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# L13 - Machine Minimization ECE 152A Fall 2006 Reading...

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1 Machine Minimization ECE 152A – Fall 2006 November 30, 2006 ECE 152A - Digital Design Principles 2 Reading Assignment Brown and Vranesic 8 Synchronous Sequential Circuits 8.6 State Minimization 8.6.1 Partitioning Minimization Procedure 8.6.2 Incompletely Specified FSMs

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2 November 30, 2006 ECE 152A - Digital Design Principles 3 Reading Assignment Roth 15 Reduction of State Tables / State Assignment 15.1 Elimination of Redundant States 15.2 Equivalent States 15.3 Determination of State Equivalence Using an Implication Table 15.4 Equivalent Sequential Circuits 15.5 Incompletely Specified State Tables November 30, 2006 ECE 152A - Digital Design Principles 4 Elimination of Redundant States Row Matching Recall CD player controller Mealy implementation contained two sets of rows with same next state and output Eliminate redundant states Row matching doesn’t identify “equivalent states” Row matching identifies “same state” Equivalent states are the more general case
3 November 30, 2006 ECE 152A - Digital Design Principles 5 Equivalent States Definitions of equivalent states Roth : 2 states equivalent iff for every single input x, outputs are the same and next states are equivalent (as opposed to row matching) Pairwise comparison using implication table Kohavi : 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 Moore reduction procedure to find equivalence partition November 30, 2006 ECE 152A - Digital Design Principles 6 Determination of State Equivalence using an Implication Table Find Equivalent Pairs 1 G C H 0 H B G 1 B F F 1 A C E 0 E A D 1 D E C 0 H F B 0 C D A z x=1 x=0 PS NS

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4 November 30, 2006 ECE 152A - Digital Design Principles 7 Determination of State Equivalence using an Implication Table (1) Construct Implication Table for Pairwise Comparison (2) First Pass Compare outputs For states to be equivalent, next state and output must be the same Put “X’s” where outputs differ November 30, 2006 ECE 152A - Digital Design Principles 8 Implication Table (first pass) B C D E F G H A B C D E F G X X X X X X X X X X X X X X X X 1 G C H 0 H B G 1 B F F 1 A C E 0 E A D 1 D E C 0 H F B 0 C D A z x=1 x=0 PS NS
5 November 30, 2006 ECE 152A - Digital Design Principles 9 Determination of State Equivalence using an Implication Table (3) One column (or row) at a time, find implied pairs November 30, 2006 ECE 152A - Digital Design Principles 10 Implication Table (second pass) B C D E F G H A B C D E F G D-F C-H X A-D C-E B-D C-H A-F E-H B-F H-H C-E A-D E-F B-D C-E D-G A-B E-H C-F A-B C-C A-G C-F B-G X X X X X X X X X X X X X X X 1 G C H 0 H B G 1 B F F 1 A C E 0 E A D 1 D E C 0 H F B 0 C D A z x=1 x=0 PS NS

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6 November 30, 2006 ECE 152A - Digital Design Principles 11 Determination of State Equivalence using an Implication Table (3) One column (or row) at a time, find implied pairs (cont) Remove self implied pairs A-D in cell A-D C-E in cell C-E
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