Behavioral Equivalence

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Presentation transcript:

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

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

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...

Behavioral Equivalence

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

Behavioral Equivalence sj R(s1) a a s1 sk R R M1 M2

Behavioral Equivalence Example

Behavioral Equivalence Example 2/1 b 1/0 b

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

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.

Equivalence of Outputs M2 a 2/1 b a 1/0 b M1 a B/1 b A/0 a b b C/0

Verifying Morphism Need also to verify relation

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

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

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

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

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.

Mealy to Moore Conversion

Mealy to Moore Conversion

Mealy State Table

Equivalent Moore State Table