MIPS

MIPS


Definition: Millions of Instructions Per Second (a measure of computer speed)
 
A Bad Joke: Meaningless Information about Processing Speed

Gates

In thinking about Moore's Law, one can think of a transistor as a gate with two possible states ON and OFF

Boolean Functions

A Boolean Function (or switching function) is a function of the form f : Bk → B, where B = {0, 1} is a Boolean domain and k is a non-negative integer. In very general terms a computer program is a sequence of Boolean Functions

Boolean Logic

We briefly describe the relationship between Boolean Functions and gates, in terms of Boolean Logic.

  1. The two Boolean constants are MATH or MATH .

  2. Boolean variables MATH etc,can take on the values $\QTR{Large}{1}$ or $\QTR{Large}{0}$ .

  3. Boolean constants and Boolean variables are Boolean expressions.

  4. If $\QTR{Large}{p}$ and $\QTR{Large}{q}$ are Boolean expressions, then $\QTR{Large}{p}$ MATH $\QTR{Large}{q}$ is a Boolean expression, whose truth table is
     
         The Gate Diagram
     
        
    .

  5. If $\QTR{Large}{p}$ and $\QTR{Large}{q}$ are Boolean expressions, then $\QTR{Large}{p}$ $\QTR{Large}{\vee }$ $\QTR{Large}{q}$ is a Boolean expression, whose truth table is
     
         The Gate Diagram
     
        

  6. If $\QTR{Large}{p}$ is a Boolean expressions, then $\lnot $ $\QTR{Large}{p}$ is a Boolean expression, whose truth table is
     
         The Gate Diagram
     
        

In forming complex Boolean expressions, the precedence levels of logical operators is:

  1. $\lnot $

  2. MATH

  3. MATH

2 The Truth Table of Complex Boolean Expressions:

Using 4-6 above, we itteratively compute the Truth Tables of any Boolean expression. As an example, consider MATH MATH



We call two Boolean expressions , $\QTR{Large}{P}$ and $\QTR{Large}{Q}$ "equal" , written $\QTR{Large}{P=Q}$ , if they have the sameTruth Tables.

One should check that if $\QTR{Large}{R\ }$ is a Boolean expression containing $\QTR{Large}{P}$ and is MATH the Boolean expression that is the result of substituting $\QTR{Large}{Q}$ for every occurrence of $\QTR{Large}{P}$ then MATH .

3 The Key Observation, The Completeness of Boolean Operations

Any Boolean Function is the truth table of a Boolean expression that is the sum of products or product of sums of Boolean variables and their negations. Hence can be realized by conjoining a collection of these three basic gates

Indeed we make do with negation and sum, or negation and product.

The Next step? - Quantum Computing