Part 1/12
Compared to a simple gote or sente, an ordinary ko is 3 times but a stage ko in the corner is 100 times as difficult.
TheoryEvery theory of ko evaluation makes some assumptions. The standard theory of ko evaluation is called 'generalised thermography', although its scope of application is not the most general. It applies to so called 'simple' cyclic positions while multiple kos or long alternating cycles require additional or alternative theory. For simple cyclic positions, we can use 'generalised thermography'.
Its major assumptions are as follows. We use territory scoring. Each local play is taxed by T points because we assume a 'rich environment' with arbitrarily many plays elsewhere at every temperature T. They can be imagined as simple gotes or coupons with this move value. Such a rich environment is a model but enables consistent evaluation of many local endgames with kos. Generalised thermography produces move values, counts and gains. Another theory might produce slightly different values but we accept the values generated by generalised thermography because their relations permit a consistent interpretation.
"Definition 3.8. Let G be a loopy game. We say that G is simple if: (i) the only loops in G are kos; and (ii) every subposition of G has at most one Left ko option and at most one Right ko option." Citation reference [18]:
https://www.lifein19x19.com/viewtopic.p ... 45#p143245In this definition, 'game' means 'position', 'subposition' means 'the position itself or a follow-up position', 'ko' means 'basic ko' or 'local position with alternating 2-play cycle', 'loop' seems to have the intended meaning 'positional cycle of plays', 'Left' means 'Black', 'Right' means 'White', 'option' means 'next move', 'ko option' means 'basic ko capture'.
In the definition, 'loop' is ambiguous. In generalised thermography, non-alternating sequences of plays are allowed. In Combinatorial Game Theory, some examples of game graphs show 'simple' loopy games with successive ko options of the same player so that non-alternating long cycles can occur when travelling on game graph. However, below we learn:
- that a player's ko option followed by the opponent's play elsewhere requires the player's local play enabling him to 'win' the ko;
- if the player's ko option is followed by the opponent's local play, it may not recapture this ko.
This implies obeying the basic ko rule that immediate recapture creating a 2-play cycle is prohibited, although condition (i) of the definition of 'simple' loopy game does not declare it. Condition (i) does, however, imply exclusion of the following:
- 0-play loop from G to G, even not if the move would incur the tax T. However, generalised thermography models a coupon stack as the environment. Unless explicitly excluded, a player also has the option of taking a coupon of the value T aka playing elsewhere.
- Suicide cycles even if the rules allow suicide.
- Long cycles without actually played 2-play cycles.
Generalised thermography applied to a simple loopy initial position allows its evaluation without simultaneously evaluating some followers as alternative initial positions. Contrarily, positions with multiple kos, such as a double ko death, might need simultaneous evaluation.
On a player's move, choose his best among the available of these options:
- best local non-ko play
- his ko option followed by the opponent's local play that does not recapture this ko
- his ko option followed by the opponent's play elsewhere and the player's local play
The opponent optimises among the second and third options. Then the player optimises among his first option and the opponent's chosen option. Ignore unavailable options. Optimising means Black maximising and White minimising.
In the theory of generalised thermography, for each local play, Black pays -T while White pays +T. Temperatures start from -1 so that the payment T = -1 for a play is the tax -1 for placing its stone. As to the tax, an equal number of local plays by Black and White cancel each other.
The player's choices do not include his initial play elsewhere. We consider it implicitly by analysing Black's and White's starts and walls and constructing the thermograph. Its mast represents both players' starts by plays elsewhere.
In a rich environment, generalised thermography calculates two pairs of trajectories: the players' walls and scaffolds. The walls are the final trajectories of a thermograph. The scaffolds are the auxiliary trajectories of a thermograph, which help finding the walls. Walls and scaffolds relate counts of a local, for example endgame, position to temperatures. Besides, a wall describes what the player should play at every temperature T while a scaffold describes what happens if the player plays then, regardless of whether he should or should not do so. The basic advice distinguishes local play and play elsewhere.
Login
Register
FAQ
Search
Gérard TAILLE
is sente for black and for a theoritical point of view this exchange must be played by black when temperature drops between 13/12 and 14/12.
tenuki in another infinitesimal