puremate.cql
// Download puremate.cql// PGN output when run on sample.pgn
/******* Find positions where black is in pure mate. There are numerous mutually incompatible definitions of "pure mate" published and on the internet. To begin the process of determining the best definition, we use the following motivation (from British Chess Magazine, August, 1898), defining a "clean mate" as: ------- a mate in in which there is no unnecessary attack by the mating side on a square in the mated King's field or the king itself. ------ The key here is that each (a) *attack* must be (b) *necessary*. The first point is (a): we are looking at *attacks* by the attacking side. Some authors have suggested in their definitions that it is the attacking *piece* that must be necessary in some cases, or in one case the attacking *move*. We disagree: we think it is more intuitive to consider actual attacks, rather than the piece causing the attacks or the move leading to the attack. (By contrast, in model mates we do consider the necessity of a piece itself). The second point is (b): the attack must be *necessary*. Many authors allow any number of attacks on attacking pieces in the king's field. For example, consider the following position: White: Kh1 Qh7 Bg6 Rh2 Ng5 Black: kh8 Here, black is in mate from the queen on h7. Each empty square in the king's field is guarded once. But the queen on h7 is attacked three times: by a bishop on g6, by a rook on h2, and by the knight on g5. Obviously, these attacks are "unnecessary" so the position should not be pure mate. Calling this pure mate in our view is not consistent with the early definitions and also not with the spirit of term. It just doesn't make intuitive sense in our opinion to allow a mate with so much superfluous attacking force and call that pure. Despite this, two widely available sources allow this kind of mate: the Oxford Companion to Chess (2d) edition ("unoccupied squares in the king's field are attacked once only") and Wikipedia ("all vacant squares in the king's field are attacked only once", last visited 10 September 2018, citing the Oxford Companion to Chess). We doubt very much that the Oxford Companion even intended to exempt attacking pieces in the king's field from being defended multiple times by pieces of their own color. We therefore take our definition of "pure mate" from the Encyclopedia of Chess Problems. We believe that in most cases this is a more authoritative source: for one thing it is exclusively focused on problem themes. It goes into far more detail and has far more themes than either Wikipedia or the Oxford Companion. More important, the Encyclopedia of Chess Problems' definition simply makes more sense to us: ----- Mate is called pure when there is only one guard or block on each square in the (mated) King's field. Double check is allowed if the mate could be parried without it. A pin-mate is allowed if pinning is needed in mate. ---- Note the key change of the word "attack" in Oxford Companion and Wikipedia to the word "guard". This difference is significant in the cases like the following: White: Ke6 Ra8 Black: ke8 Here, the square f8 is not "attacked" at all. Therefore, under the definition in the Oxford Companion (requiring squares to be attacked "once only"), this would *not* be pure mate; neither would it be pure mate under Wikipedia. Those sources do not allow squares in the king's field to be attacked 0 times. Again, we are sure this omission is unintentional, but it is there. Wikipedia and the Oxford Companion diverge from one another in the case of double check. Wikipedia does not mention double check. The Oxford Companion states that in a pure mate, "the mating move is not a double check unless this is necessary to prevent the defender from interposing a man or capturing a checking piece." Under the Oxford definition, there might be two identical positions, one of which is a pure mate, and one of which is not, depending only on the previous move (for example, if the previous move captured a piece that would otherwise have interposed). We again concur with Encyclopedia of Chess Problems and with the British Chess Magazine that pure mate should be based on the position itself, not on the move history leading up to it. Here, even Wikipedia does not quote this definition, although the Oxford Companion is the only cited reference. This motivates our treatment of double check, following the Encyclopedia of Chess Problems: We preliminarily say that a white piece x *guards* a square y in the Black King's field (which we define to exclude the king itself) if either (1) x attacks y, or (2) x would attack y if the black king were on y For example, in the position: W: Ke6 Ra8 B: Ke8 The white rook on a8 guards (but does not attack) f8. The conditions then are: 1. Every empty square in the King's field is guarded exactly once 2. Every square with a white piece in the King's field is guarded exactly once 3. Every square with a black piece in the King's field is either unguarded or guarded once by a necessary pin A necessary pin is a pin whose absence would allow the pinned piece either to affect the mate, either by interposing between a checker and the king, or by capturing a checker. Thus, if the king is in double check, there are no necessary pins. Finally, double check is allowed only if the double check itself is necessary, which we interpret to mean that each check could be refuted if not for the other check, where "refuted" means that its checker could be captured or prevented by interposition. (Of course, the mate can never be refuted by moving the black king, because removing just a check does not change the king's flight squares). The phrase "if not for the other check" means that only the attack on the king's square by that checker is counterfactually eliminated in determining whether a black piece could interpose. ======================= Implementation Notes: ====================== Global Variables: KingsField is the squares adjacent to the black king. Checkers is the set of checking pieces Functions: --------- GuardedSquares(x) ----------------- returns the set of squares in the king's field guarded by a piece x GuardedPieces(s) ---------------- returns the set of pieces guarding a square s Necessary(Checker) ------------------ Here the argument Checker is a checking piece in a double check. The function is true if that Checker's check is necessary to the mate, in the sense that removing the attack by Checker on the black king would allow a (non-pinned) black piece to prevent the remaining check either by capture or interposition *****************/ cql(input heijden.pgn) flipcolor{mate btm KingsField= . attackedby k Checkers= A attacks k function GuardedSquares(x){ KingsField attackedby x | xray (x k KingsField)} function GuardingPieces(s) { piece fy in A s & GuardedSquares(fy)} // Every flight square guarded exactly once square all FlightSquare in KingsField & [_A] GuardingPieces(FlightSquare)==1 //Every selfblocker is either unguarded, or, if it is not double check, // (a) pinned and (b) could interfere with a checker square all SelfBlocker in KingsField & a { SelfBlockerGuards=GuardingPieces(SelfBlocker) if Checkers>1 then not SelfBlockerGuards else{ //single-check case not SelfBlockerGuards or {SelfBlockerGuards==1 pin through SelfBlocker move pseudolegal from SelfBlocker to (Checkers | between (Checkers k)) or move pseudolegal enpassant from SelfBlocker } } // end the single-check case } //end considering the SelfBlocker function Necessary(Checker){ OtherChecker=Checkers & ~Checker RefutationSquares= OtherChecker | between (OtherChecker k) NonPinnedBlackPiece= {a & ~pin} & ~k NonPinnedBlackPiece attacks RefutationSquares } if(Checkers>1)then square all Checker in Checkers Necessary(Checker) }