Shortest proof games (SPG's) are simply proof games with the smallest possible number of moves.
Another name for SPG's is Help games because both Wh. and Bl. cooperate to reach a given position in a given number of moves. Thus the following three stipulations are almost equivalent: "SPG in 11.5 moves", "Help Game in 11.5 moves" and "Position after Wh. 12th. Game score?". The only difference is that the 1st stipulation tells you that, really, no shorter proof game exists. (In the 80's, many SPG's where given without a number of move, so that the stipulation was simply "Shortest proof game?".)
This kind of problem is very popular today. Of course the construction is really interesting if the (shortest) help game is unique, i.e. dual-free, and if an artistic and/or problematic element is present, e.g.
Here is a simple (?) example:
Ernest Clement Mortimer
Shortest Proof Games, 1991
Version by A. Frolkin
15+13. Position after Black's 4th move. How did the game go?
A compilation of SPGs has been published by G. Wilts and A. Frolkin.
It contains a short history of the genre: Sam Loyd was the first to ask for SPGs when he demonstrated that, starting from the initial position, it is possible to capture all 30 men in just 34 single moves. [G. Ponzetti showed in 1999 that it's possible to reach the positions Kf2/Ke7 and Kf2/Kf7 in just 16.5 moves] Then T. R. Dawson proposed the first dual-free SPG in 1913. It is only in the late 70s-early 80s that Michel Caillaud demonstrated that long, dual-free, SPGs which a rich thematic content could be composed. Since then, the genre has been very lively.
Natch is a computer program for solving SPGs. It has been written by Pascal Wassong <pascal.wassong(at)free.fr> and is freely available in the Internet. It is very helpful for composers searching for cooks.
Some "help game" problems do not ask for a given final position but rather for given moves which must be present in the game. Here is a simple (but very nice) example:
3510 - R÷sler, Peter
16+16. Find a game with 6. Pg7xf8=N mate
One-sided proof games are proof games were only one side, i.e. White, plays. Thus we get some hybrid cross of the usual SPG with the series-mover problem. Here is a nice example. Notice that, like all such problems, it has 16 white units!
8732 - Goran Wicklund
Springaren 69, May 1997
16+9. One-sided proof game in 34 moves