Conditions for efficiency in finite executions

Abstract

The theory described in the previous chapters engages in some abuse of our notions of “efficiency.” Although it is easily seen to be necessary and reasonable for our protocol to use different running time bounds for different servers in a given class, the Levin-style enumerations incur an overhead that is exponential in the length of the program used to interact with the given server. In this chapter we show that this cost is unavoidable in general. Still, the servers used to show this lower bound is intuitively designed not to allow easy access, and thus there is reason to hope that a more “open” server might not force us to pay such a prohibitively large price. To an extent, these hopes are fulfilled: we show how a different construction of a universal user can take advantage of a natural sense of “commonality” when the server is designed to efficiently aid a large collection of “natural” user strategies in achieving the goal. Another upshot of restricting our attention to such servers is that we will be able to relax our various requirements on all states of a server to a requirement merely on the “effectively reachable” states. Finally, we show that when no suitable common notion of “natural” user strategies across a class of servers exists, no common efficient universal algorithm can be designed for that class.