Based on an entirely different philosophy, I produced an alternative for my previous proposal. It would be nice if people could comment on which they like better. Where the previous extension was based on defining many different operators for chaining move steps into multi-leg moves, this new proposal retains only a single chaining operator, and puts the responsibility for describing the various conditions that must be met or things that should happen on the intermediate square to the description of the chained steps.

This seems to stay closer to the original Betza notation, and more compatible with David Howe's Bex proposal. For instance, because we are back to just one chaining operator, the trick of specifying an exact number of steps by a lading zero on the count can be embraced. I had to define a fair number of new move modalities in addition to mc, however, and not so many unused letters were left over. So I re-used the modifiers p, o and (perhaps) t in slightly different meanings. But because they can only be used in this new meaning as part of an early leg of a multi-leg move, this should not cause ambiguity with the old usage.

Betza notation 2.0

Move steps of the basic atoms

SINGLE MOVES        PAIRS OF MOVES             SETS OF FOUR MOVES

    lf rf          f f          lv rv          fh fh         lh rh
  fl     fr     fs     fs     l       r      fh     fh     lh     rh
  bl     br     bs     bs     l       r      bh     bh     lh     rh
    lb rb          b b          lv rv          bh bh         lh rh


      f                    v
    l + r                s + s
      b                    v


   fl  fr         f   f         l   r
     x              x             x
   bl  br         b   b         l   r


    f               v             v
 fl   fr         fs   fs       lv   rv
l   *   r       s   *   s     s   *   s
 bl   br         bs   bs       lv   rv
    b               v             v

Grouping

Parentheses can be used for grouping, but have no meaning by themselves. (W) is exactly the same as W.

Chaining

Atomic steps can be 'chained' into a single 'mult-leg' move. Actually W3 means (W)(W-fW)(W-fW-fW), where '-' is the chaining operator. The f means 'continuing forward', i.e. in the same direction, as directions are always measured relative to the previous leg. In the right-hand part of a chaining operator f is implied by default. To specify a bent trajectory, explicit directional prefixes are needed. Note that a bent chain can fork, and thus represent multiple moves.

Chaining implies the intermediate square should be empty, and the described move is considered blocked if it isn't. In other words, an m prefix is implied on every left-hand part of the chaining operator. Explicit use of modifiers for the move modality can override this, to create other conditions that have to be satisfied for the move to continue, or 'side effects'.

For example: a Checker is fmFfcF-mF, the forward fcF step to capture only allowed if the following (f)mF in the same direction to an empty square can also be made. The Xiangqi Horse is W-F: it can be blocked on the (m)W step, but if it isn't, it can continue to complete an N move. The Chinese Cannon is mRpR-cR: mR for the non-capture, and pR to bring it to an occupied square, from which it must continue in the same direction with an (f)cR to capture a foe. The Grasshopper, pQ-K, moves with pQ as a Queen to an occupied square, and changes stride there to a King step in the same direction (f)K to capture or non-capture. A Chu-Shogi Lion hit-and-run or double capture is cK-aK: first capture a neighboring piece with cK, then arbitrarily change direction for the aK step to possibly capture a second time.

Eruption, explosion and suicide

There is a special modifier x that specifies eruption. This means that at the start of the leg to which it applies the moving piece stays in place itself, but ejects a multitude of independent 'fragments', which will continue along the chain in every possible way simultaneously and independently. The fragments disappear when they reach the end of the chain, but when they block, it only affects themselves, not other fragments or the real piece that was left behind. For example mcB-xaK means the piece makes a Bishop move (mcB, move or capture) to square S and then erupts to emit fragments along an aK step, dispersing them in 8 directions, to capture what they find there (or silently evaporate). Thus all enemies surrounding square S will be captured, as in Tenjiku-Shogi Fire-Demon burns.

Suicide can be expressed with the aid of a null-move leg: dO suffixed behind the move. A kamikaze capture on a Bishop move would be cB-dO. This can be combined with eruption to indicate an explosion in which the piece is destroyed: So an Atomic Queen would be like mQcQ-xdO-acdK, exploding itself after capture with xdO, while the fragments move on to destroy friend and foe in all directions with acdK. [No method yet to exempt Pawn victims, though.]

Describing alien modes of capture

Unusual (= non-replacement) capture can be described by visiting the capture squares as part of a chain, but not staying there. Rifle capture can be cR-ebR, a Rook capture followed by an equally long move (ebR) in the reversed direction. The Roccoco Advancer would make a forward rifle capture from the (empty) square where it stops: mQ-cK-bK, which has the collinear rifle capture (f)cK-bK appended to the real move. Ultima Pincer-Pawn captures can be described with the aid of eruption: mR-xatD-bW. The emitted fragments move from the final square where the eruption takes place to orthogonal D-neighbors in every direction, to test if there is a friendly piece there (atD). From such a piece they bounce back to capture a pinced foe (bW), and evaporate.

Capture of the Ultima Withdrawer is cK-bK-mQ: capture an adjacent piece (cK), but only if you can then move in the opposite direction to overshoot your starting square with a non-capture Queen move. (b(m)K to get back to the starting square, and (f)mQ to make the real move of at least one step.) There is a catch, however: there might be nothing to capture, and then the Withdrawer must be able to move as Queen. This cannot be repaired by offerring an mQ move as a general alternative, because that would make the Withdrawer capture optional even when it is possible. Writing the first step as mtcK to allow it to traverse an empty square and hop over a friend does still fail at the board edge, where the detour to probe for a victim would go off board. The modifier o for allowing crossing of the board edge can cure this: cmtoK-bK-mQ.

Bifurcators and Catapults

Bifurcators are basically bent hoppers that hop over off-ray supports. By warping their trajectory it can be made to pass through the support again. The Dimachaer, which changes tack just before its initial leg hits an obstacle, would be R-pW-bW-bB, a 'rifle hop' pW-bW inserted into the trajectory to fix the deflection point, which for the 'hook mover' R-B could have been anywhere. Bifurcating after the hop would be pR-W-B, deflecting would be R-rpW-bW-rbB, etc. The usual mc modifiers can specify what it might do on the second leg.

Pieces that displace other pieces as side effect of their move are called catapults. These can be specified with the aid of the u operator. E.g. (cdW-bW-R-uW-bW)(moW-bW-R) would specify an orthogonal mover that combines Withdrawer-like capture (making the cdW-bW detour to pick off the friend or foe in its wake) with a detour to unload the captured piece just in front of where it stopped. The alternative move that doesn't drop anything first tests if there really is nothing to launch, by requiring the square behind it is empty or off-board.

Magnetic pieces? No problem! B-xaK-cdK-buK is a Bishop that attracts all pieces that are 2 squares away one square towards it. An even more compact way to write it would be B-xacdK02-buK, which combines the two King steps into one 'jumping King' step that does both the omni-directional eruption and the picking up of the pieces. Repellers? Just change it to B-xacdK-uK, which picks up the pieces on the first step, and unloads them on the next step without reversing direction.

Castling

Castling is another challenge, as it moves two pieces. But in fact it is just a sort of catapult move. One piece could be used to carry the other to its destination. E.g. a move irdiW04-buW02-bW on a Rook would sort of describe Q-side castling. The leading i would indicate it is an initial move, that can only be made when the Rook has not moved yet. The rdW04 specifies sliding to the King square (confirming the emptiness of the in-between squares), to destroy it there. That we say di in stead of just d means it can only destroy a virgin King. By default we carry along what we capture or destroy, and discard it at the end of the move, or when we capture something else. The second leg reverses direction, and unloads the King two squares further. That we use uW02 rather than uD can have consquences for passing through check. after unloading the King, the Rook doubles back again, and takes its place with a single W step.

Some nitpicking

directonality - Oblique moves come in 'chiral pairs', which behave different with respect to l and r. So rfN-rW (an overall A step) is not symmetry-equivalent to lfN-rW (an overall D step). This would make it impossible to combine all symmetry-equivalent moves to N-rW, and force us to write each move separately. This is prevented by not only expressing the directions of a later step in coordinates that rotated to fit the previous step, but also flipped (exchanging the role of l and r) to make the initial step (or the first overall step when the trajectory became chiral) aim just right of the nearest orthogonal. So lfN-rW would actually mean the mirror image of rfN-rW, and N-rW will specify an 8-fold symmetric piece with a multi-path move to the A squares.

distributivity - The chaining operator specifies a default direction f for its right-hand operand, and a default modality m for its left-hand operand. This leaves the modality of the last leg of a chain unspecified (defaulting to the normal mc), as well as the direction of the first leg (defaulting to a). Modifier prefixes applied to an entire chain, like vc(X-Y-Z), will refer to these unspecified items, i.e. vX-Y-cZ. Directional and modality modifiers thus behave differently!

Applied to a compound move set, modality modifiers apply to all moves of the set: c(XYZ) means cXcYcZ. It is in general not sensible (or meaningful) to apply directional modifiers to a compound move set: first combining the move sets just makes it harder to specify which of them you exactly mean, and the directional system is only defined to hadle sets of 4 or 8 moves. The possible exceptions are K and Q, which are convenient shortcuts. Their moves can be treated in the scheme for the non-degenerate 8-fold case. I.e. fK = fW, fsK = fF, etc. Note that for continuation legs in 45-degree rotated coordinates fK could also refer to a single F move.

exponentiation - A number behind an atom or a set of moves protected by parentheses means by default any number of the corresponding moves upto the specified value chained together. When the number starts with a zero digit, it means exactly that number of repetitions. i.e. (X-Y)03 means (X-Y)-(X-Y)-(X-Y) which by default means (mX-fY)-(mX-fY)-(mX-fY), which again defaults to m(mX-fY)-fm(mX-fY)-f(mX-fY) = mX-fmY-fmX-fmY-fmX-fY. The defaults for the explicitly written chaining operators can be overridden by explicitly writing modifiers with their operands. It would be nice if there also was a way to override the defaults of the chaining operators 'spread around' by the exponentiation. Note that you cannot do that by writing them in the base unit, as they then would also apply to first and last factor in the chain. We therefore allow notations like Arc03, to mean cA-rcA-rA. I.e. the modifiers on the exponent redefine the defaults for the chaining operators implied by the exponent. So by default A3 means Afm3.

This allows convenient notations for crooked and circular riders: Nfr8 would be upto 8 repeated Knight jumps, each next one aiming 45 degrees right of the previous one, rather than continuing straight on as a limited-range Nightrider. This is an alternative notation for qN, the Rose. And (F-lF)r0 expands to (F-lF)-r(F-lf)-r(F-lf)-... = F-lF-rF-lF-rF-lF-..., which is the Crooked Bishop, with a highly accurately defined amount of crookedness.