The power of relaxed models

Abstract

Recall that our main theorems for finite executions in Chapter 2 and Chapter 5 show that in order to construct an efficient, reliable protocol for achieving a goal, it must be possible to efficiently verify that the goal has been achieved. We know that this places limits on what kinds of goals we could hope to achieve—for example, as we saw in Section 3.3, any problem we can solve using such a communications protocol in polynomial time with the class of all helpful servers must lie in PSPACE. Some natural computational problems are outside PSPACE, though, and so it is very natural to wonder if some weaker benefit could be obtained for these problems via communication with helpful servers. Now, although we saw that strong sensing in the infinite execution model was also bound by similar limitations in Theorem 6.47, we also saw that it was possible to construct universal users from weak safety in Theorem 6.23, which was not bound by such limitations. We will see, in this chapter, that this gap can be exploited: when the reliability requirement of the protocols is relaxed in some natural ways, we can substantially extend the class of goals that we can achieve universally. We will first show, in Section 7.1.1, a protocol for deciding any computable decision problem with all helpful servers in the infinite executions model of Chapter 6. We will then show, in Section 7.1.2, how this protocol can be converted into a protocol for deciding the same problems in finite executions if we allow the protocol to err on a finite set of instances (which may depend on the server and its initial state). Finally, in Section 7.2, we return to the infinite execution setting, and consider what aspects of computational goals we used in the design of our protocols. We will introduce exploration sessions as an abstraction of the key property, and show how, together with the ability to reset the server, we can obtain a universal user that only makes a finite number of mistakes with each server that is independent of the size parameter.