Programming

Here's how I think it works

In formal language, what it means to accept a verification means does the result fall into the list of acceptable values.

Consider adding two 2-bit numbers:

• Alphabet: { 0, 1}
• Language: { x | x consists of four binary digits representing two 2-bit binary numbers where the result of adding these two numbers is a valid 2-bit binary number (i.e. falls between 00 and 11) }
• Then you have an automata that will:
  • Start from the rightmost bit
  • Add the corresponding bits (+ carry from any previous iterations)
  • Carry over to the left if needed
  • Repeat for both bits
  • Check for acceptance
• Machine as a whole simply checks did the inputs produce a valid 2-bit number, so it just accepts or rejects

The machine itself simply holds this automata and language, so all it does is take input and reject/accept end state. I think you're just getting caught up in definitions

A sum of a list of numbers I think would be something like

• Alphabet: digits 0-9 and ','
• Language: a single string of digits or a single string of digits followed by a comma and another valid string
• Automata:
  • Are we a single string of digits? If yes, accept
  • Sum the last number into the first and remove the comma
  • Repeat
• Machine: Does the some operation result in a valid string?

Machines accept a valid state or hit an error state (accept/reject). The computation happens between the input and accept/reject.

But maybe I don't understand it either. It's been a while since I poked around at this stuff.

For all possible input, only recognize the one input that's (under certain encoding scheme) equal to the sum of the given list. That's for a given list.

Another more general approach is that, only recognize the input if (under certain encoding), it's a pair of a list and a number, where the number is the sum of the list.