Relatively computably enumerable reals

A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and Y does not compute X. A real X is relatively simple and above if there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset Z of the complement of X such that Z is c.e.(Y). We prove that every nonempty Pi^0_1 class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.Comment: 5 pages. Significant changes from earlier versio

Partial Recursive Functions and Finality

Abstract. We seek universal categorical conditions ensuring the representability of all partial recursive functions. In the category Pfn of sets and partial functions, the natural numbers provide both an initial algebra and a final coalgebra for the functor 1 + −. We recount how finality yields closure of the partial functions on natural numbers under Kleene’s µ-recursion scheme. Noting that Pfn is not cartesian, we then build on work of Paré and Román, obtaining weak initiality and finality conditions on natural numbers algebras in monoidal categories that ensure the (weak) representability of all partial recursive functions. We further obtain some positive results on strong representability. All these results adapt to Kleisli categories of cartesian categories with natural numbers algebras. However, in general, not all partial recursive functions need be strongly representable.

A join theorem for the computably enumerable degrees

10.1090/S0002-9947-04-03585-8Transactions of the American Mathematical Society35672557-256

Kolmogorov complexity and the recursion theorem

10.1007/11672142_11Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)3884 LNCS149-16

Traceable sets

Abstract. We investigate systematically into the various possible notions of traceable sets and the relations they bear to each other and to other notions such as diagonally noncomputable sets or complex and autocomplex sets. We review known notions and results that appear in the literature in different contexts, put them into perspective and provide simplified or at least more direct proofs. In addition, we introduce notions of traceability and complexity such as infinitely often versions of jump traceability and of complexity, and derive results about these notions that partially can be viewed as a natural completion of the results known before. Finally, we give a result about polynomial-time bounded notions of traceability and complexity that shows that in principle the equivalences derived so far can be transferred to the time-bounded setting. 1 Introduction an