Most existing theory re nement systems are not incremental. However, any theory re nement system whose input and output theories are compatible can be used to incrementally assimilate data into an evolving theory. This is done by continually feeding its revised theory back in as its input theory. An incremental batch approach, in which the system assimilates a batch of examples at each step, se...

It is clear that Ñ would accept L (N accepts if one branch ends up in the accepting state. Ñ would notice that as rejecting state it would reject as soon as there is one rejecting state because all the states are universal. N rejects if no branch end in an accepting state, i.e., all end up in the rejecting state. But for Ñ that would mean that all the branches are accepting, therefore, it would...

Exercise 7.2. Find the fault in the following proof of P 6= NP: Assume that P = NP. Then SAT ∈ P and thus there exists a k ∈ N such that SAT ∈ DTime(n). Because every language in NP is polynomial-time reducible to SAT we have NP ⊆ DTime(n). It follows that P ⊆ DTime(n). But by the Time Hierarchy Theorem there exist languages in DTime(n) that are not in DTime(n), contradicting P ⊆ DTime(n). Ther...

Informally, Average case Complexity theory considers the average complexity with respect to a probability distribution over the set of inputs, instead of the complexity on the worst case input, as a criterion for the hardness of a problem. With regard to this theory, we explore the notion of a natural distribution. In particular, we consider different bounds on the minentropy of such distributi...