Complexity of question/answer games

AbstractQuestion/Answer games (Q/A games for short) are a generalization of the Rényi–Ulam game and they are a model for information extraction in parallel. A Q/A game, G=(D,s,(q1,…,qk)), is played on a directed acyclic graph, D=(V,E), with a distinguished start vertex s. In the ith round, Paul selects a set, Qi⊆V, of at most qi non-terminal vertices. Carole responds by choosing an outgoing edge from each vertex in Qi. At the end of k rounds, Paul wins if Carole’s answers define a unique path from the root to one of the terminal vertices in D.In this paper we analyze the complexity of Q/A games and explore the notion of fixed strategies. We show that the problem of determining if Paul wins the game played on a rooted tree via a fixed strategy is in NP. The same problem is Σ2P-complete for arbitrary digraphs. For general strategies, the problem is NP-complete if we restrict a two-round game to a digraph of depth three. An interesting aspect of this game is that it captures the even levels of the polynomial-time hierarchy when restricted to a fixed number of rounds; that is, determining if Paul wins a k-round game is Σ2k−2P-complete. The general version of the game is known to be PSPACE-complete [S. Abbasi, N. Sheikh, Some hardness results for Q/A games, Integers 7 (2007) G08]. In this paper we show that it remains PSPACE-complete even if each round consists of only two questions