[PPT] - Brzozowskis algorithm (co)algebraically Jan Rutten CWI & PowerPoint Presentation

SLIDE 1

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Brzozowski’s algorithm (co)algebraically

Jan Rutten

CWI & Radboud University

SLIDE 2

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Motivation

duality between reachability and observability:

beautiful, not very well-known.

combined use of algebra and coalgebra.
our understanding of automata is still very limited;
cf. recent research: universal automata, àtomata, weighted

automata (Sakarovitch, Brzozowski, . . . )

joint work with Bonchi, Bonsangue, Silva (CWI, 2011) and

Hansen.

based on Arbib and Manes, 1975.
Cf. Panangaden, Kupke, Koenig, Adamek, Milius, Gehrke,

Pin, Roumen, , . . .

SLIDE 3

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Motivation

duality between reachability and observability:

beautiful, not very well-known.

combined use of algebra and coalgebra.
our understanding of automata is still very limited;
cf. recent research: universal automata, àtomata, weighted

automata (Sakarovitch, Brzozowski, . . . )

joint work with Bonchi, Bonsangue, Silva (CWI, 2011) and

Hansen.

based on Arbib and Manes, 1975.
Cf. Panangaden, Kupke, Koenig, Adamek, Milius, Gehrke,

Pin, Roumen, , . . .

SLIDE 4

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Overview

1. Brzozowski’s algorithm: example
2. (Co)algebra
3. Automata (co)algebraically
4. The duality between reachability and observability
5. Reversing the automaton
6. Conclusions

SLIDE 5

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions
1. Brzozowski’s algorithm: example

SLIDE 6

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

A deterministic automaton X = {x, y, z}

x

b

a

z

b

a
y

b

a
initial state: x
final states: y and z
L(x) = {a, b}∗ a

SLIDE 7

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

A deterministic automaton X = {x, y, z}

x

b

a

z

b

a
y

b

a
initial state: x
final states: y and z
L(x) = {a, b}∗ a

SLIDE 8

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

A deterministic automaton X = {x, y, z}

x

b

a

z

b

a
y

b

a
initial state: x
final states: y and z
L(x) = {a, b}∗ a

SLIDE 9

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the automaton: rev(X)

X =

x

b

a

z

b

a
y

b

a
rev(X) =

x b

b
b
z

a

y

a

a
transitions are reversed
initial states

⇔ final states

rev(X) is non-deterministic

SLIDE 10

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the automaton: rev(X)

X =

x

b

a

z

b

a
y

b

a
rev(X) =

x b

b
b
z

a

y

a

a
transitions are reversed
initial states

⇔ final states

rev(X) is non-deterministic

SLIDE 11

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the automaton: rev(X)

X =

x

b

a

z

b

a
y

b

a
rev(X) =

x b

b
b
z

a

y

a

a
transitions are reversed
initial states

⇔ final states

rev(X) is non-deterministic

SLIDE 12

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the automaton: rev(X)

X =

x

b

a

z

b

a
y

b

a
rev(X) =

x b

b
b
z

a

y

a

a
transitions are reversed
initial states

⇔ final states

rev(X) is non-deterministic

SLIDE 13

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the automaton: rev(X)

X =

x

b

a

z

b

a
y

b

a
rev(X) =

x b

b
b
z

a

y

a

a
transitions are reversed
initial states

⇔ final states

rev(X) is non-deterministic

SLIDE 14

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Making it deterministic again: det(rev(X))

x b

b
b
z

a

y

a

a
x, y

a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
new state space: 2X = {V | V ⊆ {x, y, z} }
V

a

W

⇔ ∀w ∈ W ∃v ∈ V : v a

w

initial state:{y, z}
final states: all V with x ∈ V

SLIDE 15

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Making it deterministic again: det(rev(X))

x b

b
b
z

a

y

a

a
x, y

a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
new state space: 2X = {V | V ⊆ {x, y, z} }
V

a

W

⇔ ∀w ∈ W ∃v ∈ V : v a

w

initial state:{y, z}
final states: all V with x ∈ V

SLIDE 16

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Making it deterministic again: det(rev(X))

x b

b
b
z

a

y

a

a
x, y

a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
new state space: 2X = {V | V ⊆ {x, y, z} }
V

a

W

⇔ ∀w ∈ W ∃v ∈ V : v a

w

initial state:{y, z}
final states: all V with x ∈ V

SLIDE 17

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Making it deterministic again: det(rev(X))

x b

b
b
z

a

y

a

a
x, y

a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
new state space: 2X = {V | V ⊆ {x, y, z} }
V

a

W

⇔ ∀w ∈ W ∃v ∈ V : v a

w

initial state:{y, z}
final states: all V with x ∈ V

SLIDE 18

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Making it deterministic again: det(rev(X))

x b

b
b
z

a

y

a

a
x, y

a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
new state space: 2X = {V | V ⊆ {x, y, z} }
V

a

W

⇔ ∀w ∈ W ∃v ∈ V : v a

w

initial state:{y, z}
final states: all V with x ∈ V

SLIDE 19

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton det(rev(X)) . . .

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
. . . accepts the reverse of the language accepted by X:

L( det(rev(X)) ) = a {a, b}∗ = reverse( L(X) )

. . . and is minimal!

SLIDE 20

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton det(rev(X)) . . .

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
. . . accepts the reverse of the language accepted by X:

L( det(rev(X)) ) = a {a, b}∗ = reverse( L(X) )

. . . and is minimal!

SLIDE 21

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton det(rev(X)) . . .

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
. . . accepts the reverse of the language accepted by X:

L( det(rev(X)) ) = a {a, b}∗ = reverse( L(X) )

. . . and is minimal!

SLIDE 22

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Today’s Theorem

If: a deterministic automaton X is reachable and accepts L(X) then: det(rev(X)) is minimal and L( det(rev(X)) ) = reverse( L(X) )

SLIDE 23

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Today’s Theorem

If: a deterministic automaton X is reachable and accepts L(X) then: det(rev(X)) is minimal and L( det(rev(X)) ) = reverse( L(X) )

SLIDE 24

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Taking the reachable part of det(rev(X))

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
x, y, z

a, b

y, z

b

a
∅

a, b

reach(det(rev(X))) is reachable (by construction)

SLIDE 25

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Taking the reachable part of det(rev(X))

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
x, y, z

a, b

y, z

b

a
∅

a, b

reach(det(rev(X))) is reachable (by construction)

SLIDE 26

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Taking the reachable part of det(rev(X))

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
x, y, z

a, b

y, z

b

a
∅

a, b

reach(det(rev(X))) is reachable (by construction)

SLIDE 27

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Repeating everything, now for reach(det(rev(X)))

x, y, z a, b

y, z

b

a
∅

a, b

s

b

a

t

b

a
. . . gives us reach(det(rev(reach(det(rev(X))))))
which is (reachable and) minimal and accepts {a, b}∗ a.

SLIDE 28

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Repeating everything, now for reach(det(rev(X)))

x, y, z a, b

y, z

b

a
∅

a, b

s

b

a

t

b

a
. . . gives us reach(det(rev(reach(det(rev(X))))))
which is (reachable and) minimal and accepts {a, b}∗ a.

SLIDE 29

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Repeating everything, now for reach(det(rev(X)))

x, y, z a, b

y, z

b

a
∅

a, b

s

b

a

t

b

a
. . . gives us reach(det(rev(reach(det(rev(X))))))
which is (reachable and) minimal and accepts {a, b}∗ a.

SLIDE 30

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Repeating everything, now for reach(det(rev(X)))

x, y, z a, b

y, z

b

a
∅

a, b

s

b

a

t

b

a
. . . gives us reach(det(rev(reach(det(rev(X))))))
which is (reachable and) minimal and accepts {a, b}∗ a.

SLIDE 31

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

All in all: Brzozowski’s algorithm

x

b

a

z

b

a
y

b

a
s

b

a

t

b

a
X is reachable and accepts {a, b}∗ a
reach(det(rev(reach(det(rev(X)))))) also accepts {a, b}∗ a
. . . and is minimal!!

SLIDE 32

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

All in all: Brzozowski’s algorithm

x

b

a

z

b

a
y

b

a
s

b

a

t

b

a
X is reachable and accepts {a, b}∗ a
reach(det(rev(reach(det(rev(X)))))) also accepts {a, b}∗ a
. . . and is minimal!!

SLIDE 33

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

All in all: Brzozowski’s algorithm

x

b

a

z

b

a
y

b

a
s

b

a

t

b

a
X is reachable and accepts {a, b}∗ a
reach(det(rev(reach(det(rev(X)))))) also accepts {a, b}∗ a
. . . and is minimal!!

SLIDE 34

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

All in all: Brzozowski’s algorithm

x

b

a

z

b

a
y

b

a
s

b

a

t

b

a
X is reachable and accepts {a, b}∗ a
reach(det(rev(reach(det(rev(X)))))) also accepts {a, b}∗ a
. . . and is minimal!!

SLIDE 35

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

All in all: Brzozowski’s algorithm

x

b

a

z

b

a
y

b

a
s

b

a

t

b

a
X is reachable and accepts {a, b}∗ a
reach(det(rev(reach(det(rev(X)))))) also accepts {a, b}∗ a
. . . and is minimal!!

SLIDE 36

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions
2. (Co)algebra

algebras: F(X) f

X

coalgebras: X f

F(X)

SLIDE 37

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Examples of algebras

N × N +

N

1 + N [0, S]

N

≡ 1

N

S

N

≡ 1

N

S

N

SLIDE 38

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Examples of algebras

N × N +

N

1 + N [0, S]

N

≡ 1

N

S

N

≡ 1

N

S

N

SLIDE 39

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Examples of coalgebras

X t

P(A × X)

x a

y

⇐ ⇒ a, y ∈ t(x) X Left, label, Right

X × A × X

SLIDE 40

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Examples of coalgebras

X t

P(A × X)

x a

y

⇐ ⇒ a, y ∈ t(x) X Left, label, Right

X × A × X

SLIDE 41

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Examples of coalgebras

2ω head, tail

2 × 2ω

≡ 2ω head

tail
2

2ω ≡ 2 2ω head

tail
2ω

head((b0, b1, b2, . . .)) = b0 tail((b0, b1, b2, . . .)) = (b1, b2, b3 . . .)

SLIDE 42

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Homomorphisms

F(X) f

F(h) F(Y)

g

X

h

Y

X f

h

Y

g

F(X)

F(h)

F(Y)

SLIDE 43

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Homomorphisms

F(X) f

F(h) F(Y)

g

X

h

Y

X f

h

Y

g

F(X)

F(h)

F(Y)

SLIDE 44

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Initiality, finality

F(A) α

F(h) F(X)

f

A

∃ ! h

X

X f

∃ ! h

Z

β

F(X)

F(h)

F(Z)

initial algebras ⇐

⇒ induction

final coalgebras ⇐

⇒ coinduction

SLIDE 45

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Initiality, finality

F(A) α

F(h) F(X)

f

A

∃ ! h

X

X f

∃ ! h

Z

β

F(X)

F(h)

F(Z)

initial algebras ⇐

⇒ induction

final coalgebras ⇐

⇒ coinduction

SLIDE 46

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Initiality, finality

F(A) α

F(h) F(X)

f

A

∃ ! h

X

X f

∃ ! h

Z

β

F(X)

F(h)

F(Z)

initial algebras ⇐

⇒ induction

final coalgebras ⇐

⇒ coinduction

SLIDE 47

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions
3. Automata, (co)algebraically
Automata are complicated structures:

part of them is algebra - part of them is coalgebra

( . . . in two different ways . . . )

SLIDE 48

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions
3. Automata, (co)algebraically
Automata are complicated structures:

part of them is algebra - part of them is coalgebra

( . . . in two different ways . . . )

SLIDE 49

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

A deterministic automaton

1 i

2

X f

t
X A

where 1 = {0} 2 = {0, 1} X A = { g | g : A → X } x a

y

⇐ ⇒ t(x)(a) = y i(0) ∈ X is the initial state x is final (or accepting) ⇐ ⇒ f(x) = 1

SLIDE 50

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

A deterministic automaton

1 i

2

X f

t
X A

where 1 = {0} 2 = {0, 1} X A = { g | g : A → X } x a

y

⇐ ⇒ t(x)(a) = y i(0) ∈ X is the initial state x is final (or accepting) ⇐ ⇒ f(x) = 1

SLIDE 51

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Automata: algebra or coalgebra?

initial state: algebraic – final states: coalgebraic

1 i

2

X f

transition function: both algebraic and coalgebraic

X t

X A

X

(A X)

X × A t

X

SLIDE 52

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Automata: algebra or coalgebra?

initial state: algebraic – final states: coalgebraic

1 i

2

X f

transition function: both algebraic and coalgebraic

X t

X A

X

(A X)

X × A t

X

SLIDE 53

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Automata: algebra and coalgebra!

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

To take home: this picture!! . . . which we’ll explain next . . .

SLIDE 54

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Automata: algebra and coalgebra!

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

To take home: this picture!! . . . which we’ll explain next . . .

SLIDE 55

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Automata: algebra and coalgebra!

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

To take home: this picture!! . . . which we’ll explain next . . .

SLIDE 56

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The “automaton” of languages

2 2A∗ β

ǫ?
(2A∗)A

ǫ?(L) = 1 ⇐ ⇒ ǫ ∈ L 2A∗ = {g | g : A∗ → 2} ∼ = {L | L ⊆ A∗} β(L)(a) = La = {w ∈ A∗ | a · w ∈ L}

We say “automaton”: it does not have an initial state.

SLIDE 57

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The “automaton” of languages

2 2A∗ β

ǫ?
(2A∗)A

ǫ?(L) = 1 ⇐ ⇒ ǫ ∈ L 2A∗ = {g | g : A∗ → 2} ∼ = {L | L ⊆ A∗} β(L)(a) = La = {w ∈ A∗ | a · w ∈ L}

We say “automaton”: it does not have an initial state.

SLIDE 58

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton of languages

transitions: L

a

La where

La = {w ∈ A∗ | a · w ∈ L}

for instance:

{ab} b

a

{b}

a

b

{ǫ}

a, b

∅

a, b

a∗ b {a, b}∗

b

a
{a, b}∗

a, b

note: every state L accepts . . . . . . the language L !!

SLIDE 59

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton of languages

transitions: L

a

La where

La = {w ∈ A∗ | a · w ∈ L}

for instance:

{ab} b

a

{b}

a

b

{ǫ}

a, b

∅

a, b

a∗ b {a, b}∗

b

a
{a, b}∗

a, b

note: every state L accepts . . . . . . the language L !!

SLIDE 60

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton of languages

transitions: L

a

La where

La = {w ∈ A∗ | a · w ∈ L}

for instance:

{ab} b

a

{b}

a

b

{ǫ}

a, b

∅

a, b

a∗ b {a, b}∗

b

a
{a, b}∗

a, b

note: every state L accepts . . . . . . the language L !!

SLIDE 61

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton of languages

transitions: L

a

La where

La = {w ∈ A∗ | a · w ∈ L}

for instance:

{ab} b

a

{b}

a

b

{ǫ}

a, b

∅

a, b

a∗ b {a, b}∗

b

a
{a, b}∗

a, b

note: every state L accepts . . . . . . the language L !!

SLIDE 62

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton of languages is . . . final

2 X f

t
∃! o

2A∗

β

ǫ?
X A
A

(2A∗)A

(x)

= {w ∈ A∗ | f(xw) = 1} = the language accepted by x where: xw is the state reached after inputting the word w, and: oA(g) = o ◦ g, all g ∈ X A.

SLIDE 63

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton of languages is . . . final

2 X f

t
∃! o

2A∗

β

ǫ?
X A
A

(2A∗)A

(x)

= {w ∈ A∗ | f(xw) = 1} = the language accepted by x where: xw is the state reached after inputting the word w, and: oA(g) = o ◦ g, all g ∈ X A.

SLIDE 64

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

An aside: a proof by coinduction

Question: a∗ b {a, b}∗ = {a, b}∗ b a∗ ?? Answer: consider a∗ b {a, b}∗ b

a
{a, b}∗

a, b

{a, b}∗ b a∗

b

a
{a, b}∗ b a∗ + a∗

a, b

and conclude: yes, since the relation

R = { a∗ b {a, b}∗, {a, b}∗ b a∗ , {a, b}∗, {a, b}∗ b a∗+a∗ } is a (language!) bisimulation.

SLIDE 65

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

An aside: a proof by coinduction

Question: a∗ b {a, b}∗ = {a, b}∗ b a∗ ?? Answer: consider a∗ b {a, b}∗ b

a
{a, b}∗

a, b

{a, b}∗ b a∗

b

a
{a, b}∗ b a∗ + a∗

a, b

and conclude: yes, since the relation

R = { a∗ b {a, b}∗, {a, b}∗ b a∗ , {a, b}∗, {a, b}∗ b a∗+a∗ } is a (language!) bisimulation.

SLIDE 66

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

An aside: a proof by coinduction

Question: a∗ b {a, b}∗ = {a, b}∗ b a∗ ?? Answer: consider a∗ b {a, b}∗ b

a
{a, b}∗

a, b

{a, b}∗ b a∗

b

a
{a, b}∗ b a∗ + a∗

a, b

and conclude: yes, since the relation

R = { a∗ b {a, b}∗, {a, b}∗ b a∗ , {a, b}∗, {a, b}∗ b a∗+a∗ } is a (language!) bisimulation.

SLIDE 67

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

An aside: a proof by coinduction

Question: a∗ b {a, b}∗ = {a, b}∗ b a∗ ?? Answer: consider a∗ b {a, b}∗ b

a
{a, b}∗

a, b

{a, b}∗ b a∗

b

a
{a, b}∗ b a∗ + a∗

a, b

and conclude: yes, since the relation

R = { a∗ b {a, b}∗, {a, b}∗ b a∗ , {a, b}∗, {a, b}∗ b a∗+a∗ } is a (language!) bisimulation.

SLIDE 68

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Back to today’s picture

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

On the right: final coalgebra On the left: initial algebra . . .

SLIDE 69

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Back to today’s picture

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

On the right: final coalgebra On the left: initial algebra . . .

SLIDE 70

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Back to today’s picture

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

On the right: final coalgebra On the left: initial algebra . . .

SLIDE 71

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The “automaton” of words

1 ǫ

A∗

α

(A∗)A

ǫ is initial state α(w)(a) = w · a that is, transitions: w a

w · a

SLIDE 72

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

The automaton of words is . . . initial

1 ǫ

i
A∗

∃! r

α
X

t

(A∗)A

r A

X A

i ∈ X = initial state (to be precise: i(0)) r(w) = iw = the state reached from i after inputting w

Proof: easy exercise.
Proof: formally, because A∗ is an initial 1 + A × (−)-algebra!

SLIDE 73

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions
4. Duality
Reachability and observability are dual:

Arbib and Manes, 1975.

(here observable = minimal)

SLIDE 74

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reachability and observability

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

r(w) = state reached

n input w
(x)

= language accepted by x

We call X reachable if r is surjective.
We call X observable (= minimal) if o is injective.

SLIDE 75

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reachability and observability

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

r(w) = state reached

n input w
(x)

= language accepted by x

We call X reachable if r is surjective.
We call X observable (= minimal) if o is injective.

SLIDE 76

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reachability and observability

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

r(w) = state reached

n input w
(x)

= language accepted by x

We call X reachable if r is surjective.
We call X observable (= minimal) if o is injective.

SLIDE 77

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reachability and observability

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

r(w) = state reached

n input w
(x)

= language accepted by x

We call X reachable if r is surjective.
We call X observable (= minimal) if o is injective.

SLIDE 78

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions
5. Reversing the automaton
Reachability ⇐

⇒ observability

Being precise about homomorphisms is crucial.
Forms the basis for proof Brzozowski’s algorithm.

SLIDE 79

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Powerset construction

V g

2V

2(−) : → W 2W 2g

where 2V = {S | S ⊆ V} and, for all S ⊆ W,

2g(S) = g−1(S) ( = {v ∈ V | g(v) ∈ S} )

This construction is contravariant !!
Note: if g is surjective, then 2g is injective.

SLIDE 80

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Powerset construction

V g

2V

2(−) : → W 2W 2g

where 2V = {S | S ⊆ V} and, for all S ⊆ W,

2g(S) = g−1(S) ( = {v ∈ V | g(v) ∈ S} )

This construction is contravariant !!
Note: if g is surjective, then 2g is injective.

SLIDE 81

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Powerset construction

V g

2V

2(−) : → W 2W 2g

where 2V = {S | S ⊆ V} and, for all S ⊆ W,

2g(S) = g−1(S) ( = {v ∈ V | g(v) ∈ S} )

This construction is contravariant !!
Note: if g is surjective, then 2g is injective.

SLIDE 82

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Powerset construction

V g

2V

2(−) : → W 2W 2g

where 2V = {S | S ⊆ V} and, for all S ⊆ W,

2g(S) = g−1(S) ( = {v ∈ V | g(v) ∈ S} )

This construction is contravariant !!
Note: if g is surjective, then 2g is injective.

SLIDE 83

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing transitions

X t X A X × A

X

✤ 2(−)

2X×A 2X

(2X)A

2X

2X

2t

(2X)A

SLIDE 84

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing transitions

X t X A X × A

X

✤ 2(−)

2X×A 2X

(2X)A

2X

2X

2t

(2X)A

SLIDE 85

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing transitions

X t X A X × A

X

✤ 2(−)

2X×A 2X

(2X)A

2X

2X

2t

(2X)A

SLIDE 86

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing transitions

X t X A X × A

X

✤ 2(−)

2X×A 2X

(2X)A

2X

2X

2t

(2X)A

SLIDE 87

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing transitions

X t X A X × A

X

✤ 2(−)

2X×A 2X

(2X)A

2X

2X

2t

(2X)A

SLIDE 88

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Initial ⇐ ⇒ final

1 i

X

✤ 2(−)

2 2X 2i

2

X f

1

f

2X

SLIDE 89

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Initial ⇐ ⇒ final

1 i

X

✤ 2(−)

2 2X 2i

2

X f

1

f

2X

SLIDE 90

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Initial ⇐ ⇒ final

1 i

X

✤ 2(−)

2 2X 2i

2

X f

1

f

2X

SLIDE 91

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Initial ⇐ ⇒ final

1 i

X

✤ 2(−)

2 2X 2i

2

X f

1

f

2X

SLIDE 92

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the entire automaton

1 i

2

X f

t
X A

✤ 2(−)

1 f

2

2X 2i

2t
(2X)A
Initial and final are exchanged . . .
transitions are reversed . . .
and the result is again deterministic!

SLIDE 93

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the entire automaton

1 i

2

X f

t
X A

✤ 2(−)

1 f

2

2X 2i

2t
(2X)A
Initial and final are exchanged . . .
transitions are reversed . . .
and the result is again deterministic!

SLIDE 94

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the entire automaton

1 i

2

X f

t
X A

✤ 2(−)

1 f

2

2X 2i

2t
(2X)A
Initial and final are exchanged . . .
transitions are reversed . . .
and the result is again deterministic!

SLIDE 95

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the entire automaton

1 i

2

X f

t
X A

✤ 2(−)

1 f

2

2X 2i

2t
(2X)A
Initial and final are exchanged . . .
transitions are reversed . . .
and the result is again deterministic!

SLIDE 96

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reversing the entire automaton

1 i

2

X f

t
X A

✤ 2(−)

1 f

2

2X 2i

2t
(2X)A
Initial and final are exchanged . . .
transitions are reversed . . .
and the result is again deterministic!

SLIDE 97

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Our previous example

X =

x

b

a

z

b

a
y

b

a
2X =

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
Note that X has been reversed and determinized:

2X = det(rev(X))

SLIDE 98

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Our previous example

X =

x

b

a

z

b

a
y

b

a
2X =

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
Note that X has been reversed and determinized:

2X = det(rev(X))

SLIDE 99

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Our previous example

X =

x

b

a

z

b

a
y

b

a
2X =

x, y a b x, y, z a, b

x, z

b

a
y, z

b

a
x

b

a
y

b

a
∅

a, b

z

b

a
Note that X has been reversed and determinized:

2X = det(rev(X))

SLIDE 100

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Proving today’s Theorem

If: a deterministic automaton X is reachable and accepts L(X) then: 2X ( = det(rev(X)) ) is minimal/observable and L( 2X ) = reverse( L(X) )

SLIDE 101

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Proving today’s Theorem

If: a deterministic automaton X is reachable and accepts L(X) then: 2X ( = det(rev(X)) ) is minimal/observable and L( 2X ) = reverse( L(X) )

SLIDE 102

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Proof: by reversing A∗ r

X

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
(2X)A

(2A∗)A

X becomes 2X
initial automaton A∗ becomes (almost) final automaton 2A∗
r is surjective

⇒ 2r is injective

SLIDE 103

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Proof: by reversing A∗ r

X

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
(2X)A

(2A∗)A

X becomes 2X
initial automaton A∗ becomes (almost) final automaton 2A∗
r is surjective

⇒ 2r is injective

SLIDE 104

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Proof: by reversing A∗ r

X

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
(2X)A

(2A∗)A

X becomes 2X
initial automaton A∗ becomes (almost) final automaton 2A∗
r is surjective

⇒ 2r is injective

SLIDE 105

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Proof: by reversing A∗ r

X

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
(2X)A

(2A∗)A

X becomes 2X
initial automaton A∗ becomes (almost) final automaton 2A∗
r is surjective

⇒ 2r is injective

SLIDE 106

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Proof: by reversing A∗ r

X

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
(2X)A

(2A∗)A

X becomes 2X
initial automaton A∗ becomes (almost) final automaton 2A∗
r is surjective

⇒ 2r is injective

SLIDE 107

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reachable becomes observable

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
rev

2A∗

β

ǫ?
(2X)A

(2A∗)A (2A∗)A

If r is surjective then (2r and hence) rev ◦ 2r is injective.
That is, 2X is observable (= minimal).

SLIDE 108

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reachable becomes observable

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
rev

2A∗

β

ǫ?
(2X)A

(2A∗)A (2A∗)A

If r is surjective then (2r and hence) rev ◦ 2r is injective.
That is, 2X is observable (= minimal).

SLIDE 109

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reachable becomes observable

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
rev

2A∗

β

ǫ?
(2X)A

(2A∗)A (2A∗)A

If r is surjective then (2r and hence) rev ◦ 2r is injective.
That is, 2X is observable (= minimal).

SLIDE 110

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Reachable becomes observable

1 ǫ

i
A∗

r

α
X

t

(A∗)A

X A ✤ 2(−)

2 2X 2i

2t
2r

2A∗

2α

2ǫ
rev

2A∗

β

ǫ?
(2X)A

(2A∗)A (2A∗)A

If r is surjective then (2r and hence) rev ◦ 2r is injective.
That is, 2X is observable (= minimal).

SLIDE 111

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Summarizing

1 ǫ

i
2

A∗ r

α
X

f

t
(A∗)A

X A

1 f

2

2X 2i

2t
rev ◦ 2r 2A∗

β

ǫ?
(2X)A

(2A∗)A

If: X is reachable, i.e., r is surjective

then: rev ◦ 2r is injective, i.e., 2X is observable = minimal.

And: rev(2r(f)) = rev(o(i)), i.e., L( 2X ) = reverse( L(X) )

SLIDE 112

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Summarizing

1 ǫ

i
2

A∗ r

α
X

f

t
(A∗)A

X A

1 f

2

2X 2i

2t
rev ◦ 2r 2A∗

β

ǫ?
(2X)A

(2A∗)A

If: X is reachable, i.e., r is surjective

then: rev ◦ 2r is injective, i.e., 2X is observable = minimal.

And: rev(2r(f)) = rev(o(i)), i.e., L( 2X ) = reverse( L(X) )

SLIDE 113

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Summarizing

1 ǫ

i
2

A∗ r

α
X

f

t
(A∗)A

X A

1 f

2

2X 2i

2t
rev ◦ 2r 2A∗

β

ǫ?
(2X)A

(2A∗)A

If: X is reachable, i.e., r is surjective

then: rev ◦ 2r is injective, i.e., 2X is observable = minimal.

And: rev(2r(f)) = rev(o(i)), i.e., L( 2X ) = reverse( L(X) )

SLIDE 114

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Summarizing

1 ǫ

i
2

A∗ r

α
X

f

t
(A∗)A

X A

1 f

2

2X 2i

2t
rev ◦ 2r 2A∗

β

ǫ?
(2X)A

(2A∗)A

If: X is reachable, i.e., r is surjective

then: rev ◦ 2r is injective, i.e., 2X is observable = minimal.

And: rev(2r(f)) = rev(o(i)), i.e., L( 2X ) = reverse( L(X) )

SLIDE 115

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Summarizing

1 ǫ

i
2

A∗ r

α
X

f

t
(A∗)A

X A

1 f

2

2X 2i

2t
rev ◦ 2r 2A∗

β

ǫ?
(2X)A

(2A∗)A

If: X is reachable, i.e., r is surjective

then: rev ◦ 2r is injective, i.e., 2X is observable = minimal.

And: rev(2r(f)) = rev(o(i)), i.e., L( 2X ) = reverse( L(X) )

SLIDE 116

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Corollary: Brzozowski’s algorithm

X becomes 2X, accepting reverse(L(X))
take reachable part: Y = reachable(2X)
Y becomes 2Y, which is minimal and accepts

reverse(reverse(L(X))) = L(X)

SLIDE 117

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Corollary: Brzozowski’s algorithm

X becomes 2X, accepting reverse(L(X))
take reachable part: Y = reachable(2X)
Y becomes 2Y, which is minimal and accepts

reverse(reverse(L(X))) = L(X)

SLIDE 118

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Corollary: Brzozowski’s algorithm

X becomes 2X, accepting reverse(L(X))
take reachable part: Y = reachable(2X)
Y becomes 2Y, which is minimal and accepts

reverse(reverse(L(X))) = L(X)

SLIDE 119

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions
6. Conclusions

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

SLIDE 120

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions
6. Conclusions

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

SLIDE 121

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Generalizations

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

1 ǫ

i
B

A∗ r

α
X

f

t
BA∗
(A∗)A

r A

X A

A

(BA∗)A

A Brzozowski minimization algorithm for Moore automata.
Non-deterministic and weighted automata: under way (with

Bonsangue et al).

Probabilistic systems: under investigation.

SLIDE 122

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Generalizations

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

1 ǫ

i
B

A∗ r

α
X

f

t
BA∗
(A∗)A

r A

X A

A

(BA∗)A

A Brzozowski minimization algorithm for Moore automata.
Non-deterministic and weighted automata: under way (with

Bonsangue et al).

Probabilistic systems: under investigation.

SLIDE 123

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Generalizations

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

1 ǫ

i
B

A∗ r

α
X

f

t
BA∗
(A∗)A

r A

X A

A

(BA∗)A

A Brzozowski minimization algorithm for Moore automata.
Non-deterministic and weighted automata: under way (with

Bonsangue et al).

Probabilistic systems: under investigation.

SLIDE 124

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Generalizations

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

1 ǫ

i
B

A∗ r

α
X

f

t
BA∗
(A∗)A

r A

X A

A

(BA∗)A

A Brzozowski minimization algorithm for Moore automata.
Non-deterministic and weighted automata: under way (with

Bonsangue et al).

Probabilistic systems: under investigation.

SLIDE 125

1. Example
2. (Co)algebra
3. Automata
4. Duality
5. Reversing
6. Conclusions

Generalizations

1 ǫ

i
2

A∗ r

α
X

f

t
2A∗

β

ǫ?
(A∗)A

r A

X A

A

(2A∗)A

1 ǫ

i
B

A∗ r

α
X

f

t
BA∗
(A∗)A

r A

X A

A

(BA∗)A

A Brzozowski minimization algorithm for Moore automata.
Non-deterministic and weighted automata: under way (with

Bonsangue et al).

Probabilistic systems: under investigation.