CS 240: Architecture

Assignment: Architecture

Assign: Thursday 25 September
Due: Thursday 9 October
Policy: Individual graded synthesis assignment
Submit: Upload a PDF written on the cs240-arch-worksheet.pdf submission sheet
Reference:

Goals

To practice using combinational, arithmetic, and sequential logic.
To understand how to encode computation with low-level building block including multiplexers, flip-flops, and adders.
To master every piece of a simplified but representative instruction set architecture—the HW ISA.

Time reports

For a version of this assignment in a previous semester, students self-reported the following times:

25% of students spent <= 7 hours.
50% of students spent <= 8 hours.
75% of students spent <= 10 hours.

Assignment: Architecture
- Time reports
Contents
Exercises
Submission

Exercises

Please write your answers on the cs240-arch-worksheet.pdf submission sheet to streamline the grading process. Submit your work by scanning a PDF of all worksheet pages (in order) and uploading it to Gradescope. Remember that this course uses your Gradescope account attached to your Wellesley email address (the variety that matches your username). You can merge accounts across multiple email addresses if helpful.

Notes:

This assignment is worth 75 points.

Q1. vALUe Judgments [22 points]

In class and lab we designed an $n$-bit ALU by connecting $n$ 1-bit ALUs. In these exercises, you will improve the 4-bit version of our design.

Summary of our ALU design:

The Carry out from each bit is wired to the Carry in of the next bit, if present.
The full $n$-bit ALU has 4 lines of control input:
- Invert A controls the invert A inputs of all of the 1-bit ALUs.
- Negate B controls:
  - the invert B inputs of all of the 1-bit ALUs; and
  - the carry-in of the least-significant bit ALU.
- Two bits of Operation ID control the Operation inputs of all of the 1-bit ALUs.

Rules for extending the ALU:

Any additional logic may connect to any existing wires in the ALU and add gates or other building blocks as needed.
Minimize the number of gates or other building blocks you add. Gates such as AND can have > 2 inputs.
Choose the simplest option and clearly label new outputs.

Exercises:

[5 points] Design extra 1-bit outputs from the $n$-bit ALU for the four conditions codes. Using the provided sheet, draw and label the additional logic required for each condition code (flag), all on one ALU diagram.
1. Zero Flag: 1 if all bits of the result are 0, otherwise 0.
2. Sign Flag: 1 if the result is negative, otherwise 0.
3. Carry Flag: 1 if there is a carry out from the operation, otherwise 0.
4. Overflow Flag: 1 if there is a signed overflow, otherwise 0.
[12 points] Alice is trying to design a Less-Than flag output that computes $A < B$ for the 4-bit two’s-complement values $A$ and $B$. Her first idea is to compute $A - B$ and return a value based on the sign bit of the result: her hope is that a 0 sign bit for the result of the subtraction means $A \geq B$ and a 1 sign bit for the result means $A < B$.
1. [3 points] Give three pairs of values, $A$ and $B$, where Alice’s approach correctly calculates $A < B$.
  - In the first pair, both $A$ and $B$ should be positive numbers.
  - In the second pair, both $A$ and $B$ should be negative numbers.
  - In the third pair, one of the two numbers should be positive and one should be negative.
  - Note: Recall that 4-bit two’s complement values range from [-8, 7].
2. [2 point] Give two pairs of values, $A$ and $B$, where Alice’s approach is incorrect in calculating $A < B$ (that is, where her approach either yields 0 when $A$ actually is less than $B$, or 1 when $A$ is not actually less than $B$).
  - In the first pair, $A$ should be positive.
  - In the second pair, $A$ should be negative.
3. [1 point] Describe the key effect that causes Alice’s approach’s answer from the previous part to be incorrect.
4. [5 points] Using minimal additional logic, help Alice design a 1-bit Less-Than Flag whose value always correctly indicates $A < B$ after $A - B$ is performed. That is, the new Less-Than flag should have value 1 if $A < B$ and 0 otherwise.
  - Using the provided sheet, draw and label your design.
5. [1 point] Write how to control each of the 4 control lines of our 4-bit ALU to perform $A < B$:
  - Invert A = ?
  - Negate B = ?
  - Operation = ??
[5 points] Design a 1-bit Equality Flag output for the ALU using minimal additional logic (but no additional XOR gates). Given inputs $A$ and $B$, this output wire should carry 1 if $A == B$ and 0 otherwise.
1. [4 points] Using the provided sheet, draw and label your design.
2. [1 point] Write how to control each of the 4 control lines of our 4-bit ALU ALU to perform $A$ == $B$:
  - Invert A = ?
  - Negate B = ?
  - Operation = ??

Q2. Flop-flip-flopping [10 points]

Consider the following circuit that uses falling-edge triggered D flip-flops.

Assume the Clock input and values Q₀, Q₁, and Q₂ are all initially 0. Draw waveforms for Clock, Q₀, Q₁, and Q₂, showing at least ten Clock cycles. You do not need to turn in your waveform drawing, but it will help you reason about the circuit.

Fill in the table summarizing the values of Q₀, Q₁, and Q₂ after each Clock cycle. Note the order that the worksheet table lists Q₀, Q₁, and Q₂: this will help you identify a pattern in the next part.
Briefly explain what this circuit does at a high level, treating the Q outputs as a single multi-bit value. A phrase to a couple sentences is enough. Hint: you have seen a similar circuit behavior in lab.

Q3. Some Loopy Programs [19 points]

In this exercise you will simulate the execution of more HW ISA programs with loops.

Background

As an example, we consider a different program that has a loop to remind you of the conventions used in an execution table (as shown in the solution slides for the HW ISA architecture). Suppose that register R2 has been initialized with the signed decimal number 3 and register R3 has been initialized with the signed decimal number 7. Also suppose that the program P0 has the following Instruction Memory (IM):

Address	Contents
0x0–0x1	BEQ R2, R3, 3
0x2–0x3	ADD R2, R1, R2
0x4–0x5	SUB R3, R1, R3
0x6–0x7	JMP 0
0x8–0x9	HALT

Then here is an execution table that shows the step-by-step execution of the program P0.

Note that the way you should read the meaning of BEQ R1, R2, offset is that the change to the PC, PC <- PC + 2 + offset*2 replaces the non-branch change to the PC (PC <- PC + 2), as we designed in the HW ARCH slides. Specifically, note in the following example that the PC goes from 0 to 8 (not 10) based on the instruction BEQ R2, R3, 3, because PC + 2 + offset*2 = 0 + 2 + 3*2 = 8.

Execution table

PC	Instruction	State Changes
0x0	BEQ R2, R3, 3	PC ← PC+2 = 0+2 = 2 (because 3 = R[2] != R[3] = 7)
0x2	ADD R2, R1, R2	R[2] ← R[2]+R[1] = 3+1 = 4; PC ← PC+2 = 2+2 = 4
0x4	SUB R3, R1, R3	R[3] ← R[3]-R[1] = 7-1 = 6; PC ← PC+2 = 4+2 = 6
0x6	JMP 0	PC ← 2*0 = 0
0x0	BEQ R2, R3, 3	PC ← PC+2 = 0+2 = 2 (because 4 = R[2] != R[3] = 6)
0x2	ADD R2, R1, R2	R[2] ← R[2]+R[1] = 4+1 = 5; PC ← PC+2 = 2+2 = 4
0x4	SUB R3, R1, R3	R[3] ← R[3]-R[1] = 6-1 = 5; PC ← PC+2 = 4+2 = 6
0x6	JMP 0	PC ← 2*0 = 0
0x0	BEQ R2, R3, 3	PC ← PC+2+(2*3) = 0+2+6 = 8 (because 5 = R[2] == R[3] = 5)
0x8	HALT	program stops executing

Your Tasks

First, consider a HW ISA program P1 with the following Instruction Memory IM:

Address	Contents
0x0–0x1	ADD R0, R1, R2
0x2–0x3	ADD R1, R1, R3
0x4–0x5	ADD R3, R3, R4
0x6–0x7	BEQ R3, R4, 3
0x8–0x9	ADD R2, R4, R2
0xA–0xB	SUB R4, R1, R4
0xC–0xD	JMP 3
0xE–0xF	HALT

[8 points] In the worksheet, fill in the execution table for program P1. Use the same notational conventions used in the example execution table for P0 above. Any numbers beginning with 0x will be interpreted as hexadecimal; any numbers not beginning with 0x will be interpreted as decimal.
[3 points] Show the final values of the registers R2, R3, and R4 when the program execution halts. Again, any numbers beginning with 0x will be interpreted as hexadecimal; any numbers not beginning with 0x will be interpreted as decimal.
[3 points] Alice wants to translate program P1 (except for the final HALT instruction) into a sequence of statements in a version of C with a 16-bit int type. Fill in the rest of her program with assignments to variables named R2, R3, and R4 that represent the registers with these names in P1 and a while loop. You can use the online C compiler to check your C syntax.
```
int R0 = 0;
int R1 = 1;
int R2 = R0 + R1;
// ...
```
[5 points] Consider this new loopy program, P2, with the following instruction memory. Assume R2 and R3 hold input values when the following program starts. R4 will hold an output when the program stops. The address of each instruction in the instruction memory is shown at left. The HALT instruction stops the computer.
```
# R2 and R3 hold program inputs here.
0x0:  AND R2, R2, R4
0x2:  AND R3, R3, R5
0x4:  BEQ R5, R0, 3
0x6:  SUB R5, R1, R5
0x8:  ADD R4, R4, R4
0xA:  JMP 2
0xC:  HALT # Stops execution.
# What value is in R4?
```
1. [3 points] Execute the program with the initial state that R2 holds 2 and R3 holds 3 (you do not need to draw out the full execution table for this part). What are the final values in the registers R2, R3, R4, and R5 when the program reaches HALT? You may write your answers in either decimal or hex, but must use a 0x prefix for hex.
2. [2 points] Try a few more examples with (small) positive numbers in R2 and R3. What does this code do with R2 and R3 to compute R4? Answer with one simple line of valid C code of the form: R4 = exp;, where exp is a simple C expression using the the variables R2 and R3 (corresponding to the HW registers) and any simple C operations that accomplish the same effect as these combined HW instructions. exp must not include C statements like loops or conditionals. It should also not include any function calls.

Q4. Taking Control [8 points]

In this exercise you will design the Control Unit for the HW microarchitecture we designed in class. (Base your design on the figure below labeled “HW microarchitecture datapath”.) Our design focused on connecting the datapath components (e.g., the register file, ALU, memory, etc.), but we left the Control Unit as black box.

Instead of drawing gates for the Control Unit, we will just build out a truth table for the behavior of the control unit. The truth table should take in a 4-bit HW instruction opcodes as input. As output, the truth table should produce control bits for each output control line, as needed for the specific opcode.

The worksheet provides the Control Unit table design. The control lines we need are:

Reg Write (1 bit): controls the write-enable of the register file. If an instruction should both read and write to the register file, the write-enable should be 1 (that is, the read data lines still produce the contents of their respective registers even when the write-enable is 1).
ALU Op (4 bits): controls the ALU (with the same 4 ALU control lines we developed earlier; see the end of the Arithmetic slides).
- From left to right: Invert A, Negate B, Op₁, Op₀.
Mem Store (1 bit): controls the write-enable of the data memory.
Mem (1 bit): controls three multiplexers deciding which input to provide for:
- the register file’s write address; and
- the register file’s write data; and
- the ALU’s second operand.
Branch (1 bit, added during class): controls whether to choose the next PC based on a branch target and test.

Your task for this problem is to fill in each of the row for each of the following instructions, as shown on the worksheet: SW, ADD, SUB, AND, OR, BEQ.

For this problem, we will interpret all answers as binary (not decimal or hexadecimal).

We have completed the row for LW (“load”) as an example. For LW, the control lines are:

Opcode: 0000, as shown in the HW ISA instruction table on slide 15, “HW ISA Instructions”, of the lecture slides
Reg Write: 1, since loads do write to the register file
ALU Op: 0010, since for a LW, the design needs to add the offset to a register’s contents.
Mem Store: 0, since loads do not write to the memory.
Mem: 1, to control the 3 multiplexers for the register file’s write and the ALU’s second operand.
Branch: 0, since the design should not branch for the load instruction (LW should use the standard increment-PC-by-2 calculation).

Q5. Instruction `NOT` Missing (8 points)

The HW ISA introduced in class lacks an instruction for bitwise not. A logical choice would be NOT Rs,Rd, which takes the bitwise not of the value in source register Rs and stores into destination register Rd. While we could implement NOT with a sequence of existing HW ISA instructions (doing so would be a good practice exercise to test your understanding!), in this problem, we will add a new direct encoding for NOT to the HW ISA.

[3 points] We will first implement a related instruction, NAND, as a new ISA instruction. In the worksheet, fill in a 16-bit encoding of the NAND instruction, following the format shown in the instruction table on slide 15, “HW ISA Instructions”, of the lecture slides. Choose any unused opcode for the encoding.
[3 points] Fill in the labeled row of the Control Unit Table from earlier in the worksheet, to indicate how to control the datapath to implement NAND. Do not add extra logic to the datapath diagram—instead, use properties of boolean operations to calculate NAND using the logic already present.
[2 points] Consider how to implement NOT using NAND. Define an encoding for the NOT instruction directly in the ISA. Choose the encoding carefully to minimize additions to the Control Unit and the datapath. Assuming you have completed the previous parts of this problem, the encoding should requires no new rows or columns in the Control Unit table.

Q6. Jumping into the Unknown [8 points]

Add logic to the HW microarchitecture to implement the JMP instruction, which we did not implement in class. JMP sets the PC to the absolute instruction address given by its argument (a number) multiplied by 2. For example, JMP 3 sets the PC to 0x6, causing the instruction stored at address 0x6 (i.e., the 3rd instruction in the program) to be executed next.

[6 points] In the wiring diagram for the microarchitecture on the worksheet, you will add:
- One new Jump control line from the Control Unit (in Problem 4), carrying 1 if the current instruction is JMP and 0 otherwise.
- Wiring and logic using the Jump control line and the offset bits from the JMP instruction encoding to correctly update the PC if the current instruction is a JMP. Be sure to label any ranges of bits when splitting a wire (as done in the diagram for other operands). Note that on the worksheet, we have added the red wire and extra space toward the top left of the microarchitecture diagram for you to draw and label new logic.
Notes:
- The JMP hardware will need to use a Zero Extend unit that is different from the existing Sign Extend unit.
- Do not attempt to reuse the existing Shift unit; if you need such a shift, use a separate copy.
[1 point] In the Control Unit truth table from earlier in the worksheet, flesh out the contents of the the Jump control line in the final column.
[1 points] In the Control Unit truth table from earlier in the worksheet, flesh out the contents of the row for the JMP opcode.