Theory of Relations (2)

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Theory of Relations (2)
Sequential Machines and Finite Automata
Course of Mathematics
Pusan National University
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Yoshhiro Mizoguchi

Institute of Mathematics for Industry
Kyushu University, JAPAN
ym@imi.kyushu-u.ac.jp

November 3-4, 2011

Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 1 / 34

Table of Contents

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1 Sequential Machine
Preliminary
Reachable and Observable
Mimimal Realization

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2 Finite Automata
Introduction to Theory of Automata
The Myhill-Nerode theorem
Minimal Realization

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3 Applications of Relational Calculus to Theory of Automata
Nondeterministic Finite Automaton
Coproduct and Product Automaton
Reverse, Concatenate, Closure
Examples

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4 References

November 3-4, 2011 2 / 34

Sequential Circuit (1)

1 2 0 1 0 2 0
0 1 0 0 1 1 0
dc S
c
Q
0 1 0 0 0 1 0

1 0 0 1 0 0 0
dc c
R

One of the elementary units of sequential circuits, the RS Flip-ﬂop circuit
produces an output signal sequence according to the sequence of input
signals. Output signals are 1(on) or 0(o f f ) so we write the outputs as
Y = {0, 1}. There are two input signals S(et) and R(eset). We consider the
pair (SR) of S and R, and we denote 0 = (00), 1 = (01) and 2 = (10). So
we can consider the inputs as X = {0, 1, 2}.
The model of a sequential circuit consists of the state set Q = {a, b}, the
state transition function δ : Q × X → Q, and the output function
β : Q → Y.
November 3-4, 2011 3 / 34


The table below is the value of δ and β. The ﬁgure is the state transition
diagram of the RS Flip-ﬂop circuit.

q x δ(q, x)
a 0 a
a 1 a q β(q) 0or1 0or2

‡ ‡
a 2 b a 0 a/0 2E b/1

b 0 b b 1 i1
b 1 a
b 2 b

The labels of a vertex consists of a state and an output symbol. An edge
means a state transition and the label on an edge corresponds to the input
symbol.

November 3-4, 2011 4 / 34


An sequential circuit can be considered as a function from an input word to
an output word. That is we can consider an sequential circuit as a function
f : X∗ → Y ∗ where X∗ is the set of words over X including an empty
string ε.
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Problem .
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Construct an algebraic model (Sequential Machine) of sequential
circuits using a state set, a state transition function and an output
function.
What kind of function from input words to output words is realizable
by a ﬁnite state sequential machine?
How to construct an efﬁcient sequential machine with small number of
. states.
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November 3-4, 2011 5 / 34

Epi-Mono Factorization

A function F : X → Y from a set X to a set Y can be represented by a
composition F(x) = m(e(x)) (x ∈ X) of an injective function m : Z → Y
and an surjective function e : X → Z . The set Z is uniquely determined by
f up to isomorphism. That is Z F(X) = {F(x) ∈ Y | x ∈ X} and
Z X/ ∼ = {[x] | x ∈ X}, where the equivalent relation ∼ on X is deﬁned
by [ x ∼ x′ ⇔ F(x) = F(x′ )] and [x] = {x′ ∈ X | x ∼ x′ } is a set of
equivalence class including x.
(Note)
A function e : X → Z is surjective if there exists an element x ∈ X satisfying
e(x) = z for any element z ∈ Z . A function m : Z → Y is injective if m(z1 ) m(z2 )
for any two elements z1 , z2 ∈ Z satisfying z1 z2 . We denote an injection by an
arrow with tail , and a surjective by an arrow with head .

November 3-4, 2011 6 / 34

Sequential Machine

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Definition (Sequential Machine) .
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A sequential mmachine is a sextuple M = (X, Q, δ, q0 , Y, β) where

X is the set of inputs,
Q is the set of states,
δ:Q×X →Q is the transition function,
q0 ∈ Q is the initial state,
Y is the set of outputs, and
. β:Q→Y is the output map.
.. .

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(Note) This sequential machine (SM) is called Moore style SM. The Mealy style
SM is defined by an alternate output map λ : Q × X → Y insted of β. These two
model are equivalent. Mealy style does not have an output for the initial state, but
the rest of relations between inputs and outputs are mutually transformable.
We sometime define SM by pentad without an initial state.

November 3-4, 2011 7 / 34

Run map and Response map

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Definition (Run map) .
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Let δ : Q × X → Q be a transition function. We define its run map to be
the unique map δ∗ : Q × X∗ → Q defined inductively by δ∗ (q, ε) = q, and
δ∗ (q, xw) = δ∗ (δ(q, x), w) ( q ∈ Q, x ∈ X, w ∈ X∗ ).
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Let f : X∗ → Y be a function. We also define the function f∗ : X∗ → Y ∗
defined inductively by f∗ (ε) = f (ε), and f∗ (wx) = f∗ (w) f (wx) ( x ∈ X,
w ∈ X∗ ).
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Definition (Response map) .
..
Let M = (X, Q, δ, q0 , Y, β) be a sequential machine. We define its
response map to be the map ( f M )∗ : X∗ → Y ∗ where f M : X∗ → Y is
defined by f M (w) = β(δ∗ (q0 , w)) (w ∈ X∗ ).
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November 3-4, 2011 8 / 34

Realization (1)

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Deﬁnition .
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Let t : X ∗ → Y ∗ be a function. If there exists a sequential machine M

such that t = ( f M )∗ , then M is called a realization of t .
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We call t : X∗ → Y ∗ is realizable if there exisits a function f : X∗ → Y
such that t = f∗ .
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Proposition .
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A function t : X ∗ → Y ∗ is realizable if and only if for any w ∈ X ∗ , x ∈ X

there exists y ∈ Y such that t(wx) = t(w)y.
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The condition is equivalent that the value t(wx) is depending on only w
and is not depending on x. We call a function t : X∗ → Y ∗ satisfying the
condition as a sequential function.

November 3-4, 2011 9 / 34

Realization (2)

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Proposition .
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Let t : X∗ → Y ∗ be a sequential function. Then there exist a sequential
machine M which is a realization of t .
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Let f : X∗ → Y be a function and t = f∗ . We introduce two kinds of
sequential machines which is a realization of t .
M I = (X, X∗ , δ I , ε, Y, f )
δ I (w, x) = wx (w ∈ X∗ , x ∈ X).
∗
MT = (X, Y X , δT , f, Y, βT )
∗
Y X is the set of all maps from X∗ to Y , that is { f | f : X∗ → Y},
δT ( f, x) : X∗ → Y is defined by δT ( f, x)(w) = f (xw) ( x ∈ X, w ∈ X∗ ),
∗
and βT ( f ) = f (ε) ( f ∈ Y X ).
We can verify easily f = f MI = f MT . We note that both M I and MT is not
a finite sequential machine. That is the state set is not a finite set.

November 3-4, 2011 10 / 34

Finite Realization

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Problem .
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What kind of sequential function f : X ∗ → Y which have a finite state

realization?
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The state sets of MT is infinite but the most of states are unreachable from
the initial state f . If the states reachable from f is finite, then we can
construct a finite sequential machine from MT . That is the condition to
∗
have a finite representation is that the set Z = {δ∗ ( f, w) ∈ Y X | w ∈ X∗ } is
T
finite.
∗
We define a function F : X∗ → Y X by F(w) = δ∗ ( f, w) (w ∈ X∗ ). F is
T
divided to the composition of a surjection and an injection. Since
Z = F(X∗ ), we have Z = X∗ / ∼ by the equivalence relation ∼ defined by
[w ∼ w′ ⇔ δ∗ ( f, w) = δ∗ ( f, w′ )]. We note δ∗ ( f, w)(z) = δ∗ ( f, w′ )(z) is
T T T T
f (wz) = f (w′ z) for any z ∈ X∗ .
If the number of the equivalence class is finite, then there exists a finite
realization.
November 3-4, 2011 11 / 34

Reachable and Observable

Let M = (X, Q, δ, q0 , Y, β) be a sequential machine and its response map
∗
f M : X∗ → Y . We deﬁne F : X∗ → Y X by F(w) = δ∗ ( f M , w).
T ∗
F is divided into a composition of fe : X∗ → Q and f m : Q → Y X such
that
F(w) = f m( fe (w)),
where

fe (w) = δ∗ (q0 , w), and
f m(q)(w) = β(δ∗ (q, w)) (w ∈ X∗ , q ∈ Q).

If fe is a surjection then we call M as reachable.
If f m is an injection then we call M as obserbable or reduced.
We note that a reachable and obserbable sequential machine is
minimal representation of f M .

November 3-4, 2011 12 / 34

Minimal Realization

Let M = (X, Q, δ, q0 , Y, β) be a sequential machine. and F = f m ◦ fe .
∗
Assume fe : X∗ → Q is a surjection. If f m : Q → Y X is not an injection,
then we can construct a minimal realization using the epi-mono
factorization of f m.
The equivalence relation [ q ∼ q′ ⇔ f m(q) = f m(q′ )] on Q is that
f m(q)(w) = f m(q′ )(w) for any w ∈ X∗ . That is β(δ∗ (q, w)) = β(δ∗ (q′ , w)). If
Q is a finite set and |Q| = n, then it is sufficient to check the condition
β(δ∗ (q, w)) = β(δ∗ (q′ , w)) for finite number of words w with |w| n. So we
can check q ∼ q′ for any state q and q′ in finite steps, and we can
construct a minimal state sequential machine.

November 3-4, 2011 13 / 34

Introduction to Theory of Automata

The first published study on automata [6] is the Automata studies in 1956
edited by C. E. Shanon who is famous as an originator of the information
theory and J. McCarthy who is a famous researcher of the fields artificial
intelligence.
At that time, they formalized an abstract model of a sequentil circuit and
investigated relationship between inputs and outputs analyzing a state
transition functions.
Once the notion of accept states is introduced, a machine is considered as
an acceptor and investigations of recognized language are started. This is
the origin of the theory of language and automata.
The first paper [4] about finite state automata is ’finite Automata and Their
Decision Problems’ by M. O. Rabin and D. Scott pubshed in 1959. They
were awarded an ACM Turing aword in 1976 for this research.

November 3-4, 2011 14 / 34

Alphabet, Word, Concatenation

An alphabet is a finite, nonempty set. The elements of an plphabet are
feferred to as letters, or symbols. A word over an alphabet is a finite
string consisting of zero or more letters of the alphabet, in which the same
letter may occur several times. The string consisting of zero letters is
called the empty word, written ε. The length of a word w, denoted by |w|,
is the number of letters in w. Again by definition, |ε| = 0.
Let Σ be an alphabet. The set of all words over an alphabet Σ is denoted
by Σ∗ . The sets Σ∗ is infinite for any Σ. Algebraically speaking, Σ∗ is the
free monoid with the identity ε generated by Σ.
For words w1 and w2 , the juxtaposition w1 w2 is called the concatenation
of w1 and w2 . The empty word is an identity with respect to concatenation,
εw = wε = w holds for all word w.

November 3-4, 2011 15 / 34

Kleene Closure

Subsets of Σ∗ are referred to as formal languages, or briefly, languages
over Σ.
The concatenation (or product) of two languages L1 and L2 is defined by
L1 L2 = {w1 w2 | w1 ∈ L1 , w2 ∈ L2 }.
The (Kleene) closure L∗ of a language L is defined to be the union of all
powers of L, that is L∗ = ∪∞ L n where L0 = {ε}, and L n = L n−1 · L
n=0
( n ≥ 1).
The closure of all words Σ is Σ∗ and there is no confusions defined as a
set of all words over Σ.

November 3-4, 2011 16 / 34

Deterministic Automata (1)

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Definition .
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Deterministic Automata A determinisiic finite automaton is a pentad
M = (Σ, Q, δ, q0 , F) where
Σ is the alphabet,
Q is the finite set of states,
δ:Q×Σ→Q is the transition function,
q0 ∈ Q is the initial state, and
. F⊂Q is the set of accept states.
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November 3-4, 2011 17 / 34

Deterministic Automata (2)

s1 s2 · · · s n inputs
T E move head to right after a state transition

Controler( q)
Finite Automaton M

A finite automaton is illustrated as above figure. The input letters are on
input tape. The first state of controller is the initial state q0 . If the state is q
and input letter is s then the state is changed to δ(q, s). After changing the
state the head is moved to right. Repeating these procedures until the end
of an input word. If the head reached to the end of an input word, then
check the state is accept state or not.

November 3-4, 2011 18 / 34

Recognized Language

Let δ : Q × Σ → Q be a state transition function. The function
δ∗ : Q × Σ∗ → Q is uniquely determined by δ∗ (q, ε) = q, and
δ∗ (q, wa) = δ(δ∗ (q, w), a) (w ∈ Σ∗ , a ∈ Σ).
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Deﬁnition .
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The recognized language L(M) ⊂ Σ∗ is deﬁned by

L(M) = {w ∈ Σ∗ | δ∗ (q0 , w) ∈ F}.

L(M) is referred to as the language accepted by M.
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November 3-4, 2011 19 / 34

Example

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Example .
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Let M = (Q, Σ, δ, q0 , F) be a ﬁnite automaton where Q = {q0 , q1 , q2 },
F = {q1 }, q0 ∈ Q. The state transition function δ : Q × Σ → Q is deﬁned
as follows
δ(q0 , a) = q1 , δ(q0 , b) = q2 ,
δ(q1 , a) = q2 , δ(q1 , b) = q0 ,
δ(q2 , a) = q2 , δ(q2 , b) = q2 .
For example, the word w = aba is acceptable δ∗ (q0 , aba)
= δ∗ (δ(q0 , a), ba) = δ∗ (q1 , ba) = δ∗ (δ(q1 , b), a) = δ∗ (q0 , a) = δ∗ (δ(q0 , a), ε)
= ∗
. δ (q1 , ε) = q1 ∈ F.
.. .

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November 3-4, 2011 20 / 34

State Transition Diagram

A ﬁnite automaton M is denoted by a following ﬁgure called state transition
diagram. Vertices are states and the initial state has an arrow without
label. According to the input letters follow arrows with same label with
input letter. The vertex corresponding to an accept state has double circle
and if the following the input letters ended at the vertices with double circle
then the input word is accepted.

a, b c

q0 ' b

q2 i
A
b q E
a
1
a

November 3-4, 2011 21 / 34

The Myhill-Nerode theorem (1)

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Definition .
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An equivalence relation ∼ on Σ∗ is said to be right invariant if
”w1 ∼ w2 ⇒ w1 z ∼ w2 z (∀z ∈ Σ∗ )” for any w1 , w2 ∈ Σ∗ .
An equivalence relation ∼ is finite index if the number of equivalence
classes is finite. That is {[x] | x ∈ Σ∗ } is finite set where
. [x] = {x′ | x ∼ x′ }.
.. .

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Let L Σ∗ . We define a relation w1 ∼ L w2 on Σ∗ by

w1 z ∈ L ⇔ w2 z ∈ L (∀z ∈ Σ∗ ).

The relation ∼ L is a right invariant equivalent relation.

November 3-4, 2011 22 / 34

The Myhil-Nerode theorem (2)

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Theorem (Nerode) .
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Let L Σ∗ be a language on Σ. Then, the following three conditions (1),
(2) and (3) are equivalent.
(1) The set L is acpted by some finite automaton.
(2) L is the union of some of the equivalent classes of a right invariant
equivalence relation of finite index.
(3) The right invariant equivalence relation ∼ L induced by L is of finite
. index.
.. .

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Let L be the language L in (3) and Q = {[w] | w ∈ Σ ∗ }, δ([w], a) = [wa],

q0 = [ε], F = {[w] | w ∈ L}. Then M = (Q, Σ, δ, q0 , F) is a finite automaton
and L = L(M). Further, if L = L(M′ ) for some finite automata
M′ = (Q′ , Σ, δ′ , q′ , F′ ) then |Q| ≤ |Q′ |. That is M is a minimal state
0
automaton with L = L(M).

November 3-4, 2011 23 / 34

Minimal Realization

Let M = (X, Q, X, δ, q0 , Y, β) be a sequential machine and Y = {0, 1}.
There is an one-to-one coresspondence between an output map
β : Q → {0, 1} and a subset F = {q ∈ Q | β(q) = 1} of Q. That is a finite
automata is considered as a sequential machine with Y = {0, 1}.
A finite sequential machine is exactly a finite automaton. Further, a
function f : X∗ → Y is corresponde to a subset of X∗ that is a recognized
language.
In the previous section, we construct a minimal realization of f using an
equivalence relation w ∼ w′ defined by f (wz) = f (wz′ ) (∀z ∈ X∗ ). This is
the equivalence relation ∼ L induced by L.

November 3-4, 2011 24 / 34

Example

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Example (Minimalize) .
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The automton M′ in right ﬁgure is the minimalized automaton of M in left
ﬁgure. We note p0 = {q0 , q2 }, p1 = {q1 , q3 }, and p2 = {q4 , q5 }.
.
.. .

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E
y‚ ‡
b
q0 ‡ b$
X
b a, b

q2 $
r

a
j
r
ˆ q5 a rr ‡
p1 '

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