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

# Boolean Functions & Truth Tables

#### Thursday 10/31, 2002

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**Assigned Reading in Murdocca & Heuring:** A.5-A.9

**Assigned Reading in Neveln:**

**Assigned:** HW3

**Due:** DigSim1

**Topics Covered:**

- Sum-Of-Products (SOP) and Product-Of-Sums (POS) circuits
for the MAJ3 function.
- Truth tables, Boolean formulas and Combinational circuits are
equivalent in the sense that they represent the same set of functions.
- Simplifying formulas and circuits using algebraic techniques.
- In general, simplifying formulas is a hard problem. There is
no known fast algorithm that can determine if a Boolean formula is
unsatisfiable (all inputs result in the formula evaluating to 0).
The existence of such an algorithm would solve the famous P versus NP
problem (which is open and is IMHO the most important problem in
computer science) and would imply, among other things, a fast algorithm
for cracking encryption schemes like RSA.
- Postulates and theorems of Boolean Algebra. (BTW, Associativity
should be a postulate instead of a theorem.) In general, a Boolean
Algebra is a mathematical structure with two operations + and *. The
operations should be closed (a + b and a * b must be members of the
underlying set), commutative, associative and distributive. In addition,
unique identity elements 0 and 1 must exist and unique complements
must exist. Notice that the definition of a Boolean Algebra does not
make reference to the number of elements in the underlying set nor
to an "addition table" for + and a "multiplication table" for *.
Nevertheless, the theorems for Boolean Algebra (e.g., DeMorgan's Law)
follow from the postulates.
- For Boolean Algebras over the two-element set {0,1} these
postulates and theorems can simply be verified using truth tables.
- Discussed Homework 3.

Last Modified:
14 Nov 2003 10:22:44 EST
by
Richard Chang
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