93 research outputs found
Random strings and tt-degrees of Turing complete C.E. sets
We investigate the truth-table degrees of (co-)c.e.\ sets, in particular,
sets of random strings. It is known that the set of random strings with respect
to any universal prefix-free machine is Turing complete, but that truth-table
completeness depends on the choice of universal machine. We show that for such
sets of random strings, any finite set of their truth-table degrees do not meet
to the degree~0, even within the c.e. truth-table degrees, but when taking the
meet over all such truth-table degrees, the infinite meet is indeed~0. The
latter result proves a conjecture of Allender, Friedman and Gasarch. We also
show that there are two Turing complete c.e. sets whose truth-table degrees
form a minimal pair.Comment: 25 page
Computable Categoricity of Trees of Finite Height
We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ03-condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆0n+1-categorical but not ∆0n-categorical
The Borel complexity of the class of models of first-order theories
We investigate the descriptive complexity of the set of models of first-order
theories. Using classical results of Knight and Solovay, we give a sharp
condition for complete theories to have a -complete
set of models. We also give sharp conditions for theories to have a
-complete set of models. Finally, we determine the Turing
degrees needed to witness the completeness
A jump operator on the Weihrauch degrees
A partial order admits a jump operator if there is a map that is strictly increasing and weakly monotone. Despite its name, the
jump in the Weihrauch lattice fails to satisfy both of these properties: it is
not degree-theoretic and there are functions such that
. This raises the question: is there a jump operator
in the Weihrauch lattice? We answer this question positively and provide an
explicit definition for an operator on partial multi-valued functions that,
when lifted to the Weihrauch degrees, induces a jump operator. This new
operator, called the totalizing jump, can be characterized in terms of the
total continuation, a well-known operator on computational problems. The
totalizing jump induces an injective endomorphism of the Weihrauch degrees. We
study some algebraic properties of the totalizing jump and characterize its
behavior on some pivotal problems in the Weihrauch lattice
The Complements of Lower Cones of Degrees and the Degree Spectra of Structures
We study Turing degrees a for which there is a countable structure whose degree spectrum is the collection {x : x ≰ a}. In particular, for degrees a from the interval [0′, 0″], such a structure exists if a′ = 0″, and there are no such structures if a″ \u3e 0‴
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