1 State Reduction: Row Matching Example 1, Section 14.3 is reworked, setting up enough states to remember the first three bits of every possible input sequence.

2 E D H J State Reduction: Row Matching

3 Reduced State Table and Graph

4 Theorem 15.1 Two states p and q of a sequential network are equivalent iff for every single input X, the outputs are the same and the next states are equivalent, that is,  (p,X) =  (q,X)  and  (p,X ) =  (q,X ) where  (p,X) is the output given the present state p and input X and  (p,X ) is the next state given the present state p and input X. The row matching procedure is a special case of this theorem in which the next states are actually the same instead of just being equivalent Equivalent States Table 13-4

5 Implication Chart Method Self-implied pairs redundant

6 Initial Chart First Pass -eliminating implied pairs d-f square has an X

7 First Pass Second Pass -e.g. place X in square a-g since square b-d has an X.

8 Original State Table Reduced State Table -rows d, e eliminated Second Pass

9 1. Construct a chart which contains a square for each pair of states. 2. Compare each pair of rows in the state table. If the outputs associated with states i and j are different, place an X in square i-j to indicate non- equivalence. If the outputs are the same, place the implied pairs in square i-j. (If the next states of i,j are m,n resp. then m-n is an implied pair.) Eliminate any self-implied pairs which are redundant by crossing then out. If the outputs and next states are the same (or if i-j only implies itself) place a check mark in square i-j to indicate i  j. 3. Second Pass: Go through the table column by column. Eliminate (place an X) the square with implied pair m-n, if square m-n contains an X. 4. If any X’s were added on a previous pass, repeat with an additional pass. 5. In the final chart, each square with co-ords i-j which does not contain an X implies the equivalence of i and j. If desired, row matching can be used to partially reduce the state table before constructing the implication chart. Implication Chart Method: Summary

10 Equivalent Sequential Networks Equivalent by inspection of State Graphs

11 Equivalent Sequential Networks Second Pass e.g. Column A: compare row A in state table for N 1 with each of the rows S 0, S 1 …S 3 in state table for N 2