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UMBC CMSC 313, Computer Organization & Assembly Language, Spring 2002, Section 0101

# Circuit Simplification II

#### Thursday 05/02, 2001

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**Assigned Reading in Murdocca & Heuring:**

**Assigned Reading in Neveln:**

**Assigned:**

**Due:**

**Topics Covered:**

- More examples of circuit simplification using Karnaugh maps:
We use the truth table for the sequence detector discussed
earlier. The circuits in SOP/POS
form were smaller than those we derived from simplification
using Boolean algebra, because we did not take full advantage
of the "don't cares" previously.
- The Quine-McCluskey algorithm is a generalization of
Karnaugh maps for simplifying combinational circuits. While
Karnaugh maps for up to 4 variables have a visual/graphical
appeal, they are not suitable for more than 6 variables and
are difficult to implement in a program. The Quine-McCluskey
algorithm can be used for any number of variables, but
the running time is exponential in the number of variables.
- Finite state machines can also be simplified through
state reduction. We used the following definition
of distinguished states.
- States X and Y of a finite state machine M are
distinguished if there exists an input r such that
the output of M in state X reading input r is
different from the output of M in state Y reading
input r.
- States X and Y of a finite state machine are
distinguished if there exists an input r such that
M in state X reading input r goes to state X',
M in state Y reading input r goes to state Y' and
we already know that X' and Y' are distinguished states.

- The state reduction algorithm makes passes through
all pairs of states (X,Y) that we have not already found
to be distinguished. For each pair (X,Y), we check if
X and Y are distinguished using the definition above.
If no new distinguished pairs are found during a pass,
then the algorithm terminates. At the end of the
algorithm, states that are not found to be distinguished
are in fact equivalent. (The proof of this "fact" is
usually given in an Automata Theory class like CMSC 451.
Though not hard, we'll dispense with the proof here.)
Equivalent states can be combined to form an equivalent
finite state machine with fewer states.
- The finite state machine constructed in this manner
has the smallest number of states possible. (Another
fact proven in CMSC 451.)

Last Modified:
14 Nov 2003 10:17:29 EST
by
Richard Chang
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