Preface- Acknowledgments- Introduction- I Background- Preliminaries- Computability Theory- Kolmogorov Complexity of Finite Strings- Relating Plain and Prefix-Free Complexity- Effective Reals- II Randomness of Sets- Martin-Lof Randomness- Other Notions of Effective Randomness- Algorithmic Randomness and Turing Reducibility- III Relative Randomness- Measures of Relative Randomness- The Quantity of K- and Other Degrees- Randomness-Theoretic Weakness- Lowness for Other Randomness Notions- Effective Hausdorff Dimension- IV Further Topics- Omega as an Operator- Complexity of CE Sets- References- Index

Algorithmic Randomness and Complexity

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Infinite Abelian Groups

The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. Covering the basics as well as recent research results, this book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.

Computability and randomness

Classification theory and the number of non-isomorphic models

We study the proof–theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RTkn denote Ramsey's theorem for k–colorings of n–element sets, and let RT<∞n denote (∀k)RTkn. Our main result on computability is: For any n ≥ 2 and any computable (recursive) k–coloring of the n–element sets of natural numbers, there is an infinite homogeneous set X with X″ ≤T 0(n). Let IΣn and BΣn denote the Σn induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models of arithmetic enables us to show that RCA0 + IΣ2 + RT22 is conservative over RCA0 + IΣ2 for Π11 statements and that RCA0 + IΣ3 + RT<∞2 is Π11-conservative over RCA0 + IΣ3. It follows that RCA0 + RT22 does not imply BΣ3. In contrast, J. Hirst showed that RCA0 + RT<∞2 does imply BΣ3, and we include a proof of a slightly strengthened version of this result. It follows that RT<∞2 is strictly stronger than RT22 over RCA0.

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On the strength of Ramsey's theorem for pairs

The d.r.e. degrees are not dense

We study computably enumerable equivalence relations (ceers), under the reducibility if there exists a computable function f such that if and only if , for every . We show that the degrees of ceers under the equivalence relation generated by form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if , where denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are -complete (the former answering an open question of Gao and Gerdes).

Universal computably enumerable equivalence relations

We show that the principle PART from Hirschfeldt and Shore is equivalent to the ∑0 2 -Bounding principle B∑ 0 2 over RCA 0 , answering one of their open questions Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that D 2 2 implies B∑ 0 2 and is thus indeed equivalent to Stable Ramsey's Theorem for Pairs (SRT 2 2 ) This also allows us to conclude that the combinatorial principles IPT 2 2 , SPT 2 2 and SIPT 2 2 defined by Dzhafarov and Hirst all imply B∑ 0 2 and thus that SPT 2 2 and SIPT 2 2 are both equivalent to SRT 2 2 as well Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA 0 , to the failure of B∑ 0 2 

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On the role of the collection principle for Sigma^0_2-formulas in second-order reverse mathematics

In Reverse Mathematics, the axiom system DNR, asserting the existence of diagonally non-recursive functions, is strictly weaker than WWKL0 (weak weak Konig's Lemma).

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Comparing DNR and WWKL

We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a question of Mileti by showing that every computable stable $2$-coloring of pairs has an incomplete $\Delta^0_2$ infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT$^2_2$ does not imply RT$^2_2$.

Steffen Lempp

Papers

The d.r.e. degrees are not dense

Universal computably enumerable equivalence relations

On the role of the collection principle for Sigma^0_2-formulas in second-order reverse mathematics

Comparing DNR and WWKL

The Strength of Some Combinatorial Principles Related to Ramsey's Theorem for Pairs