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5 - State Reduction

# 5 - State Reduction - State Reduction 79 Equivalent...

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79 State Reduction

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80 Equivalent: Definition: Two states A and B are said to be equivalent (i.e., A ° B) iff, for every possible input sequence, the sequential machine produces the same output regardless of whether A or B is the initial state. Implication Table 1. Reduce the following state table. a. Implication Table Present State X=0 X=1 A G/0 F/0 B E/0 C/1 C G/0 G/0 D A/1 G/0 E B/1 A/0 F D/0 E/1 G H/0 E/1 H C/1 F/0 Next State/output B C FG D E AB AG F ED EC G EH EC DH H AC FG BC AF A B C D E F G
81 b. Merger Diagram Equivalent Classes (AC)(DH)(FG) (B) (E) A D F B E c. Reduced Table Present State X=0 X=1 A F/0 F/0 B E/0 A/1 D A/1 F/0 E B/1 A/0 F D/0 E/1 Next State/output

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82 Example 41: (State Reduction) Reduce the next state table shown below. X 1 X 2 Present State 00 01 11 10 A B/0 C/0 B/1 A/0 B E/0 C/0 B/1 D/1 C A/0 B/0 C/1 D/1 D C/0 D/0 A/1 B/0 E C/0 C/0 C/1 E/0 Next State/output Step A: Implication Table. B C AE D BC CD AB E BC CD AC BE A B C D Step B: Merger Diagram. Step C: Equivalent Classes. (AE)(BC)(D) A B D
83 Step D: Reduced Table. X 1 X 2 X 1 X 2 Present State 00 01 11 10 Present State 00 01 11 10 A B/0 BC /0 B/1 A/0 A B/0 B/0 B/1 A/0 B AE /0 BC /0 B/1 D/1 B A/0 B/0 B/1 D/1 C A/0 B/0 C/1 D/1 ° D B/0 D/0 A/1 B/0 D BC /0 D/0 A/1 B/0 Next State/output E C/0 C/0 C/1 E/0 Next State/output

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84 State Reduction Example 42: Reduce the following next state table. Input Present State X=0 X=1 A A/0 A/0 B A/1 D/1 C A/1 E/0 D E/0 E/0 E A/0 D/0 Next State/Output Step 1: Implication Table. B C D AE E AD AE A B C D Step 2: Merger Diagram. Step 3: Equivalent Classes. (ADE)(B)(C) A B C Step 4: Reduced Table. Input Present State X=0 X=1 A A/0 A/0 B A/1 A/1 C A/1 A/0 Next State/Output
85 Incompletely Specified Machine When a state machine is used as part of a large digital system, it frequently happens that certain sequences will never occur as inputs to the sequential network. In other cases, the output of the state machine is only observed at certain times rather than at every clock time. Such restrictions lead the unspecified next states or outputs in the state table. When such don’t cares are present, we say that the state table incompletely specified. Example 43: Network A can only generate two possible output sequences, X=100 and X=110. Thus, the state machine B has only two possible input sequences. When the third input in the sequence is received, the output of B is to be Z=0 if 100 was received and Z=1 if 110 was received. Assume that network C ignores the value of Z at other times so that we don’t care what Z is during the first two inputs in the X sequence. Design a minimal state machine B. State Design. Next State Table. Present State X=0 X=1 S 0 -/- S 1 /- S 1 S 2 /- S 3 /- S 2 S 0 /0 -/- S 3 S 0 /1 -/- Next State/Output

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86 Definition 1 : Two states are compatible if for each possible input they have the same output whenever specified and their next states are compatible whenever they are specified.
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