How is transaction fee calculated in details?

Question

There's a similar question here. I've read it but still got some questions.

I know the transaction fee is simply lamports_per_signature * number_of_signatures (plus a little miner tip). However, what is number_of_signatures exactly?

Inside the transaction data, we should specify an array of accounts, denoting which accounts are to be used in this transaction. I'm wondering what number_of_signatures is in each of the following cases:

1. There are 2 accounts in the array and both are signers. However, these 2 accounts are in fact the same account. i.e., I'm passing duplicated accounts.
2. There are 2 accounts in the array, and the first is signer, the second is not. These 2 accounts are actually the same too.
3. There are 3 different accounts in the array. The first is signer, the second is not and it's a PDA. The third is a program account. Inside one of the instructions, the program performs a CPI to the third account, and the second account (PDA) becomes a signer.

Answer 1

number_of_signatures is the number of unique signatures in the transaction that must be verified by the runtime. An account can only sign a transaction once. Given that, here are the answers:

1. The two accounts are deduplicated when the transaction's "message" is compiled, so only one signature is produced
2. Same thing here, only one signature is produced
3. PDAs do not produce signatures, which is why you'll sometimes see that a PDA's "signature" is designated in quotes. This means you'll only have one signature.