A Uniform Theory of Hybrid Automata Renato Neves and Lu s S. - - PowerPoint PPT Presentation

A Uniform Theory of Hybrid Automata

Renato Neves and Lu´ ıs S. Barbosa November 6, 2018

INESC TEC (HASLab) & University of Minho 1

Introduction

Hybrid Systems

Computational devices that interact with their physical environment

2

Preliminaries

Hybrid Automata The standard formalism for designing hybrid systems

3

Preliminaries

Hybrid Automata The standard formalism for designing hybrid systems They are classical automata enriched with machinery to specify continuous evolutions and discrete resets [Henzinger, 1996]

3

Preliminaries

Hybrid Automata The standard formalism for designing hybrid systems They are classical automata enriched with machinery to specify continuous evolutions and discrete resets [Henzinger, 1996] Example Water level regulator; it raises the water level (l) periodically ˙ l = 2 ˙ t = 1 t ≤ c

t≥c t:=0

• ˙

l = 0 ˙ t = 1 t ≤ c

t≥c t:=0

• 3

Motivation

The notion of hybrid automata has several variants, e.g.

• Deterministic [Henzinger, 1996]
• Non-deterministic [Henzinger, 1996]
• Probabilistic [Sproston, 2000]
• Reactive [Liu et al., 1999]
• Weighted [Bouyer, 2006]

4

Motivation

The notion of hybrid automata has several variants, e.g.

• Deterministic [Henzinger, 1996]
• Non-deterministic [Henzinger, 1996]
• Probabilistic [Sproston, 2000]
• Reactive [Liu et al., 1999]
• Weighted [Bouyer, 2006]

No uniform framework for hybrid automata

4

Coalgebras

Coalgebras can help us solving this issue

5

Coalgebras

Coalgebras can help us solving this issue For this particular case we will see that they provide,

• 1. generic semantics,

5

Coalgebras

Coalgebras can help us solving this issue For this particular case we will see that they provide,

• 1. generic semantics,
• 2. generic notions of bisimulation

5

Coalgebras

Coalgebras can help us solving this issue For this particular case we will see that they provide,

• 1. generic semantics,
• 2. generic notions of bisimulation
• 3. and observational behaviour,

5

Coalgebras

Coalgebras can help us solving this issue For this particular case we will see that they provide,

• 1. generic semantics,
• 2. generic notions of bisimulation
• 3. and observational behaviour,
• 4. and ‘regular expression’-like languages.

5

Hybrid Automata as Coalgebras

Coalgebras

Definition Given a functor F : C → C, an F-coalgebra is a C-morphism of the type X → FX Key idea The functor F determines the branching type

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Coalgebras

Definition Given a functor F : C → C, an F-coalgebra is a C-morphism of the type X → FX Key idea The functor F determines the branching type Examples

• 1. (Kripke Frames) Powerset functor P : Set → Set

6

Coalgebras

Definition Given a functor F : C → C, an F-coalgebra is a C-morphism of the type X → FX Key idea The functor F determines the branching type Examples

• 1. (Kripke Frames) Powerset functor P : Set → Set
• 2. (Deterministic Automata) (−)Σ × 2 : Set → Set

6

Coalgebras

Definition Given a functor F : C → C, an F-coalgebra is a C-morphism of the type X → FX Key idea The functor F determines the branching type Examples

• 1. (Kripke Frames) Powerset functor P : Set → Set
• 2. (Deterministic Automata) (−)Σ × 2 : Set → Set
• 3. (Markov Chain) Distribution functor D : Set → Set

6

Hybrid Automata

Definition ([Henzinger, 1996])

A hybrid automaton is a tuple (M, E, X, dyn, inv, asg, grd) where

• M is a finite set of modes, E is a transition relation E ⊆ M × M,

and X is a finite set of real-valued variables {x1, . . . , xn}.

• dyn is a function that associates to each mode a predicate over the

variables in X ∪ ˙ X, where ˙ X = { ˙ x1, . . . , ˙ xn} represents the first derivatives of the variables in X.

• inv is a function that associates to each mode a predicate over the

variables in X.

• asg is a function that given an edge returns an assignment over X.

The function grd associates each edge with a guard.

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Hybrid Automata

Example The bouncing ball ˙ p = v ˙ v = g p ≥ 0 p = 0 ∧ v ≤ 0, v := v × −0.5

• 8

Hybrid Automata

Example The bouncing ball ˙ p = v ˙ v = g p ≥ 0 p = 0 ∧ v ≤ 0, v := v × −0.5

• Its observable behaviour consists of continuous evolutions

intercalated with resets

8

Hybrid Automata

Example Water level regulator ˙ l = 2 ˙ t = 1 t ≤ c

t≥c t:=0

• ˙

l = 0 ˙ t = 1 t ≤ c

t≥c t:=0

• 9

Hybrid Automata

Example Water level regulator ˙ l = 2 ˙ t = 1 t ≤ c

t≥c t:=0

• ˙

l = 0 ˙ t = 1 t ≤ c

t≥c t:=0

• Its observable behaviour also consists of continuous evolutions

intercalated with resets

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A Surprisingly Useful Remark

Hybrid automata are classical automata but with decorated states and edges.

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A Surprisingly Useful Remark

Hybrid automata are classical automata but with decorated states and edges. M → P(M × Asg × Grd) × DifEq × StInv

10

A Surprisingly Useful Remark

Hybrid automata are classical automata but with decorated states and edges. M → P(M × Asg × Grd) × DifEq × StInv This immediately gives,

• a uniform notion of hybrid automata,
• bisimulation and languages

10

A Surprisingly Useful Remark

Hybrid automata are classical automata but with decorated states and edges. M → P(M × Asg × Grd) × DifEq × StInv This immediately gives,

• a uniform notion of hybrid automata,
• bisimulation and languages

We can now start studying hybrid automata in a uniform way

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A Zoo of Hybrid Automata

M → F(M × Asg × Grd) × DifEq × StInv

11

A Zoo of Hybrid Automata

M → F(M × Asg × Grd) × DifEq × StInv Id ⇒ Deterministic hybrid automata

11

A Zoo of Hybrid Automata

M → F(M × Asg × Grd) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata

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A Zoo of Hybrid Automata

M → F(M × Asg × Grd) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata D ⇒ Markov hybrid automata

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A Zoo of Hybrid Automata

M → F(M × Asg × Grd) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata D ⇒ Markov hybrid automata PD ⇒ Probabilistic hybrid automata

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A Zoo of Hybrid Automata

M → F(M × Asg × Grd) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata D ⇒ Markov hybrid automata PD ⇒ Probabilistic hybrid automata W ⇒ Weighted hybrid automata

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A Zoo of Hybrid Automata

M → F(M × Asg × Grd) × DifEq × StInv Id ⇒ Deterministic hybrid automata P ⇒ Classical hybrid automata D ⇒ Markov hybrid automata PD ⇒ Probabilistic hybrid automata W ⇒ Weighted hybrid automata We can additionally consider an input dimension

11

Semantics of Hybrid Automata

Semantics of Hybrid Automata

We will show how to build a ‘semantics’ functor − : HybAt(F) → Category of coalgebras

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Semantics of Hybrid Automata

We will show how to build a ‘semantics’ functor − : HybAt(F) → Category of coalgebras Reminder The observable behaviour of hybrid automata consists of continuous evolutions intercalated with resets

12

Semantics of Hybrid Automata

Notation Let X be a topological space. The set HX denotes

• r∈[0,∞)

X [0,r] the set of all continuous trajectories over intervals [0, r].

13

Semantics of Hybrid Automata

Notation Let X be a topological space. The set HX denotes

• r∈[0,∞)

X [0,r] the set of all continuous trajectories over intervals [0, r]. Assumption The function dyn only outputs differential equations with exactly

• ne solution. This induces a function

flow : M × Rn × [0, ∞) → Rn

13

Semantics of Hybrid Automata

Assumption (for simplicity) As soon as an edge is enabled the current state must switch Let us omit state invariants; they complicate the theory and can be added straightfowardly later on

14

Semantics of Hybrid Automata

Assumption (for simplicity) As soon as an edge is enabled the current state must switch Let us omit state invariants; they complicate the theory and can be added straightfowardly later on Semantics M × Rn → F(M × Asg × Grd) × DifEq ⇒ M × Rn → F(M × Asg × Grd) × (Rn)[0,∞) ⇒ M × Rn → F

• M × Asg × Grd × (Rn)[0,∞)

⇒ M × Rn → F (M × Asg × (H(Rn) + 1)) ⇒ M × Rn → F (M × Asg × H(Rn) + M × Asg × 1) ⇒ M × Rn → F (M × Rn × H(Rn) + 1)

14

Semantics of Hybrid Automata

Assumption (for simplicity) As soon as an edge is enabled the current state must switch Let us omit state invariants; they complicate the theory and can be added straightfowardly later on Semantics M × Rn → F(M × Asg × Grd) × DifEq ⇒ M × Rn → F(M × Asg × Grd) × (Rn)[0,∞) ⇒ M × Rn → F

• M × Asg × Grd × (Rn)[0,∞)

⇒ M × Rn → F (M × Asg × (H(Rn) + 1)) ⇒ M × Rn → F (M × Asg × H(Rn) + M × Asg × 1) ⇒ M × Rn → F (M × Rn × H(Rn) + 1) We obtain a coalgebra for F(− × H(Rn) + 1)

14

Semantics of Hybrid Automata

The previous calculation determines a ‘semantics’ functor − : HybAt(F) → CoAlg (F(− × H(Rn) + 1))

15

Semantics of Hybrid Automata

The previous calculation determines a ‘semantics’ functor − : HybAt(F) → CoAlg (F(− × H(Rn) + 1)) It generalises the semantics of deterministic, classical and probabilistic hybrid automata

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Observable Behaviour

Observable Behaviour

Final Coalgebra A coalgebra νF → FνF is final if for every coalgebra c : X → FX there exists a unique coalgebra morphism of the type fc : X → νF

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Observable Behaviour

Final Coalgebra A coalgebra νF → FνF is final if for every coalgebra c : X → FX there exists a unique coalgebra morphism of the type fc : X → νF Each state x ∈ X is mapped into its behaviour fc(x) ∈ νF

16

Observable Behaviour

Final Coalgebra A coalgebra νF → FνF is final if for every coalgebra c : X → FX there exists a unique coalgebra morphism of the type fc : X → νF Each state x ∈ X is mapped into its behaviour fc(x) ∈ νF Example The carrier of the final coalgebra for (− × H(Rn) + 1) is the set H(Rn)∗ + H(Rn)ω

16

Observable Behaviour

Final Coalgebra A coalgebra νF → FνF is final if for every coalgebra c : X → FX there exists a unique coalgebra morphism of the type fc : X → νF Each state x ∈ X is mapped into its behaviour fc(x) ∈ νF Example The carrier of the final coalgebra for (− × H(Rn) + 1) is the set H(Rn)∗ + H(Rn)ω The observable behaviour of deterministic hybrid automata thus consists of continuous evolutions intercalated with resets

16

The Bouncing Ball Revisited (Deterministic case)

Example Using the semantics functor − we obtain the following picture, ˙ p = v ˙ v = g p ≥ 0 p = 0 ∧ v > 0, v := v × −0.5

• fb(∗, 5, 0)

f− : M × Rn → H(Rn)∗ + H(Rn)ω

0.2 0.4 0.6 0.8 1 1 2 3 4 5

time pos 1st element

0.2 0.4 0.6 0.8 1 1 2 3 4 5

time pos 2nd element

0.2 0.4 0.6 0.8 1 1 2 3 4 5

time pos 3rd element

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Bisimulation

Coalgebraic Bisimulation

Definition Consider a functor F : Set → Set and two F-coalgebras, (X, c), (Y , d). A relation R ⊆ X × Y is called a bisimulation if there is an F-coalgebra (R, r) that makes the following diagram commute X

c

• R

π1

• π2
• r
• Y

d

• FX

FR

Fπ1

• Fπ2

FY

18

Coalgebraic Bisimulation

Definition Consider a functor F : Set → Set and two F-coalgebras, (X, c), (Y , d). A relation R ⊆ X × Y is called a bisimulation if there is an F-coalgebra (R, r) that makes the following diagram commute X

c

• R

π1

• π2
• r
• Y

d

• FX

FR

Fπ1

• Fπ2

FY

Definition Two states x ∈ X, y ∈ Y are called bisimilar (x ∼ y) if there exists a bisimulation R such that (x, y) ∈ R

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Bisimulation

Example Take c : X → X × A and d : Y → Y × A. The condition x ∼ y holds iff π2 · c(x) = π2 · d(y) and π1 · c(x) ∼ π1 · d(y)

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Bisimulation

Example Take c : X → X × A and d : Y → Y × A. The condition x ∼ y holds iff π2 · c(x) = π2 · d(y) and π1 · c(x) ∼ π1 · d(y) Example Take c : X → P(X × A) and d : Y → P(Y × A). The condition x ∼ y holds iff x

a

− → x′ entails y

a

− → y′ and x′ ∼ y′; and vice-versa

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Bisimulation

Example Take c : X → X × A and d : Y → Y × A. The condition x ∼ y holds iff π2 · c(x) = π2 · d(y) and π1 · c(x) ∼ π1 · d(y) Example Take c : X → P(X × A) and d : Y → P(Y × A). The condition x ∼ y holds iff x

a

− → x′ entails y

a

− → y′ and x′ ∼ y′; and vice-versa Coalgebraic bisimilarity coincides with the usual notions of bisimilarity for several types of transitions systems

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Coalgebraic Bisimulation

Example Take c : X → X × H(Rn) + 1 and d : Y → Y × H(Rn) + 1. The condition x ∼ y holds iff x

(d,e)

− → x′ entails y

(d,e)

− → y′ and x′ ∼ y′; and vice-versa

20

Coalgebraic Bisimulation

Example Take c : X → X × H(Rn) + 1 and d : Y → Y × H(Rn) + 1. The condition x ∼ y holds iff x

(d,e)

− → x′ entails y

(d,e)

− → y′ and x′ ∼ y′; and vice-versa Coalgebraic bisimilarity and the traditional notion of bisimilarity for hybrid automata do not coincide

20

Coalgebraic Bisimulation

Example Take c : X → X × H(Rn) + 1 and d : Y → Y × H(Rn) + 1. The condition x ∼ y holds iff x

(d,e)

− → x′ entails y

(d,e)

− → y′ and x′ ∼ y′; and vice-versa Coalgebraic bisimilarity and the traditional notion of bisimilarity for hybrid automata do not coincide In the latter case, two bisimilar states need not output exactly the same trajectory

20

Coalgebraic Bisimulation

Example Take c : X → X × H(Rn) + 1 and d : Y → Y × H(Rn) + 1. The condition x ∼ y holds iff x

(d,e)

− → x′ entails y

(d,e)

− → y′ and x′ ∼ y′; and vice-versa Coalgebraic bisimilarity and the traditional notion of bisimilarity for hybrid automata do not coincide In the latter case, two bisimilar states need not output exactly the same trajectory But must be equal up to an equivalence relation on M × Rn

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Coalgebraic Φ-bisimulation

The classical definition of bisimilarity for hybrid automata ([Henzinger, 1996]) it is defined at the semantic level.

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Coalgebraic Φ-bisimulation

The classical definition of bisimilarity for hybrid automata ([Henzinger, 1996]) it is defined at the semantic level. Definition Take a coalgebra c : X → X × H(Rn) + 1 and two states x, y ∈ X. The condition x ∼Φ y holds iff x

(d,e,x)

− → x′ entails y

(d,e′,y))

− → y′ such that x′ ∼ y′ and (d, e, x)Φ(d, e′, y) hold; and vice-versa

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Coalgebraic Φ-Bisimulation

The Starting Point Each equivalence relation Φ : (M × Rn) × (M × Rn) induces a quotient map q : M × Rn ։ Q

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Coalgebraic Φ-Bisimulation

The Starting Point Each equivalence relation Φ : (M × Rn) × (M × Rn) induces a quotient map q : M × Rn ։ Q And then . . . HybAt(F)

semantics

• CoAlg(F(− × H(Rn) + 1))

colour

• CoAlg(F(− × H(Rn × M) + 1))

forget

• quotient
• CoAlg(F(− × HQ + 1))

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Coalgebraic Φ-Bisimulation

The colouring process M × Rn → F (M × Rn × H(Rn) + 1) ⇒ M × Rn → F (M × Rn × H(Rn) + 1) × M ⇒ M × Rn → F (M × Rn × H(Rn) × M + M) ⇒ M × Rn → F (M × Rn × H(Rn × M) + 1) A transition system will now disclose the current mode in the form

• f a constant trajectory.

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Coalgebraic Φ-Bisimulation

Theorem Coalgebraic Φ-bisimilarity covers the classic notions of bisimilarity for,

• 1. deterministic,
• 2. non-deterministic,
• 3. and probabilistic hybrid automata.

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Regular Expressions

Languages for Hybrid Automata

No Kleene’s theorem for hybrid automata in the literature (only for timed automata [Asarin et al., 1997])

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Languages for Hybrid Automata

No Kleene’s theorem for hybrid automata in the literature (only for timed automata [Asarin et al., 1997]) Kleene’s theorem (coalgebraically) Under mild conditions, every state x ∈ X and finite coalgebra X → FX yield a ‘regular expression’ and vice-versa [Silva, 2010]

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Languages for Hybrid Automata

No Kleene’s theorem for hybrid automata in the literature (only for timed automata [Asarin et al., 1997]) Kleene’s theorem (coalgebraically) Under mild conditions, every state x ∈ X and finite coalgebra X → FX yield a ‘regular expression’ and vice-versa [Silva, 2010] Corollary Under mild conditions, every mode m ∈ M and finite element of HybAt(F) yield a ‘regular expression’ and vice-versa [Silva, 2010].

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Languages for Hybrid Automata

When F = Id regular expressions for hybrid automata are given by, ǫ ∋ ∅ | x | (ǫ | a) | b | ǫ ∧ ǫ | µx.γ γ ∋ ∅ | (ǫ | a) | b | ǫ ∧ ǫ | µx.γ a is the set of pairs of guards and assignments; b is the set of systems of differential equations. Example The bouncing ball revisited ˙ p = v ˙ v = g p = 0 ∧ v > 0, v := v × −0.5

• µx.(x | ψ) ∧ ( ˙

p = v, ˙ v = g) where ψ = ( ˙ p = v, ˙ v = g).

26

Languages for Hybrid Automata

When F = Id regular expressions for hybrid automata are given by, ǫ ∋ ∅ | x | (ǫ | a) | b | ǫ ∧ ǫ | µx.γ γ ∋ ∅ | (ǫ | a) | b | ǫ ∧ ǫ | µx.γ a is the set of pairs of guards and assignments; b is the set of systems of differential equations. Example The bouncing ball revisited ˙ p = v ˙ v = g p = 0 ∧ v > 0, v := v × −0.5

• µx.(x | ψ) ∧ ( ˙

p = v, ˙ v = g) where ψ = ( ˙ p = v, ˙ v = g). Note that fa(∗, −) = fǫ(b.b. expression, −).

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Calculi for Hybrid Automata

We also wish to obtain a calculus of regular expressions such that ǫ1 ≡ ǫ2 ⇔ ǫ1 ∼ ǫ2

27

Calculi for Hybrid Automata

We also wish to obtain a calculus of regular expressions such that ǫ1 ≡ ǫ2 ⇔ ǫ1 ∼ ǫ2 Kleene’s theorem 2 (coalgebraically) Under mild conditions, the set of regular expressions for a functor F : Set → Set can be given a complete calculus [Silva, 2010]

27

Calculi for Hybrid Automata

We also wish to obtain a calculus of regular expressions such that ǫ1 ≡ ǫ2 ⇔ ǫ1 ∼ ǫ2 Kleene’s theorem 2 (coalgebraically) Under mild conditions, the set of regular expressions for a functor F : Set → Set can be given a complete calculus [Silva, 2010] But the calculus obtained for hybrid automata HybAt(F) is not good enough:

27

Calculi for Hybrid Automata

We also wish to obtain a calculus of regular expressions such that ǫ1 ≡ ǫ2 ⇔ ǫ1 ∼ ǫ2 Kleene’s theorem 2 (coalgebraically) Under mild conditions, the set of regular expressions for a functor F : Set → Set can be given a complete calculus [Silva, 2010] But the calculus obtained for hybrid automata HybAt(F) is not good enough: it is missing equivalence rules for the differential equations, assignments, and guards

27

Calculi for Hybrid Automata

Example The bouncing ball The two expressions below are not equivalent µx.(x | ψ) ∧ ( ˙ p = v, ˙ v = g) and µx.(x | ψ) ∧ ( ˙ p = v + 0, ˙ v = g)

28

Calculi for Hybrid Automata

Example The bouncing ball The two expressions below are not equivalent µx.(x | ψ) ∧ ( ˙ p = v, ˙ v = g) and µx.(x | ψ) ∧ ( ˙ p = v + 0, ˙ v = g) The calculus is not taking the semantics functor − : HybAt(F) → CoAlg (F(− × H(Rn) + 1)) into account.

28

Calculi for Hybrid Automata

Example The bouncing ball The two expressions below are not equivalent µx.(x | ψ) ∧ ( ˙ p = v, ˙ v = g) and µx.(x | ψ) ∧ ( ˙ p = v + 0, ˙ v = g) The calculus is not taking the semantics functor − : HybAt(F) → CoAlg (F(− × H(Rn) + 1)) into account. Ideally, we would like to have ǫ1 ≡ ǫ2 ⇔ fδ(ǫ1, −) = fδ(ǫ2, −)

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Conclusions and Future work

Conclusions

Our goal (Revisited) A uniform framework for designing hybrid systems

29

Conclusions

Our goal (Revisited) A uniform framework for designing hybrid systems Coalgebras seem to be powerful tool for this:

29

Conclusions

Our goal (Revisited) A uniform framework for designing hybrid systems Coalgebras seem to be powerful tool for this: we could reconstruct part of the theory of hybrid automata in a uniform way

29

Conclusions

Our goal (Revisited) A uniform framework for designing hybrid systems Coalgebras seem to be powerful tool for this: we could reconstruct part of the theory of hybrid automata in a uniform way and obtained new results for them as well.

29

Future Work

Further develop a uniform theory of automata. In particular,

• we wish to explore new variants of hybrid automata (but this

time just by switching functors F),

30

Future Work

Further develop a uniform theory of automata. In particular,

• we wish to explore new variants of hybrid automata (but this

time just by switching functors F),

• develop new notions of bisimulation (e.g. approximate),

30

Future Work

Further develop a uniform theory of automata. In particular,

• we wish to explore new variants of hybrid automata (but this

time just by switching functors F),

• develop new notions of bisimulation (e.g. approximate),
• establish complete calculi for regular expressions.

30

References i

Asarin, E., Caspi, P., and Maler, O. (1997). A kleene theorem for timed automata. In Proceedings, 12th Annual IEEE Symposium on Logic in Computer Science, Warsaw, Poland, June 29 - July 2, 1997, pages 160–171. IEEE Computer Society. Bouyer, P. (2006). Weighted timed automata: Model-checking and games. Electronic Notes in Theoretical Computer Science, 158:3–17.

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References ii

Henzinger, T. A. (1996). The theory of hybrid automata. In LICS96’: Logic in Computer Science, 11th Annual Symposium, New Jersey, USA, July 27-30, 1996, pages 278–292. IEEE. Liu, J., Liu, X., Koo, T.-K. J., Sinopoli, B., Sastry, S., and Lee, E. A. (1999). A hierarchical hybrid system model and its simulation. In Decision and Control, 38th IEEE Conference, Phoenix, USA, December 7–10, 1999, volume 4, pages 3508–3513. IEEE.

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References iii

Silva, A. (2010). Kleene Coalgebra. PhD thesis, Radboud Universiteit Nijmegen. Sproston, J. (2000). Decidable model checking of probabilistic hybrid automata. In Joseph, M., editor, FTRTFT’00: International Symposium

• n Formal Techniques in Real-Time and Fault-Tolerant

Systems, Pune, India, September 20–22, 2000, volume 1926 of Lecture Notes in Computer Science, pages 31–45. Springer.

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