State Reduction: Row Matching

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
State Reduction Row Matching
D
E
J
H
3
State Reduction Row Matching
Reduced State Table and Graph
4
Equivalent States
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, l (p,X) l
(q,X) and d (p,X) d (q,X) where l (p,X) is
the output given the present state p and input X
and d (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
Table 13-4
5
Implication Chart Method
Self-implied pairs redundant
6
Initial Chart
d-f square has an X
First Pass -eliminating implied pairs
7
First Pass
Second Pass -e.g. place X in square a-g since
square b-d has an X.
8
Original State Table
Second Pass
Reduced State Table -rows d, e eliminated
9
Implication Chart Method Summary
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 ºj. 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 Xs 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.
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
N1 with each of the rows S0, S1S3 in state table
for N2