The puzzle in the cave of the middle island is a boolean logic calculation with the symbols for ON/OFF and AND/OR swapped out.

To solve the puzzle you must use the clues to determine which characters represent ON/OFF and AND/OR, solve the calculation in the data-pad closest to the buttons, and use it to input the correct on/off sequence on the buttons.

There are 3 complete examples in data-pads on the opposite side of the room that can be used to decipher the symbols.

Quick guide on how to do Boolean Algebra[en.wikipedia.org]:

AND: All conditions must be true for the result to be true.

• true AND true = true
  
• true AND false = false
  
• false AND false = false

OR: At least one of the conditions must be true for the result to be true.

• true OR true = true
  
• true OR false = true
  
• false OR false = false