Noncomputable Spectral Sets

It is a basic fact that, given a computer language and a computable integer function, there exists a shortest program in that language which computes the desired function. Once a programmer establishes the correctness of a program, she then need only verify that the program is the shortest possible in order to declare complete victory. Unfortunately, she can’t. A creature that could identify minimal programs would not only be able to decide the halting problem, but she could even decide the halting problem for machines with halting set oracles. Such a creature exceeds the powers of ordinary machine cognition, and must therefore be a divine jument-numen. Suppose, however, the programmer would be satisfied to know just whether or not her program is minimal up to finitely many errors. In this case, even the jument-numen is helpless: something much stronger is needed. Just as we can associate equality with the ordinary notion of “minimal,” we can associate an equivalence relation, =∗, with the principle of “minimal up to finitely many errors.” This thesis is the first to explore the extensive realm of minimal indices beyond the =∗ relation. Every equivalence relation gives rise to a notion of minimality, modulo that relation. We call the resulting collection of minimal indices a spectral set, because it selects exactly one function from each equivalence class. Spectral sets are rare, natural examples of non-index sets which are neither Σn nor Πn-complete. In this thesis, we classify spectral sets according to their thinness and information content. We give optimal immunity results for the spectral sets (that is, we identify types of sets which are not contained in spectral sets), and we place these sets in the arithmetic hierarchy (which quantitatively measures their information contents). Some lower bounds in the arithmetic hierarchy follow from immunity properties alone, but we further refine viii these immunity bounds using direct techniques. We also measure information content with Turing equivalences. In fact, the spectral sets become canonical iterations of the halting set when we list our computer programs in the right order. Regardless of the particular numbering, a reasonable amount of information is always present in such sets. We now informally describe the contents of some spectral sets. The Π3-Separation Theorem says that the spectral sets pertaining to ≡1, = ∗, and ≡m each have the same complexity with respect to the arithmetic hierarchy, yet each of these sets is immune against a different level of the arithmetic hierarchy. We can thus quantitatively compare the strength of equivalence relations by way of immunity. We also prove a result which we call the Forcing Lowness Lemma. Using this lemma, we show that ∅′′′′ is decidable in MIN (the spectral set for ≡T) together with ∅ ′′. This result is probably optimal, and we apply the Forcing Lowness Lemma again to show that, in some formal sense, this fact will be difficult to prove. Armed with this new machinery, we highlight its utility with some new applications. First, we prove the Peak Hierarchy Theorem: there exists a set which neither contains nor is disjoint from any infinite arithmetic set, yet the set is majorized by a computable function. Furthermore, the set that we construct is natural in the sense that it contains a spectral set. Along the way, we construct a computable sequence of c.e. sets in which no set can be computed from the join of the others, for any iteration of the jump operator. We use the machinery of spectral sets to quantitatively compare the power of nondeterminism with the power of the jump operator. We show that in the world of computably enumerable sets, nondeterminism contributes nothing to immunity. In this respect, the jump operator outshines nondeterminism. Nonetheless, we can ascend naturally from the lowest level of the spectral hierarchy using nondeterminism. Finally, we present connections to the Arslanov Completeness Criterion which stand as immediate consequences of immunity properties for spectral sets.