Let's say all the AI knows about it's surroundings is a pixel-map that it has which clearly shows walkable terrain and obstacles. I want the AI to be able to traverse this terrain until it finds an exit point.

There are some restrictions:

• There is always a way to the exit in the entire map that the AI walks around in, but there may be dead ends. The path to the exit is always pretty random, meaning that if you stand at crossroads, nothing indicates which direction would be the right one to go.

• It doesn't matter if the AI reaches a dead end, but it has to be able walk back out of it to a previously not inspected location and continue its search there.

• Initially, the AI starts out knowing only the starting area of the whole map. As it walks around, new points will be added to the pixel-map as the AI corresponding to the AIs range of sight (think of it like the AI is clearing the fog of war)
• The problem is in 2D space. All I have is the pixel map.
• There are no paths in the pixel map which are "too narrow". The AI fits through everything.
• It shouldn't be a brute force solution. E.g. it would be possible to simply find a path to each pixel in the pixel map that is yet undiscovered (with A*, for example), which will lead to the AI discovering new pixels. This could be repeated until the end is reached.
• The path doesn't have to be the shortest path (this is impossible without knowing the entire map beforehand), but when movements within the visible area are calculated, the shortest and from a human standpoint most logical path should be taken (e.g. if you can see a way out of your room into a hallway, you would obviously go there instead of exploring the corner of your current room).

What kind of approaches to solve this problem are there?