Randomness and Computability

<p>This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus on randomness for finite strings. Levin introduced two functions which measure the randomness of finite strings. One function is derived from a universal monotone machine and the other function is derived from an optimal computably enumerable semimeasure. Gacs proved that infinitely often, the gap between these two functions exceeds the inverse Ackermann function (applied to string length). This thesis improves this result to show that infinitely often the difference between these two functions exceeds the double logarithm. Another separation result is proved for two different kinds of process machine. Information about the randomness of finite strings can be used as a computational resource. This information is contained in the overgraph. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not truth-table complete. This question is answered in the negative. Related results are also established. This thesis makes advances in the theory of randomness for infinite binary sequences. A variant of process machines is used to characterise computable randomness, Schnorr randomness and weak randomness. This result is extended to give characterisations of these types of randomness using truthtable reducibility. The computable Lipschitz reducibility measures both the relative randomness and the relative computational power of real numbers. It is proved that the computable Lipschitz degrees of computably enumerable sets are not dense. Infinite binary sequences can be regarded as elements of Cantor space. Most research in randomness for Cantor space has been conducted using the uniform measure. However, the study of non-computable measures has led to interesting results. This thesis shows that the two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent. Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is shown that every PA degree computes a neutral measure. Work of Miller is used to show that the set of atoms of a neutral measure is a countable Scott set and in fact any countable Scott set is the set of atoms of some neutral measure. Neutral measures are used to prove new results in computability theory. For example, it is shown that the low computable enumerable sets are precisely the computably enumerable sets bounded by PA degrees strictly below the halting problem. This thesis applies ideas developed in the study of randomness to computability theory by examining indifferent sets for comeager classes in Cantor space. A number of results are proved. For example, it is shown that there exist 1-generic sets that can compute their own indifferent sets.</p>

Randomness and Computability

<p>This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus on randomness for finite strings. Levin introduced two functions which measure the randomness of finite strings. One function is derived from a universal monotone machine and the other function is derived from an optimal computably enumerable semimeasure. Gacs proved that infinitely often, the gap between these two functions exceeds the inverse Ackermann function (applied to string length). This thesis improves this result to show that infinitely often the difference between these two functions exceeds the double logarithm. Another separation result is proved for two different kinds of process machine. Information about the randomness of finite strings can be used as a computational resource. This information is contained in the overgraph. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not truth-table complete. This question is answered in the negative. Related results are also established. This thesis makes advances in the theory of randomness for infinite binary sequences. A variant of process machines is used to characterise computable randomness, Schnorr randomness and weak randomness. This result is extended to give characterisations of these types of randomness using truthtable reducibility. The computable Lipschitz reducibility measures both the relative randomness and the relative computational power of real numbers. It is proved that the computable Lipschitz degrees of computably enumerable sets are not dense. Infinite binary sequences can be regarded as elements of Cantor space. Most research in randomness for Cantor space has been conducted using the uniform measure. However, the study of non-computable measures has led to interesting results. This thesis shows that the two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent. Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is shown that every PA degree computes a neutral measure. Work of Miller is used to show that the set of atoms of a neutral measure is a countable Scott set and in fact any countable Scott set is the set of atoms of some neutral measure. Neutral measures are used to prove new results in computability theory. For example, it is shown that the low computable enumerable sets are precisely the computably enumerable sets bounded by PA degrees strictly below the halting problem. This thesis applies ideas developed in the study of randomness to computability theory by examining indifferent sets for comeager classes in Cantor space. A number of results are proved. For example, it is shown that there exist 1-generic sets that can compute their own indifferent sets.</p>

WEAKLY 2-RANDOMS AND 1-GENERICS IN SCOTT SETS

Let ${\cal S}$ be a Scott set, or even an ω-model of WWKL. Then for each A ε S, either there is X ε S that is weakly 2-random relative to A, or there is X ε S that is 1-generic relative to A. It follows that if A1,…,An ε S are noncomputable, there is X ε S such that each Ai is Turing incomparable with X, answering a question of Kučera and Slaman. More generally, any ∀∃ sentence in the language of partial orders that holds in ${\cal D}$ also holds in ${{\cal D}^{\cal S}}$, where ${{\cal D}^{\cal S}}$ is the partial order of Turing degrees of elements of ${\cal S}$.

Scott's problem for Proper Scott sets

Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω1. Here, I show that assuming the Proper Forcing Axiom (PFA), every A-proper Scott set is the standard system of a model of PA. I define that a Scott set is proper if the quotient Boolean algebra /Fin is a proper partial order and A-proper if is additionally arithmetically closed. I also investigate the question of the existence of proper Scott sets.

A note on standard systems and ultrafilters

Let (M, )⊨ ACA0 be such that , the collection of all unbounded sets in , admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in such that M thinks T is consistent. We prove that there is an end-extension N ⊨ T of M such that the subsets of M coded in N are precisely those in . As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T.

Degrees of unsolvability of continuous functions

We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f ∈ [0,1] computes a non-computable subset of ℕ there is a non-total degree between Turing degrees a <Tb iff b is a PA degree relative to a; ⊆ 2ℕ is a Scott set iff it is the collection of f-computable subsets of ℕ for some f ∈ [0,1] of non-total degree; and there are computably incomparable f, g ∈ [0,1] which compute exactly the same subsets of ℕ. Proofs draw from classical analysis and constructive analysis as well as from computability theory.

Completions of PA: Models and enumerations of representable sets

We generalize a result on True Arithmetic (ℐA) by Lachlan and Soare to certain other completions of Peano Arithmetic (PA). If ℐ is a completion of PA, then Rep(ℐ) denotes the family of sets X ⊆ ω for which there exists a formula φ(x) such that for all n ∈ ω, if n ∈ X, then ℐ ⊢ φ(S(n) (0)) and if n ∉ X, then ℐ ⊢ ┐φ(S(n)(O)). We show that if S, J ⊆ P(ω) such that S is a Scott set, J is a jump ideal, S ⊂ J and for all X ∈ J, there exists C ∈ S such that C is a “coding” set for the family of subtrees of 2<ω computable in X, and if ℐ is a completion of PA Such that Rep(ℐ) = S, then there exists a model A of ℐ such that J is the Scott set of A and no enumeration of Rep(ℐ) is computable in A. The model A of ℐ is obtained via a new notion of forcing.Before proving our main result, we demonstrate the existence of uncountably many different pairs (S, J) satisfying the conditions of our theorem. This involves a new characterization of 1-generic sets as coding sets for the computable subtrees of 2<ω. In particular, C C ⊆ ω is a coding set for the family of subtrees of 2<ω computable in X if and only if for all trees T ⊆ 2<ω computable in X, if χc is a path through T, then there exists σ ∈ T such that σ ⊂ χc and every extension of σ is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2<ω. We show they are identical.

Requirement systems

Methods for carrying out transfinitely nested priority constructions have been developed by Harrington [7] and by Ash [2, 1, 3, 4]. Ash's method has different versions, with later ones becoming simpler. Lemmp and Lerman [11] have also developed a method, for finitely nested constructions. Ash formulated abstractly the object of a nested priority construction, and he proved a metatheorem for what he called “α-systems”, listing conditions which guarantee the success of the construction. Harrington's method of “workers”, at least in its original, informal state, seems more flexible than Ash's α-systems.In [10, 9], there are finite and transfinite versions of a metatheorem for workers. The statements are complicated, and these metatheorems have not proved to be very useful. The present paper gives a new transfinite metatheorem. The statement is considerably simpler than the one in [9], although not so simple as that in [3]. The new metatheorem grew, in part, out of an effort to find a new proof of Ash's metatheorem. The new metatheorem yields the one in [3], and it seems more flexible. A different generalization of Ash's metatheorem will be given in [5].Ash's metatheorem is easier to use than the one in the present paper, and the result in [3] is certainly the one to use wherever it applies. Here we give one application of the new metatheorem which does not seem to follow from the result in [3]. This is a theorem on models “representing” a given Scott set, which implies one half of a recent result of Solovay [18], on Turing degrees of models of particular completions of Peano arithmetic (PA).