This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC-509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995

We show, roughly speaking, that it requires ! iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if ff is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has ff th -jump degree d if d is the least degree which is the ff-th jump of some degree c such there is an isomorphic copy of A with universe ! in which the functions and relations have degree at most c. We show that every degree d 0 (!) is the ! th - jump degree of a Boolean algebra, but that, for n ! !, no Boolean algebra has n th -jump degree d ? 0 (n) . The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight, together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribe...

We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of Cantor-Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite Cantor-Bendixson rank.

this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25

Abstract. The objective of this paper is to uncover the structure of the back-andforth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the back-and-forth equivalence classes of a Boolean algebra for finite levels. This result has implications for characterizing the relatively intrinsically Σ 0 n relations of Boolean algebras as existential formulas over a finite set of relations. 1.

Abstract. In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e. strongly jump traceables is Π 0 4-complete. §1. Introduction. One of the fundamental concerns of computability theory is in understanding the relative difficulty of computational problems as measured by Turing reducubility (≤T). The equivalence classes of the preordering ≤T are called Turing degrees, and it is long recognized that the fundamental operator on the structure of the Turing degrees is the jump operator. For a set A, the

Abstract. Knight and Stob proved that every low4 Boolean algebra is 0 (6)-isomorphic to a computable one. Furthermore, for n = 1, 2, 3, 4, every lown Boolean algebra is 0 (n+2)-isomorphic to a computable one. We show that this is not true for n = 5: there is a low5 Boolean algebra that is not 0 (7)-isomorphic to any computable Boolean algebra. It is worth remarking that, because of the machinery developed, the proof uses at most a 0 ′ ′-priority argument. The technique used to construct this Boolean algebra is new and might be useful in other applications, such as to solve the lown Boolean algebra problem either positively or negatively. 1.

In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set

A countable structure (with finite signature) is computable if its universe can be identi-fied with ω in such a way as to make the relations and operations computable functions. In this thesis, I study which Boolean algebras and linear orders are computable. Making use of Ketonen invariants, I study the Boolean algebras of low Ketonen depth, both classically and effectively. Classically, I give an explicit characterization of the depth zero Boolean algebras; provide continuum many examples of depth one, rank ω Boolean algebras with range ω + 1; and provide continuum many examples of depth ω, rank one Boolean algebras. Effectively, I show for sets S ⊆ ω + 1 with greatest element, the depth zero Boolean algebras Bu(S) and Bv(S) are computable if and only if S \{ω} is Σ 0 n↦→2n+3 in the Feiner Σ-hierarchy. Making use of the existing notion of limitwise monotonic functions and the new notion of limit infimum functions, I characterize which shuffle sums of ordinals below ω + 1 have computable copies. Additionally, I show that the notions of limitwise monotonic functions relative to 0 ′ and limit infimum functions coincide.

Abstract. Fraïssé studied countable structures S through analysis of the age of S, i.e., the set of all finitely generated substructures of S. We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.