Consider a machine with a byte addressable main memory of $2^{32}$ bytes divided into blocks of size 32 bytes. Assume that a direct mapped cache having 512 cache lines is used with this machine. The size of the tag field in bits is _________.

For a direct-mapped cache architecture:

1. Main Memory Address Bits:

   $$\text{Memory Size} = 2^{32}\text{ bytes} \implies 32\text{ address bits}$$

2. Block Offset (Word Offset) Bits:

   $$\text{Block Size} = 32\text{ bytes} = 2^5\text{ bytes} \implies 5\text{ bits}$$

3. Cache Line Index Bits:

   $$\text{Number of Lines} = 512 = 2^9 \implies 9\text{ bits}$$

4. Tag Field Bits:

   $$\text{Tag Size} = \text{Total Address Bits} - (\text{Index Bits} + \text{Block Offset Bits})$$

   $$\text{Tag Size} = 32 - (9 + 5) = 32 - 14 = 18\text{ bits}$$