On Extensions of Embeddings into the Enumeration Degrees of the Σ2-sets

We give an algorithm for deciding whether an embedding of a finite partial order P into the enumeration degrees of the Σ2-sets can always be extended to an embedding of a finite partial order Q ⊃ P. 1. The theorem. Reducibilities are relations on the power set of the natural numbers, conveying that a set A ⊆ ω can in some sense be “computed” or, in more generality, “defined”, from another set B ⊆ ω, usually denoted as A ≤r B (where r specifies what “computations” are allowed). Reducibilities are assumed to be reflexive and transitive but usually not antisymmetric, i. e., they give a partial preordering on P(ω). They induce equivalence relations (defined by A ≡r B iff A ≤r B and B ≤r A), and the equivalence class of a set is called its (r-)degree. The degree of a set thus captures the computational complexity of a set of natural numbers while stripping away the information about the set irrelevant from a computational point of view (such as membership of a particular number, etc.). The most important reducibility of classical computability theory is the Turing reducibility, denoting that a set A can be computed from a set B by means of an oracle Turing machine (i. e., by a hypothetical computer with unlimited resources and access to membership information about the “oracle” set B). In addition, there are various other reducibilities using models of computation which in various ways restrict the run time, memory space, or oracle access, or which allow infinite schemes of computation. All the reducibilities mentioned in the above paragraph are based on the following model of computing a set A from a set B: A query about membership in A is reduced to an effectively generated sequence of queries about membership in B. Enumeration reducibility introduces a different concept: One produces an enumeration of a set A from an arbitrary enumeration of a set B, i. e., in the above Turing 1991 Mathematics Subject Classification. 03D30.