theory of computationmath.sharif.ir/faculties/uploads/daneshgar/courses/... · theory of...

THEORY OF COMPUTATION (Automata as Algebras)

THEORY OF COMPUTATION

Chapter7Regular Languages

(Automata as Algebras)

A. Daneshgar(Slides prepared by Z. Ghafouri)

Department of Mathematical SciencesSharif University of Technology

2020, March, 1st (1998, Esfand, 10)

THEORY OF COMPUTATION (Automata as Algebras), A. Daneshgar , (Slides prepared by Z. Ghafouri)


An algebra

Definition: An algebra is a pair A = (A,O) where A is a set and Ois a set of operations on A.

I An n−ary operation τ on A is a function that takes nelements of A and returns a single element of A; i.e.

τ : A× . . .× A︸︷︷︸n

→ A,

I A 0−ary operation (nullary operation) is simply an element ofA, or a constant,

I A 1−ary operation (unary operation) is simply a function fromA to A,

I A 2−ary operation (binary operation) is simply a functionfrom A× A to A.



A pre-automaton

Definition: A pre-automaton

A =(Q, q0 ∈ Q, {τi : Q → Q}i∈Σ

)with input symbols Σ of n elements, is an algebra with one nullaryand n unary operations on a set of states A.



HomomorphismsA =

(Q, q0 ∈ Q, {τi : Q → Q}i∈Σ

)B =

(Q ′, q0

′ ∈ Q ′, {τi ′ : Q ′ → Q ′}i∈Σ

)A B

x

y

σ(x)

σ(y)

σ : Q → Q ′τi τ ′i

∀i τ ′i ◦ σ = σ ◦ τiσ(q0) = q′0



Isomorphisms

Definition: Two algebras

A =(Q, q0 ∈ Q, {τi : Q → Q}i∈Σ

)and

B =(Q ′, q0

′ ∈ Q ′, {τi ′ : Q ′ → Q ′}i∈Σ

)are said to be isomorphic if there exists a bijective homomorphismσ : A → B such that its inverse, σ−1 : B → A, is also a bijectivehomomorphism.



Subalgebras

I Let A = (A,O) be an algebra and let τ ∈ O be an n−ary operation

of A. A subset B of A is τ− closed if a0, . . . , an−1 ∈ B imply

τ(a0, . . . , an−1) ∈ B.

I B is a closed subset of A if B is τ− closed for each operation τ of

A.

I A subalgebra of the algebra A = (A,O) is an algebra B = (B,OB),where B ⊆ A is a closed subset of A and

OB = {τ |Bn : τ ∈ O is an n−ary operation}.

In other words, a subalgebra is a subset of an algebra which is

closed under all its operations and carrying the induced operations.

I A reduced algebra is an algebra whose only subalgebras are the

empty set and itself.



Reduced pre-automata

A reduced pre-automaton

A =(Q, q0 ∈ Q, {τi : Q → Q}i∈Σ

)is a pre-automaton which is reduced as an algebra. Note that thisis equivalent to the condition that each state of Q is reachablefrom the initial state q0.



Equivalence relations

I An equivalence relation on X is a binary relation ∼ on X suchthat

I ∀x ∈ X (x ∼ x) (reflexivity),

I ∀x , y ∈ X (x ∼ y → y ∼ x) (simmetry),

I ∀x , y , z ∈ X (x ∼ y ∧ y ∼ z → x ∼ z) (transitivity).

I The equivalence class of an element x in X with respect tothe equivalence relation ∼ , denoted by [x ]∼, is the set of allelements of X which are equivalent to x , i.e.

[x ]∼ = {y ∈ X | x ∼ y}.



I Every two equivalence classes [x ]∼ and [y ]∼ are either equalor disjoint, so the set of equivalence classes is a partition of X.

I The set of all equivalence classes of X by ∼, denoted byX/ ∼, is called the quotient set of X .

I For any partition P of a set X , the relation ∼ on X defined by

x ∼ y if and only if x and y belong to the same element of P,

is an equivalence relation on X .

Hence, talking about equivalence relations on a set is essentially the

same as talking about partitions of a set: An equivalence relation

determines a partition (in which the subsets are the equivalence

classes), and a partition determines an equivalence relation (in

which being equivalent means belonging to the same subset).



Equivalence coming from an onto-map

Any onto-map σ : Aonto−→ B gives rise to an equivalence relation on

A according to which

a1 ∼σ a2 ⇔ σ(a1) = σ(a1).

Hence, the collection of all σ−inverse images of the elements of Bforms a partition of A.



Congruence relations

Definition: Let A = (A,O) be an algebra. A congruence relation isan equivalence relation on A which is compatible with the algebraoperations. In the case of a pre-automaton where each operation iseither nullary or unary, it means that for every x and y in A andeach unary operation τi :

[x ] = [y ]⇔ [τi (x)] = [τi (y)].



The quotient pre-automaton

Definition: Let A =(Q, q0 ∈ Q, {τi : Q → Q}i∈Σ

)be a

pre-automaton and ∼ be a congruence relation on A. Thequotient pre-automaton of A by ∼ is

A/ ∼=(Q/ ∼, [q0], {τ̃i : Q/ ∼ → Q/ ∼}i∈I

)where τ̃i ([x ]) = [τi (x)].



The canonical homomorphismConsider the map σ : A → A/ ∼ defined as σ(x) = [x ]; then σ iscalled the natural or the canonical map. Obviously σ is ahomomorphism, or equivalently:

x

τi (x)

σ(x) = [x ]

[τi (x)] = τ̃i ([x ])def τ̃i

σ

τi

σ

τ̃i

In other words

σ(τi (x)) = [τi (x)] = τ̃i (σ(x)).

andσ(q0) = [q0 ].



LetA =

(Q, q0 ∈ Q, {τi : Q → Q}i∈Σ

)and

B =(Q ′, q0

′ ∈ Q ′, {τi ′ : Q ′ → Q ′}i∈Σ

)be two pre-automata and σ : A → B be an onto homomorphism.Since σ maps A onto B, the collection of all σ−inverse images ofthe elements of B forms a partiotion of A. Let ∼σ be theequivalence relation corresponding to this partition. Since σ is ahomomorphism, ∼σ is a congruence relation on A:

[x ]∼σ = [y ]∼σ ⇔ σ(x) = σ(y)

⇔ τi′(σ(x)) = τi

′(σ(y))

⇔ σ(τi (x)) = σ(τi (y))

⇔ [τi (x)]∼σ = [τi (y)]∼σ

⇔ τ̃i ([x ]∼σ) = τ̃i ([x ]∼σ).



Hence, having an onto homomorphism σ : A → B, one can makethe quotient pre-automaton A by ∼σ

A/∼σ =(Q/∼σ, [q0], {τ̃i : Q/∼σ → Q/∼σ}i∈Σ

)where τ̃i ([x ]) = [τi (x)].

Question:A B

A/∼σ

σ

onto hom

canonicalhom

?



Consider the bijective map

σ̃ : A/∼σ −→ B

[x ] 7→ σ(x)

[q0] 7→ σ(q0) = q0′

Then σ̃ is a homomorphism:

[x ]

[τi (x)]

σ(x)

σ(τi (x)) = τi′(σ(x))

σ : hom

σ̃

τ̃i

σ̃

τi′

Consequently,A/∼σ ' B.



Without loss of genelarity, assume that all pre-automata are defined on {0, 1}.

So a pre-automaton is an algebra of type (0,1,1).

Consider the following fixed pre-automaton:

Ur =({0, 1}∗, λ ∈ {0, 1}∗, {ri : {0, 1}∗ → {0, 1}∗}i=0,1

)where ri (w1 . . .wn) = w1 . . .wni .

Claim: For any reduced pre-automaton

B =(P, p0 ∈ P, {τi : P → P}i=0,1

)there exists an onto homomorphism σB : Ur

onto−→ B such thatσB(w) = B(w). In other words

σB(w1 . . .wn) = τwn ◦ . . . ◦ τw2 ◦ τw1(p0).



Sketch of Proof.

B : a reduced pre-automaton⇒ σB : an onto-map

and w

wi

B(w)

σB(wi) = τi (B(w))

def σB ,B(w)

σB

ri

σB

τi

Hence,σB is a homomorphism.

Therefore,

Ur/∼σB ' B.



Example

The following two pre-automata are isomorphic:

q0start

q2

q1

1

0

0

1

0, 1

{λ}start

S

{10n}1

0

0

1

0, 1

where S = {0, 1}∗ \ ({λ} ∪ {10n | n ∈ N}).



Ur B

λ

10n

S

q0

q1

q2

σB

σB

σB



The minimal automatonLet L ⊆ {0, 1}∗ be a language. We are trying to construct aminimal pre-automaton M

Lto accept L. We will be needing some

ingredients as follows:

I Definition: For a language L ⊆ {0, 1}∗ and any x ∈ {0, 1}∗,define:

x\L = {u ∈ {0, 1}∗ | xu ∈ L}.

From this definition, it follows that for every x , y ∈ {0, 1}∗:

x\(y\L) = {u ∈ {0, 1}∗ | xu ∈ y\L}= {u ∈ {0, 1}∗ | yxu ∈ L}= (yx)\L.



I Definition: For a language L ⊆ {0, 1}∗, define a relation ∼ on{0, 1}∗ as follows: for x , y ∈ {0, 1}∗,

x ∼ y ⇔ x\L = y\L⇔ {u ∈ {0, 1}∗ | xu ∈ L} = {u ∈ {0, 1}∗ | yu ∈ L}.

In other words, x ∼ y if and only if for anyu ∈ {0, 1}∗, xu and yu are either both in L or bothare not in L.

I Note: The relation ∼ is indeed an equivalence relation; It iseasy to see that the relation ∼ is reflexive, symmetric, andtransitive, because the equality relation has these properties.



Now we are ready to construct the minimal pre-automatonaccepting L.

Claim: For a language L ⊆ {0, 1}∗, ML

= (Q , q0 , τ0 , τ1) is theminimal pre-automaton w.r.t. right concatenation accepting L,where

1. Q = {x\L | x ∈ {0, 1}∗},

2. q0 = λ\L,

3. τi (x\L) = i\(x\L) = (xi)\L, i = 0, 1.



ExampleL = {1 0n | n ∈ N}

λ\L = L

1\L = {0n | n ∈ N} = Z

1 0j\L = Z j ∈ N

0\L = ∅

0u\L = ∅ u ∈ {0, 1}∗

11u\L = ∅ u ∈ {0, 1}∗

101u\L = ∅ u ∈ {0, 1}∗

...

Lstart

∅

Z1

0

0

1

0, 1



Relations

Equivalence Relations

Congruence Relations

∅

{(x, x) | x ∈ {0, 1}∗}

{0, 1}∗ × {0, 1}∗

∼L

'L



I A language L ⊆ {0, 1}∗ determines an equivalence relation ∼L

on {0, 1}∗ in a usual way, i.e. for x , y ∈ {0, 1}∗

x ∼L y ⇔ x , y ∈ L ∨ x , y ∈ Lc .

I Now we define a congruence relation 'L on {0, 1}∗ as follows:for x , y ∈ {0, 1}∗,

x 'L y ⇔ x\L = y\L.

I Claim: 'L is the greatest congruence relation less than ∼L.



Claim1: The relation 'L is a congruence relation on Ur .

Proof. It is easy to see that 'L is an equivalence relation on{0, 1}∗, so it suffices to show that 'L is compatible with ri , fori=0,1:

x 'L y ⇔ x\L = y\L⇔ ∀u ∈ {0, 1}∗ (xu, yu ∈ L ∨ xu, yu 6∈ L)

⇔ ∀u ∈ {0, 1}∗ ∀i ∈ {0, 1} (xiu, y iu ∈ L ∨ xiu, y iu 6∈ L)

⇔ xi\L = y i\L⇔ xi 'L y i

⇔ ri (x) 'L ri (y).



Claim2: 'L ⊆ ∼L.

Proof.

x 'L y ⇒ x\L = y\L⇒ ∀u ∈ {0, 1}∗ (xu, yu ∈ L ∨ xu, yu 6∈ L)

⇒ for λ ∈ {0, 1}∗ (xλ, yλ ∈ L ∨ xλ, yλ 6∈ L)

⇒ x , y ∈ L ∨ x , y 6∈ L

⇒ x , y ∈ L ∨ x , y ∈ Lc

⇒ x ∼L y .



...

Claim3: For any congruence relation ρ on {0, 1}∗,

ρ ⊆ ∼L =⇒ ρ ⊆ 'L .

Proof.

(x , y) ∈ ρ ⇒ ∀i ∈ {0, 1} (ri (x), ri (y)) ∈ ρ⇒ ∀i ∈ {0, 1} (xi , y i) ∈ ρ⇒ ∀u ∈ {0, 1}∗ (xu, yu) ∈ ρ⇒ ∀u ∈ {0, 1}∗ xu ∼L yu⇒ ∀u ∈ {0, 1}∗ (xu, yu ∈ L ∨ xu, yu ∈ Lc)⇒ ∀u ∈ {0, 1}∗ (xu, yu ∈ L ∨ xu, yu 6∈ L)⇒ x\L = y\L⇒ x 'L y .



Now we can form the quotient pre-automaton Ur/'L and showthat

Ur/'L ' ML.

Therefore, it follows from the construction of Ur/'L that

ML

is the minimal pre-automaton w.r.t. right concatenationaccepting L.

In order to show this isomorphism, consider the map σ : Ur →Mby σ([x ]'L

) =M(x) = x\L for any x ∈ {0, 1}∗.



I σ is a well-defined map:

1. For any x = x1 . . . xn:

M(x) = M(x1 . . . xn)

= τxn ◦ . . . ◦ τx2 ◦ τx1 (λ\L)

= τxn ◦ . . . ◦ τx2 (x1\L)

= τxn ◦ . . . ◦ τx3 (x1x2\L)

...

= x1 . . . xn\L= x\L.

2. For any x , y ∈ {0, 1}∗:

[x ]'L= [y ]'L

⇔ x 'L y ⇔ x\L = y\L.



I σ is a homomorphism:

[x ]'L

[ri (x)]'L= [xi ]'L

x\L

xi\L

σ

r̃i

σ

τi

τi ◦ σ = σ ◦ r̃i .

I Obviously, σ is a bijective map.



In a similar way, we can treat L w.r.t. left concatenation as follows:

Definition: For a language L ⊆ {0, 1}∗ and any x ∈ {0, 1}∗, define:

L\x = {u ∈ {0, 1}∗ | ux ∈ L}.

It follows from the definition that for every x , y ∈ {0, 1}∗:

(L\x)\y = {u ∈ {0, 1}∗ | uy ∈ L\x}= {u ∈ {0, 1}∗ | uyx ∈ L}= L\(yx).



Hence, for a language L ⊆ {0, 1}∗ the minimal pre-automaton w.r.t.left concatenation accepting L is N

L= (Q , q0 , τ0 , τ1) , where

1. QL = {L\x | x ∈ {0, 1}∗},

2. q0 = L\λ,

3. τi (L\x) = (L\x)\i = L\(ix). i = 0, 1.



Example

L = {1 0n | n ∈ N}

L\λ = L

L\0 = L

L\0j = L j ∈ N

L\1 = {λ}

L\1 0j = {λ} j ∈ N

L\u = ∅

u 6= λ, u 6= 0j , 1 0j j ∈ N

Lstart

∅

{λ}1

0

0, 1

0, 1


theory of computationmath.sharif.ir/faculties/uploads/daneshgar/courses/... · theory of...

Documents