# 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.
1. 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.
2. 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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