From heaps of matches to the limits of computability
Artikel i vetenskaplig tidskrift, 2013

We study so-called invariant games played with a fixed number d of heaps of matches. A game is described by a finite list M of integer vectors of length d specifying the legal moves. A move consists in changing the current game-state by adding one of the vectors in M, provided all elements of the resulting vector are nonnegative. For instance, in a two-heap game, the vector (1, -2) would mean adding one match to the first heap and removing two matches from the second heap. If (1, -2) is an element of M, such a move would be permitted provided there are at least two matches in the second heap. Two players take turns, and a player unable to make a move loses. We show that these games embrace computational universality, and that therefore a number of basic questions about them are algorithmically undecidable. In particular, we prove that there is no algorithm that takes two games M and M' (with the same number of heaps) as input, and determines whether or not they are equivalent in the sense that every starting-position which is a first player win in one of the games is a first player win in the other.

Combinatorial Games

Computational Complexity

Logic in Computer Science

P-POSITIONS

SUBTRACTION GAMES

INVARIANT

Författare

Urban Larsson

Chalmers, Matematiska vetenskaper, Matematik

Göteborgs universitet

Johan Wästlund

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Electronic Journal of Combinatorics

1097-1440 (ISSN) 1077-8926 (eISSN)

Vol. 20 3 Paper 41-

Ämneskategorier

Diskret matematik

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2020-01-31