We study two-player take-away games whose outcomes emulate two-state one-dimensional cellular automata, such as Wolfram’s rules 60 and 110. Given an initial string consisting of a central data pattern and periodic left and right patterns, the rule 110 cellular automaton was recently proved Turing-complete by Matthew Cook. Hence, many questions regarding its behavior are algorithmically undecidable. We show that similar questions are undecidable for our rule 110 game. 1

Abstract: We consider the following modification of annihilation game called node blocking. Given a directed graph, each vertex can be occupied by at most one token. There are two types of tokens, each player can move his type of tokens. The players alternate their moves and the current player i selects one token of type i and moves the token along a directed edge to an unoccupied vertex. If a player cannot make a move then he loses. We consider the problem of determining the complexity of the game: given an arbitrary configuration of tokens in a directed acyclic graph, does the current player has a winning strategy? We prove that the problem is PSPACE-complete. Keywords: annihilation game, node blocking, PSPACE-completeness 1