Finitary reducibility on equivalence relations

We introduce the notion of finitary computable reducibility on equivalence relations on the natural numbers. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be $\Pi_{n+2}$-complete under computable reducibility, we show that, for every $n$, there does exist a natural equivalence relation which is $\Pi_{n+2}$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy

Complexity of equivalence relations and preorders from computability theory

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. Here $f$ is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and $f$ must be computable. We show that there is a $\Pi_1$-complete equivalence relation, but no $\Pi k$-complete for $k \ge 2$. We show that $\Sigma k$ preorders arising naturally in the above-mentioned areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets, which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma 4$.Comment: To appear in J. Symb. Logi

Some properties of D.C.E. reals and their degrees

Master'sMASTER OF SCIENC

Foundations of Online Structure Theory II: The Operator Approach

We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the input to be a countable object of an arbitrary complexity. We give a new framework which (i) ties online algorithms with computable analysis, (ii) shows how to use modifications of notions from computable analysis, such as Weihrauch reducibility, to analyse finite but uniform combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine structure of finite analogs of infinite combinatorial problems, and (iv) see how similar ideas can be amalgamated from areas such as EX-learning, computable analysis, distributed computing and the like. One of the key ideas is that online algorithms can be viewed as a sub-area of computable analysis. Conversely, we also get an enrichment of computable analysis from classical online algorithms

Computably enumerable Turing degrees and the meet property

Working in the Turing degree structure, we show that those degrees which contain computably enumerable sets all satisfy the meet property, i.e. if a is c.e. and b < a, then there exists non-zero m < a with b ^m = 0. In fact, more than this is true: m may always be chosen to be a minimal degree. This settles a conjecture of Cooper and Epstein from the 80s

Foundations of Online Structure Theory II: The Operator Approach

We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the input to be a countable object of an arbitrary complexity. We give a new framework which (i) ties online algorithms with computable analysis, (ii) shows how to use modifications of notions from computable analysis, such as Weihrauch reducibility, to analyse finite but uniform combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine structure of finite analogs of infinite combinatorial problems, and (iv) see how similar ideas can be amalgamated from areas such as EX-learning, computable analysis, distributed computing and the like. One of the key ideas is that online algorithms can be viewed as a sub-area of computable analysis. Conversely, we also get an enrichment of computable analysis from classical online algorithms

Counting the changes of random Δ20 sets

We study the number of changes of the initial segment Zs ↾n for computable approximations of a Martin-Löf random Δ02Δ20 set Z. We establish connections between this number of changes and various notions of computability theoretic lowness, as well as the fundamental thesis that, among random sets, randomness is antithetical to computational power. We introduce a new randomness notion, called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that Zs ↾n changes more than c2n times. We establish various connections with ω-c.e. tracing and omega;-c.e. jump domination, a new lowness property. We also examine some relationships to randomness theoretic notions of highness, and give applications to the study of (weak) Demuth cuppability.Fil: Figueira, Santiago. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Hirschfeldt, Denis R.. University of Chicago; Estados UnidosFil: Miller, Joseph S.. University of Wisconsin; Estados UnidosFil: Ng, Keng Meng. Nanyang Technological University; SingapurFil: Nies, André. The University Of Auckland; Nueva Zeland