The error complexity of strategies in infinite executions

Abstract

The basic theory of semantic communication in infinite executions introduced in Chapter 6 suffers from some of the same defects as the theory for finite executions from Chapter 2: namely, as discussed in Chapter 4, in that setting, the universal strategies we constructed suffered from an exponential overhead (in the length of the target strategy) in their running time, and we saw that such overhead was unavoidable in general. In the present chapter, we consider the analogous overhead in the number of errors incurred by a universal strategy in infinite executions. We will see that this overhead is also, in general, unavoidable by a similar argument. Again, similar to the developments of Chapter 4, we wish to lay down some natural conditions under which the overhead incurred by our generic constructions can be avoided. We will see that when the user strategies are assumed to come from some class of sufficiently simple strategies, there exist universal strategies that only incur a number of errors that is polynomial in the length of the description of the relevant user strategy. In fact, it turns out that the problem of constructing such a universal user for multi-session goal in which each session is a single round and a sensing function that provides immediate feedback is precisely the problem of learning the class of concepts corresponding to the class of strategies in the on-line learning model introduced by Bārzdiņš and Freivalds (1972) and investigated by Littlestone (1988). Thus, each solution to the on-line learning problem for a concept class yields a generic construction of a universal user from a sensing function that is viable with the corresponding class of strategies, and vice-versa. The reverse connection, together with the lower bounds of Section 4.4 suggest limits to the power of universal users based on the kind of sensing we have discussed thus far. We will observe that some natural kinds of feedback allow the construction of universal users for richer user strategies, and we suggest the exploration of such feedback as a next step towards the construction of universal users of suitable efficiency for real problems.