 We would like to consider the behavior of the successor module described above as an element of a set \(G_\texttt{num}\multimap G_\texttt{num}\) of strategies that compute functions over the natural numbers. If we consider only the sequences of interactions of \(S\) taking place either on the left or on the right side of the module of  Figure 1,  we see that they describe positions of \(G_\texttt{num}\), with an inversion of polarity (active/passive) depending on which side the interactions take place: the module is initially passive, and becomes active upon receiving a request from the environment. Such inversion, represented by the complementary labeling of the moves \(\overline{\lambda_G}\), assigning to Proponent the moves of the Opponent in \(G\) and conversely, is essential to the definition of a game \(G \multimap H\):