Impartial Games and Recursive Functions
Doktorsavhandling, 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.

Turing complete

Heap game

Splitting sequences

Rule 110

Game complexity

Dual game

Invariant subtraction game

Cellular automaton


Impartial game

Beatty sequences

Take-away game

Blocking maneuver

Comply maneuver

Complementary sequences


Game reflexivity

Move-size dynamic

Dictionary process


Algorithmically undecidable

Wythoff Nim

Game convergence

Subtraction game

Opponent: Professor Vladimir Gurvich


Urban Larsson

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

The star-operator and invariant subtraction games

Theoretical Computer Science,; Vol. 422(2012)p. 52-58

Artikel i vetenskaplig tidskrift

Blocking Wythoff Nim

Electronic Journal of Combinatorics,; Vol. 18(2011)p. 18-18

Artikel i vetenskaplig tidskrift

Invariant and dual subtraction games resolving the Duchene-Rigo conjecture

Theoretical Computer Science,; Vol. 412(2011)p. 729-735

Artikel i vetenskaplig tidskrift

2-Pile Nim With a Restricted number of Move-size Imitations (with an appendix by Peter Hegarty)

Integers Journal,; Vol. 9(2009)p. 671-690

Artikel i vetenskaplig tidskrift

A Generalized Diagonal Wythoff Nim

Integers,; Vol. 12(2012)p. 1-24

Artikel i vetenskaplig tidskrift

Impartial games emulating one-dimensional cellular automata and undecidability

Journal of Combinatorial Theory - Series A,; Vol. 120(2013)p. 1116-1130

Artikel i vetenskaplig tidskrift

From heaps of matches to the limits of computability

Electronic Journal of Combinatorics,; Vol. 20(2013)p. Paper 41-

Artikel i vetenskaplig tidskrift

Maharaja Nim


The first part of the introduction is written in a relaxed style that I hope will attract a broader audience.


Diskret matematik



Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 3524


Opponent: Professor Vladimir Gurvich

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