CSCE 312 Fall 2023
Computer Organization
Lecture 3

Digital Logic Design II

Topics for today:

Examples of Building Circuits

Let's look at some of the logical and arithmetic operations a computer does, and see how they might be implemented as combinational circuits.
  1. Bit equality. How do we test whether two bits a and b are equal?

    The truth table looks like this: 

    a  b  a==b
0  0   1
0  1   0
1  0   0
1  1   1
    What does the Karnaugh map for this function look like?

    What is a minimal sum-of-products (SOP) for this function?

    What would a combinational circuit for this function look like?

  2. Integer equality (word equality). For an n-bit integer, there are 2n inputs: n-bit integers a and b represented by bits a0...an-1 and b0...bn-1. The output is 1 if a==b, 0 otherwise. How would we construct such a circuit? For instance, if n=2, then the truth table looks like this: 
    a1 a0 b1 b0  a==b
0  0  0  0    1
0  0  0  1    0
0  0  1  0    0
0  0  1  1    0
0  1  0  0    0
0  1  0  1    1
0  1  1  0    0
0  1  1  1    0
1  0  0  0    0
1  0  0  1    0
1  0  1  0    1
1  0  1  1    0
1  1  0  0    0
1  1  0  1    0
1  1  1  0    0
1  1  1  1    1
    How might this function be useful in a computer? It can be used to implement the == operator in C++.

    What does the Karnaugh map for this function look like?

    What is a minimal SOP for this function?

    One way to implement this function is to build a circuit based on the minimal SOP. But what if n=3? Then there are 6 variables. The K-map would have 64 0s and 1s. If n=4, then there are eight variables. The K-map would have 256 0s and 1s. On modern machines, this circuit would have to work for n=32. That would yield a K-map with 18,446,744,073,709,551,616 0s and 1s, and keeping track of the adjacencies would be difficult on two-dimensional paper.

    Another way to implement this function is to link together many 1-bit equality circuits. How would you do that?

  3. Bit multiplexor (MUX). A multiplexor is a device for choosing one of two inputs based on a third control input. So there are three inputs: a, b, and s, and one output which is a, if s==1, or b, if s==0.

    The truth table looks like this: 

    s  a  b  MUX(a,b,s)
0  0  0      0
0  0  1	     1
0  1  0      0
0  1  1      1
1  0  0      0
1  0  1      0
1  1  0      1
1  1  1      1
    How might this function be useful in a computer? Think of it as a logic-level version of the if/else statement. It basically says if (s) output = a; else output = b;
  4. Word-level multiplexor. This function is a multi-output version of the bit multiplexor, using a control input to pass either an n-bit word a or an n-bit word b. There are 2n inputs: n-bit words a and b represented by bits a0...an-1 and b0...bn-1. The outputs c0...cn-1 are the corresponding bits of a if s==1, or b if s==0.

    For n=2, the truth table looks like this: 

    s  a1 a0 b1 b0  c0  c1
0  0  0  0  0   0  0
0  0  0  0  1   0  1
0  0  0  1  0   1  0
0  0  0  1  1   1  1
0  0  1  0  0   0  0
0  0  1  0  1   0  1
0  0  1  1  0   1  0
0  0  1  1  1   1  1
0  1  0  0  0   0  0
0  1  0  0  1   0  1
0  1  0  1  0   1  0
0  1  0  1  1   1  1
0  1  1  0  0   0  0
0  1  1  0  1   0  1
0  1  1  1  0   1  0
0  1  1  1  1   1  1
1  0  0  0  0   0  0
1  0  0  0  1   0  0
1  0  0  1  0   0  0
1  0  0  1  1   0  0
1  0  1  0  0   0  1
1  0  1  0  1   0  1
1  0  1  1  0   0  1
1  0  1  1  1   0  1
1  1  0  0  0   1  0
1  1  0  0  1   1  0
1  1  0  1  0   1  0
1  1  0  1  1   1  0
1  1  1  0  0   1  1
1  1  1  0  1   1  1
1  1  1  1  0   1  1
1  1  1  1  1   1  1
    The general n-bit multiplexor function would be used in a computer for multiplexing words (e.g. integers) rather than just bits.

    Since this function has multiple outputs, we don't know yet how to make a K-map for it. One way would be to draw two 5-variable K-maps, with the first representing c0 and the second representing c1. This drawing would have 64 0s and 1s.

    An easier way to make a circuit out of this mess would be to combine two 1-bit multiplexors.

  5. 4-input multiplexor. A 4-input multiplexor chooses from one of four inputs based on two control inputs. The control inputs can be thought of as a 2-bit number, from 0 through 3, that chooses one of the four inputs. So there are six inputs: i0, i1, i2, i3, s0, and s1. The truth table looks like this: 
    s0 s1 i0 i1 i2 i3  output |    s0 s1 i0 i1 i2 i3  output
0  0  0  0  0  0     0    |    1  0  0  0  0  0     0
0  0  0  0  0  1     0    |    1  0  0  0  0  1     0
0  0  0  0  1  0     0    |    1  0  0  0  1  0     1
0  0  0  0  1  1     0    |    1  0  0  0  1  1     1
0  0  0  1  0  0     0    |    1  0  0  1  0  0     0
0  0  0  1  0  1     0    |    1  0  0  1  0  1     0
0  0  0  1  1  0     0    |    1  0  0  1  1  0     1
0  0  0  1  1  1     0    |    1  0  0  1  1  1     1
0  0  1  0  0  0     1    |    1  0  1  0  0  0     0
0  0  1  0  0  1     1    |    1  0  1  0  0  1     0
0  0  1  0  1  0     1    |    1  0  1  0  1  0     1
0  0  1  0  1  1     1    |    1  0  1  0  1  1     1
0  0  1  1  0  0     1    |    1  0  1  1  0  0     0
0  0  1  1  0  1     1    |    1  0  1  1  0  1     0
0  0  1  1  1  0     1    |    1  0  1  1  1  0     1
0  0  1  1  1  1     1    |    1  0  1  1  1  1     1
0  1  0  0  0  0     0    |    1  1  0  0  0  0     0
0  1  0  0  0  1     0    |    1  1  0  0  0  1     1
0  1  0  0  1  0     0    |    1  1  0  0  1  0     0
0  1  0  0  1  1     0    |    1  1  0  0  1  1     1
0  1  0  1  0  0     1    |    1  1  0  1  0  0     0
0  1  0  1  0  1     1    |    1  1  0  1  0  1     1
0  1  0  1  1  0     1    |    1  1  0  1  1  0     0
0  1  0  1  1  1     1    |    1  1  0  1  1  1     1
0  1  1  0  0  0     0    |    1  1  1  0  0  0     0
0  1  1  0  0  1     0    |    1  1  1  0  0  1     1
0  1  1  0  1  0     0    |    1  1  1  0  1  0     0
0  1  1  0  1  1     0    |    1  1  1  0  1  1     1
0  1  1  1  0  0     1    |    1  1  1  1  0  0     0
0  1  1  1  0  1     1    |    1  1  1  1  0  1     1
0  1  1  1  1  0     1    |    1  1  1  1  1  0     0
0  1  1  1  1  1     1    |    1  1  1  1  1  1     1
    That is truly a horrible truth table, but sometimes we have to solve truth tables like that, so let's see what a 6-variable K-map would look like:

    Hey, that's not so bad when you look at it as a K-map.

    Why would you want such a circuit? This is the basis of how array indexing is implemented in a high-level programming language like C++. You provide a k-bit array index and get one of 2k items from memory. Something has to do that conversion from the index to the location of the items, and in a computer that something is a multiplexor.

  6. Full adder. We did this one last time, but let's see it again. How would we do this using NANDs instead of ANDs, ORs, and NOTs?

Memories

So far, we have seen only combinational circuits, where the inputs completely determined the outputs. There's no notion of memory, i.e., state. What if we wanted to design a circuit such that the output is a stored bit, rather than some function of the inputs? That can't be done with combinational, i.e. acyclic, circuits, but it can be done if we allow circuits to have cycles.

R-S Latch

The follow circuit is called an R-S latch:

When both R and S are set to 1, the output of the circuit is a single stored bit that can be thought of as constantly looping through the circuit.

To store a one, we bring R down to 0.

To store a zero, we keep R at 1 and bring S down to 0.

We can build arrays of latches to hold words (e.g. integers). An array of latches is called a register. Special registers in the CPU are made visible to the architecture, i.e., machine language instructions can access special CPU registers.

6T SRAM Cell

Note that an R-S latch consumes 8 transistors, 4 for each NAND gate. This is fine for registers, but for large memories we might want a smaller circuit. Think of a large memory as a 2-dimensional array of single-bit memory cells. It's 2-dimensional because chips are 2-dimensional. Using multiplexors, we translate a memory address into one of many possible columns and one of many possible rows. The bit corresponding to the given column and row can be picked out and read or modified.

A 6-transistor SRAM cell looks like this:

The two inverters (i.e., the four middle transistors) provide storage for a single bit. Think of the bit as continously going around and around the two inverters in a counter-clockwise manner. The bit is the output of the top inverter, and the complement of the bit is the output of the bottom inverter as well as the input to the top inverter. The following occurs when we want to read the bit stored in this SRAM cell:

  1. There are two column select lines. Each one is precharged to 1 by the column selector logic. (All other column select lines in the memory array would be reset to 0.)
  2. The row select line for this cell is set to 1 by the row decoder logic. (All other row select lines would also be reset to 0.)
  3. At this point, the current from the row select line flows to two NMOS transistors which begin to allow the bit and its inverse to flow out of the SRAM cell to the column select lines. One of the column select lines is slowly pulled to 0. The process is slow because the tiny SRAM transistors have to drive the substantial column select lines that have been precharged to 1. An analog device called a senseamp at the bottom of the column select lines quickly detects any difference in current and decides that the bit is a 0 if the left column select line is going to 0, or a 1 if the right column select line is going to 0.
The following occurs if we want to write a bit x to the SRAM cell:
  1. The row select line is set to 1, allowing current to flow between the column select lines and the inverters.
  2. The left column select line is driven to x and the right line is driven to the complement of x. The beefier drivers are stronger than the inverters, so the value currently going around the inverters is replaced with the new value.

C++ program for doing ripple-carry addition

This C++ program does ripple-carry addition, as well as subtraction, on 32-bit integers. 
#include <stdio.h>
#include <iostream>

void int2bin (int, bool[], int);
int bin2int (bool[], int);
void ripple_carry_add (bool[], bool[], bool[], int);
bool parity (bool, bool, bool);
bool majority (bool, bool, bool);
void subtract (bool[], bool[], bool[], int n);

int main (void) {
	int a = 1234;
	int b = 5678;
	int c;
	bool A[32], B[32], C[32];

	int2bin (a, A, 32);
	int2bin (b, B, 32);
	ripple_carry_add (A, B, C, 32);
	c = bin2int (C, 33);
	std::cout << c << "\n";

	subtract (A, B, C, 32);
	c = bin2int (C, 33);
	std::cout << c << "\n";
}

// convert integer x to n-bit Boolean array v

void int2bin (int x, bool v[], int n) {
	for (int i=0; i<n; i++) {
		v[i] = (x & 1) == 1;
		x /= 2;
	}
}

// convert n-bit Boolean array v to integer and return it

int bin2int (bool v[], int n) {
	int x = 0;
	for (int i=n-1; i>=0; i--) {
		x *= 2;
		if (v[i]) x++;
	}
	return x;
}

// do a ripple-carry add on n-bit Boolean arrays A and B, storing the result
// in n+1-bit Boolean array C

void ripple_carry_add (bool A[], bool B[], bool C[], int n) {
	bool carry;

	// no carry in; 2-bit parity is XOR

	C[0] = A[0] ^ B[0];

	// two-bit carry is AND

	carry = A[0] && B[0];

	// add corresponding bits of the array, along
	// with the carry generated initially or from previous
	// iterations of the loop

	for (int i=1; i<n; i++) {
		C[i] = parity (A[i], B[i], carry);
		carry = majority (A[i], B[i], carry);
	}

	// n'th bit is the carry out of the loop

	C[n] = carry;
}

// return the parity of x, y, and z

bool parity (bool x, bool y, bool z) {
	return x ^ y ^ z;
}

// return the majority of x, y, and z

bool majority (bool x, bool y, bool z) {
	return (x && y) || (x && z) || (y && z);
}

// subtract n-bit Boolean array B from A, placing result in n+1-bit array C
// i.e. C = A - B

void subtract (bool A[], bool B[], bool C[], int n) {
	bool complementB[n];
	bool negativeB[n+1];
	bool one[n];
	int i;

	// we'll subtract B from A by adding -B to A
	// first we have to find -B, which is the bitwise complement
	// of B plus 1

	// find the logical complement of B

	for (i=0; i<n; i++) complementB[i] = ! B[i];

	// make a 1

	one[0] = true;
	for (i=1; i<n; i++) one[i] = false;

	// find the negation of B

	ripple_carry_add (complementB, one, negativeB, n);

	// find A + (- B)

	ripple_carry_add (A, negativeB, C, n);
}