[PPT] - GoBack On Synthesizing Controllers from Bounded-Response Properties PowerPoint Presentation

SLIDE 1

GoBack

SLIDE 2

On Synthesizing Controllers from Bounded-Response Properties 1 / 23

On Synthesizing Controllers from Bounded-Response Properties

Oded Maler Verimag Dejan Niˇ ckovi´ c Verimag Amir Pnueli Weizmann Institute NYU

SLIDE 3

Overview

On Synthesizing Controllers from Bounded-Response Properties 2 / 23

Introduction
Property-based Synthesis

✦

Bounded-response Properties

MTL-B

✦

Syntax and Semantics

✦

Non-Determinism

From MTL-B to Deterministic Temporal Testers

✦

Pastification of MTL-B formulae

✦

Bounded-variability assumption

Application to Synthesis: Arbiter Example

✦

Specification in MTL-B

✦

Experimental Results

Conclusion

SLIDE 4

Introduction

On Synthesizing Controllers from Bounded-Response Properties 3 / 23

Controller

· · · · · · r1 r2 rm gn g2 g1 Environment variables Controller variables BAD r1 r2 r2 r1 g1 g2 g2 g1 l0 l1 l2 . . . . . . r3 g1

Automatic controller synthesis from high-level specifications

✦

Problem posed in [Chu63]

✦

Theoretically solved in [BL69,TB73]

SLIDE 5

Introduction

On Synthesizing Controllers from Bounded-Response Properties 4 / 23

BAD r1 r2 r2 r1 g1 g2 g2 g1 l0 l1 l2 . . . . . . r3 g1

(r1 → r1Sg1) (g1 →

1

[0,1] r1)

(g1 →

1

[0,2] r1)

. . .

Synthesizing controllers from temporal logic formulae [PR89]

✦

Recent improvements [PPS06,PP06]

Property-based synthesis problem:

Given a temporal property ϕ defined over two distinct alphabets A and B, build a finite-state transducer (controller) from Aω to Bω such that all of its behaviors satisfy ϕ.

We are interested in controller synthesis from real-time temporal logic specifications

SLIDE 6

Introduction

On Synthesizing Controllers from Bounded-Response Properties 4 / 23

BAD r1 r2 r2 r1 g1 g2 g2 g1 l0 l1 l2 . . . . . . r3 g1

(r1 → r1Sg1) (g1 →

1

[0,1] r1)

(g1 →

1

[0,2] r1)

. . .

Synthesizing controllers from temporal logic formulae [PR89]

✦

Recent improvements [PPS06,PP06]

Property-based synthesis problem:

Given a temporal property ϕ defined over two distinct alphabets A and B, build a finite-state transducer (controller) from Aω to Bω such that all of its behaviors satisfy ϕ.

We are interested in controller synthesis from real-time temporal logic specifications

SLIDE 7

Introduction

On Synthesizing Controllers from Bounded-Response Properties 4 / 23

BAD r1 r2 r2 r1 g1 g2 g2 g1 l0 l1 l2 . . . . . . r3 g1

(r1 → r1Sg1) (g1 →

1

[0,1] r1)

(g1 →

1

[0,2] r1)

. . .

Synthesizing controllers from temporal logic formulae [PR89]

✦

Recent improvements [PPS06,PP06]

Property-based synthesis problem:

Given a temporal property ϕ defined over two distinct alphabets A and B, build a finite-state transducer (controller) from Aω to Bω such that all of its behaviors satisfy ϕ.

We are interested in controller synthesis from real-time temporal logic specifications

SLIDE 8

Temporal Logic and Controller Synthesis

On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Specification Temporal Logic Non−Deterministic Game Automaton Deterministic Game Automaton Controller translation determinization controller synthesis alg.

SLIDE 9

Temporal Logic and Controller Synthesis

On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Specification Temporal Logic Non−Deterministic Game Automaton Deterministic Game Automaton Controller acceptance conditions non−determinism translation determinization controller synthesis alg. timed automata

SLIDE 10

Temporal Logic and Controller Synthesis

On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Specification Temporal Logic Deterministic Game Automaton Controller safety deterministic controller synthesis alg. translation Non−Deterministic Game Automaton Past

SLIDE 11

Temporal Logic and Controller Synthesis

On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Specification Temporal Logic Non−Deterministic Game Automaton Deterministic Game Automaton Controller translation determinization controller synthesis alg. timed automata Bounded Response non−determinism safety

SLIDE 12

Temporal Logic and Controller Synthesis

On Synthesizing Controllers from Bounded-Response Properties 5 / 23

Specification Temporal Logic Deterministic Game Automaton Controller safety deterministic Eliminate sources of non−determinism controller synthesis alg. translation Non−Deterministic Game Automaton Bounded Response

SLIDE 13

Motivation for Bounded-Response Properties

On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Bounded-response correspond to safety properties

✦

→ Limited scope wrt more general liveness properties

Liveness properties abstract away the upper bound requirement of occurrence of

events

✦

But many applications require specifying explicitly such upper bound:

■

Hard real-time systems

■

Scheduling problems

■

. . .

We choose Bounded Response Metric Temporal Logic - MTL-B as the specification

formalism

✦

MTL [Koy90] without unbounded until

✦

Punctual operators (unlike MITL [AFH96])

✦

Allows specifying non-trivial properties

✦

Can be interpreted both in discrete and dense time

✦

We consider specifications of type ϕ where ϕ is an MTL-B formula

SLIDE 14

Motivation for Bounded-Response Properties

On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Bounded-response correspond to safety properties

✦

→ Limited scope wrt more general liveness properties

Liveness properties abstract away the upper bound requirement of occurrence of

events

✦

But many applications require specifying explicitly such upper bound:

■

Hard real-time systems

■

Scheduling problems

■

. . .

We choose Bounded Response Metric Temporal Logic - MTL-B as the specification

formalism

✦

MTL [Koy90] without unbounded until

✦

Punctual operators (unlike MITL [AFH96])

✦

Allows specifying non-trivial properties

✦

Can be interpreted both in discrete and dense time

✦

We consider specifications of type ϕ where ϕ is an MTL-B formula

SLIDE 15

Motivation for Bounded-Response Properties

On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Bounded-response correspond to safety properties

✦

→ Limited scope wrt more general liveness properties

Liveness properties abstract away the upper bound requirement of occurrence of

events

✦

But many applications require specifying explicitly such upper bound:

■

Hard real-time systems

■

Scheduling problems

■

. . .

We choose Bounded Response Metric Temporal Logic - MTL-B as the specification

formalism

✦

MTL [Koy90] without unbounded until

✦

Punctual operators (unlike MITL [AFH96])

✦

Allows specifying non-trivial properties

✦

Can be interpreted both in discrete and dense time

✦

We consider specifications of type ϕ where ϕ is an MTL-B formula

SLIDE 16

Motivation for Bounded-Response Properties

On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Bounded-response correspond to safety properties

✦

→ Limited scope wrt more general liveness properties

Liveness properties abstract away the upper bound requirement of occurrence of

events

✦

But many applications require specifying explicitly such upper bound:

■

Hard real-time systems

■

Scheduling problems

■

. . .

We choose Bounded Response Metric Temporal Logic - MTL-B as the specification

formalism

✦

MTL [Koy90] without unbounded until

✦

Punctual operators (unlike MITL [AFH96])

✦

Allows specifying non-trivial properties

✦

Can be interpreted both in discrete and dense time

✦

We consider specifications of type ϕ where ϕ is an MTL-B formula

SLIDE 17

Motivation for Bounded-Response Properties

On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Bounded-response correspond to safety properties

✦

→ Limited scope wrt more general liveness properties

Liveness properties abstract away the upper bound requirement of occurrence of

events

✦

But many applications require specifying explicitly such upper bound:

■

Hard real-time systems

■

Scheduling problems

■

. . .

We choose Bounded Response Metric Temporal Logic - MTL-B as the specification

formalism

✦

MTL [Koy90] without unbounded until

✦

Punctual operators (unlike MITL [AFH96])

✦

Allows specifying non-trivial properties

✦

Can be interpreted both in discrete and dense time

✦

We consider specifications of type ϕ where ϕ is an MTL-B formula

SLIDE 18

Motivation for Bounded-Response Properties

On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Bounded-response correspond to safety properties

✦

→ Limited scope wrt more general liveness properties

Liveness properties abstract away the upper bound requirement of occurrence of

events

✦

But many applications require specifying explicitly such upper bound:

■

Hard real-time systems

■

Scheduling problems

■

. . .

We choose Bounded Response Metric Temporal Logic - MTL-B as the specification

formalism

✦

MTL [Koy90] without unbounded until

✦

Punctual operators (unlike MITL [AFH96])

✦

Allows specifying non-trivial properties

✦

Can be interpreted both in discrete and dense time

✦

We consider specifications of type ϕ where ϕ is an MTL-B formula

SLIDE 19

Motivation for Bounded-Response Properties

On Synthesizing Controllers from Bounded-Response Properties 6 / 23

Bounded-response correspond to safety properties

✦

→ Limited scope wrt more general liveness properties

Liveness properties abstract away the upper bound requirement of occurrence of

events

✦

But many applications require specifying explicitly such upper bound:

■

Hard real-time systems

■

Scheduling problems

■

. . .

We choose Bounded Response Metric Temporal Logic - MTL-B as the specification

formalism

✦

MTL [Koy90] without unbounded until

✦

Punctual operators (unlike MITL [AFH96])

✦

Allows specifying non-trivial properties

✦

Can be interpreted both in discrete and dense time

✦

We consider specifications of type ϕ where ϕ is an MTL-B formula

SLIDE 20

MTL-B: Syntax and Semantics

On Synthesizing Controllers from Bounded-Response Properties 7 / 23

Syntax:

ϕ := p | ¬ϕ | ϕ1 ∨ ϕ2 | ϕ1U[a,b]ϕ2 | ϕ1S[a,b]ϕ2 | ϕ1Sϕ2 | ϕ1P[a,b]ϕ2

Semantics:

. . . (ξ, t) | = ϕ1 U[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊕ [a, b] (ξ, t′) | = ϕ2 and ∀t′′[t, t′], (ξ, t′′) | = ϕ1 (ξ, t) | = ϕ1 P[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊖ [0, b − a] (ξ, t′) | = ϕ2 and ∀t′′ ∈ [t − b, t′], (ξ, t′′) | = ϕ1 (ξ, t) | = ϕ1 S[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊖ [a, b] (ξ, t′) | = ϕ2 and ∀t′′ ∈ [t, t′], (ξ, t′′) | = ϕ1 . . .

SLIDE 21

MTL-B: Syntax and Semantics

On Synthesizing Controllers from Bounded-Response Properties 7 / 23

Syntax:

ϕ := p | ¬ϕ | ϕ1 ∨ ϕ2 | ϕ1U[a,b]ϕ2 | ϕ1S[a,b]ϕ2 | ϕ1Sϕ2 | ϕ1P[a,b]ϕ2

Semantics:

. . . (ξ, t) | = ϕ1 U[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊕ [a, b] (ξ, t′) | = ϕ2 and ∀t′′[t, t′], (ξ, t′′) | = ϕ1 (ξ, t) | = ϕ1 P[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊖ [0, b − a] (ξ, t′) | = ϕ2 and ∀t′′ ∈ [t − b, t′], (ξ, t′′) | = ϕ1 (ξ, t) | = ϕ1 S[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊖ [a, b] (ξ, t′) | = ϕ2 and ∀t′′ ∈ [t, t′], (ξ, t′′) | = ϕ1 . . .

ϕ2 ϕ1 ϕ1P[a,b]ϕ2 ϕ2 ϕ1 ϕ1S[a,b]ϕ2 ϕ2 ϕ1 ϕ1U[a,b]ϕ2

t t + a t + b t − a t − b t − (b − a)

SLIDE 22

MTL-B: Syntax and Semantics

On Synthesizing Controllers from Bounded-Response Properties 7 / 23

Syntax:

ϕ := p | ¬ϕ | ϕ1 ∨ ϕ2 | ϕ1U[a,b]ϕ2 | ϕ1S[a,b]ϕ2 | ϕ1Sϕ2 | ϕ1P[a,b]ϕ2

Semantics:

. . . (ξ, t) | = ϕ1 U[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊕ [a, b] (ξ, t′) | = ϕ2 and ∀t′′[t, t′], (ξ, t′′) | = ϕ1 (ξ, t) | = ϕ1 P[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊖ [0, b − a] (ξ, t′) | = ϕ2 and ∀t′′ ∈ [t − b, t′], (ξ, t′′) | = ϕ1 (ξ, t) | = ϕ1 S[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊖ [a, b] (ξ, t′) | = ϕ2 and ∀t′′ ∈ [t, t′], (ξ, t′′) | = ϕ1 . . .

Notes:

✦

“Handshake” semantics of bounded until

✦

Precedes operator ∼ past equivalent of bounded until

Derived operators:

Q

[a,b],

`

[a,b],

1

[a,b], [a,b]

SLIDE 23

MTL-B: Syntax and Semantics

On Synthesizing Controllers from Bounded-Response Properties 7 / 23

Syntax:

ϕ := p | ¬ϕ | ϕ1 ∨ ϕ2 | ϕ1U[a,b]ϕ2 | ϕ1S[a,b]ϕ2 | ϕ1Sϕ2 | ϕ1P[a,b]ϕ2

Semantics:

. . . (ξ, t) | = ϕ1 U[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊕ [a, b] (ξ, t′) | = ϕ2 and ∀t′′[t, t′], (ξ, t′′) | = ϕ1 (ξ, t) | = ϕ1 P[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊖ [0, b − a] (ξ, t′) | = ϕ2 and ∀t′′ ∈ [t − b, t′], (ξ, t′′) | = ϕ1 (ξ, t) | = ϕ1 S[a,b] ϕ2 ↔ ∃ t′ ∈ t ⊖ [a, b] (ξ, t′) | = ϕ2 and ∀t′′ ∈ [t, t′], (ξ, t′′) | = ϕ1 . . .

Notes:

✦

“Handshake” semantics of bounded until

✦

Precedes operator ∼ past equivalent of bounded until

Derived operators:

Q

[a,b],

`

[a,b],

1

[a,b], [a,b]

SLIDE 24

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 25

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 26

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

✸[a,b]p p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 27

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 28

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 29

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 30

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 31

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 32

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

SLIDE 33

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

p

Q

1 p

t t − 1

SLIDE 34

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

p

Q

1 p

t

x0 := 0

t − 1

SLIDE 35

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

p

Q

1 p

t t − 1

x0 := 0 x1 := 0

SLIDE 36

MTL-B and Non-Determinism

On Synthesizing Controllers from Bounded-Response Properties 8 / 23

Two sources of non-determinism
Acausality

✦

Semantics

f

future temporal logics acausal

■

Satisfiability

f

ϕ at time t depends on the input signal value at time t′ ≥ t

t t + a t + b p ✸[a,b]p

✦

Past fragments of temporal logics have causal semantics

Unbounded Variability

✦

No bound on the variability of input signals

✦

→ remember unbounded number

f events

■

Example:

Q

1 p - perfect shift

register for p

p

Q

1 p

t t − 1

x0 := 0 · · · x2 := 0 x1 := 0

SLIDE 37

From MTL-B to Deterministic Timed Automata: Overview

On Synthesizing Controllers from Bounded-Response Properties 9 / 23

MTL-B Property

SLIDE 38

From MTL-B to Deterministic Timed Automata: Overview

On Synthesizing Controllers from Bounded-Response Properties 9 / 23

non−determinism Eliminates acausality−based Pastification MTL-B Property MTL-B Property Past

SLIDE 39

From MTL-B to Deterministic Timed Automata: Overview

On Synthesizing Controllers from Bounded-Response Properties 9 / 23

non−determinism Eliminates acausality−based Assumption Bounded−variability

Input−deterministic Timed Game Automaton

Pastification Eliminates unbounded variability−based non−determinism Translation to DTA [MNP05] MTL-B Property MTL-B Property Past

SLIDE 40

From MTL-B to Deterministic Timed Automata: Overview

On Synthesizing Controllers from Bounded-Response Properties 9 / 23

non−determinism Eliminates acausality−based Assumption Bounded−variability

Input−deterministic Timed Game Automaton

Pastification Controller Synthesis Algorithm Eliminates unbounded variability−based non−determinism Translation to DTA [MNP05] [AMP95]

Real−time Controller

[CDF+05] MTL-B Property MTL-B Property Past

SLIDE 41

From MTL-B to Deterministic Timed Automata: Overview

On Synthesizing Controllers from Bounded-Response Properties 9 / 23

non−determinism Eliminates acausality−based Assumption Bounded−variability

Input−deterministic Timed Game Automaton

Pastification Controller Synthesis Algorithm Eliminates unbounded variability−based non−determinism Translation to DTA [MNP05] [AMP95]

Real−time Controller

[CDF+05] MTL-B Property MTL-B Property Past

SLIDE 42

Pastification of MTL-B formulae

On Synthesizing Controllers from Bounded-Response Properties 10 / 23

Key idea: Change the time direction from future to past

✦

MTL-B formula fully determined withing a bounded horizon

✦

→ Eliminate the “predictive” aspect of the semantics

Example: ϕ = p →

1

[1,2] [0,2] q

What would be the “equivalent” past formula ψ that describes the same pattern from

t + 4?

✦

ψ =

Q

4 p →

Q

[0,1]

`

[0,2] q

SLIDE 43

Pastification of MTL-B formulae

On Synthesizing Controllers from Bounded-Response Properties 10 / 23

Key idea: Change the time direction from future to past

✦

MTL-B formula fully determined withing a bounded horizon

✦

→ Eliminate the “predictive” aspect of the semantics

Example: ϕ = p →

1

[1,2] [0,2] q

What would be the “equivalent” past formula ψ that describes the same pattern from

t + 4?

✦

ψ =

Q

4 p →

Q

[0,1]

`

[0,2] q

SLIDE 44

Pastification of MTL-B formulae

On Synthesizing Controllers from Bounded-Response Properties 10 / 23

Key idea: Change the time direction from future to past

✦

MTL-B formula fully determined withing a bounded horizon

✦

→ Eliminate the “predictive” aspect of the semantics

Example: ϕ = p →

1

[1,2] [0,2] q

→ t t + 1 t + 2 t + 3 t + 4 → p∗ ∗∗ ∗∗ ∗∗ ∗∗ . . . p∗ ∗q ∗q ∗q ∗∗ . . . p∗ ∗∗ ∗q ∗q ∗q

What would be the “equivalent” past formula ψ that describes the same pattern from

t + 4?

✦

ψ =

Q

4 p →

Q

[0,1]

`

[0,2] q

SLIDE 45

Pastification of MTL-B formulae

On Synthesizing Controllers from Bounded-Response Properties 10 / 23

Key idea: Change the time direction from future to past

✦

MTL-B formula fully determined withing a bounded horizon

✦

→ Eliminate the “predictive” aspect of the semantics

Example: ϕ = p →

1

[1,2] [0,2] q

→ t t + 1 t + 2 t + 3 t + 4 → p∗ ∗∗ ∗∗ ∗∗ ∗∗ . . . p∗ ∗q ∗q ∗q ∗∗ . . . p∗ ∗∗ ∗q ∗q ∗q

What would be the “equivalent” past formula ψ that describes the same pattern from

t + 4?

✦

ψ =

Q

4 p →

Q

[0,1]

`

[0,2] q

SLIDE 46

Pastification of MTL-B formulae

On Synthesizing Controllers from Bounded-Response Properties 10 / 23

Key idea: Change the time direction from future to past

✦

MTL-B formula fully determined withing a bounded horizon

✦

→ Eliminate the “predictive” aspect of the semantics

Example: ϕ = p →

1

[1,2] [0,2] q

→ t t + 1 t + 2 t + 3 t + 4 → p∗ ∗∗ ∗∗ ∗∗ ∗∗ . . . p∗ ∗q ∗q ∗q ∗∗ . . . p∗ ∗∗ ∗q ∗q ∗q ← t − 4 t − 3 t − 2 t − 1 t ←

What would be the “equivalent” past formula ψ that describes the same pattern from

t + 4?

✦

ψ =

Q

4 p →

Q

[0,1]

`

[0,2] q

SLIDE 47

Temporal Depth of an MTL-B formula

On Synthesizing Controllers from Bounded-Response Properties 11 / 23

Each future MTL-B formula admits a number D(ϕ) indicating its temporal depth

✦

The satisfaction of ϕ by a signal ξ from any position t is fully determined within the interval [t, t + D(ϕ)] D(p) = D(¬ϕ) = D(ϕ) D(ϕ1 ∨ ϕ2) = max{D(ϕ1), D(ϕ2)} D(ϕ1U[a,b]ϕ2) = b + max{D(ϕ1), D(ϕ2)}

Syntax-dependent upper-bound on the actual depth

✦

Example: D(✷[a,b]T) = b

SLIDE 48

Temporal Depth of an MTL-B formula

On Synthesizing Controllers from Bounded-Response Properties 11 / 23

Each future MTL-B formula admits a number D(ϕ) indicating its temporal depth

✦

The satisfaction of ϕ by a signal ξ from any position t is fully determined within the interval [t, t + D(ϕ)] D(p) = D(¬ϕ) = D(ϕ) D(ϕ1 ∨ ϕ2) = max{D(ϕ1), D(ϕ2)} D(ϕ1U[a,b]ϕ2) = b + max{D(ϕ1), D(ϕ2)}

Syntax-dependent upper-bound on the actual depth

✦

Example: D(✷[a,b]T) = b

SLIDE 49

Pastify Operator

On Synthesizing Controllers from Bounded-Response Properties 12 / 23

Relation between ϕ and ψ = Π(ϕ, d):

(ξ, t) | = ϕ ↔ (ξ, t + d) | = ψ

Definition: The operator Π on future MTL-B formulae ϕ and a displacement d ≥ D(ϕ)

is defined recursively as: Π(p, d) =

Q

d p

Π(¬ϕ, d) = ¬Π(ϕ, d) Π(ϕ1 ∨ ϕ2, d) = Π(ϕ1, d) ∨ Π(ϕ2, d) Π(ϕ1U[a,b]ϕ2, d) = Π(ϕ1, d − b)P[a,b]Π(ϕ2, d − b) Π(1

[a,b] ϕ, d)

=

Q

[0,b−a] Π(ϕ, d − b)

Equisatisfaction of

ϕ and ψ: ξ | = ϕ ↔ ξ | = ψ

SLIDE 50

Pastify Operator

On Synthesizing Controllers from Bounded-Response Properties 12 / 23

Relation between ϕ and ψ = Π(ϕ, d):

(ξ, t) | = ϕ ↔ (ξ, t + d) | = ψ

Definition: The operator Π on future MTL-B formulae ϕ and a displacement d ≥ D(ϕ)

is defined recursively as: Π(p, d) =

Q

d p

Π(¬ϕ, d) = ¬Π(ϕ, d) Π(ϕ1 ∨ ϕ2, d) = Π(ϕ1, d) ∨ Π(ϕ2, d) Π(ϕ1U[a,b]ϕ2, d) = Π(ϕ1, d − b)P[a,b]Π(ϕ2, d − b) Π(1

[a,b] ϕ, d)

=

Q

[0,b−a] Π(ϕ, d − b)

Equisatisfaction of

ϕ and ψ: ξ | = ϕ ↔ ξ | = ψ

SLIDE 51

Pastify Operator

On Synthesizing Controllers from Bounded-Response Properties 12 / 23

Relation between ϕ and ψ = Π(ϕ, d):

(ξ, t) | = ϕ ↔ (ξ, t + d) | = ψ

Definition: The operator Π on future MTL-B formulae ϕ and a displacement d ≥ D(ϕ)

is defined recursively as: Π(p, d) =

Q

d p

Π(¬ϕ, d) = ¬Π(ϕ, d) Π(ϕ1 ∨ ϕ2, d) = Π(ϕ1, d) ∨ Π(ϕ2, d) Π(ϕ1U[a,b]ϕ2, d) = Π(ϕ1, d − b)P[a,b]Π(ϕ2, d − b) Π(1

[a,b] ϕ, d)

=

Q

[0,b−a] Π(ϕ, d − b)

Equisatisfaction of

ϕ and ψ: ξ | = ϕ ↔ ξ | = ψ

SLIDE 52

Bounded Variability of Input Signals

On Synthesizing Controllers from Bounded-Response Properties 13 / 23

Definition:
A signal ξ is of (∆, k)-bounded variability if for every interval of the form [t, t + ∆] the

number of changes in the value of ξ is at most k

1 2 3 4 5 6 k k−2 k−1 t t + ∆ ξ

The bounded variability is preserved by MTL-B operators

SLIDE 53

Bounded Variability of Input Signals

On Synthesizing Controllers from Bounded-Response Properties 13 / 23

Definition:
A signal ξ is of (∆, k)-bounded variability if for every interval of the form [t, t + ∆] the

number of changes in the value of ξ is at most k

1 2 3 4 5 6 k k−2 k−1 t t + ∆ ξ

The bounded variability is preserved by MTL-B operators

SLIDE 54

Temporal testers for MTL-B formulae

On Synthesizing Controllers from Bounded-Response Properties 14 / 23

Temporal testers for LTL proposed in [KP05]

✦

Compositional basis for automata construction corresponding to LTL formulae

✦

Extension to real-time temporal logics

■

Past-MITL [MNP05]

■

MITL [MNP06]

Temporal testers for Past-MITL are deterministic

✦

Under the bounded variability assumption, deterministic temporal tester construction naturally extends to past MTL-B operators such as

Q

d or Sd

How to build a deterministic temporal tester for P[a,b] operator?

SLIDE 55

Temporal testers for MTL-B formulae

On Synthesizing Controllers from Bounded-Response Properties 14 / 23

Temporal testers for LTL proposed in [KP05]

✦

Compositional basis for automata construction corresponding to LTL formulae

✦

Extension to real-time temporal logics

■

Past-MITL [MNP05]

■

MITL [MNP06]

Temporal testers for Past-MITL are deterministic

✦

Under the bounded variability assumption, deterministic temporal tester construction naturally extends to past MTL-B operators such as

Q

d or Sd

How to build a deterministic temporal tester for P[a,b] operator?

SLIDE 56

Temporal testers for MTL-B formulae

On Synthesizing Controllers from Bounded-Response Properties 14 / 23

Temporal testers for LTL proposed in [KP05]

✦

Compositional basis for automata construction corresponding to LTL formulae

✦

Extension to real-time temporal logics

■

Past-MITL [MNP05]

■

MITL [MNP06]

Temporal testers for Past-MITL are deterministic

✦

Under the bounded variability assumption, deterministic temporal tester construction naturally extends to past MTL-B operators such as

Q

d or Sd

How to build a deterministic temporal tester for P[a,b] operator?

SLIDE 57

Deterministic Temporal Tester for

Q

[a,b] ϕ

On Synthesizing Controllers from Bounded-Response Properties 15 / 23

Event recorder [MNP05]

✦

The core

f

the tester-based translation from Past MITL to timed automata

✦

Takes ϕ as input and

Q

[a,b] ϕ as

utput

✦

The automaton

utputs

1 whenever x1 ≥ a

Trivial extension for

Q

b ϕ with the

bounded variability assumption

y1 ≤ b ϕ 010101 y1 ≤ b 01010 ¬ϕ y1 ≥ b/s ϕ y1 ≤ b y1 ≤ b ¬ϕ ¬ϕ ϕ 01 010 0101 y1 ≥ b/s y1 ≥ b/s y1 ≥ b/s ¬ϕ/y1 := 0 ¬ϕ/y2 := 0 ¬ϕ y1 ≤ b (01)m0 . . . ϕ/x1 := 0 ϕ/x2 := 0 ϕ/x3 := 0

SLIDE 58

Deterministic Temporal Tester for

Q

[a,b] ϕ

On Synthesizing Controllers from Bounded-Response Properties 15 / 23

Event recorder [MNP05]

✦

The core

f

the tester-based translation from Past MITL to timed automata

✦

Takes ϕ as input and

Q

[a,b] ϕ as

utput

✦

The automaton

utputs

1 whenever x1 ≥ a

Trivial extension for

Q

b ϕ with the

bounded variability assumption

y1 ≤ b ϕ 010101 y1 ≤ b 01010 ¬ϕ y1 ≥ b/s ϕ y1 ≤ b y1 ≤ b ¬ϕ ¬ϕ ϕ 01 010 0101 y1 ≥ b/s y1 ≥ b/s y1 ≥ b/s ¬ϕ/y1 := 0 ¬ϕ/y2 := 0 ¬ϕ y1 ≤ b (01)m0 . . . ϕ/x1 := 0 ϕ/x2 := 0 ϕ/x3 := 0

SLIDE 59

Deterministic Temporal Tester for ϕ1Pϕ2

On Synthesizing Controllers from Bounded-Response Properties 16 / 23

Observation [MN04]: If p is a signal of (b, 1)-bounded variability, then

✦

(ξ, t) | = p U[a,b]q iff (ξ, t) | = p ∧

1

[a,b](p ∧ q)

✦

(ξ, t) | = p P[a,b]q iff (ξ, t) | =

Q

b p ∧

Q

[0,b−a](p ∧ q)

p q

SLIDE 60

Deterministic Temporal Tester for ϕ1Pϕ2

On Synthesizing Controllers from Bounded-Response Properties 16 / 23

Observation [MN04]: If p is a signal of (b, 1)-bounded variability, then

✦

(ξ, t) | = p U[a,b]q iff (ξ, t) | = p ∧

1

[a,b](p ∧ q)

✦

(ξ, t) | = p P[a,b]q iff (ξ, t) | =

Q

b p ∧

Q

[0,b−a](p ∧ q)

p q t t + a t + b t′

SLIDE 61

Deterministic Temporal Tester for ϕ1Pϕ2

On Synthesizing Controllers from Bounded-Response Properties 16 / 23

Observation [MN04]: If p is a signal of (b, 1)-bounded variability, then

✦

(ξ, t) | = p U[a,b]q iff (ξ, t) | = p ∧

1

[a,b](p ∧ q)

✦

(ξ, t) | = p P[a,b]q iff (ξ, t) | =

Q

b p ∧

Q

[0,b−a](p ∧ q)

p q t t + a t + b t′ p p ∧ q

SLIDE 62

Deterministic Temporal Tester for ϕ1Pϕ2

On Synthesizing Controllers from Bounded-Response Properties 16 / 23

Observation [MN04]: If p is a signal of (b, 1)-bounded variability, then

✦

(ξ, t) | = p U[a,b]q iff (ξ, t) | = p ∧

1

[a,b](p ∧ q)

✦

(ξ, t) | = p P[a,b]q iff (ξ, t) | =

Q

b p ∧

Q

[0,b−a](p ∧ q)

p

q t t + a t + b t′ p p ∧ q p

SLIDE 63

Deterministic Temporal Tester for ϕ1Pϕ2

On Synthesizing Controllers from Bounded-Response Properties 17 / 23

Any signal p of (b, k) variability (k > 1), can be decomposed into k signals

p1, p2, . . . , pk, such that:

✦

p = p1 ∨ p2 ∨ . . . ∨ pk

✦

pi ∧ pj always false for every i = j

✦

pi is of (b, 1)-variability

p1 p3 p2 p

For such pi’s we have:

(ξ, t) | = p U[a,b]q ↔ (ξ, t) | = Wk

i=1 pi U[a,b]q

(ξ, t) | = p P[a,b]q ↔ (ξ, t) | = Wk

i=1 pi P[a,b]q

The splitting of p can be achieved trivially using an automaton realizing a counter

modulo k.

SLIDE 64

Deterministic Temporal Tester for ϕ1Pϕ2

On Synthesizing Controllers from Bounded-Response Properties 17 / 23

Any signal p of (b, k) variability (k > 1), can be decomposed into k signals

p1, p2, . . . , pk, such that:

✦

p = p1 ∨ p2 ∨ . . . ∨ pk

✦

pi ∧ pj always false for every i = j

✦

pi is of (b, 1)-variability

p1 p3 p2 p

For such pi’s we have:

(ξ, t) | = p U[a,b]q ↔ (ξ, t) | = Wk

i=1 pi U[a,b]q

(ξ, t) | = p P[a,b]q ↔ (ξ, t) | = Wk

i=1 pi P[a,b]q

The splitting of p can be achieved trivially using an automaton realizing a counter

modulo k.

SLIDE 65

Deterministic Temporal Tester for ϕ1Pϕ2

On Synthesizing Controllers from Bounded-Response Properties 17 / 23

Any signal p of (b, k) variability (k > 1), can be decomposed into k signals

p1, p2, . . . , pk, such that:

✦

p = p1 ∨ p2 ∨ . . . ∨ pk

✦

pi ∧ pj always false for every i = j

✦

pi is of (b, 1)-variability

p1 p3 p2 p

For such pi’s we have:

(ξ, t) | = p U[a,b]q ↔ (ξ, t) | = Wk

i=1 pi U[a,b]q

(ξ, t) | = p P[a,b]q ↔ (ξ, t) | = Wk

i=1 pi P[a,b]q

The splitting of p can be achieved trivially using an automaton realizing a counter

modulo k.

SLIDE 66

Synthesis of an Arbiter

On Synthesizing Controllers from Bounded-Response Properties 18 / 23

Architecture of an arbiter

Arbiter

r1 rn gn g1 · · · · · ·

Typical timed interaction between the

arbiter an a client i

Communication protocol between the

arbiter an a client i

SLIDE 67

Synthesis of an Arbiter

On Synthesizing Controllers from Bounded-Response Properties 18 / 23

Architecture of an arbiter

Arbiter

r1 rn gn g1 · · · · · ·

Typical timed interaction between the

arbiter an a client i

Communication protocol between the

arbiter an a client i

rigi rigi rigi rigi

SLIDE 68

Synthesis of an Arbiter

On Synthesizing Controllers from Bounded-Response Properties 18 / 23

Architecture of an arbiter

Arbiter

r1 rn gn g1 · · · · · ·

Typical timed interaction between the

arbiter an a client i

gi ri d2 d1 d3

Communication protocol between the

arbiter an a client i

rigi rigi rigi rigi

SLIDE 69

Synthesis of an Arbiter: MTL-B Specification

On Synthesizing Controllers from Bounded-Response Properties 19 / 23

Initial conditions

✦

IE : V

i ri

✦

IC : V

i gi

Safety requirements

✦

SE : V

i ri → riS (ri ∧ gi) ∧ V i(ri → riB(ri ∧ gi)

✦

SC : V

i(gi → giS(ri ∧ gi)) ∧ V i(gi → giB(ri ∧ gi))

Bounded liveness requirements

✦

LE : V

i(gi →

1

[0,d1] ri)

✦

LC : V

i(ri →

1

[0,d2] gi) ∧ V i(ri →

1

[0,d3] gi)

Main formula

✦

(IE → IC) ∧ ( ` (Π(SE) ∧ Π(LE)) → (Π(SE) ∧ Π(LC)))

SLIDE 70

Synthesis of an Arbiter: MTL-B Specification

On Synthesizing Controllers from Bounded-Response Properties 19 / 23

Initial conditions

✦

IE : V

i ri

✦

IC : V

i gi

Safety requirements

✦

SE : V

i ri → riS (ri ∧ gi) ∧ V i(ri → riB(ri ∧ gi)

✦

SC : V

i(gi → giS(ri ∧ gi)) ∧ V i(gi → giB(ri ∧ gi))

Bounded liveness requirements

✦

LE : V

i(gi →

1

[0,d1] ri)

✦

LC : V

i(ri →

1

[0,d2] gi) ∧ V i(ri →

1

[0,d3] gi)

Main formula

✦

(IE → IC) ∧ ( ` (Π(SE) ∧ Π(LE)) → (Π(SE) ∧ Π(LC)))

SLIDE 71

Synthesis of an Arbiter: MTL-B Specification

On Synthesizing Controllers from Bounded-Response Properties 19 / 23

Initial conditions

✦

IE : V

i ri

✦

IC : V

i gi

Safety requirements

✦

SE : V

i ri → riS (ri ∧ gi) ∧ V i(ri → riB(ri ∧ gi)

✦

SC : V

i(gi → giS(ri ∧ gi)) ∧ V i(gi → giB(ri ∧ gi))

Bounded liveness requirements

✦

LE : V

i(gi →

1

[0,d1] ri)

✦

LC : V

i(ri →

1

[0,d2] gi) ∧ V i(ri →

1

[0,d3] gi)

Main formula

✦

(IE → IC) ∧ ( ` (Π(SE) ∧ Π(LE)) → (Π(SE) ∧ Π(LC)))

SLIDE 72

Synthesis of an Arbiter: MTL-B Specification

On Synthesizing Controllers from Bounded-Response Properties 19 / 23

Initial conditions

✦

IE : V

i ri

✦

IC : V

i gi

Safety requirements

✦

SE : V

i ri → riS (ri ∧ gi) ∧ V i(ri → riB(ri ∧ gi)

✦

SC : V

i(gi → giS(ri ∧ gi)) ∧ V i(gi → giB(ri ∧ gi))

Bounded liveness requirements

✦

LE : V

i(gi →

1

[0,d1] ri)

✦

LC : V

i(ri →

1

[0,d2] gi) ∧ V i(ri →

1

[0,d3] gi)

Main formula

✦

(IE → IC) ∧ ( ` (Π(SE) ∧ Π(LE)) → (Π(SE) ∧ Π(LC)))

SLIDE 73

Synthesis of an Arbiter: MTL-B Specification

On Synthesizing Controllers from Bounded-Response Properties 19 / 23

Initial conditions

✦

IE : V

i ri

✦

IC : V

i gi

Safety requirements

✦

SE : V

i ri → riS (ri ∧ gi) ∧ V i(ri → riB(ri ∧ gi)

✦

SC : V

i(gi → giS(ri ∧ gi)) ∧ V i(gi → giB(ri ∧ gi))

Bounded liveness requirements

✦

LE : V

i(gi →

1

[0,d1] ri)

✦

LC : V

i(ri →

1

[0,d2] gi) ∧ V i(ri →

1

[0,d3] gi)

Main formula

✦

(IE → IC) ∧ ( ` (Π(SE) ∧ Π(LE)) → (Π(SE) ∧ Π(LC)))

SLIDE 74

Synthesis of an Arbiter: Experimental Results

On Synthesizing Controllers from Bounded-Response Properties 20 / 23

Discrete time synthesis
d3 = 1

N d1 d2 Size Time d1 d2 Size Time d1 d2 Size Time 2 2 4 466 0.00 3 5 654 0.01 4 6 946 0.02 3 2 8 1382 0.14 3 10 2432 0.34 4 12 4166 0.51 4 2 12 4323 0.63 3 15 7402 1.12 4 18 16469 2.33 5 2 16 13505 1.93 3 20 26801 4.77 4 24 50674 10.50 6 2 20 43366 8.16 3 25 84027 22.55 4 30 168944 64.38 7 2 24 138937 44.38 3 30 297524 204.56 4 36 700126 1897.56

Exponential growth of BDD nodes in N and d2

✦

Expected using discrete time

SLIDE 75

Synthesis of an Arbiter: Experimental Results

On Synthesizing Controllers from Bounded-Response Properties 20 / 23

Discrete time synthesis
d3 = 1

N d1 d2 Size Time d1 d2 Size Time d1 d2 Size Time 2 2 4 466 0.00 3 5 654 0.01 4 6 946 0.02 3 2 8 1382 0.14 3 10 2432 0.34 4 12 4166 0.51 4 2 12 4323 0.63 3 15 7402 1.12 4 18 16469 2.33 5 2 16 13505 1.93 3 20 26801 4.77 4 24 50674 10.50 6 2 20 43366 8.16 3 25 84027 22.55 4 30 168944 64.38 7 2 24 138937 44.38 3 30 297524 204.56 4 36 700126 1897.56

Exponential growth of BDD nodes in N and d2

✦

Expected using discrete time

SLIDE 76

Conclusion

On Synthesizing Controllers from Bounded-Response Properties 21 / 23

Complete chain that allows to synthesize controllers automatically from real-time

bounded-response temporal specifications

✦

Bounded-response temporal property → deterministic timed automaton

■

Pastification of MTL-B formulae

■

Bounded-variability assumption

Future work

✦

Focus on efficient symbolic algorithms in the spirit of [CDF+05]

✦

Apply the synthesis algorithm to more complex specifications of real-time scheduling problems

SLIDE 77

Conclusion

On Synthesizing Controllers from Bounded-Response Properties 21 / 23

Complete chain that allows to synthesize controllers automatically from real-time

bounded-response temporal specifications

✦

Bounded-response temporal property → deterministic timed automaton

■

Pastification of MTL-B formulae

■

Bounded-variability assumption

Future work

✦

Focus on efficient symbolic algorithms in the spirit of [CDF+05]

✦

Apply the synthesis algorithm to more complex specifications of real-time scheduling problems

SLIDE 78

Conclusion

On Synthesizing Controllers from Bounded-Response Properties 21 / 23

Complete chain that allows to synthesize controllers automatically from real-time

bounded-response temporal specifications

✦

Bounded-response temporal property → deterministic timed automaton

■

Pastification of MTL-B formulae

■

Bounded-variability assumption

Future work

✦

Focus on efficient symbolic algorithms in the spirit of [CDF+05]

✦

Apply the synthesis algorithm to more complex specifications of real-time scheduling problems

SLIDE 79

References

On Synthesizing Controllers from Bounded-Response Properties 22 / 23

[AFH96]

R. Alur, T. Feder, and T.A. Henzinger, The Benefits of Relaxing Punctuality,

Journal of the ACM 43, 116–146, 1996 (first published in PODC’91). [AMP95] E. Asarin, O. Maler and A. Pnueli, Symbolic Controller Synthesis for Discrete and Timed Systems, Hybrid Systems II, 1–20, LNCS 999, 1995. [BL69] J.R. B¨ uchi and L.H. Landweber, Solving Sequential Conditions by Finite-state Operators, Trans. of the AMS 138, 295–311, 1969. [CDF+05] F. Cassez, A. David, E. Fleury, K.G. Larsen and D. Lime, Efficient On-the-Fly Algorithms for the Analysis of Timed Games, CONCUR’05, 66–80, 2005. [Chu63]

A. Church, Logic, Arithmetic and Automata, in Proc. of the Int. Cong. of

Mathematicians 1962, 23–35, 1963. [KP05]

Y. Kesten and A. Pnueli, A Compositional Approach to CTL∗ Verification,

Theoretical Computer Science 331, 397–428, 2005. [Koy90]

R. Koymans, Specifying Real-time Properties with Metric Temporal Logic,

Real-time Systems 2, 255–299, 1990.

SLIDE 80

References

On Synthesizing Controllers from Bounded-Response Properties 23 / 23

[MN04]

O. Maler and D. Nickovic, Monitoring Temporal Properties of Continuous

Signals, FORMATS/FTRTFT’04, 152–166, LNCS 3253, 2004. [MNP05] O. Maler, D. Nickovic and A. Pnueli, Real Time Temporal Logic: Past, Present, Future, FORMATS’05, 2–16, LNCS 3829, 2005. [MNP06] O. Maler, D. Nickovic and A. Pnueli, From MITL to Timed Automata, FORMATS’06, 274–289, LNCS 4202, 2006. [MPS95] O. Maler, A. Pnueli and J. Sifakis, On the Synthesis of Discrete Controllers for Timed Systems, STACS’95, 229–242, LNCS 900, 1995. [PPS06]

N. Piterman, A. Pnueli and Y. Sa’ar, Synthesis of Reactive(1) Designs,

VMCAI’06, 364–380, 2006. [PP06]

N. Piterman and A. Pnueli, Faster Solutions of Rabin and Streett Games,

LICS’06, 275–284, 2006. [RW89] P .J. Ramadge and W.M. Wonham, The Control of Discrete Event Systems,

Proc. of the IEEE 77, 81–98, 1989.