Multi-leg moves in simple Betza notation

Again I had an idea for an ad-hoc extension to Betza notation, which would be totally redundant in the much more general (and complex) Betza 2.0 proposal. The latter distinguishes itself by allowing chaining of different atoms into multi-leg moves, which is very flexible, but rarely needed. In original Betza a move always only involves a single atom, and lots of modifiers on it. This is often good enough. Many pieces can make multiple steps, but then they almost always repeat the same step, and original Betza notation provides for this with numeric suffixes like W3.

I consider it useful to see how far one can stretch this single-atom paradigm. It will never be able to do as much as Betza 2.0, but it has the virtue of great simplicity. And simplicity is IMO the key to success. So there definitely is room for a Betza '1.5', as it were.

So I wanted to add a modifier 'a' (still one of the few unused lower-case letters!), standing for 'Again', allowing a repeat of the atom to which it applies with an Arbitrary direction change. So aK is a bit like K02: exactly two King steps, but unlike K02 the second step does not have to go into the exact same direction as the first, but can go in Any direction. Like in K02 the intermediate square must be unoccupied for the move to be possible. In Tenjiku Shogi this would be called an 'Area move'. You see, lots of reasons to use 'a' for this! :-)

Now a unique thing about 'a' is that it is an ordering modifier. Normally Betza modifiers can be arbitrarily permuted, but w.r.t. 'a' it really makes a difference whether they are appearing left or right of it. Because what appears left of it pertains to the first step, and what appears right to the second. Using this convention to write the default move modality, aK = mamcK: the first step cannot capture, the second can move and capture. (Having plain 'm' as default modality for non-final legs might not be optimal, though; See below!)

Multiple repeats can be indicated by simply repeating the 'a' prefix: aaK = mamamcK would be a range-3 area move, as the Tenjiku-Shogi Vice General and Fire Demon have. Now if we want to describe the Chu Shogi Lion, the usual way is to say KNAD+something, where the 'something' must account for the two-leg moves that affect the intermediate square. (Without affecting that square, they would be equivalent to the direct jumps already given by KNAD.) Using the 'a' modifier this could be written as caK, a 2-step area move where the first step must capture, and the second step can move or capture. In fact, we could add the good old hopper prefix 'p' here, to indicate that the move could also hop over occupied squares: pmcaK. This would make the KNAD completely redundant! Note that it is most useful to interpret the 'p' as applied only to the following 'a', rather than to the move as a whole. This would allow you more precise specification of where the hop can be taken when there are more than 2 steps. I.e. paaK would be different from apaK, as the first must traverse occupied-empty-empty/foe, while the latter needs empty-occupied-empty/foe. (Note that aapK makes as much sense as pK: none.)

Now we did not account for the Lion's null move, when it uses the second step to retract the first. The consensus is that a Lion can only do that if there is a neighboring empty square. (Which is a bit illogical: if it can make a D move when completely surrounded by pieces on all K squares, why would it not be able to hop back and forth over them?) So the O atom that is sometimes proposed for a null-step cannot be used for it, as it would convey an unconditional null move. Unfortunately the pmcaK notation also includes an unconditional null move, as the direction arbitrariness implied by the 'a' operator does not exclude moving exactly back. And we would not want to exclude that, as caK should include the 'rifle capture' (called 'igui' in Shogi), and maK the null move. The problem is in the paK. An ad-hoc solution to that would be to make use of that other hopping modifier, 'g', to indicate the difference. This modifier specifies you should land immediately behind the obstacle. We could make a liberal interpretation of 'behind' in the context of bent trajectories, arguing that exactly retracing your step is in fact stopping in front of it, rather than behind. So 'pa' would be a hop that could change direction truly arbitrarily, while 'ga' would exclude a 180-degree reversal of the path. With this convention, the Chu Shogi Lion would be completely described by gmcaK. (Note that the single K steps can always be made through a detour, as gmc effectively allows you to ignore the square altogether.)

Now Chu Shogi also has pieces (Eagle and Falcon) which can do a linear two-leg move, more general than W02 or F02 because it is allowed to reverse direction: out-out or out-in. Like with the Lion's two-leg move, they don't have to stop when the first leg is a capture. The linearity restiction can be indicated by putting directional modifiers on the second step, i.e. behind the 'a', with the interpretation that 'f' means outward (i.e. relative to the previous step, like in Betza 2.0 continuation steps). A Checker's capture would be fcafmF this way: fcF for the first leg that must capture forward, followed by an F step in the same direction to an empty square. The Eagle could also reverse direction, and would thus have fcavF (as v = fb = out or in, and there is no restriction in this case that it ends on a formerly empty square). Incorporating also the fA leaps and the conditional null move would make it fgmcavF. (Where the 'g' modality would suppress the 'b' component of the next leg's 'v', reducing to an over-all jA, but the 'mc' modalities would both allow it, for null move and igui, respectively.)

The large Shogi variants (Dai Dai and up) have 'hook movers': Rooks or Bishops that can turn one corner. This can be written as asR or asB, the second leg being sideway (i.e. at 90 degrees). Turning a corner happens to be optional, so the full description has to be RasR or BasB. (Unfortunately the simpler aR does not do, as the afR that includes does miss the W steps. WaR would do it, though!)

Another complex piece there is the Lion Dog: three in/out steps along the same ray, which could capture or jump on the way. The (exactly) 3-step component of that would be pmcafpmcavK: the first two steps would go outward over empty or occupied squares, optionally cleaning them out if they contained a foe. Then you would either step back to the first or go on to the third square, with full move/capture permission. This includes the one-step moves, by taking the second step as 'pm' and the third as 'b'. So it has to be supplemented only by the Eagle-like two-leg moves gmcavK.

As a final application there is the Free Eagle of Tenjiku Shogi, which moves as Queen, or can make a two-leg Ferz area move with 'Lion powers' (i.e. continue after capture). This would be QgmcaF.

Hmm, after having written all this, I cannot help noticing that this combination gmc/pmc occurs very frequently. That starts me wondering if it would not be a more-convenient default for the non-final legs than a plain m. With gmc as default modality we would have:

Lion:       (aK)
Checker:    fmF(fcafmF)
Falcon:     BbsR(favW)
Eagle:      RbB(favF)
Lion Dog:   K(avKafavK)
Hook Mover: (WmaR)
Capricorn:  (FmaB)
Vice Gen.:  BcppB(mamaK)
Free Eagle: Q(aF)

where I wrote the parentheses only to highlight the multi-leg moves we discussed. That looks really compact! Despite the need for an extra plain K on the Lion dog, as the g/p convention would forbid the realization out-outhop-in of the afavK move to mimic the one-step move, and it would take more writing to overrule the default gmc to pmc than just writing the one-step moves separately. (Which is less contrived anyway.) The extra 'm' needed on the hook movers and Vice General is just a very small price to pay.

So let's give 'a' moves Lion power (gmca) by default!