Taking-and-merging games as rewrite games

This work is a contribution to the study of rewrite games. Positions are finite words, and the possible moves are defined by a finite number of local rewriting rules {u(i) -> v(i)}(i is an element of)(I): a move consists in the substitution of one occurrence of u(i) by v(i), for some i. We introduce and investigate taking-and-merging games, that is, where each rule is of the form a(k) -> epsilon. We give sufficient conditions for a game to be such that the losing positions (resp. the positions with a given Grundy value) form a regular language or a context-free language. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games. Finally we show that more general rewrite games quickly lead to undecidable problems. Namely, it is undecidable whether there exists a winning position in a given regular language, even if we restrict to games where each move strictly reduces the length of the current position.