Equivalences and Congruences on Infinite Conway Games

The increasing use of games as a convenient metaphor for modeling interactions has spurred the growth of a broad variety of game definitions in Computer Science. Furthermore, in the presentation of games many related concepts are None has a unique definition. Some, but not always the same, are taken as primitive while the others are reduced to them. And many more properties need to be specified before the kind of game one is interested in is actually pinned down, e.g.: perfect knowledge, zero-sum, chance, number of players, finiteness, determinacy, etc. All this together with the wide gamut of games arising in real life calls for a unifying foundational approach to games. In [HL09], we started such a programme using very unbiased foundational tools, namely algebras and coalgebras. We build upon Conway's notion of game. It provides an elementary but sufficiently abstract notion of game amenable to a rich algebraic-coalgebraic treatment because of the special role that sums of games play in this theory. Conway games [Con01] are combinatorial games, namely no chance 2-player games, the two players being conventionally called Left (L) and Right (R). Such games have positions, and in any position there are rules which restrict L to move to any of certain positions, called the Left positions, while R may similarly move only to certain positions, called the Right positions. L and R move in turn, the player who plays first is denoted by I, while the one playing second is denoted by II. Notice that L or R can be either I or II and conversely. The need for this extra generality is due to the fact that in most games each player has a different set of options. Moreover, as we will see in the definition of sums of games, there may not be a strict alternation of moves in any given component. The game is of perfect knowledge, i.e. all positions are public to both players. The game ends when one of the players has no move. In normal play the other player is the winner, while in misère play, the winner is the very player himself. The payoff function yields only 0 or 1. Many games played on boards are combi-L and R may have different sets of moves are called partizan. Many notions of games such as those which arise in Set Theory, in Automata Theory, or in Semantics of Programming Languages …