CS 2011 Test 3 Practice Problems

CS 2011, A Term 1999
Professor Sergio A. Alvarez
Practice Problems for Test 3

Instructions: Read each problem carefully. Write your answers clearly. Include explanations. Unexplained answers may not receive full credit. Good luck!

A C++ program contains the following declaration of an external assembly procedure:
```
   extern "C" void doublebyref(int x, int& y);
```
- (a) Give the full proc / endp directives for the assembly procedure. Do not include any explicit parameter declarations.
```
   _doublebyref proc
      ...
   _doublebyref endp
```
- (b) Write assembly instructions that set up a stack frame within the assembly procedure and then copy the value of the parameter x to register ax. Previous contents of ax should be saved for future use.
```
   push bp
   mov bp,sp
   push ax
   mov ax,[bp+4]
```
- (c) Write assembly instructions that double the value of x and store the result in the location indicated by y. You may assume that the stack frame has previously been set up and that x has been copied to registers ax as described in part (b) of this problem.
```
   add ax,ax
   mov di,[bp+6]
   mov [di],ax
```
- (d) Write assembly instructions for the assembly procedure that finalize the stack frame of part (b), perform any necessary final tasks, and return control to the calling program.
```
   pop ax
   pop bp
   ret
```

Which of the following Boolean operators are complete in the sense that each of them alone (in sufficient quantities) suffices to compute any Boolean expression? For each operator that you claim is complete, give a general argument in support of your claim. For each operator that you claim is not complete, give a specific truth table that cannot be reproduced by any combination of operators of the given type.

(a) AND

Not complete. Example: NOT:

   A   NOT A 

   T     F
   F     T

(b) OR

Not complete. Example: NOT:

   A   NOT A 

   T     F
   F     T

Not complete. Example: AND

   A   B   A AND B

   T   T      T
   F   T      F
   T   F      F
   F   F      F

(d) NOR

Complete. Enough to show that all disjunctive normal forms
may be expressed in terms of NOR. In turn, enough to show
that NOT, AND, OR may be expressed in terms of NOR.
Here goes:

   NOT A = A NOR A
   A OR B = NOT (A NOR B) = (A NOR B) NOR (A NOR B)
   A AND B = (NOT A) NOR (NOT B) = (A NOR A) NOR (B NOR B)

Consider the Boolean function g(A,B,C,D) that returns 1 if the 4-bit unsigned binary integer ABCD equals 0, 1, 2, 3, 8, or 10 (decimal), and returns 0 otherwise.

(a) Give the disjunctive normal form (standard sum of products expression) for g, based on its truth table. Include the truth table in your answer.

   Truth table:

   A B C D       g

   0 0 0 0       1   A'B'C'D'
   0 0 0 1       1   A'B'C'D
   0 0 1 0       1   A'B'CD'
   0 0 1 1       1   A'B'CD
   0 1 0 0       0   
   0 1 0 1       0
   0 1 1 0       0
   0 1 1 1       0
   1 0 0 0       1   AB'C'D'
   1 0 0 1       0   
   1 0 1 0       1   AB'CD'
   1 0 1 1       0
   1 1 0 0       0  
   1 1 0 1       0 
   1 1 1 0       0  
   1 1 1 1       0

   Expression: A'B'C'D' + A'B'C'D + A'B'CD' + A'B'CD + AB'C'D' + AB'CD'

(b) Draw a Karnaugh map for g and use it to find a simplified equivalent Boolean expression for g as a sum of exactly two products, each containing exactly two factors.

     CD: 00 01 11 10   Group four 1's in top row (A'B')
   AB                  Group four 1's in corners (B'D')
   00     1  1  1  1   
   01     0  0  0  0
   11     0  0  0  0   Final expression: A'B' + B'D'
   10     1  0  0  1

(a) Using only standard two-input NOR gates, construct a logic circuit that computes the Boolean function

   f(A,B,C) = A'B + C

Your answer may be given either as a circuit diagram, or else as a symbolic expression using the notation X NOR Y. In any case, explain your answer.

   Ingredients:
   X' = X NOR X
   X AND Y = X' NOR Y'
   X OR Y = (X NOR Y)' = (X NOR Y) NOR (X NOR Y)

   Therefore,
      f(A,B,C) = ((A NOR B') NOR C)'
               = ((A NOR B') NOR C) NOR ((A NOR B') NOR C)

   This translates easily to a circuit.

(b) Use a Karnaugh map to find the simplest possible Boolean sum of products expression for the function h(A,B,C,D) whose value is 1 if the unsigned binary integer ABCD is greater than or equal to 11d, and is 0 otherwise.

Here's the Karnaugh map:

      CD 00 01 11 10
   AB
   00    
   01
   11     1  1  1  1
   10           1

Cover the horizontal row of four 1's using the expression AB.
Cover the vertical column of two 1's using the expression ACD.

answer: AB + ACD

Consider the basic latch circuit discussed in the lectures, containing two NOR gates, the first having output Q and inputs Q~, R, the second having output Q~ and inputs Q, S.
- (a) Draw the latch circuit. Label each input and output for each of the two NOR gates.
  Click for diagram
- (b) Fill in the output values:
```
Q = 1, Q~ = 0
```
  if the input values are R=0, S=1.
- (c) Fill in the output values:
```
Q = 0, Q~ = 1
```
  if the input values are R=1, S=0.
- (d) Fill in the output values:
```
Q = 0, Q~ = 0
```
  if the input values are R=1, S=1.
Draw the circuit diagram for a 4-bit binary counter. Your circuit should have 4 outputs D0, D1, D2, D3, a clock input C, and a reset input R. The outputs should be set to 0 on the first rising clock edge occurring while the reset input is 1. The outputs should change on each rising clock input occurring while the reset output is 0, in such a way that the new outputs represent the result of incrementing the old outputs by 1 in the usual binary positional notation, with D0 as the least significant bit and D3 as the most significant bit. You may use standard flip-flops as building blocks.
```
 
   This was done in class.
```

CS 2011, A Term 1999 Professor Sergio A. Alvarez Practice Problems for Test 3

CS 2011, A Term 1999
Professor Sergio A. Alvarez
Practice Problems for Test 3