

1.1

− − → (ℓ0, 1.1)

c1

− − → (ℓ1, 1.1)

2.1

− − → (ℓ1, 3.2)

c2

− − → (ℓ2, 3.2)

0.1

− − → (ℓ2, 3.3)

u

− → (ℓ0, 0) · · ·

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 5 / 30

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SLIDE 11

Control of Timed Systems Controllability with Lν

Model for Control: Timed Game Automata

TGA = TA + controllable and uncontrollable actions

Actions partitioned as Actc = {c1, c2, c3} Actu = {u}

x := 0 ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u Control Objective = subset of the runs of a TGA

Safety objective

“Avoid the Bad state”

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 6 / 30

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SLIDE 12

Control of Timed Systems Controllability with Lν

Solving Timed Games (1/2)

x := 0 ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u A general controller is defined by a strategy f if ρ is a run from the initial state: f (ρ) = do a controllable action or do nothing

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30

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SLIDE 13

Control of Timed Systems Controllability with Lν

Solving Timed Games (1/2)

x := 0 ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u A general controller is defined by a strategy f

A Partial Strategy f

f (each run ending in ℓ0, x < 2) = do nothing f (each run ending in ℓ0, x = 2) = c1

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30

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SLIDE 14

Control of Timed Systems Controllability with Lν

Solving Timed Games (1/2)

x := 0 ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u A general controller is defined by a strategy f A strategy restricts the set of runs of the TGA

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30

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SLIDE 15

Control of Timed Systems Controllability with Lν

Solving Timed Games (1/2)

x := 0 ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u A general controller is defined by a strategy f A strategy restricts the set of runs of the TGA (G f ) = G controlled by strategy f

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30

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SLIDE 16

Control of Timed Systems Controllability with Lν

Solving Timed Games (1/2)

x := 0 ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u A general controller is defined by a strategy f A strategy restricts the set of runs of the TGA (G f ) = G controlled by strategy f Given φ a control objective, s a state, The strategy f is winning from s if s | = φ in (G f ) The state s is winning if there is a winning strategy fs from s

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 7 / 30

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SLIDE 17

Control of Timed Systems Controllability with Lν

Solving Timed Games (2/2)

Input: a TGA G and a control objective φ Problem: is there a strategy f s.t. (G f ) | = φ ? Solution: compute the set of winning states

1

define a controllable predecessors operator

2

compute a fixed point that gives the set of winning states

3

check whether the initial state is winning

Fundamental Results for Timed Control

[Maler et al., 95, De Alfaro et al., 01] Control Problem is EXPTIME-Complete for TA and reachability

  • bjectives

Controller Synthesis is effective Memoryless strategies are sufficient to win

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 8 / 30

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SLIDE 18

Control of Timed Systems Controllability with Lν

Our Contribution

Control objective in Lν (safety and bounded liveness)

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 9 / 30

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SLIDE 19

Control of Timed Systems Controllability with Lν

Our Contribution

Control objective in Lν (safety and bounded liveness) Reduction of the Control Problem for (TA, Lν) to a Model-Checking Problem for (TA, Lc

ν) :

there is a strategy f s.t. (G f ) | = φ ⇐ ⇒ G | = φ

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 9 / 30

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SLIDE 20

Control of Timed Systems Controllability with Lν

Our Contribution

Control objective in Lν (safety and bounded liveness) Reduction of the Control Problem for (TA, Lν) to a Model-Checking Problem for (TA, Lc

ν) :

there is a strategy f s.t. (G f ) | = φ ⇐ ⇒ G | = φ Properties of the new logic Lc

ν

Expressiveness Model Checking over TA Compositionality

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 9 / 30

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SLIDE 21

Control of Timed Systems Controllability with Lν

Our Contribution

Control objective in Lν (safety and bounded liveness) Reduction of the Control Problem for (TA, Lν) to a Model-Checking Problem for (TA, Lc

ν) :

there is a strategy f s.t. (G f ) | = φ ⇐ ⇒ G | = φ Properties of the new logic Lc

ν

Expressiveness Model Checking over TA Compositionality

Implementation The tool CMC [Laroussinie et al., 98]

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 9 / 30

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SLIDE 22

Control of Timed Systems Controllability with Lν

Outline

◮

Control of Timed Systems

◮

Controllability with Lν

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 10 / 30

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SLIDE 23

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 24

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or Discrete step properties: a ϕ, [a] ϕ, a an action

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 25

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or Discrete step properties: a ϕ, [a] ϕ, a an action Time step properties: δ ϕ, [δ] ϕ

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 26

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: a ϕ, [a] ϕ, a an action Time step properties: δ ϕ, [δ] ϕ Time guarded properties: x in ϕ with x a formula clock

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 27

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: a ϕ, [a] ϕ, a an action Time step properties: δ ϕ, [δ] ϕ Time guarded properties: x in ϕ with x a formula clock Greatest fixed point properties: Z =ν ϕ(Z)

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 28

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: a ϕ, [a] ϕ, a an action Time step properties: δ ϕ, [δ] ϕ Time guarded properties: x in ϕ with x a formula clock Greatest fixed point properties: Z =ν ϕ(Z)

Syntax of Lν

Lν ∋ ϕ ::= p | t t | ff | ϕ ∧ ϕ | ϕ ∨ ϕ | x in ϕ | x ⊲ ⊳ c | [a] ϕ | a ϕ | [δ] ϕ | δ ϕ | Z =ν φ where p an atomic prop., a ∈ Act, x a formula clock, ⊲ ⊳ ∈ {<, ≤, =, ≥, >}, c ∈ Q≥0, Z an identifier.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 29

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: a ϕ, [a] ϕ, a an action Time step properties: δ ϕ, [δ] ϕ Time guarded properties: x in ϕ with x a formula clock Greatest fixed point properties: Z =ν ϕ(Z)

Some Lν formulas

Σ the alphabet of all actions, x a formula clock, s a state of a TA

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 30

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: a ϕ, [a] ϕ, a an action Time step properties: δ ϕ, [δ] ϕ Time guarded properties: x in ϕ with x a formula clock Greatest fixed point properties: Z =ν ϕ(Z)

Some Lν formulas

Σ the alphabet of all actions, x a formula clock, s a state of a TA “No a is enabled in s”: (s, x) | = [a] ff

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 31

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: a ϕ, [a] ϕ, a an action Time step properties: δ ϕ, [δ] ϕ Time guarded properties: x in ϕ with x a formula clock Greatest fixed point properties: Z =ν ϕ(Z)

Some Lν formulas

Σ the alphabet of all actions, x a formula clock, s a state of a TA “No a is enabled in s”: (s, x) | = [a] ff “At most 5 t.u. can elapse from s”: (s, x) | = x in [δ] (x ≤ 5)

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 32

Control of Timed Systems Controllability with Lν

The Timed Modal Logic Lν

Atomic propositions + and, or + Clock Constraints x ≤ c Discrete step properties: a ϕ, [a] ϕ, a an action Time step properties: δ ϕ, [δ] ϕ Time guarded properties: x in ϕ with x a formula clock Greatest fixed point properties: Z =ν ϕ(Z)

Some Lν formulas

Σ the alphabet of all actions, x a formula clock, s a state of a TA “No a is enabled in s”: (s, x) | = [a] ff “At most 5 t.u. can elapse from s”: (s, x) | = x in [δ] (x ≤ 5) “The states that avoid Bad”: (s, x) ∈ Z, Z =ν BAD ∧ [Σ] Z ∧ [δ] Z

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 11 / 30

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SLIDE 33

Control of Timed Systems Controllability with Lν

Semantics of Lν

Given A a TA, φ an Lν formula, ρ an assignment for identifiers (Z) Interpretation of φ in context ρ is a set of extended states (s, w) with: s = (ℓ, v) a state of A and w a valuation of the formula clocks ρ assigns to each identifier a set of extended states

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 12 / 30

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SLIDE 34

Control of Timed Systems Controllability with Lν

Semantics of Lν

Given A a TA, φ an Lν formula, ρ an assignment for identifiers (Z) Interpretation of φ in context ρ is a set of extended states (s, w) with: s = (ℓ, v) a state of A and w a valuation of the formula clocks ρ assigns to each identifier a set of extended states [ [x ⊲ ⊳ c] ] ρ

def

= {(s, u) | u(x) ⊲ ⊳ c} [ [ϕ1 ∨ ϕ2] ] ρ

def

= [ [ϕ1] ] ρ ∪ [ [ϕ2] ] ρ (∩ for ∧) [ [a ϕ] ] ρ def = {(s, u) | ∃ s

a

− − → s′ and (s′, u) ∈ [ [ϕ] ] ρ} [ [[a] ϕ] ] ρ def = {(s, u) | ∀ s

a

− − → s′ , (s′, u) ∈ [ [ϕ] ] ρ}

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 12 / 30

slide-35
SLIDE 35

Control of Timed Systems Controllability with Lν

Semantics of Lν

Given A a TA, φ an Lν formula, ρ an assignment for identifiers (Z) Interpretation of φ in context ρ is a set of extended states (s, w) with: s = (ℓ, v) a state of A and w a valuation of the formula clocks ρ assigns to each identifier a set of extended states [ [δ ϕ] ] ρ def = {(s, u) | ∃s

d

− − → s′ and (s′, u + d) ∈ [ [ϕ] ] ρ} [ [[δ] ϕ] ] ρ def = {(s, u) | ∀s

d

− − → s′, (s′, u + d) ∈ [ [ϕ] ] ρ} [ [x in ϕ] ] ρ

def

= {(s, u) | (s, u[x ← 0]) ∈ [ [ϕ] ] ρ} [ [X] ] ρ

def

= ρ(X) [ [X =ν ϕ] ] ρ

def

= {S | S ⊆ [ [ϕ] ] (ρ[X → S])}

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 12 / 30

slide-36
SLIDE 36

Control of Timed Systems Controllability with Lν

Semantics of Lν

Given A a TA, φ an Lν formula, ρ an assignment for identifiers (Z) Interpretation of φ in context ρ is a set of extended states (s, w) with: s = (ℓ, v) a state of A and w a valuation of the formula clocks ρ assigns to each identifier a set of extended states [ [δ ϕ] ] ρ def = {(s, u) | ∃s

d

− − → s′ and (s′, u + d) ∈ [ [ϕ] ] ρ} [ [[δ] ϕ] ] ρ def = {(s, u) | ∀s

d

− − → s′, (s′, u + d) ∈ [ [ϕ] ] ρ} [ [x in ϕ] ] ρ

def

= {(s, u) | (s, u[x ← 0]) ∈ [ [ϕ] ] ρ} [ [X] ] ρ

def

= ρ(X) [ [X =ν ϕ] ] ρ

def

= {S | S ⊆ [ [ϕ] ] (ρ[X → S])} For closed formula, [ [φ] ] does not depend on ρ A | = φ ⇐ ⇒ ((ℓ0, 0), 0) ∈ [ [φ] ]

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 12 / 30

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SLIDE 37

Control of Timed Systems Controllability with Lν

Results for Lν

[Laroussinie et al., 95a, Laroussinie et al.,95b]

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 13 / 30

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SLIDE 38

Control of Timed Systems Controllability with Lν

Results for Lν

[Laroussinie et al., 95a, Laroussinie et al.,95b] Model Checking over TA is EXPTIME-Complete

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 13 / 30

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SLIDE 39

Control of Timed Systems Controllability with Lν

Results for Lν

[Laroussinie et al., 95a, Laroussinie et al.,95b] Model Checking over TA is EXPTIME-Complete Lν is compositional for TA: if φ ∈ Lν then (A B) | = φ ⇐ ⇒ A | = φ/B with the quotient formula φ/B ∈ Lν

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 13 / 30

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SLIDE 40

Control of Timed Systems Controllability with Lν

Results for Lν

[Laroussinie et al., 95a, Laroussinie et al.,95b] Model Checking over TA is EXPTIME-Complete Lν is compositional for TA: if φ ∈ Lν then (A B) | = φ ⇐ ⇒ A | = φ/B with the quotient formula φ/B ∈ Lν Lν allows to express timed bisimilarity via characteristic formula

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 13 / 30

slide-41
SLIDE 41

Control of Timed Systems Controllability with Lν

Results for Lν

[Laroussinie et al., 95a, Laroussinie et al.,95b] Model Checking over TA is EXPTIME-Complete Lν is compositional for TA: if φ ∈ Lν then (A B) | = φ ⇐ ⇒ A | = φ/B with the quotient formula φ/B ∈ Lν Lν allows to express timed bisimilarity via characteristic formula Model Checker for Lν: CMC [Laroussinie et al., 98] compute quotient formula φ and check nil | = φ

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 13 / 30

slide-42
SLIDE 42

Control of Timed Systems Controllability with Lν

Results for Lν

[Laroussinie et al., 95a, Laroussinie et al.,95b] Model Checking over TA is EXPTIME-Complete Lν is compositional for TA: if φ ∈ Lν then (A B) | = φ ⇐ ⇒ A | = φ/B with the quotient formula φ/B ∈ Lν Lν allows to express timed bisimilarity via characteristic formula Model Checker for Lν: CMC [Laroussinie et al., 98] compute quotient formula φ and check nil | = φ

Open Problem for Lν

Satisfiability for Timed Automata

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 13 / 30

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SLIDE 43

Control of Timed Systems Controllability with Lν

Sampling Control with Lν

G(∆) = G with all controllable actions separated by k · ∆ t.u., k ∈ N

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 14 / 30

slide-44
SLIDE 44

Control of Timed Systems Controllability with Lν

Sampling Control with Lν

G(∆) = G with all controllable actions separated by k · ∆ t.u., k ∈ N

Sampling Control Problem (SCP)

Input: G a TGA, φ an Lν objective, ∆ ∈ Q≥0 a sampling rate SCP: “Is there a controller f s.t. G(∆) f | = φ ?”

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 14 / 30

slide-45
SLIDE 45

Control of Timed Systems Controllability with Lν

Sampling Control with Lν

G(∆) = G with all controllable actions separated by k · ∆ t.u., k ∈ N

Sampling Control Problem (SCP)

Input: G a TGA, φ an Lν objective, ∆ ∈ Q≥0 a sampling rate SCP: “Is there a controller f s.t. G(∆) f | = φ ?”

Model for G(∆)

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 14 / 30

slide-46
SLIDE 46

Control of Timed Systems Controllability with Lν

Sampling Control with Lν

G(∆) = G with all controllable actions separated by k · ∆ t.u., k ∈ N

Sampling Control Problem (SCP)

Input: G a TGA, φ an Lν objective, ∆ ∈ Q≥0 a sampling rate SCP: “Is there a controller f s.t. G(∆) f | = φ ?”

Model for G(∆)

ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 14 / 30

slide-47
SLIDE 47

Control of Timed Systems Controllability with Lν

Sampling Control with Lν

G(∆) = G with all controllable actions separated by k · ∆ t.u., k ∈ N

Sampling Control Problem (SCP)

Input: G a TGA, φ an Lν objective, ∆ ∈ Q≥0 a sampling rate SCP: “Is there a controller f s.t. G(∆) f | = φ ?”

Model for G(∆)

ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u cε cε cε

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 14 / 30

slide-48
SLIDE 48

Control of Timed Systems Controllability with Lν

Sampling Control with Lν

G(∆) = G with all controllable actions separated by k · ∆ t.u., k ∈ N

Sampling Control Problem (SCP)

Input: G a TGA, φ an Lν objective, ∆ ∈ Q≥0 a sampling rate SCP: “Is there a controller f s.t. G(∆) f | = φ ?”

Model for G(∆)

ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u cε cε cε ℓ0 [z ≤ ∆] z = ∆; Actc; z := 0 Automaton A∆

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 14 / 30

slide-49
SLIDE 49

Control of Timed Systems Controllability with Lν

Reduction of Sampling Control Problem to a Model Checking Problem

Avoid Bad: Z =ν Bad ∧ [Σ] Z ∧ [δ] Z

“Is there a controller f s.t. (G(∆) f ) | = φ ?” with φ = Z and Z =ν Bad ∧ [Σ] Z ∧ [δ] Z

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 15 / 30

slide-50
SLIDE 50

Control of Timed Systems Controllability with Lν

Reduction of Sampling Control Problem to a Model Checking Problem

Avoid Bad: Z =ν Bad ∧ [Σ] Z ∧ [δ] Z

“Is there a controller f s.t. (G(∆) f ) | = φ ?” with φ = Z and Z =ν Bad ∧ [Σ] Z ∧ [δ] Z amounts to checking G(∆) | = φ with φ = Y and Y =ν Bad ∧ [Actu] Y ∧ [δ] Y ∧ ([Actc] ff ∨ Actc Y )

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 15 / 30

slide-51
SLIDE 51

Control of Timed Systems Controllability with Lν

Reduction of Sampling Control Problem to a Model Checking Problem

Avoid Bad: Z =ν Bad ∧ [Σ] Z ∧ [δ] Z

“Is there a controller f s.t. (G(∆) f ) | = φ ?” with φ = Z and Z =ν Bad ∧ [Σ] Z ∧ [δ] Z amounts to checking G(∆) | = φ with φ = Y and Y =ν Bad ∧ [Actu] Y ∧ [δ] Y ∧ ([Actc] ff ∨ Actc Y )

Theorem

Given G a TGA, φ a control objective in Ldet

ν

⊆ Lν, ∆ ∈ Q≥0 ∃f s.t. G(∆) f | = φ ⇐ ⇒ G(∆) | = φ ⇐ ⇒ G A∆ | = φ φ can be built automatically (syntactic translation of φ), φ is in Lν.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 15 / 30

slide-52
SLIDE 52

Control of Timed Systems Controllability with Lν

∆-Dense-Time Control

G([∆, +∞[) all controllable actions separated by at least ∆ t.u.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 16 / 30

slide-53
SLIDE 53

Control of Timed Systems Controllability with Lν

∆-Dense-Time Control

G([∆, +∞[) all controllable actions separated by at least ∆ t.u.

∆-Dense-Time Control Problem

Input: G a TGA, φ an Lν objective, ∆ ∈ Q≥0 a minimum delay ∆-CP: “Is there a controller f s.t. G([∆, +∞[) f | = φ ?”

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 16 / 30

slide-54
SLIDE 54

Control of Timed Systems Controllability with Lν

∆-Dense-Time Control

G([∆, +∞[) all controllable actions separated by at least ∆ t.u.

∆-Dense-Time Control Problem

Input: G a TGA, φ an Lν objective, ∆ ∈ Q≥0 a minimum delay ∆-CP: “Is there a controller f s.t. G([∆, +∞[) f | = φ ?”

Model for G([∆, +∞[)

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 16 / 30

slide-55
SLIDE 55

Control of Timed Systems Controllability with Lν

∆-Dense-Time Control

G([∆, +∞[) all controllable actions separated by at least ∆ t.u.

∆-Dense-Time Control Problem

Input: G a TGA, φ an Lν objective, ∆ ∈ Q≥0 a minimum delay ∆-CP: “Is there a controller f s.t. G([∆, +∞[) f | = φ ?”

Model for G([∆, +∞[)

ℓ0 [x ≤ 4] ℓ1 [x ≤ 5] ℓ2 [x ≤ 5] Bad x ≤ 4; c1 c2 c3; x := 0 x < 2;u x > 3;u ℓ0 z ≥ ∆; Actc; z := 0 Automaton B∆

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 16 / 30

slide-56
SLIDE 56

Control of Timed Systems Controllability with Lν

Dense-Time Control with Lν

Reduction of ∆-Control Problem to Model-Checking

Aim: Given φ in Lν, prove the following reduction: ∃f s.t. G([∆, +∞[) f | = φ ⇐ ⇒ G([∆, +∞[) | = φ with φ built syntactically.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 17 / 30

slide-57
SLIDE 57

Control of Timed Systems Controllability with Lν

Dense-Time Control with Lν

Reduction of ∆-Control Problem to Model-Checking

Aim: Given φ in Lν, prove the following reduction: ∃f s.t. G([∆, +∞[) f | = φ ⇐ ⇒ G([∆, +∞[) | = φ with φ built syntactically. Lν is not expressive enough for φ Objective φ given by Z =ν Bad ∧ [u]Z ∧ [c]Z ∧ [δ] Z

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 17 / 30

slide-58
SLIDE 58

Control of Timed Systems Controllability with Lν

Dense-Time Control with Lν

Reduction of ∆-Control Problem to Model-Checking

Aim: Given φ in Lν, prove the following reduction: ∃f s.t. G([∆, +∞[) f | = φ ⇐ ⇒ G([∆, +∞[) | = φ with φ built syntactically. Lν is not expressive enough for φ Objective φ given by Z =ν Bad ∧ [u]Z ∧ [c]Z ∧ [δ] Z ℓ0 Bad 1 ≤ x ≤ 2 c; x := 0 x > 2; u Intuition: φ will contain [δ] [δ] = ⇒ “after all delays” (ℓ0, x = 0) will not sat. [δ] [u] Bad

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 17 / 30

slide-59
SLIDE 59

Control of Timed Systems Controllability with Lν

Dense-Time Control with Lν

Reduction of ∆-Control Problem to Model-Checking

Aim: Given φ in Lν, prove the following reduction: ∃f s.t. G([∆, +∞[) f | = φ ⇐ ⇒ G([∆, +∞[) | = φ with φ built syntactically. Lν is not expressive enough for φ Need some until operator: [δ

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 17 / 30

slide-60
SLIDE 60

Control of Timed Systems Controllability with Lν

Dense-Time Control with Lν

Reduction of ∆-Control Problem to Model-Checking

Aim: Given φ in Lν, prove the following reduction: ∃f s.t. G([∆, +∞[) f | = φ ⇐ ⇒ G([∆, +∞[) | = φ with φ built syntactically. Lν is not expressive enough for φ Need some until operator: [δ We need to restrict the set of control objectives (φ)

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 17 / 30

slide-61
SLIDE 61

Control of Timed Systems Controllability with Lν

Dense-Time Control with Lν

Reduction of ∆-Control Problem to Model-Checking

Aim: Given φ in Lν, prove the following reduction: ∃f s.t. G([∆, +∞[) f | = φ ⇐ ⇒ G([∆, +∞[) | = φ with φ built syntactically. Lν is not expressive enough for φ Need some until operator: [δ We need to restrict the set of control objectives (φ) s ∃f1 s.t. s | = φ1 ∃f2 s.t. s | = c φ2 Objective: φ1 ∧ c φ2 Build a strategy f from f1 and f2 to ensure φ1 ∧ c φ2 f2(s) = c, but f1(s) ?

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 17 / 30

slide-62
SLIDE 62

Control of Timed Systems Controllability with Lν

Dense-Time Control with Lν

Reduction of ∆-Control Problem to Model-Checking

Aim: Given φ in Lν, prove the following reduction: ∃f s.t. G([∆, +∞[) f | = φ ⇐ ⇒ G([∆, +∞[) | = φ with φ built syntactically. Lν is not expressive enough for φ Need some until operator: [δ We need to restrict the set of control objectives (φ) Define a sublogic Ldet

ν

⊂ Lν s.t. strategies can be merged

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 17 / 30

slide-63
SLIDE 63

Control of Timed Systems Controllability with Lν

The logics Lc

ν and Ldet ν

Extension of Lν for Timed Control

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 18 / 30

slide-64
SLIDE 64

Control of Timed Systems Controllability with Lν

The logics Lc

ν and Ldet ν

Extension of Lν for Timed Control

Lc

ν = Lν + new modality [δ

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 18 / 30

slide-65
SLIDE 65

Control of Timed Systems Controllability with Lν

The logics Lc

ν and Ldet ν

Extension of Lν for Timed Control

Lc

ν = Lν + new modality [δ

ϕ [δ ψ ∼ ϕ Weak Until ψ

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 18 / 30

slide-66
SLIDE 66

Control of Timed Systems Controllability with Lν

The logics Lc

ν and Ldet ν

Extension of Lν for Timed Control

Lc

ν = Lν + new modality [δ

ϕ [δ ψ ∼ ϕ Weak Until ψ

Semantics of ϕ [δ ψ

(s, u) | = ϕ [δ ψ ⇐ ⇒

either ∀t ∈ R≥0, s

t

− − → s′ = ⇒ (s′, u + t) | = ϕ

  • r ∃t ∈ R≥0 s.t. s

t

− − → s′ and (s′, v + t) | = ψ and ∀0 ≤ t′ < t, s

t′

− − → s′′ we have (s′′, v + t′) | = ϕ

Allows to express prevention of time-elapsing

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 18 / 30

slide-67
SLIDE 67

Control of Timed Systems Controllability with Lν

The logics Lc

ν and Ldet ν

Extension of Lν for Timed Control

Lc

ν = Lν + new modality [δ

ϕ [δ ψ ∼ ϕ Weak Until ψ

Allows to express prevention of time-elapsing Restriction of control objectives to Ldet

ν

Syntax CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 18 / 30

slide-68
SLIDE 68

Control of Timed Systems Controllability with Lν

The logics Lc

ν and Ldet ν

Extension of Lν for Timed Control

Lc

ν = Lν + new modality [δ

ϕ [δ ψ ∼ ϕ Weak Until ψ

Allows to express prevention of time-elapsing Restriction of control objectives to Ldet

ν

Syntax

rule out conjunctions of the type (c ψ) ∧ φ for arbitrary φ

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 18 / 30

slide-69
SLIDE 69

Control of Timed Systems Controllability with Lν

The logics Lc

ν and Ldet ν

Extension of Lν for Timed Control

Lc

ν = Lν + new modality [δ

ϕ [δ ψ ∼ ϕ Weak Until ψ

Allows to express prevention of time-elapsing Restriction of control objectives to Ldet

ν

Syntax

rule out conjunctions of the type (c ψ) ∧ φ for arbitrary φ Allow only conjunctions like c1 φ1 ∧ c2 φ2

Allows to merge strategies

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 18 / 30

slide-70
SLIDE 70

Control of Timed Systems Controllability with Lν

The logics Lc

ν and Ldet ν

Extension of Lν for Timed Control

Lc

ν = Lν + new modality [δ

ϕ [δ ψ ∼ ϕ Weak Until ψ

Allows to express prevention of time-elapsing Restriction of control objectives to Ldet

ν

Syntax

rule out conjunctions of the type (c ψ) ∧ φ for arbitrary φ Allow only conjunctions like c1 φ1 ∧ c2 φ2

Allows to merge strategies

Theorem

Given G a TGA, φ a control objective in Ldet

ν

⊆ Lν, ∆ ∈ Q≥0 ∃f s.t. G([∆, +∞[) f | = φ ⇐ ⇒ G([∆, +∞[) | = φ ⇐ ⇒ G B∆ | = φ φ can be built automatically (syntactic translation of φ), φ is in Lc

ν.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 18 / 30

slide-71
SLIDE 71

Control of Timed Systems Controllability with Lν

How to build the control formula φ ?

ϕ =

σ∈Actc∪{λ} ϕ σ

ϕ σ holds in s if there is a strategy prescribing σ in s which can enforce ϕ.

  • α∈A

α

σ def

=

  • α∈A

α σ

  • α∈A

α

σ def

=

  • α∈A

α σ a ϕ

σ def

=    ff if σ, a ∈ Actc ∧ σ = a a ϕ ∧ σ t t if a ∈ Actu a ϕ

  • therwise

δ ϕ

σ def

= δ ϕ if σ = λ ϕ σ if σ ∈ Actc

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 19 / 30

slide-72
SLIDE 72

Control of Timed Systems Controllability with Lν

How to build the control formula φ ? (cont.)

[ac] ϕ

σ def

= σ t t if ac = σ ac ϕ if ac = σ [au] ϕ

σ def

= [au] ϕ ∧ σ t t [δ] ϕ

σ def

=    ϕ σ if σ ∈ Actc ϕ λ [δ

ac∈Actc

ϕ ac

  • therwise

x ∼ c σ

def

= x ∼ c ∧ σ t t r in ϕ

σ def

= r in ϕ σ X

σ def

= Xσ ∧ σ t t

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 20 / 30

slide-73
SLIDE 73

Control of Timed Systems Controllability with Lν

Outline

◮

Control of Timed Systems

◮

Controllability with Lν

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 21 / 30

slide-74
SLIDE 74

Control of Timed Systems Controllability with Lν

Properties of the new operator [δ

Expressivity

The logic Lc

ν is strictly more expressive than Lν over timed automata.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 22 / 30

slide-75
SLIDE 75

Control of Timed Systems Controllability with Lν

Properties of the new operator [δ

Expressivity

The logic Lc

ν is strictly more expressive than Lν over timed automata.

([a] ff ) [δ (b t t) cannot be expressed with Lν

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 22 / 30

slide-76
SLIDE 76

Control of Timed Systems Controllability with Lν

Properties of the new operator [δ

Expressivity

The logic Lc

ν is strictly more expressive than Lν over timed automata.

([a] ff ) [δ (b t t) cannot be expressed with Lν

Model checking

Computation

The model-checking of Lc

ν over timed automata is EXPTIME-complete.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 22 / 30

slide-77
SLIDE 77

Control of Timed Systems Controllability with Lν

Properties of the new operator [δ

Expressivity

The logic Lc

ν is strictly more expressive than Lν over timed automata.

([a] ff ) [δ (b t t) cannot be expressed with Lν

Model checking

Computation

The model-checking of Lc

ν over timed automata is EXPTIME-complete.

Compositionality

Quotient

The logic Lc

ν is compositional for the class of timed automata.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 22 / 30

slide-78
SLIDE 78

Control of Timed Systems Controllability with Lν

Properties of the new operator [δ

Expressivity

The logic Lc

ν is strictly more expressive than Lν over timed automata.

([a] ff ) [δ (b t t) cannot be expressed with Lν

Model checking

Computation

The model-checking of Lc

ν over timed automata is EXPTIME-complete.

Compositionality

Quotient

The logic Lc

ν is compositional for the class of timed automata.

(A1 A2) | = ϕ ⇐ ⇒ A1 | = ϕ/A2

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 22 / 30

slide-79
SLIDE 79

Control of Timed Systems Controllability with Lν

Outline

◮

Control of Timed Systems

◮

Controllability with Lν

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 23 / 30

slide-80
SLIDE 80

Control of Timed Systems Controllability with Lν

Conclusion & Further Work

Results

Control Objectives in Ldet

ν

Reduction of Control Problem for (TA, Ldet

ν ) to a Model Checking

Problem for (TA, Lc

ν)

∃f s.t. (G f ) | = φ ⇐ ⇒ G | = φ Properties of the new logic Lc

ν

Strictly more expressive than Lν Model-Checking is EXPTIME-Complete Lc

ν is compositional for TA

Implementation: The tool CMC [Laroussinie et al., 98]

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 24 / 30

slide-81
SLIDE 81

Control of Timed Systems Controllability with Lν

Conclusion & Further Work

Results

Control Objectives in Ldet

ν

Reduction of Control Problem for (TA, Ldet

ν ) to a Model Checking

Problem for (TA, Lc

ν)

∃f s.t. (G f ) | = φ ⇐ ⇒ G | = φ Properties of the new logic Lc

ν

Strictly more expressive than Lν Model-Checking is EXPTIME-Complete Lc

ν is compositional for TA

Implementation: The tool CMC [Laroussinie et al., 98]

Further Work

Extend Ldet

ν

Synthesize Controllers Extend to Partial Observation Use More general notion of strategies

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 24 / 30

slide-82
SLIDE 82

Control of Timed Systems Controllability with Lν

Related Work

Discrete Time Case

ATL [Alur et al., 02] Reduction of CP to MC Problem with µ-calculus: loop µ-calculus [Arnold et al., 03] Quantified µ-calculus [Riedweg et al., 03]

Timed Case External specifications = TA [D’Souza et al., 02] TCTL control objective [Faella et al., 02]

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 25 / 30

slide-83
SLIDE 83

Control of Timed Systems Controllability with Lν

References

Rajeev Alur, Thomas A. Henzinger, and Orna Kupferman. Alternating-time temporal logic. Journal of the ACM, 49:672–713, 2002. Luca De Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Symbolic algorithms for infinite-state games. In Proc. 12th International Conference on Concurrency Theory (CONCUR’01), volume 2154 of LNCS, pages 536–550. Springer, 2001. Eugene Asarin, Oded Maler, Amir Pnueli, and Joseph Sifakis. Controller synthesis for timed automata. In Proc. IFAC Symposium on System Structure and Control, pages 469–474. Elsevier Science, 1998. André Arnold, Aymeric Vincent, and Igor Walukiewicz. Games for synthesis of controllers with partial observation. Theoretical Computer Science, 303(1):7–34,2003.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 26 / 30

slide-84
SLIDE 84

Control of Timed Systems Controllability with Lν

References (cont.)

Luca Aceto, Patricia Bouyer, Augusto Burgueño, and Kim G. Larsen. The power of reachability testing for timed automata. Theoretical Computer Science (TCS), 300(1–3):411–475, 2003. Patricia Bouyer, Franck Cassez, and François Laroussinie. Modal logics for timed control. Research Report LSV-05-04, Laboratoire Spécification & Vérification, ENS de Cachan, France, 2005. Deepak D’Souza and P. Madhusudan. Timed control synthesis for external specifications. In Proc. 19th Int. Symp. Theoretical Aspects of Computer Science (STACS’2002), volume 2285 of LNCS, pages 571–582, Springer, 2002.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 27 / 30

slide-85
SLIDE 85

Control of Timed Systems Controllability with Lν

References (cont.)

Marco Faella, Salvatore La Torre, and Aniello Murano. Dense real-time games. In Proc. 17th IEEE Symposium on Logic in Computer Science (LICS’02), pages 167–176. IEEE Computer Society Press, 2002. François Laroussinie and Kim G. Larsen. Compositional model-checking of real-time systems. In Proc. 6th International Conference on Concurrency Theory (CONCUR’95), volume 962 of Lecture Notes in Computer Science, pages 27–41. Springer, 1995. François Laroussinie and Kim G. Larsen. CMC: A tool for compositional model-checking of real-time systems. In Proc. IFIP Joint International Conference on Formal Description Techniques & Protocol Specification, Testing, and Verification (FORTE-PSTV’98), pages 439–456. Kluwer Academic, 1998.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 28 / 30

slide-86
SLIDE 86

Control of Timed Systems Controllability with Lν

References (cont.)

François Laroussinie, Kim G. Larsen, and Carsten Weise. From timed automata to logic – and back. In Proc. 20th International Symposium on Mathematical Foundations

  • f Computer Science (MFCS’95), volume 969 of Lecture Notes in

Computer Science, pages 529–539. Springer, 1995. Oded Maler, Amir Pnueli, and Joseph Sifakis. On the synthesis of discrete controllers for timed systems. In Proc. 12th Annual Symposium on Theoretical Aspects of Computer Science (STACS’95), volume 900, pages 229–242. Springer, 1995. Stéphane Riedweg and Sophie Pinchinat. Quantified µ-calculus for control synthesis. In Proc. 28th International Symposium on Mathematical Foundations

  • f Computer Science (MFCS’03), volume 2747 of Lecture Notes in

Computer Science, pages 642–651. Springer, 2003.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 29 / 30

slide-87
SLIDE 87

Timed Automata

A Timed Automaton A is a tuple (L, ℓ0, Act, X, inv, − →) where: L is a finite set of locations ℓ0 is the initial location X is a finite set of clocks Act is a finite set of actions − → is a set of transitions of the form ℓ

g,a,R

− − − − → ℓ′ with:

ℓ, ℓ′ ∈ L, a ∈ Act a guard g which is a clock constraint over X a reset set R which is the set of clocks to be reset to 0

Clock constraints are boolean combinations of x ∼ k with x ∈ C and k ∈ Z and ∼∈ {≤, <}.

Back CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 30 / 30

slide-88
SLIDE 88

Semantics of Timed Automata

Let A = (L, ℓ0, Act, X, inv, − →) be a Timed Automaton. A state (ℓ, v) of A is in L × RX

≥0

The semantics of A is a Timed Transition System SA = (Q, q0, Act ∪ R≥0, − →) with: Q = L × RX

≥0

q0 = (ℓ0, 0) − → consists in: (discrete transition): (ℓ, v)

a

− − → (ℓ′, v′) ⇐ ⇒        ∃ ℓ

g,a,r

− − − − → ℓ′ ∈ A v | = g v′ = v[r ← 0] v′ | = inv(ℓ′) (delay transition): (ℓ, v)

d

− − → (ℓ, v + d) ⇐ ⇒ d ∈ R≥0 ∧ v + d | = inv(ℓ)

Back CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 31 / 30

slide-89
SLIDE 89

Syntax of Ldet

ν

Ldet

ν

∋ ϕ, ψ ::= X | ϕ ∨ ψ |

  • α∈A

α | Z =ν φ where A denotes a deterministic set of basic terms {α1, α2, . . . , αn}: Basic terms: α ::= t t | ff | x ⊲ ⊳ c | r in σ ϕ | r in [σ] ϕ with σ ∈ Act ∪ {λ} Deterministic set of basic terms: for all σ ∈ Act ∪ {λ} there is at most one i s.t. αi = r in σ ϕ or αi = r in [σ] ϕ.

Back CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 32 / 30

slide-90
SLIDE 90

The following results are taken from [Aceto et al., 03].

Test Automaton

Let T be a timed automaton with a set of rejecting locations N. T is a test automaton for the property φ if for all timed automata B: B | = φ ⇐ ⇒ ReachableStatesOf(B T) ∩ N = ∅ A property φ can be tested if there is a test automaton Tφ for φ.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 33 / 30

slide-91
SLIDE 91

The following results are taken from [Aceto et al., 03].

Test Automaton

Let T be a timed automaton with a set of rejecting locations N. T is a test automaton for the property φ if for all timed automata B: B | = φ ⇐ ⇒ ReachableStatesOf(B T) ∩ N = ∅ A property φ can be tested if there is a test automaton Tφ for φ.

The Logic L∀S

L∀S is a strict subset of Lν: no δ , restricted or, restricted a .

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 33 / 30

slide-92
SLIDE 92

The following results are taken from [Aceto et al., 03].

Test Automaton

Let T be a timed automaton with a set of rejecting locations N. T is a test automaton for the property φ if for all timed automata B: B | = φ ⇐ ⇒ ReachableStatesOf(B T) ∩ N = ∅ A property φ can be tested if there is a test automaton Tφ for φ.

The Logic L∀S

L∀S is a strict subset of Lν: no δ , restricted or, restricted a .

Test Automata and Logics

A property φ can be tested iff φ is a formula of L∀S.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 33 / 30

slide-93
SLIDE 93

The following results are taken from [Aceto et al., 03].

Test Automaton

Let T be a timed automaton with a set of rejecting locations N. T is a test automaton for the property φ if for all timed automata B: B | = φ ⇐ ⇒ ReachableStatesOf(B T) ∩ N = ∅ A property φ can be tested if there is a test automaton Tφ for φ.

Test Automata and Logics

A property φ can be tested iff φ is a formula of L∀S.

Ldet

ν

is more expressive than L∀S

The formula X =ν [δ] X ∧ [a] X ∧ δ b t t cannot be expressed in L∀S.

CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 33 / 30

slide-94
SLIDE 94

Model-Checking and Compositionality for Ldet

ν

Computation of [ [ϕ [δ ψ] ] given [ [ϕ] ] and [ [ψ] ] :

← − − [ [ϕ] ]cc ∪ ← − − − − − − − − − − → [ [ψ] ] ∪ [ [ϕ] ] cc ∩

  • [

[ψ] ] ∪

  • [

[ϕ] ] ∩ ← − − − − − − − [ [ϕ] ]+ ∩ [ [ψ] ]

  • Back

Compositionality

  • ϕ1 [δ ϕ2
  • /ℓ

def

=

  • inv(ℓ) =

⇒ (ϕ1/ℓ)

  • [δ
  • inv(ℓ) ∧ (ϕ2/ℓ)
  • This is what is implemented in CMC.

Back CONCUR’05 (San Francisco, CA) Modal Logics for Timed Control 34 / 30