Impartial Games and Recursive Functions

Doctoral thesis, 2013

Interest in 2-player impartial games often concerns the famous theory of Sprague-Grundy. In this thesis we study other aspects, bridging some gaps between combinatorial number theory, computer science and combinatorial games. The family of heap games is rewarding from the point of view of combinatorial number theory, partly because both the positions and the moves are represented simply by finite vectors of nonnegative integers. For example the famous game of Wythoff Nim on two heaps of tokens has a solution originating in Beatty sequences with modulus the Golden ratio. Sometimes generalizations of this game have similar properties, but mostly they are much harder to grasp fully. We study a spectrum of such variations, and our understanding of them ranges from being complete in the case of easier problems, to being very basic in the case of the harder ones. One of the most far reaching results concerns the convergence properties of a certain $\star\star$-operator for invariant subtraction games, introduced here to resolve an open problem in the area. The convergence holds for any game in any finite dimension. We also have a complete understanding of the reflexive properties of such games. Furthermore, interesting problems regarding computability can be formulated in this setting. In fact, we present two Turing complete families of impartial (heap) games. This implies that certain questions regarding their behavior are algorithmically undecidable, such as: Does a given finite sequence of move options alternate between N- and P-positions? Do two games have the same sets of P-positions? The notion of N- and P-positions is very central to the class of normal play impartial games. A position is in P if and only if it is safe to move there. This is virtually the only theory that we need. Therefore we hope that our material will inspire even advanced undergraduate students in future research projects. However we would not consider it impossible that the universality of our games will bridge even more gaps to other territories of mathematics and perhaps other sciences as well. In addition, some of our findings may apply as recreational games/mathematics.