Probabilistic game automata

Abstract

We define a probabilistic game automaton, a general model of a two-person game. We show how this model includes as special cases the games against nature of Papadimitriou [9], the Arthur-Merlin games of Babai [1] andthe interactive proof systems of Goldwasser, Micali and Rackoff [5]. We prove a number of results about another special case, games against unknown nature, which is a generalization of games against nature. In our notation, we let UP(UC, resp.) denote the class of two-person games with unbounded two-sided error where one player plays randomly with partial information (complete information, resp.) and the otherplayer plays existentially. Hence, the designation UC refers to games against known nature andUP refers to games against unknown nature. We show that

$$\begin{gathered}ATIME(t(n)) = UC - TIME(t(n)) \subseteq UP - TIME(t(n)) \subseteq UC - TIME(t^2 (n)) \hfill \\ASPACE(s(n)) = UC - SPACE(s(n)) \subseteq UP - SPACE(log(s(n))) \hfill \\\end{gathered}$$

where ATIME and ASPACE refer to alternating time and spacerespectively. We assume that all the space and time bounds are deterministically constructible and s(n)=Ω(n). The equalityATIME(t(n))=UC-TIME(t(n)) is due to Papadimitriou[9]. All the other inclusions above except one involve the simulation of one game by another. The exception is the result that UC-SPACE(s(n))⊑ASPACE(s(n)) which is shown byreducing a certain game theoretic problem to linear programming.

Supported by an IBM fellowship.

Supported by National Science Foundation grant number DCR-8402565. Some of the research for this paper was done at the Mathematical Sciences Research Institute, Berkeley, California.