A Congruence-based Perspective on Automata Minimization Algorithms - - PowerPoint PPT Presentation

Pierre Ganty, Elena Gutiérrez, Pedro Valero

IMDEA Software Institute, Madrid, Spain

A Congruence-based Perspective on Automata Minimization Algorithms

Séminaire MF - LaBRI

June, 23rd, 2020

• E. Gutiérrez, IMDEA Software, Madrid

2

Motivation

a a a b b b b

a b

+ +

all words with at least one ‘a’ followed by at least one ‘b’ Finite-state automaton Regular language

Automata Minimization Algorithms

• E. Gutiérrez, IMDEA Software, Madrid

2

Motivation

a a a b b b b

a b

+ +

all words with at least one ‘a’ followed by at least one ‘b’

Find the finite-state automaton with the least number of states for the language

Finite-state automaton Regular language

Automata Minimization Algorithms

• E. Gutiérrez, IMDEA Software, Madrid

2

Motivation

a a b b

a a a b b b b

a b

+ +

all words with at least one ‘a’ followed by at least one ‘b’

Find the finite-state automaton with the least number of states for the language

Finite-state automaton Regular language Minimal (deterministic) finite-state automaton

Automata Minimization Algorithms

• E. Gutiérrez, IMDEA Software, Madrid

3

Automata Minimization Algorithms

Motivation

• E. Gutiérrez, IMDEA Software, Madrid

3

Hopcroft’s algorithm Moore's algorithm Revuz’s algorithm Double-reversal method

Automata Minimization Algorithms

Motivation

• E. Gutiérrez, IMDEA Software, Madrid

3

Hopcroft’s algorithm Moore's algorithm Revuz’s algorithm Double-reversal method

Partition of the set of states

Automata Minimization Algorithms

Motivation

• E. Gutiérrez, IMDEA Software, Madrid

3

Hopcroft’s algorithm Moore's algorithm Revuz’s algorithm Double-reversal method

Partition of the set of states Combination of automata constructions

1.Reverse 3.Reverse 2.Determinization 4.Determinization

Automata Minimization Algorithms

Motivation

• E. Gutiérrez, IMDEA Software, Madrid

3

Hopcroft’s algorithm Moore's algorithm Revuz’s algorithm Double-reversal method

Partition of the set of states Combination of automata constructions

1.Reverse 3.Reverse 2.Determinization 4.Determinization

• the double-reversal method, and
• its connection with the partition-based methods

Give new language-theoretical insights on:

Goal

Automata Minimization Algorithms

Motivation

• E. Gutiérrez, IMDEA Software, Madrid

Σ∗

set of all words over the alphabet Σ

def

=

4

∼

Common purpose: Build Nerode’s equivalence relation on words

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

Σ∗

set of all words over the alphabet Σ

def

=

4

Σ∗

set of all words over the alphabet Σ

def

=

∼

Common purpose: Build Nerode’s equivalence relation on words

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼ v

u

v

u, v ∈ Σ∗

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼ v

u

v

u ∼L v ⇔ u−1L = v−1L

Given an automaton and its language :

L

u, v ∈ Σ∗

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼ v

u

v

u ∼L v ⇔ u−1L = v−1L

{w | uw ∈ L}

Given an automaton and its language :

L

u, v ∈ Σ∗

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼ v

u

v

u ∼L v ⇔ u−1L = v−1L

Given an automaton and its language :

L

u, v ∈ Σ∗

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼ v

u

v

u ∼L v ⇔ u−1L = v−1L

• Finite number of equivalence classes

Given an automaton and its language :

L

u, v ∈ Σ∗

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼L v ⇔ u−1L = v−1L

• Finite number of equivalence classes

Given an automaton and its language :

L

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼L v ⇔ u−1L = v−1L

• Finite number of equivalence classes
• Congruence

Given an automaton and its language :

L

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼L v ⇔ u−1L = v−1L

• Finite number of equivalence classes
• Congruence
• Precisely represents L

Given an automaton and its language :

L

∼

Σ∗

set of all words over the alphabet Σ

def

=

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼L v ⇔ u−1L = v−1L

• Finite number of equivalence classes
• Congruence
• Precisely represents L

Given an automaton and its language :

L

∼

Σ∗

set of all words over the alphabet Σ

def

=

L

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼L v ⇔ u−1L = v−1L

• Finite number of equivalence classes
• Congruence
• Precisely represents L

Build a deterministic automaton for

Given an automaton and its language :

L

∼

Σ∗

set of all words over the alphabet Σ

def

=

L

Language-theoretical Perspective

L

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼L v ⇔ u−1L = v−1L

• Finite number of equivalence classes
• Congruence
• Precisely represents L

Build a deterministic automaton for

Given an automaton and its language :

L

∼

Σ∗

set of all words over the alphabet Σ

def

=

L

Language-theoretical Perspective

L

• E. Gutiérrez, IMDEA Software, Madrid

5

Common purpose: Build Nerode’s equivalence relation on words

u ∼L v ⇔ u−1L = v−1L

• Finite number of equivalence classes
• Congruence
• Precisely represents L

Build a deterministic automaton for

• Coarsest congruence satisfying the latter

properties

Build the minimal deterministic automaton for L

Given an automaton and its language :

L

∼

Σ∗

set of all words over the alphabet Σ

def

=

L

Language-theoretical Perspective

L

• E. Gutiérrez, IMDEA Software, Madrid

6

Hopcroft’s algorithm Revuz’s algorithm Double-reversal method Moore’s algorithm

Partition of the set of states

Combination of automata constructions

Refinement Fusion

Σ∗

Nerode’s equivalence relation on words

∼

1.Reverse 3.Reverse 2.Determinization 4.Determinization

L

Language-theoretical Perspective

• E. Gutiérrez, IMDEA Software, Madrid

7

Hopcroft’s algorithm Revuz’s algorithm

Double-reversal method

Moore’s algorithm Partition of the set of states Combination of automata constructions

Refinement Fusion

1.Reverse 3.Reverse 2.Determinization 4.Determinization

Language-theoretical Perspective

Σ∗

∼

L

• E. Gutiérrez, IMDEA Software, Madrid

7

Hopcroft’s algorithm Revuz’s algorithm

Double-reversal method

Moore’s algorithm Partition of the set of states Combination of automata constructions

Refinement Fusion

1.Reverse 3.Reverse 2.Determinization 4.Determinization

Language-theoretical Perspective

Contributions

• Automata constructions based on equivalences
• New simple proof of double-reversal method
• Revisit generalization of the double-reversal

method

• Invariant of Moore’s algorithm

Σ∗

∼

L

• E. Gutiérrez, IMDEA Software, Madrid

8

Language-theoretical Perspective

Hopcroft’s algorithm Revuz’s algorithm

Double-reversal method

Moore’s algorithm Partition of the set of states Combination of automata constructions

Refinement Fusion

1.Reverse 3.Reverse 2.Determinization 4.Determinization

In this talk

• Automata constructions based on equivalences
• New simple proof of double-reversal method

Σ∗

∼

L

• Revisit generalization of the double-reversal

method

• Invariant of Moore’s algorithm

• E. Gutiérrez, IMDEA Software, Madrid

9

a a b b

Deterministic (DFA)

Σ = {

}

a b

,

a b

+ +

The Basics

• E. Gutiérrez, IMDEA Software, Madrid

9

a a b b a a a a b b b

Nondeterministic (NFA) Deterministic (DFA)

Σ = {

}

a b

,

a b

+ +

The Basics

• E. Gutiérrez, IMDEA Software, Madrid

a a b b a

10

a a b b a

N

The Basics

Reverse Construction

• E. Gutiérrez, IMDEA Software, Madrid

Reverse Construction

a a b b a

11

N R

Reverse Automaton

a a b b

bbaaa

. . .

,

,

,

,

L(N R) = {

}

baa bbaa b a ba

=

+ +

N

The Basics

• E. Gutiérrez, IMDEA Software, Madrid

12

The Basics

Co-deterministic automaton

a a b b a

N R

Reverse Automaton

a a b b

bbaaa

. . .

,

,

,

,

L(N R) = {

}

baa bbaa b a ba

=

+ +

N

12

• E. Gutiérrez, IMDEA Software, Madrid

12

The Basics

Co-deterministic automaton

a a b b a

N R

Reverse Automaton Co-deterministic (co-DFA)

a a b b

bbaaa

. . .

,

,

,

,

L(N R) = {

}

baa bbaa b a ba

=

+ +

N

12

• E. Gutiérrez, IMDEA Software, Madrid

13

Congruences

Equivalences on words with good properties w.r.t. concatenation of symbols

• E. Gutiérrez, IMDEA Software, Madrid

13

Congruences

Equivalences on words with good properties w.r.t. concatenation of symbols

Right congruences:

u ∼r v ⇒ ua ∼r va

Σ∗

u

∼r

, ∀a ∈ Σ

v

ua

va

:

• E. Gutiérrez, IMDEA Software, Madrid

13

Congruences

Equivalences on words with good properties w.r.t. concatenation of symbols

av

au

Σ∗

u

∼`

u ∼` v ⇒ au ∼` av

Left congruences:

, ∀a ∈ Σ

v

:

Right congruences:

u ∼r v ⇒ ua ∼r va

Σ∗

u

∼r

, ∀a ∈ Σ

v

ua

va

:

• E. Gutiérrez, IMDEA Software, Madrid

14

How to build a deterministic automaton from a right congruence

• is a finite right congruence
• ( precisely represents )

P∼r(L) = L

∼r

∼r

Σ∗

The DFA accepts L

[Gutiérrez et. al, MFCS 2019]

[Hopcroft and Ullman, 1979]

L

∼r

L

• E. Gutiérrez, IMDEA Software, Madrid

( precisely represents )

15

How to build a co-deterministic automata from a left congruence

• is a finite left congruence
• ∼`

L0

The co-DFA accepts L0

P⇠`(L0) = L0

[Gutiérrez et. al, MFCS 2019]

Σ∗

∼`

∼`

L0

• E. Gutiérrez, IMDEA Software, Madrid

16

∼`

∼r

Lemma:

[Gutiérrez et. al, MFCS 2019]

A property of dual congruences and ∼r

Σ∗

L

Σ∗

∼`

LR

• E. Gutiérrez, IMDEA Software, Madrid

16

∼`

∼r

Lemma:

Given

u ∼r v

⇔ uR ∼` vR

[Gutiérrez et. al, MFCS 2019]

A property of dual congruences and ∼r

Σ∗

L

Σ∗

∼`

LR

• E. Gutiérrez, IMDEA Software, Madrid

17

then

( )

=

Lemma:

[Gutiérrez et. al, MFCS 2019]

∼r

Σ∗

L

Σ∗

∼`

LR

∼`

∼r

A property of dual congruences and

R

⇔ uR ∼` vR

Given

u ∼r v

• E. Gutiérrez, IMDEA Software, Madrid

Instances of right congruences

• is a finite right congruence
• P∼r(L) = L

∼r

18

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

Instances of right congruences

[Büchi, 1989; Khoussainov and Nerode, 2001]

∼r

L

Language-based

u ∼r

L v ⇔ u−1L = v−1L

Given a regular language L

The minimal DFA for L

Σ∗

L

• is a finite right congruence
• P∼r(L) = L

∼r

18

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

Instances of right congruences

[Büchi, 1989; Khoussainov and Nerode, 2001]

∼r

L

Language-based

u ∼r

L v ⇔ u−1L = v−1L

Given a regular language L

The minimal DFA for L

Σ∗

L

Minr(L)

• is a finite right congruence
• P∼r(L) = L

∼r

18

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

Instances of right congruences

[Büchi, 1989; Khoussainov and Nerode, 2001]

∼r

L

Language-based

u ∼r

L v ⇔ u−1L = v−1L

Given a regular language L

The minimal DFA for L

Σ∗

L

Minr(L)

• is a finite right congruence
• P∼r(L) = L

∼r

Σ∗

L

∼r

N

Automata-based N

Given an NFA with L(N) = L

A DFA for :

L

u ∼r

N v ⇔ postN (u) = postN (v)

18

[Gutiérrez et. al, MFCS 2019]

the usual “determinization”

• peration

• E. Gutiérrez, IMDEA Software, Madrid

Instances of right congruences

[Büchi, 1989; Khoussainov and Nerode, 2001]

∼r

L

Language-based

u ∼r

L v ⇔ u−1L = v−1L

Given a regular language L

The minimal DFA for L

Σ∗

L

Minr(L)

Detr(N)

• is a finite right congruence
• P∼r(L) = L

∼r

Σ∗

L

∼r

N

Automata-based N

Given an NFA with L(N) = L

A DFA for :

L

u ∼r

N v ⇔ postN (u) = postN (v)

18

[Gutiérrez et. al, MFCS 2019]

the usual “determinization”

• peration

• E. Gutiérrez, IMDEA Software, Madrid

Instances of right congruences

⌫

[Büchi, 1989; Khoussainov and Nerode, 2001]

∼r

L

Language-based

u ∼r

L v ⇔ u−1L = v−1L

Given a regular language L

The minimal DFA for L

Σ∗

L

Minr(L)

Detr(N)

• is a finite right congruence
• P∼r(L) = L

∼r

Σ∗

L

∼r

N

Automata-based N

Given an NFA with L(N) = L

A DFA for :

L

u ∼r

N v ⇔ postN (u) = postN (v)

18

[Gutiérrez et. al, MFCS 2019]

the usual “determinization”

• peration

• E. Gutiérrez, IMDEA Software, Madrid

Instances of right congruences

[Büchi, 1989; Khoussainov and Nerode, 2001]

∼r

L

Language-based

u ∼r

L v ⇔ u−1L = v−1L

Given a regular language L

Σ∗

L

• is a finite right congruence
• P∼r(L) = L

∼r

Σ∗

L

∼r

N

Automata-based N

Given an NFA with L(N) = L

u ∼r

N v ⇔ postN (u) = postN (v)

19

[Gutiérrez et. al, MFCS 2019]

=

?

• E. Gutiérrez, IMDEA Software, Madrid

Instances of right congruences

[Büchi, 1989; Khoussainov and Nerode, 2001]

∼r

L

Language-based

u ∼r

L v ⇔ u−1L = v−1L

Given a regular language L

Σ∗

L

• is a finite right congruence
• P∼r(L) = L

∼r

Σ∗

L

∼r

N

Automata-based N

Given an NFA with L(N) = L

u ∼r

N v ⇔ postN (u) = postN (v)

19

[Gutiérrez et. al, MFCS 2019]

=

?

∼r

L=∼r N iff u−1L = v−1L ⇔ postN (u) = postN (v)

Lemma:

• E. Gutiérrez, IMDEA Software, Madrid

20

Instances of left congruences

• is a finite left congruence
• Automata-based

N

Given an NFA with L(N) = L

A co-DFA for L

u ∼`

N v ⇔ preN (u) = preN (v)

Language-based

Given a regular language L

The minimal co-DFA for L

u ∼`

L v ⇔ Lu−1 = Lv−1

P∼`(L) = L

∼`

Min`(L) Det`(N)

⌫

[Gutiérrez et. al, MFCS 2019]

Σ∗

∼`

L

LR

LR

∼`

N

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R

Double-reversal Method

21

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr

Double-reversal Method

21

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr

Double-reversal Method

21

[Gutiérrez et. al, MFCS 2019]

then

( )

R

=

Lemma:

Given u ∼r v

⇔ uR ∼` vR

∼r

Σ∗

L

Σ∗

∼`

LR

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr

R

Double-reversal Method

21

[Gutiérrez et. al, MFCS 2019]

then

( )

R

=

Lemma:

Given u ∼r v

⇔ uR ∼` vR

∼r

Σ∗

L

Σ∗

∼`

LR

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr

R

Double-reversal Method

21

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr

R

Double-reversal Method

∼r

L=∼r N iff u−1L = v−1L ⇔ postN (u) = postN (v)

Lemma:

21

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr

R

Double-reversal Method

∼r

L=∼r N iff u−1L = v−1L ⇔ postN (u) = postN (v)

Lemma:

21

[Gutiérrez et. al, MFCS 2019]

True if is a co-DFA

N

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr Detr

R

Detr(Det`(N))

Double-reversal Method

∼r

L=∼r N iff u−1L = v−1L ⇔ postN (u) = postN (v)

Lemma:

21

[Gutiérrez et. al, MFCS 2019]

True if is a co-DFA

N

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr Detr

R

Minr

Detr(Det`(N))

Double-reversal Method

∼r

L=∼r N iff u−1L = v−1L ⇔ postN (u) = postN (v)

Lemma:

21

[Gutiérrez et. al, MFCS 2019]

True if is a co-DFA

N

• E. Gutiérrez, IMDEA Software, Madrid

N N R

R Det`

Det`(N) Detr(N R)

Detr Detr

Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

Double-reversal Method

∼r

L=∼r N iff u−1L = v−1L ⇔ postN (u) = postN (v)

Lemma:

21

[Gutiérrez et. al, MFCS 2019]

True if is a co-DFA

N

• E. Gutiérrez, IMDEA Software, Madrid

22

N N R

Det`

Det`(N) Detr(N R)

Detr Detr

Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

Double-reversal Method

R

Detr((Detr(N R))R)

Thm: [Brzozowski, 1962] Let be an NFA. Then

N

L(N)

is isomorphic to the minimal DFA for

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

23

N N R

Det`

Detr(N R)

Detr Detr

Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

Double-reversal Method

Det`(N)

∼r

L=∼r N iff u−1L = v−1L ⇔ postN (u) = postN (v)

R

Lemma:

True if is a co-DFA

N

• E. Gutiérrez, IMDEA Software, Madrid

23

N N R

Det`

Detr(N R)

Detr Detr

Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

Double-reversal Method

⌫

⌫

∼`

N

∼`

L

?

Language-based Automata-based

Det`(N)

R

• E. Gutiérrez, IMDEA Software, Madrid

23

N N R

Det`

Detr(N R)

Detr Detr

Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

Double-reversal Method

⌫

⌫

∼`

N

∼`

L

Language-based Automata-based Simulation-based

Det`(N)

∼`

←

R

• E. Gutiérrez, IMDEA Software, Madrid

23

N N R

Det`

Detr(N R)

Detr Detr

Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

Double-reversal Method

⌫

⌫

∼`

N

∼`

L

Language-based Automata-based Simulation-based

Sim`(N, ←)

∼`

←

R

• E. Gutiérrez, IMDEA Software, Madrid

24

Contributions of This Work

Double-reversal Method Moore’s algorithm Generalization of the Double-reversal Method

[Brzozowski, 1962]

Detr(Det`(N))

≡

Minimal DFA for

[Brzozowski and Tamm, 2014]

L

N

L

: NFA : language of

[Moore, 1956]

N

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

24

Contributions of This Work

Double-reversal Method Moore’s algorithm Generalization of the Double-reversal Method

[Brzozowski, 1962]

Detr(Det`(N))

≡

Minimal DFA for

[Brzozowski and Tamm, 2014]

L

N

L

: NFA : language of : left language of w.r.t.

N

Lq

q

Detr(N)

≡

Minimal DFA for L

iff

P∼r

L(Lq) = Lq

∀q :

[Moore, 1956]

N

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

24

Contributions of This Work

Double-reversal Method Moore’s algorithm Generalization of the Double-reversal Method

[Brzozowski, 1962]

Detr(Det`(N))

≡

Minimal DFA for

[Brzozowski and Tamm, 2014]

L

N

L

: NFA : language of : left language of w.r.t.

N

Lq

q

Detr(N)

≡

Minimal DFA for L

iff

P∼r

L(Lq) = Lq

∀q :

[Moore, 1956]

At each step of Moore’s partition refinement:

P (n)

∼r

L (Lq) = Lq

∀q ∈ Q(n) :

n

N

[Gutiérrez et. al, MFCS 2019]

• E. Gutiérrez, IMDEA Software, Madrid

25

Conclusions

• Congruences as language abstractions
• Left-right duality between our

automata-based congruences to explain double-reversal method

N N R

Det`

Det`(N) Detr(N R)

Detr Detr Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

R

P∼r

L(Lq) = Lq

, ∀q ∈ Q

• More general view on the method

• E. Gutiérrez, IMDEA Software, Madrid

25

Conclusions

• Congruences as language abstractions
• Left-right duality between our

automata-based congruences to explain double-reversal method

N N R

Det`

Det`(N) Detr(N R)

Detr Detr Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

R

P∼r

L(Lq) = Lq

, ∀q ∈ Q

• More general view on the method

Sim`(N, ←)

• E. Gutiérrez, IMDEA Software, Madrid

25

Conclusions

• Congruences as language abstractions
• Left-right duality between our

automata-based congruences to explain double-reversal method

N N R

Det`

Det`(N) Detr(N R)

Detr Detr Det` R R

Minr

Detr(Det`(N)) Det`(Detr(N R))

Min`

R

P∼r

L(Lq) = Lq

, ∀q ∈ Q

• More general view on the method

Questions?

Sim`(N, ←)

• E. Gutiérrez, IMDEA Software, Madrid

26

Auxiliary Material

27

u−1L = v−1L ⇐ ⇒ postN

u (I) = postN v (I)

Let N be a co-DFA. Then, for each u, v ∈ Σ∗ :

Theorem:

Proof:

⇒) ⇐)

Assume postN

u (I) 6= postN v (I) . Then,

Trivial ( since always finer than ).

u v =

=

But

contradicts that is a co-DFA!

N

postN

u (I) = postN v (I)

Therefore,

• E. Gutiérrez (IMDEA Software Institute)

∼r

N

∼r

L w0 w0

There cannot be one word w0 that allows me to reach the final state from two different states in a co-DFA