Games with 1-backtracking

We associate with any game G another game, which is a variant of it, and which we call bck ( G ) . Winning strategies for bck ( G ) have a lower recursive degree than winning strategies for G : if a player has a winning strategy of recursive degree 1 over G , then it has a recursive winning strategy over bck ( G ) , and vice versa. Through bck ( G ) we can express in algorithmic form, as a recursive winning strategy, many (but not all) common proofs of non-constructive Mathematics, namely exactly the theorems of the sub-classical logic Limit Computable Mathematics (Hayashi (2006) [6], Hayashi and Nakata (2001) [7]).