Strongly Bounded Turing Reducibilities and Computably Enumerable Sets

Preface In this course we survey some recent work on the strongly bounded Turing re-ducibilities on the computably enumerable sets. Bounded Turing reducibilities are obtained from classical Turing reducibility by imposing upper bounds on the use functions (i.e., on the size of the oracle queries) of the reductions. The most popular bounded Turing reducibility which has been intensively studied in the past decades is bounded Turing (bT) reducibility-also called weak truth-table (wtt) reducibility-where the use function is bounded by a computable function. Quite recently some reducibilities based on more strict bounds have been introduced , called computable Lipschitz reducibility and identity bounded Turing reducibility. Here the size of the queries is bounded by the input up to an additive constant or by the input itself: A set A is computable-Lipschitz or cl-reducible to a set B if A is Turing reducible to B by a Turing functional Φ where the use function ϕ of Φ is bounded by the identity function up to an additive constant, i.e., ϕ(n) ≤ n + O(1). The special case of a cl-reduction where the use function is bounded by the identity function (i.e., where the additive constant is 0) is called an identity bounded Turing reduction (ibT-reduction). In the following we will refer to these two reducibilities as the strongly bounded Turing (sbT) reducibilities. Computable Lipschitz reducibility (also called strong weak truth-table (sw) oder linear reducibility) was introduced by Downey, Hirschfeldt and LaForte in 2001 [DHL01, DHL04] in the context of some investigations in algorithmic ran-domness. Note that, for a set A which is cl-reducible to a set B, the finite initial segment A n of A can be computed from the corresponding initial segment B n of B with the help of a constant number of additional bits. So, in particular, the Kolmogorov complexity of A n is bounded by the Kolmogorov complexity of B n up to an additive constant. Moreover, Downey, Hirschfeldt and LaForte have shown that, on the computably enumerable (c.e.) sets, cl-reducibility coincides with Solovay reducibility which may be viewed as a relative measure of the speed by which a real number can be effectively approximated by rational numbers. Identity bounded Turing reducibility was introduced by Soare in 2004 [So04] in the context of some applications of computability theory to some problems in differential geometry. In the past years the partial orderings of the degrees induced by the strongly …