Deterministic finite automata

Deterministic Finite Automata
Definition: A deterministic finite automaton (DFA) consists of
1. a finite set of states (often denoted Q)
2. a finite set  of symbols (alphabet)
3. a transition function that takes as argument a state and a
symbol and returns a state (often denoted )
4. a start state often denoted q0
5. a set of final or accepting states (often denoted F )
We have q 0  Q and F  Q

So a DFA is mathematically represented as a 5-uple
(Q, , , q 0, F )
The transition function  is a function in
Q ×   Q
Q ×  is the set of 2-tuples (q, a) with q  Q and a  

How to present a DFA? With a transition table
 0q
 1q
q2
0
q2
q1
q2
1
q0
q1
q1
The  indicates the start state: here q0
The  indicates the final state(s) (here only one final state q 1)
This defines the following transition diagram
1 0
0 1q0 q
2
q1 0 1,

For this example
Q = {q 0, q 1, q 2}
start state q0
F = {q 1}
 = {0, 1}
 is a function from Q ×  to Q
 : Q ×   Q
 (q 0, 1) = q0
 (q 0, 0) = q2

Example: password
When does the automaton accepts a word??
It reads the word and accepts it if it stops in an accepting state
t h e nq0
6 t
q5
=
q
1
q2 q3 q4
6 h= 6 e=
6 n=
Only the word then is accepted
Here Q = { 0q , q 1, q 2, q 3, q 4}
 is the set of all characters
F = {q 4}
We have a "stop" or "dead" state q 5, not accepting

How a DFA Processes Strings
Let us build an automaton that accepts the words that contain 01
as a subword
 = {0, 1}
L = {x01y | x, y   }
We use the following states
A: start
B: the most recent input was 1 (but not 01 yet)
C: the most recent input was 0 (so if we get a 1 next we should go
to the accepting state D)
D: we have encountered 01 (accepting state)

We get the following automaton
1
1
A B
0
0
0
C
1
D 0 1,
Transition table
A
B
C
D
0
C
C
C
D
1
B
B
D
D
Q = {A,B,C,D},  = {0,1}, start state A, final state(s) {D}

Extending the Transition Function to Strings
In the previous example, what happens if we get 011? 100? 10101?
We define ˆ(q, x) by induction
ˆ : Q ×    Q
BASIS ˆ(q, ,) = q for |x| = 0
INDUCTION suppose x = ay (y is a string, a is a symbol)
 (ˆq , ay) = ˆ((q, a), y)
Notice that if x = a we have
 (ˆq , a) = (q, a) since a = a, and ˆ((q, a), ,) = (q, a)

ˆ
Extending the Transition Function to Strings
: Q ×    Q
We write q.x instead of ˆ(q, x)
We can now define mathematically the language accepted by a
given automaton Q, , , q 0, F
L = {x    | q 0.x  F }
On the previous example 100 is not accepted and 10101 is accepted

Minimalisation
The same language may be represented by diferent DFA
1
0
1 0 1
A B
0
C D 0 1,
and
1
0
A
0
B
1
C 0 1,

Minimalisation
Later in the course we shall show that there is only one machine
with the minimum number of states (up to renaming of states)
Furthermore, there is a (clever) algorithm which can find this
minimal automaton given an automaton for a language

Mn
Example
the "cyclic" automaton with n states on  = {1} such that
l
L (M n) = {1 | n divides l}

Functional representation: Version 1
Q = A|B|C and E = 0|1 and W = [E]
One function next : Q × E  Q
next (A, 1) = A, next (A, 0) = B
next (B, 1) = C, next (B, 0) = B
next (C, b) = C
One function run : Q × W  Q
run (q, b : x) = run (next (q, b), x),
accept x = final (run (A, x)) where
final A = final B = F alse,
run (q, []) = q
final C = T rue

E = 0|1,
W = [E]
Three functions F A, F B, FC
FA
FB
FC
(1 : x) = FA x,
(1 : x) = FC x,
(1 : x) = FC x,
FA
FB
FC
: W  Bool
(0 : x) = FB x,
(0 : x) = FB x,
(0 : x) = FC x,
FA
FB
FC
[] = F alse
[] = F alse
[] = T rue
We have a mutual recursive definition of 3 functions

data Q = A | B | C
data E = O | I
next :: Q -> E -> Q
next A I = A
next A O = B
next B I = C
next B O = B
next C _ = C
run :: Q -> [E] -> Q
run q (b:x) = run (next q b) x
run q [] = q

accept :: [E] -> Bool
accept x = final (run A x)
final :: Q -> Bool
final A = False
final B = False
final C = True

We have
Q -> E -> Q ~
~
Q x E -> Q
E -> (Q -> Q)

data Q = A | B | C
data E = O | I
next :: E -> Q -> Q
next I A = A
next O A = B
next I B = C
next O B = B
next _ C = C
run :: Q -> [E] -> Q
run q (b:x) = run (next b q) x
run q [] = q

-- run q [b1,...,bn] is
-- next bn (next b(n-1) (... (next b1 q)...))
-- run = foldl next

A proof by induction
A very important result, quite intuitive, is the following.
Theorem: for any state q and any word x and y we have
q.(xy) = (q.x).y
Proof by induction on x. We prove that: for all q we have
q.(xy) = (q.x).y (notice that y is fixed)
Basis: x = , then q.(xy) = q.y = (q.x).y
Induction step: we have x = az and we assume q 0.(zy) = (q 0.z).y
for all q0

The other definition of ˆ
Recall that a(b(cd)) = ((ab)c)d; we have two descriptions of words
We define ˆ0 (q, ,) = q and
 (0ˆq, xa) = (ˆ0 (q, x), a)
Theorem: We have q.x = ˆ(q, x) = ˆ0 (q, x) for all x

The other definition of ˆ
Indeed we have proved
q., = q and q.(xy) = (q.x).y
As a special case we have q.(xa) = (q.x).a
This means that we have two functions f(x) = q.x and
g (x) = ˆ0 (q, x) which satisfy
f (,) = g(,) = q and
f (xa) = f(x).a g (xa) = g(x).a
Hence f(x) = g(x) for all x that is q.x = ˆ0 (q, x)

Automatic Theorem Proving
f(0) = h(0) = 0, g(0) = 1
f (n + 1) = g(n), g(n + 1) = f(n), h(n + 1) = 1  h(n)
We have f(n) = h(n)
We can prove this automatically using DFA

We have 8 states: Q = {0, 1} × {0, 1} × {0, 1}
We have only one action  = {1} and ((a, b, c), s) = (b, a, 1  c)
The initial state is (0, 1, 0) = (f(0), g(0), h(0))
Then we have (0, 1, 0).1n = (f(n), g(n), h(n))
We check that all accessible states satisfy a = c (that is, the
property a = c is an invariant for each transition of the automata)

A more complex example
f(0) = 0
f(2) = 1
f(1) = 1
f(3) = 0
f (n + 2) = f(n) + f(n + 1)  f(n)f(n + 1)
f(4) = 1 f(5) = 1 . . .
Show that f(n + 3) = f(n) by using Q = {0, 1} × {0, 1} × {0, 1}
and the transition function (a, b, c) 7 (b, c, b + c  bc) with the
initial state (0, 1, 1)

Product of automata
How do we represent interaction between machines?
This is via the product operation
There are diferent kind of products
We may then have combinatorial explosion: the product of n
automata with 2 states has 2n states!

Product of automata (example)
p0
The product of p1 A B p1 and
p0
p1 p0
p0 C D p0 is
p1
p1
A, C
p1
A, D
p0
B, C
p1
B, D
p1
p0
p0
If we start from A, C and after the word w we are in the state A,D
we know that w contains an even number of p 0s and odd number of
p 1s

Model of a system of users that have three states I(dle),
R(equesting) and U(sing). We have two users for k = 1 or k = 2
Each user is represented by a simple automaton
rk
ik
uk

The complete system is represented by the product of these two
automata; it has 3 × 3 = 9 states
i1 , i2 r1 , i2 u1 , i2
i1 , r2 r1 , r2 u1 , r2
i1 , u2 r1 , u2 u1 , u2

The Product Construction
Given A 1 = (Q 1, ,  1, q 1, F 1) and A 2 = (Q 2, ,  2, q 2, F 2) two DFAs
with the same alphabet  we can define the product A = A 1 × A2
set of state Q = Q 1 × Q2
transition function (r 1, r 2).a = (r 1.a, r 2.a)
intial state q 0 = (q 1, q 2)
accepting states F = F 1 × F2

Lemma: (r 1, r 2).x = (r 1.x, r 2.x)
We prove this by induction
BASE: the statement holds for x = ,
STEP: if the statement holds for y it holds for x = ya

Theorem: L(A 1 × A 2) = L(A 1)  L(A 2)
Proof: We have ( 1q , q 2).x = (q 1.x, q 2.x) in F if q 1.x  F1 and
q 2.x  F 2, that is x  L(A 1) and x  L(A 2)
Example: let Mk be the "cyclic" automaton that recognizes
multiple of k, such that L(M k) = {an
M 6 × M 9 ' M18
| k divides n}, then
Notice that 6 divides k and 9 divides k if 18 divides k

Product of automata
It can be quite difcult to build automata directly for the
intersection of two regular languages
Example: build a DFA for the language that contains the subword
ab twice and an even number of a's

Variation on the product
We define A 1 ⊕ A2
accepting state
as A 1 × A2 but we change the notion of
(r 1, r 2) accepting if r 1  F1
Theorem: If A1 and A2
or r 2  F2
are DFAs, then
L (A 1 ⊕ A 2) = L(A 1)  L(A 2)
Example: multiples of 3 or of 5 by taking M 3 ⊕ M5

Complement
If A = (Q, , , q 0, F ) we define the complement A ¯ of A as the
automaton
¯ = (Q, , , q 0, Q  F )
Theorem: If A is a DFA, then L( ¯A) =    L(A)
Remark: We have A ⊕ A 0 = A × A0
A

Languages
Given an alphabet 
A language is simply a subset of 
Common languages, programming languages, can be seen as sets of
words
Definition: A language L   is regular if there exists a DFA
A , on the same alphabet  such that L = L(A)
Theorem: If L 1, L2 are regular then so are
L 1  L 2, L 1  L 2,    L1

Remark: Accessible Part of a DFA
Consider the following DFA
0 0
q0
0
1
q1
1
q2
0
0
q3
1
it is clear that it accepts the same language as the DFA
0
q0
0
1
q1
1
which is the accessible part of the DFA
The remaining states are not accessible from the start state and
can be removed

The set
Acc = {q 0.x | x   }
is the set of accessible states of the DFA (states that are accessible
from the state q 0)

Proposition: If A = (Q, , , q 0, F ) is a DFA then and
A0
Proof: It is clear that A 0 is well defined and that L(A 0)  L(A).
If x  L(A) then we have q 0.x  F and also q 0.x  Acc. Hence
q 0.x  F  Acc and x  L(A 0).
= (Q  Acc, , , q 0, F  Acc) is a DFA such that L(A) = L(A 0).

Take  = {a, b}.
Define L set of x    such that any a in x is followed by a b
Define L 0 set of x    such that any b in x is followed by a a
Then L  L 0 = {,}
Intuitively if x 6= , in L we have
. . . a . . .  . . . a . . . b . . .
if x in L 0 we have
. . . b . . .  . . . b . . . a . . .

We should have L  L 0 = {,} since a nonempty word in L  L0
should be infinite
We can prove this automatically with automata!
L is regular: write a DFA A for L
L 0 is regular: write a DFA A 0 for L
We can then compute A × A 0 and check that
L  L 0 = L(A × A ) = {,} 0

Application: control system
We have several machines working concurrently
We need to forbid some sequence of actions. For instance, if we
have two machines MA and MB, we may want to say that MB
cannot be on when MA is on. The alphabets will contain: onA,
ofA, onB, ofB
Between onA, on2 there should be at least one ofA
The automaton expressing this condition is
6=onA,onB
p0
onB ofB
p3
ofA
onA
6=onB,ofA
p1
onB
p2
onA

Application: control system
What is interesting is that we can use the product construction to
combine several conditions
For instance, another condition maybe that onA should appear
before onB appear. One automaton representing this condition is
6=onA,onB
q0
onB
q2
onA
q1
We can take the product of the two automata to express the two
conditions as one automaton, which may represent the control
system

Deterministic finite automata

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Deterministic finite automata