This is a neat game that portrays that exact concept. It isn't really your system, but the idea is that the in game objects are supposed to visually represent data in a fictional system, and your job is to fix the corrupted bits to progress.
For one that's actually your system, there's Lose/Lose. Each alien is a file on your system, and the file gets deleted when you kill it. When your ship dies, the game erases itself. But it doesn't reflect the system structure beyond using your files as enemies, so it's more on par with the Doom thing in that regard.
http://www.brokendimension.com/