GATE CS 2024 Set 2 Q.61: Machine Instructions and Addressing Modes

A processor uses a 32-bit instruction format and supports byte-addressable memory access. The ISA of the processor has 150 distinct instructions. The instructions are equally divided into two types, namely R-type and I-type, whose formats are shown below.

R-type Instruction Format:

Field 1	Field 2	Field 3	Field 4	Field 5
OPCODE	UNUSED	DST Register	SRC Register1	SRC Register 2

I-type Instruction Format:

Field 1	Field 2	Field 3	Field 4
OPCODE	DST Register	SRC Register	# Immediate value/address

In the OPCODE, 1 bit is used to distinguish between I-type and R-type instructions and the remaining bits indicate the operation. The processor has 50 architectural registers, and all register fields in the instructions are of equal size.

Let $X$ be the number of bits used to encode the UNUSED field, $Y$ be the number of bits used to encode the OPCODE field, and $Z$ be the number of bits used to encode the immediate value/address field. The value of $X + 2Y + Z$ is ________

Your answer

Official answer and solution

Official answer: 34
From: Official GATE 2024 answer key

Solution

Compute the bit-widths of the OPCODE field ( $Y$ ), each register field, the UNUSED field ( $X$ ), and the Immediate field ( $Z$ ).

OPCODE field size ( $Y$ ):
- The $150$ instructions are equally divided into $75$ R-type and $75$ I-type instructions.
- Encoding $75$ operations of each type requires $\lceil \log_2(75) \rceil = 7$ bits, plus $1$ bit to distinguish R-type vs. I-type: $Y = 1 + 7 = 8\text{ bits}$
Register field size:
- To address $50$ architectural registers, each register field requires $\lceil \log_2(50) \rceil = 6$ bits.
UNUSED field size ( $X$ ) in 32-bit R-type format (which has 3 register fields): $X = 32 - Y - 3 \times 6 = 32 - 8 - 18 = 6\text{ bits}$
Immediate value/address field size ( $Z$ ) in 32-bit I-type format (which has 2 register fields): $Z = 32 - Y - 2 \times 6 = 32 - 8 - 12 = 12\text{ bits}$
Final value: $X + 2Y + Z = 6 + 2(8) + 12 = 34$

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 2 (CS2)
Question: 61 of 65
Section: Computer Science and Information Technology

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