Conway games, algebraically and coalgebraically

Details Conway games, algebraically and coalgebraically Conway games, algebraically and coalgebraically

2011-07-07

arxiv.org/abs/1107.1351v3

LMCS 7 (3:8) 2011

Computer Science

Logic in Computer Science

30 pages

Scientific paper

10.2168/LMCS-7(3:8)2011

Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on non-losing strategies. Hypergames are a fruitful metaphor for non-terminating processes, Conway's sum being similar to shuffling. We develop a theory of hypergames, which extends in a non-trivial way Conway's theory; in particular, we generalize Conway's results on game determinacy and characterization of strategies. Hypergames have a rather interesting theory, already in the case of impartial hypergames, for which we give a compositional semantics, in terms of a generalized Grundy-Sprague function and a system of generalized Nim games. Equivalences and congruences on games and hypergames are discussed. We indicate a number of intriguing directions for future work. We briefly compare hypergames with other notions of games used in computer science.

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