Undecidability and initial segments of the (r.e.) tt-degrees

Abstract

A notion of reducibility ≤r between sets is specified by giving a set of procedures for computing one set from another. We say that a set A is r-reducible to a set B, A ≤rB, if one of the procedures applied to B gives A. Associated with any such reducibility notion is the structure of r-degrees, the equivalence classes of sets with respect to this reducibility, with the induced ordering. The most general notion of a computable reducibility is that of Turing, ≤T. Here we say that A ≤TB if there is a Turing machine φe which, when equipped with an oracle for B, computes A: φeB = A. Such Turing degree computations are characterized by the phenomenon that only during the computation itself do we discover which questions about B need to be answered to compute A(x). In contrast, for nearly all other computable reducibilities the set of questions needed is given in advance by a recursive procedure. Perhaps the most common example of such a procedure is many-one reducibility, ≤m: A ≤mB if there is a recursive function f such that x ∈ A ⇔ f(x) ∈ B.Reducibilities with the property that the output, A(x), is determined by the answers that B gives to a set of questions calculated recursively from x are said to be of tabular type. The most general tabular reducibility is called truth-table reducibility, ≤tt. The procedures [e] associated with this reducibility are specified by a recursive function f (= {e}) which, for each x, gives a set of n questions about the oracle and, for each of the possible 2n sets of answers, gives the corresponding output. As usual this defines A ≤ttB as “there is an e such that [e]B = A”. It is with this notion of reducibility and the associated tt-degrees that we shall be concerned in this paper. Basic information on several such strong reducibilities can be found in Rogers [28]. For more information we recommend the survey articles by Odifreddi [25] and Degtev [2] as well as Odifreddi's book [26].

Effective embeddings into strong degree structures

“…Unfortunately, there are no general techniques for constructing a non-computable c. e. set that is truth-table reducible to a given non-computable c. e. set. This is evidenced by the fact that the c. e. truth-table degrees contain minimal elements [11]. This is the main obstacle to using the techniques in this paper to solve this problem.…”

“…Among such sentences are those that assert that a certain lattice can be embedded as a finite initial segment. Much is known about which lattices can be embedded as finite initial segments of ØØ (see, for example [11]). Unfortunately, not much is known about which lattices can not be so embedded into ØØ .…”

The undecidability of the Π4-theory for the r.e. wtt and Turing degrees

“…Thus p = 1 and the n r+ i-theory of the degree structure is undecidable with the same value for r. For the component (R), in the case of the r.e. degree structures one can rely on the constructions in the original undecidability proofs ([La72], [HtS90] and [N92] for the r.e. m-, tt-and btt-degrees, respectively).…”