1. Behavioral Equivalence
Sequential Machine Theory
Prof. K. J. Hintz
Department of Electrical and Computer Engineering
Lecture 7
Modifications and updates by Marek Perkowski

2. Behavioral Equivalence
“Black Box” View of Machines M1 I1 O1 M2 I2 O2

3. Behavioral 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...

4. Behavioral Equivalence

5. Behavioral Equivalence
To show that R is a behavioral Equivalence, it is necessary to show that:

6. Behavioral Equivalencesj
R(s1) a a s1 sk R R M1 M2

7. Behavioral Equivalence Example

8. Behavioral Equivalence Example
2/1 b
1/0 b

9. Behavioral Equivalence Example
2/1 b a
1/0 b M1 a
B/1 b
A/0 a b b
C/0

10. Behavioral 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.

11. Equivalence of OutputsM2 a
2/1 b a
1/0 b M1 a
B/1 b
A/0 a b b
C/0

12. Verifying Morphism
Need also to verify relation

13. Verifying Morphism a M2 a
2/1 b a
1/0 b M1 a
B/1 b
A/0 a b b
C/0

14. How to check Behavioral Equivalence of Moore/Mealy
Constructive Approach
Moore to Mealy (easy)

15. Moore State a a b
Sk/0k b c c

16. Mealy State Sk
a/oo
b/ok
c/ok
a/ok
b/op
c/oq

17. Mealy 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.

18. Mealy to Moore Conversion

19. Mealy to Moore Conversion

20. Mealy State Table

21. Equivalent Moore State Table