A proof game

The absurd accomplishments of the top proof game composers in embedding huge amounts of fireworks within the format of unique proof games, and the capability of specialized software tools in proving soundness, mean that there's less interest in non-unique ones than a few years ago.

However, there is a small group of mathematically oriented folks who like to compose proof games where the set of moves is fixed but the order of the moves can vary. The key thing here is to count the number of different solutions, and to try to reach exactly a special number (e.g. current year number) or embed interesting combinatorial functionality. These are called queue problems.

Fascinating discussion with great camaraderie apparent. I've done most of my PG assembly by myself with just an engine for company. Just driving forward until I find something sound, reversing if I'm cooked. The approach that you all seem to be taking is much more analytic: not a mention of Popeye or Euclide. Perhaps the kinds of positions I have tended to end up with are less amenable to human solving, I don't know. I can't think of any I've composed which use even a block, for example. Probably I can learn usefully from your styles of composition. I know Joachim and I think I know Arisktotle. No idea about the others: what I would say to cobra21 is don't give up on the exactness. It may seem hard but honestly you are way better than one guy I remember when he began who is now one of the top composers in the world. And it's in the details that a lot of satisfaction comes, for both composers and solvers: knowing that the solution spans not just the big picture but also the little details. Top composers, of whom I am not one, delight not just in having fantastic matrices to act as a foundation, but also through having the leisure to put in curlicues and widgets which are not just decorative, but serve to allow opportunistic uniqueness in parts of the design space where uniqueness has not historically been found. This allows for new parts of the chess tree to be accessed, and even more originality found. I don't know if this makes sense, but all I am trying to say is that uniqueness is not about perfectionism - it's something fundamental. Orthodox retros, with no additional information provided, must perforce be looser in their control. A composer of a retro has to manufacture his own tempo mechanism and embed that in the position. But there is a price to be exacted for the proof game constraint of minimum number of moves, and that price is the demand for precision. Anyway, enough rambling.

The future of the SPG is the "reflex-SPG" (I believe) and the same is even more true for the pure analytical retros. Although the SPG-form is currently alive and kicking, my feeling is it needs breathing space to attain longevity. The 1- and 2-reflexes are the perfect bridges from the various retro-types to the traditional chess game. I am not as willing as cobra91 to give up uniqueness but retro-chess could make good use of a generic mechanism to control its history space. Chess is just a bit too free to easily manage its inverse function and it is kind of a miracle that so much retro-stuff for it has been developed. From that viewpoint, I sympathize with Cobra's call for a bigger creative retro-space!

Note: Obviously, the fairy dimension makes all composition space unlimited, but I refer to the standard retro-varieties where most composers meet one another.

Interesting - surely freedom cuts both ways though? The freedom to reach neat positions is the freedom for bugs to emerge. Why does your idea actually provide a net help?

There are several reasons why it would help:

(a) much of retro-chess is about kings and king checks. Reflex chess emphasizes that point and creates endless new opportunities to play with that property.

(b) the need to check every move against the obligation to checkmate in 1 or 2 moves is a multiplier for the search steps to find appropriate solution moves. Note that retraction in reflex chess is the algorithmic sibling of the popular defensive proca retractor type. There you must retract to find a place to mate in one, here you must retract to avoid all places with a mate in one in order to "release a position" or find an SPG.

(c) The logic behind a type like the defensive proca retraction is "weird". There is no storyline to sell defensive retractions. The logic for reflex chess is completely transparent. Rather than assuming the completely moronic play of most retros, you assume that the players are intelligent primates with the ability to at least calculate a checkmate in 1 or 2 moves. It will therefore be easy to attract a large audience for this type, also amongst newbies.

Note that reflex-chess is a real bridge from retro chaos to normal chess. For instance, a (theoretical) retro problem under 200-reflex rules could only deliver Magnus Carlsen level proof games. Every solution move must avoid the brute force "mate-in-200" (if possible) which equates to almost perfect play.

There are other ways to boast the basic playing strength in retro-problems like "force capture of unprotected units" or "the obligation to capture any higher valued unit" but I predict that the reflex version will become dominant in the next 25 years and overtake traditional random play in most retro types.

Need to produce a few staggering examples but at my standard composition speed this will take years. Hope somebody takes that burden off me!

Various unconnected thoughts on this:

1) It's certainly new design space to be exploited, and it's always nice to have problems that combine competitive and and retro play, although I'm far from convinced about the superiority of the competitive kind for composition.

(2) I am often troubled in proof game design by checks: frequently the check-response pair can commute easily with other moves. Often a cook comes from the response being played before the check. Reflex might help here.

(3) But I would say the main source of bad cooks are due to imposters: one pawn impersonating its neighbour, insuring the rooks are distinguishable from one another. At least kings don't have imposters! Pawns are also difficult to manage during their advance: either they have to march forward without interaction, or they have to be busy capturing at every step. Reflex chess doesn't particularly help here.

(4) I am reminded of the retro in "The Flanders Panel" by Arturo Perez-Reverte, where part of the retro logic is that someone wouldn't randomly sacrifice a queen. Obviously the issue with this is the subjectivity of how bad an error someone might make, given that the diagram itself was the usual heap of nonsense from a play perspective. Setting a rigid definition of what "good play" comprises (i.e. people don't miss mates in one) would clear up this kind of swamp and allow some great compositions.

(5) There are implications for analytic engines: basically this means that search has to be forwards through the game, rather than tying together chains of moves that have logical dependencies.

(6) This could have much greater impact in the conventional retro world than in the proof games. There is a huge amount of looseness in much retro play, and having the additional constraints to avoid mates in 1 could serve to nail the down the solutions. Also, the mine for traditional retros appears much closer to being exhausted than that of proof games.

(7) Final thought: could leverage an endgame tablebase, to produce some retro problems that always adopt one of the best moves at every point. This would achieve the mythical "200 move look-ahead" right off, but for a subset of the game.

Thanks for your free associations!

Any trouble the reflex type would cause to engine-analysis, I would count as a top prize. Even better if it would become the source of a new class of composition bugs by lack of efficient proof game checkers. It may even resurrect the honorable profession of soundness-testing which was in its final preparatory steps for being sunk six feet undergound. I have no allegiance to the machine of any kind.

I agree the reflex type has immediate drastic effects on the retro-analytical type since almost any knot has a king or both kings in its center. The automatic effects on SPGs are much smaller until composers start focusing on activating the reflex rule. I predict we will see many more proof games with king walks and exposed kings which require elaborate detours to avoid mating the kings.

Of course, my prediction is about as reliable as the prediction for a presidential election in the USA, but I think I have a good intuition about the power of this device. One day a (possibly new) composer will amaze the retro-community with incredible reflex-creations and will find followers. I hope that, by championing this type, I help bringing this day a bit closer! I posted one (simple) example on chess.com but I can't find it anymore and failed to reconstruct it.

@EvinSung: It's a good thing you already posted the same diagram twice. One of them for your PG 5.5 solution, the other one for mine! Yes there are 2 - almost symmetrical - solutions. In mine, white lost castling right. Can you find it?

Ah, I just read through the whole topic and found a post which says there are indeed 2 solutions. Commonly a unique solution to an SPG is required but it's different when the composer intentionally designed 2 (or more) solutions with a special relationship. Here it is the symmetry of both lines.

Personally I think the composer cheated. He wanted just 1 solution but couldn't get rid of the second one. Therefore he made it part of the "design".

Can't say I agree with Aristotle's assessment of this fine proof game with two precise solutions. Very few moves are repeated across the two parts, and there is an excellent change of white switchback piece at the end. Nice features like these rarely happen by accident!

Well it's possible of course. I found the 2nd solution (before knowing there were 2) from the assumption that one might exist with a similar switchback to the one in EvinSungs post. The horizontal pawn formation of the PAS allows many different move permutation to get rid of some - a familiar hurdle when composing SPGs. Probably a 100 years ago I would believe it is all by design but today with engines testing proof games it is just is likely they would surprise you with the gift of a logical "dual" after analysis. Which may then be rebaptized "twin".

Another argument  for "cheating" is worth mentioning independent of PG computer programs. The "at home" SPG type is very restrictive and getting such a position is not accidental. Which means that the author started knowing he wanted an "at home" outcome. When he found one answering to the other design requirements (like switchback), he was very happy. The next day he found the dual (with or without engine) and he was very unhappy.

Now what commonly happens is that the composer starts fiddling and tweaking the proof game until the dual is gone while retaining his design concept. However, the "at home" type is so restrictive that it won't let you tweak a lot. There is practically nothing you can do beyond  a complete redesign with different requirements. Which means you give up on the original idea.

....Or you can decide that Mohammed comes to the mountain when the mountain doesn't come to him - or as we say in the west "go with the flow". The dual arose from the fact that much in the PAS (and also in chess moves) is symmetrical and therefore an enemy of the unique "at home" SPG. Not surprising that the dual looks similar to the firstborn. So the author decides to escape by emphasizing the composition concepts which support that: "echo", "twin", "symmetry". Forgetting it is not hard to get an echo, it is hard not to get an echo. Well he may, but I won't fall for that!