FINITE STATE AUTOMATA WITH OUTPUT

1

• FINITE STATEAUTOMATAWITH OUTPUT

2
????????????
TRANSDUCER

• The question of finite automata
• represent physical machines
• Investigate two models
• Mealy machine (G.H.Mealy, 1955)
• Moore machine (E.F. Moore, 1956)
• Original purpose design model for sequential
  circuits.

3
???????????????????
TRANSDUCER
MOORE MACHINE

• ?????
• A Moore machine is a collection of five things
• Finite set of states Q where q0 initial
  state.
• Input alphabet ? a, b, c, (letters).
• Output alphabet ? x, y, z, (characters).
• Transition function ? Q ? ? ? Q.
• Output function ? Q ? ?.

4
???????????????????
TRANSDUCER
MOORE MACHINE
EXAMPLE Input ? a, b Output ? 0, 1
States q0 (initial state), q1, q2, q3.
b
a
q1
0
q0
1
b
a
a
a
b
q2
0
q3
1
b
10010
EXAMPLE Given an input abab, the output is
5
???????????????????
TRANSDUCER
MOORE MACHINE
EXAMPLE Input ? a, b Output ? 0, 1
States q0 (initial state), q1, q2, q3.
b
b
a
q1
0
q0
0
a
b
a
Counting aab
a
q3
1
q2
0
b
Given an input aaababbaabb, the output is
000010000010
6
????????????????????
TRANSDUCER
MEALY MACHINE

• ?????
• A Mealy machine is a collection of five things
• Finite set of states Q where q0 initial
  state.
• Input alphabet ? a, b, c, (letters).
• Output alphabet ? x, y, z, (characters).
• Transition function ? Q ? ? ? Q.
• Output function ? Q ? ? ? ?.

7
????????????????????
TRANSDUCER
MEALY MACHINE
EXAMPLE Input ? a, b Output ? 0, 1
States q0 initial state.
q1
b/1
a/0
a/1
b/0
q0
q2
q3
a/0, b/1
b/1
a/1
EXAMPLE Given an input aaabb, the output is
01110
8
????????????????????
TRANSDUCER
MEALY MACHINE
EXAMPLE Input ? 0, 1 Output ? 0, 1
States q0 initial state
q1
0/0,1/1
0/1
0/1
INCREMENT MACHINE
1/0
q0
q2
1/0
1011
EXAMPLE Given an input 0011, the output is
9
?????????????????????????
TRANSDUCER
EQUIVALENCE

• ?????
• Two machines are equivalent if for the same
  input string, two machines result in the same
  output string.
• ????????
• Mealy machine can be equivalent to a Moore
  machine with deleting its automatic start output.

10
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS

• ???????
• Given a Moore machine, there is a Mealy machine
  that is equivalent to it.

a
a/t
b
qi
b/t
qi
t
becomes
c
c/t
11
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS

• EXAMPLE Counting aab

b
b
a
q1
/0
q0
/0
a
b
a
a
q3
/1
q2
/0
b
12
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS

• EXAMPLE Counting aab

b/0
b/0
a/0
q1
q0
a/0
b/0
a/0
a/0
q3
q2
b/1
13
?????????????????????????
TRANSDUCER
EQUIVALENCE

• ???????
• Given a Mealy machine, there is a Moore machine
  that is equivalent to it.

SOME RESULTS
a/1
a/0
a/1
b/1
b
b/1
qi1
1
b/1
a/1
qi
becomes
b/1
c
b/0
a
qio
0
b
c/0
c
14
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS
b/1
q1
a/0
a/1
b/0
q0
a/1
q2
b/0
q3
a/0
b/1
15
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS
b/1
0
a
q1
q00
b/0
a/1
b/0
1
a/1
q01
a/1
q2
b/0
q3
a/0
b
16
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS
b
0
1
a
q00
q11
a
b/0
b/0
1
a/1
q01
a/1
q2
b/0
q3
a/0
b
17
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS
b
0
1
a
q00
q11
a
a
b/0
a
1
a
q01
b
1
q21
b/0
q3
b
0
a
q20
b
18
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS
b
0
1
a
q00
q11
a
a
b
a
1
a
q01
b
1
q21
b
0
b
0
q30
a
q20
b
19
?????????????????????????
TRANSDUCER
EQUIVALENCE
SOME RESULTS

• LEFT (RIGHT) SEQUENTIAL STATE MACHINE
• A state machine M is said to be a left (right)
  sequential state machine if the input of M is
  taken into account serially in the most left
  (right) character first mode.
• Note
• Automata with output is usually called a
  transducer or 2-tape automata.

20
????? ??????
?
Define a finite automaton for binary division by 7
21
????? ??????
MULTIPLICATION BY 3
0/0
?
1/1
0/0
q0
q2
1/1
q1
0/1
1/0
10
1
22
????? ??????
DIVISION BY 3
0/0
?
1/1
0/0
q0
q2
1/1
q1
1/0
0/1
/10
/1
RIGHT TRANSDUCER