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# From heaps of matches to the limits of computability

Urban Larsson (Institutionen för matematiska vetenskaper, matematik) ; Johan Wästlund (Institutionen för matematiska vetenskaper, matematik)
(2013)
[Preprint]

We study so-called invariant games played with a fixed number $d$ of heaps of matches. A game is described by a finite list $\mathcal{M}$ 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 $\mathcal{M}$, 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 $(1,-2) \in \mathcal{M}$, 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 $\mathcal{M}$ and $\mathcal{M}'$ (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.

Nyckelord: Algorithmically undecidable, Cellular Automata, Game complexity, Heap game, Impartial game, P-equivalence

CPL Pubid: 175717

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# Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)