[Retros] request for real definition of PRA

[Valery Liskovets]

RVs with 2 special moves

Dear retro-friends,

A.Buchanan properly identified non-dominated vectors of "game
state variables" in general, although he considered them only
in application to pRA. But such mutually exclusive vectors,
which I call "retro-variants" (RVs), are presented (at least
implicitly) in the problems of ALL recognized controversial
genres including "retro-strategy" (where, unlike pRA, not every
RV is accompanied with a solution)! Instead of general analysis,
let us restrict ourselves to the simplest and rather transparent
(and very well-known in principle) particular case of TWO
interacting special moves in orthodox chess: castling and/or e.p.
My aim here is to sketchily observe and interpret uniformly all
possible logical interconnections between them, as I understand
this subject.

In general, for an arbitrary set of such (potentially possible)
moves in a chess position, an RV is defined more precisely as
  a maximal (by inclusion) compatible set of legal castlings
  and illegal starting captures e.p.
Of course, this notion makes sense only in the case when there is
more than one RV, i.e., the legalities are not independent (and
all moves are significant). The set of all RVs in the position
is called its retro-relationship. Designations:
 w,w1,w2 - legal w. castling;   b,b1,b2 - legal b. castling;
 e,e1,e2 - legal starting e.p.;  ~r = NOT r;   rs = r AND s;
 [r,s] = r~s OR ~rs - exactly one element is valid (XOR).

In our binary case, a position-implementable retro-relationship
is always a pair of alternate RVs: RV1 and RV2 = NOT RV1. Every
type of binary retro-relationship can be represented by the
formula [r,s] where r and s are the two moves taken as legal or
illegal. We identify 8 possible types of retro-relationships.
They are listed below together with their implementation genres
in DIRECT mate compositions:

  T1 [w1,w2] RV1 = w1~w2 RV2 = ~w1w2 Implementation: AL(pRA)
 T2 [w,b]   RV1 = w~b   RV2 = ~wb   Implementation: PF(RS)
 T3 [b1,b2] RV1 = b1~b2 RV2 = ~b1b2 Implementation: PR(pRA)
 T4 [e,w]   RV1 = e~w   RV2 = ~ew   Implementation: AL(pRA)
 T5 [e,b]   RV1 = e~b   RV2 = ~eb   Implementation:   -
 T6 [e,~w]  RV1 = ew    RV2 = ~e~w  Implementation: AP(RS)
 T7 [e,~b]  RV1 = eb    RV2 = ~e~b  Implementation: PR(pRA)
 T8 [e1,e2] RV1 = e1~e2 RV2 = ~e1e2 Implementation: PR(pRA)

Verbal descriptions:
 T1, T2, T3: two mutually incompatible castlings
 T4, T5: incompatible castling & e.p.
 T6, T7: castling implies e.p.
 T8: two mutually incompatible e.p.

Of course, by the adopted conventions on castling/e.p.,
the formulas [w1,~w2], [w,~b], [b1,~b2] and [e1,~e2] are
(position-)unimplementable.

Controversial genres (classic):
 pRA = partial RA (known also as CRAC; often called RV, but
here we use the latter notion in another, more concrete sense)
 RS = retro-synthesis (retro-strategy)
 AL = "ad libitum" (Keym (Dawson)): pRA [least familiar genre]
 AP = "a posteriori" (Petrovic): RS
 PF = "post factum" (Adamson (Hoeg), subordination): RS
 PR = "a priori" (Oeffner (Loyd, Langstaff)): pRA.

Under implementation I mean a SOLUTION-implementation, i.e. a
sound (in its OWN genre) direct mate problem in which the position
contains a given retro-relationship and the solution (including
tries and set play) contains these moves and depends on the retro-
relationship. Implementation genres of these 8 types deserve a
detailed discussion elsewhere. I don't claim to embrace all the
possible implementations, and they should be revised considerably
(even among the four main genres) for help-mate and other
stipulations. Below I illustrate only several types.

T1. We look for a problem with two incompatible w castlings
any of which results in a mate in the stipulated number of moves.
These solutions are not cooks for each other (as one could think
if the problem would not be stipulated "pRA") since each of them
is only a "partial" solution that is legal only for one of two
alternate RVs. This is a simplest sample of "typ Keym" pRA, and
there exist many sound problems of this content. Almost the same
can be said about T4 with starting e.p. instead of one castling.
Without the mark "pRA" in the stipulation, however, such a problem
would be sound outside RA as having one solution (with castling).

T2. We look for a problem with incompatible w and b castlings.
Such are well-known subordination problems, in which, typically,
W castles immediately (without any doubt as to "what if..."),
thereby choosing RV1 and excluding RV2. In subtler samples, W
postpones castling and provides B the possibility to castle first
instead of W. Now, the full solution contains lines with each of
two castlings (i.e. it joins both retro-variants, rather than
consists of "partial" solutions!), and it is solvable only because
they are incompatible. This analysis does show that the notion of
RV, as it was introduced above, is quite applicable to retro-
synthesis (for AP, too) and is adequate for its interpretation.

T5. In such a direct-mate problem, if there existed any, a
solution that is legal for RV2 (W cannot start with e.p. and B
may castle) would be, a fortiori, legal for RV1. Moreover I cannot
imagine a more or less natural logic under which both RVs would
be significant for the solution but one of them be excluded at all
(just as it takes place with RV2 in the RS-implementations of T2
and T6).

The following observation is important: (at least) in our
particular case, all four genres exploit (almost exclusively)
their own types. Say, sound problems of "typ Oeffner" pRA exist
only of types T4, T7 and T8, and no other genre pretends to
use creatively any of these types. Therefore the solver can,
in principle, recognize the intrinsic genre (one of four) of
a sound controversial problem without any other stipulated prompt
(in reality this is not the only way to properly recognize
problem's genre).

I haven't researched other retro-interacting element in this
respect, even those with which I've much practised, and invented
something: "turn to move" in ordinary chess (AP-Keym) and "change
of the color" in retro-volages chess. Instead, I analysed the
ternary retro-variants (ones that combine legalities of 3 special
moves: castling/e.p.). I identified 37 potential types of ternary
retro-relationships and can described them in terms of the same
bracketing operations, binary and ternary. A much more
sophisticated picture... I believe that all types are position-
implementable. Unfortunately, I still have no definite opinion on
the implementability of many of them in direct mate problems in
the main or "hybrid" (do exist) controversial genres.

Any comments are welcome.
 Valery Liskovets

andrew buchanan wrote:


> Dear Retro chums,

>

> I would really appreciate a *definition* of PRA (aka RV, aka "Ceriani

> Ethics").

>

> It's all very well to hear "well it sort of works like this in *this*

> problem". I get a hint of a whiff what it might be about. But that

> gives me little predictive power when it comes to the next PRA problem.

> So I thought I would send this email once and then I will know.

> Hopefully.

>

> Here is my best guess so far as to what PRA might be.

>

> Suppose we have a position. There are a number of game state variables

> (who has the move, which castlings rights are retained, what en

> passants might be on). These variables give a set of vectors, e.g. they

> might be:

>

> W,C1Y,C2Y,E1N

> W,C1Y,C2N,E1N

> B,C1N,C2N,E1Y

> ...

>

> where the first one means there is a proof game which results in the

> position, with White to play, both castlings OK, but en passant not

> working. Etc.

>

> Now some subset of this set is the twinset. How do we pick them? I

> think we define a relation ">" (pronounced "dominates") on the vectors.

> We say that u>v if every piece in v has all its powers (and possibly

> more) in u. (So in the list above, the first vector dominates the

> second.)

>

> The PRA twins are those vectors which are not dominated by any other.

> So in the above list it would be the first & third vectors.

>

> Am I right so far?

>

> Note that this definition of PRA is independent of words like

> "castling" & "en passant". Hence it can be applied to fairy chess with

> state as well...