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.
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State Reduction: Row Matching
D
E
J
H
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State Reduction: Row Matching
Reduced State Table and Graph
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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
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Table 13-4
Implication Chart Method
Self-implied pairs
redundant
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Initial Chart
d-f square has an X
First Pass
-eliminating
implied pairs
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First Pass
Second Pass
-e.g. place X in square
a-g since square b-d
has an X.
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Original State Table
Second Pass
Reduced
State Table
-rows d, e
eliminated
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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 nonequivalence. 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
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state table before constructing the implication chart.
Equivalent Sequential Networks
Equivalent by
inspection of
State Graphs
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Equivalent Sequential Networks
Second Pass
e.g. Column A:
compare row A in state table for N1 with each of the
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rows S0, S1…S3 in state table for N2