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 M1M1 I1I1 O1O1 M2M2 I2I2 O2O2

3. Behavioral Equivalence Two machines, M 1 and M 2, are behaviorally equivalent iff –The sets of inputs are the same, I 1 = I 2, –The sets of outputs are the same, O 1 = O 2 and, there exists a behavioral equivalence relation (which is not necessarily injective) between the states...

4. Behavioral Equivalence

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

6. Behavioral Equivalence a s1s1 sjsj R (s 1 ) sksk R R a M1M1 M2M2

7. Behavioral Equivalence Example M1M1 A/0 C/0 B/1 a a a b b b

8. Behavioral Equivalence Example M2M2 1/0 2/1 a a b b

9. Behavioral Equivalence Example M1M1 A/ 0 C/ 0 B/ 1 a a a b b b M2M2 1/0 2/1 a a b b

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 Outputs M1M1 A/ 0 C/ 0 B/ 1 a a a b b b M2M2 1/0 2/1 a a b b

12. Verifying Morphism Need also to verify relation

13. M1M1 A/ 0 C/ 0 B/ 1 a a a b b b M2M2 1/0 2/1 a a b b Verifying Morphism

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

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

16. Mealy State SkSk a/o o b/o k c/o k a/o k b/o p c/o q

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

20. Mealy State Table

21. Equivalent Moore State Table