Innocence in 2-dimensional games

In this note, we reformulateHyland-Ong-Nickau pointer game semantics for PCF, as a more concurrent style of games, called 2-dimensional games. The main novelty of these games is that they are played on concurrent graphs, instead of trees. This enables to permute moves, and to decompose the innocence constraint on strategies, as local permutation properties. 1 Step one: reformulate pointer games as tree games Thanks to linear logic, the task of designing a game model of PCF may be separated in two steps: first, design the linear part of the model; then, break the “exponential barrier” by solving the following “interleaving” problem. Recall that a play s of a game !A is obtained by interleaving (possibly several) plays s1; :::; sn of the game A. The problem is to interleave the plays s1; :::; sn in such a way that each of the threads si may be distinguished in s, and later extracted from it. Two solutions to that problem arose in the litterature, each one inducing a particular “comonoidal” construction of the game !A: index threads, as in [1, 7]. use pointers, as in [6, 13]. The first solution leads to tree game models, where plays are strings of moves; the second solution leads to pointer game models, where plays are justified strings of moves, where justification indicates for every movem of the play, whetherm is initial, or justified by a previous move of the play. Several constraints on strategies may be expressed in pointer games, like wellbracketedness, innocence, visibility or single-threadedness, each one corresponding to a particular computational paradigms, in a fully abstract fashion [3, 2, 5]. Pointers are fundamental to define these constraints on strategies. For instance, innocence is derived from the concept of Player view of an alternated (justified) play s, that is, the alternated (justified) play psq defined by induction: ps m nq = ps mq n if A(n) = +1, ps m t nq = psq m n ifm justifies n, and A(n) = -1, ps nq = n if no move of s justifies n, and A(n) = -1, p q = where is the empty word.