What notations are there to describe a logic puzzle format? Here, I'm talking about logic puzzles like Sudoku, Bridges, Kakuro, Tents; games with exact rules and an exact & finite number of solutions (possibly even zero solutions).
I'm not talking about puzzle notations within a game, such as chess notation or Rubik's Cube configuration notation. I'm talking about how to represent the puzzle rules, as well as the puzzle itself within those rules.
I do know about the Stanford General Game Playing project, which includes solving logic games, but that's also not quite what I'm looking for. The representation only provides for a single puzzle. I'm looking for a sort-of function from "puzzle description format" to puzzle, plus the rules of the puzzle.
In principle, this should exist. Many logical puzzles are of the format "find a predicate (solution) such that the given structure (puzzle instance) satisfies these clauses (puzzle rules)." So it seems that finite second order existential logic should suffice, and that would also seem to mesh with how hard these puzzles tend to be (as they're often in NP). I wanted to know if there are alternative descriptions, with puzzle examples.