behavioral equivalence sequential machine theory prof. k. j. hintz department of electrical and...
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Behavioral EquivalenceBehavioral Equivalence
Sequential Machine Theory
Prof. K. J. HintzDepartment of Electrical and Computer
Engineering
Lecture 7
Modifications and updates by Marek Perkowski
Behavioral EquivalenceBehavioral EquivalenceBehavioral EquivalenceBehavioral Equivalence
“Black Box” View of Machines
M1I1 O1
1
M2I2 O2
2
Behavioral EquivalenceBehavioral Equivalence
Two machines, M1 and M2, are behaviorally equivalent iff– The sets of inputs are the same, I1 = I2,
– The sets of outputs are the same, O1 = O2
and, there exists a behavioral equivalence relation (which is not necessarily injective) between the states...
Behavioral EquivalenceBehavioral Equivalence
xsxs
x
,,
* then, if
,and
=)( Range
=)( Domain
: thatsuch
:
2
*
21
*
1
21
2
1
21
ISS
S
S
SS
R
R
R
R
Behavioral EquivalenceBehavioral Equivalence
To show that R is a behavioral Equivalence, it is necessary to show that:
kj ss
asasas
asasas
R
RR
R
IS
and
OO
IS
,,,
,,,
11211
21
11211
Behavioral EquivalenceBehavioral Equivalence
a
s1
sj R(s1)
sk
RR
a
M1 M2
kj ss
asasas
asasas
R
RR
R
IS
and
OO
IS
,,,
,,,
11211
21
11211
Behavioral Equivalence ExampleBehavioral Equivalence Example
M1
A/0
C/0
B/1a
a
a
b
b
b
PS a b o/p
A B A 0
B B C 1
C B A 0
Behavioral Equivalence ExampleBehavioral Equivalence Example
M2
1/0
2/1aa
b
PS a b o/p
1 2 1 0
2 2 1 1
b
Behavioral Equivalence ExampleBehavioral Equivalence Example
122*21
*1
2
1
21
21
where,,
2,1 Range
Domain
1,,2,,1,
1,01,0
,,
ssxsxs
CB,A,
CBA
baba
R
R
R
=R
S
S
OO
II
M1
A/0
C/0
B/1
a
a
a
b
b
b
PS a b o/p
A B A 0
B B C 1
C B A 0
M2
1/0
2/1a a
b
bPS a b o/p
1 2 1 0
2 2 1 1
Behavioral Equivalence ExampleBehavioral Equivalence Example
• Since This Is a Moore Machine, It Is Only Necessary to Check Each State, i.e.,
• For Mealy Machine, Also Need to Check For Each Input.
ii ss R21
Equivalence of OutputsEquivalence of Outputs
checks state so ,01 0
1
checks state so ,12 1
2
checks state so ,01 0
1
21
21
21
21
CC
C
BB
B
AA
A
R
R
R
MM
M1
A/0
C/0
B/1
a
a
a
b
b
b
PS a b o/p
A B A 0
B B C 1
C B A 0
M2
1/0
2/1a a
b
bPS a b o/p
1 2 1 0
2 2 1 1
Verifying MorphismVerifying Morphism
Need also to verify relation
inputs all and states allfor .
binput for checks state so, ,11
,1
,,
ainput for checks state so, ,22
,1
,,
,,,
2
21
2
21
121
etc
A
bA
bAbA
A
aB
aAaA
asasas iii
R
RR
R
RR
RR IS
M1
A/0
C/0
B/1
a
a
a
b
b
bPS a b o/p
A B A 0
B B C 1
C B A 0
M2
1/0
2/1a a
b
bPS a b o/p
1 2 1 0
2 2 1 1
inputs all and states allfor .
binput for checks state so, ,11
,1
,,
ainput for checks state so, ,22
,1
,,
,,,
2
21
2
21
121
etc
A
bA
bAbA
A
aB
aAaA
asasas iii
R
RR
R
RR
RR IS
Verifying MorphismVerifying Morphism
How to check Behavioral How to check Behavioral Equivalence of Moore/MealyEquivalence of Moore/Mealy
How to check Behavioral How to check Behavioral Equivalence of Moore/MealyEquivalence of Moore/Mealy
• Constructive Approach– Moore to Mealy (easy)
21
21
2
1
Let
, , , , =Mealy
, , , , = Moore
OO
II
OIS
OIS
M
M
Moore StateMoore State
Sk/0k
a
b
c
c
b
a
a b c
sk ok
sl sk
sm sk
sn sk
Mealy StateMealy State
a b c
sk (so,oo) (sp,op) (sq,oq)
sl (sk,ok)
sm (sk,ok)
sn (sk,ok)
Sk
a/oob/ok
c/ok
a/ok
b/op
c/oq
Mealy to Moore ConversionMealy to Moore Conversion
• Construct a New Set of States Consisting of Each Present State Combined With Each of the Possible Outputs. Some of the States With Outputs May Not Be Reachable.
• A More Sophisticated Method Would Generate a New Set of States Consisting Only of Those State/Output Combinations Reached by the Head of an Arrow.
Mealy to Moore ConversionMealy to Moore Conversion
Mealy to Moore ConversionMealy to Moore Conversion
Mealy State TableMealy State Table
a b c d
R ( R, x ) ( R, 0 )
P ( R, 0 )
Q ( R, 1 )
Equivalent Moore State TableEquivalent Moore State Table
a b c d o/p
R1 R3 R1 0
R2 R3 R1 1
R3 R3 R1 x
P R1
Q R2