# Combinational logic

#### 4/25/00

Combinational logic

Possible logic functions of two variables

Cost of different logic functions

Minimal set of functions

An algebraic structure

Boolean algebra

Logic functions and Boolean algebra

Axioms and theorems of Boolean algebra

Axioms and theorems of Boolean algebra (contĺd)

Axioms and theorems of Boolean algebra (contĺ)

Axioms and theorems of Boolean algebra (contĺ)

Proving theorems (rewriting)

Proving theorems (perfect induction)

A simple example

Apply the theorems to simplify expressions

From Boolean expressions to logic gates

From Boolean expressions to logic gates (contĺd)

From Boolean expressions to logic gates (contĺd)

Waveform view of logic functions

Choosing different realizations of a function

Which realization is best?

Which is the best realization? (contĺd)

Are all realizations equivalent?

Implementing Boolean functions

Canonical forms

Sum-of-products canonical forms

Sum-of-products canonical form (contĺd)

Product-of-sums canonical form

Product-of-sums canonical form (contĺd)

S-o-P, P-o-S, and de Morganĺs theorem

Four alternative two-level implementations of F = AB + C

Waveforms for the four alternatives

Mapping between canonical forms

Incompleteley specified functions

Notation for incompletely specified functions

Simplification of two-level combinational logic

The uniting theorem

Boolean cubes

Mapping truth tables onto Boolean cubes

Three variable example

Higher dimensional cubes

m-dimensional cubes in a n-dimensional Boolean space

Karnaugh maps

Karnaugh maps (contĺd)

Karnaugh map examples

More Karnaugh map examples

Karnaugh map: 4-variable example

Karnaugh maps: donĺt cares

Karnaugh maps: donĺt cares (contĺd)

Design example: two-bit comparator

Design example: two-bit comparator (contĺd)

Design example: two-bit comparator (contĺd)

Design example: 2x2-bit multiplier

Design example: 2x2-bit multiplier (contĺd)

Design example: BCD increment by 1

Design example: BCD increment by 1 (contĺd)

Definition of terms for two-level simplification

Examples to illustrate terms

Algorithm for two-level simplification

Algorithm for two-level simplification (example)

Combinational logic summary

Author: Gaetano Borriello