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Lecture 29

# Lecture 29 - ECE52 Spring 11 Lecture 29 Midterm 2 1 State...

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1 ECE52 Spring 11 Lecture 29 3/28/11 Midterm 2 4/8/11

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2 State Minimization In general, except for extremely simple machines, the first state diagram we write down will not be minimal – viz vending machine example. S 0 S 1 S 2 S 3 S 7 [Open] S 4 [Open] S 5 [Open] S 6 [Open] S 8 [Open] N D N D N D N D Reset Reset 10¢ 15+¢ [Open] N N N,D D D
3 In that case, we used intuition to minimize. To be more systematic: Two states S i and S j are said to be equivalent iff for every possible input sequence, the same output sequence will be produced regardless of whether S i or S j is the initial state straightforward but tedious to do manually automated in CAD tools An alternative approach can be used manually with some success: determine which states are NOT equivalent – simplifies subsequent equivalence problem!

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4 State Partitioning k-successors: for a FSM with one input w, state S j is a 1(0)- successor of state S i if the machine moves from S i to S j on input 1(0). For multiple inputs, generalize this to k input combinations. If 2 states are equivalent, all their k-successors for all k must also be equivalent State partition: a partition consists of one or more blocks, where each block comprises a subset of states that may be equivalent, but are definitely not equivalent to states in other blocks Step 1: assume all states might be equivalent – one block. Step 2: Partition such that states in each block generate same output values Continue forming new partitions by testing k-successors of states in each block – are all k-successors in same block? if not subdivide
5 Partition Example (Moore) P1= (ABCDEFG) – now observe outputs P2= (ABD)(CEFG) Now: 0-successors of ABD are BDB – all in same block 1-successors of ABD are CFG – also all in same block Present Next state Output state w = 0 w = 1 z A B C 1 B D F 1 C F E 0 D B G 1 E F C 0 F E D 0 G F G 0

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6 Partition Example: (ABD)(CEFG) CEFG 0-successors are FFEF – ok CEFG 1-successors are ECDG – NOT OK: state F different P3 = (ABD)(CEG)(F): (ABD) 0-successors BDB ok (ABD) 1 successors CFG – NOT ok – B not equivalent P4 = (AD)(B)(CEG)(F) – no further violations
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Lecture 29 - ECE52 Spring 11 Lecture 29 Midterm 2 1 State...

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