arXiv subjects

Andre Nies

Publications and source records attributed to Andre Nies.

42 records · Page 3Linked to original sources

arXiv2013

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 $Π_1$-complete equivalence relation, but no $Πk$-complete for $k \ge 2$. We show that $Σk$ preorders arising naturally in the above-mentioned areas are $Σk$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $Σ2$, almost inclusion on r.e.\ sets, which is $Σ3$, and Turing reducibility on r.e.\ sets, which is $Σ4$.

math.LO↗

arXiv2013

Coding Methods in Computability Theory and Complexity Theory

A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees, it has been used to determine the complexity of the elementary theory, to provide restrictions on automorphisms, and even to obtain definability results (work of Harrington, Nies, Shore, Slaman, and others). We describe how a similar program can be carried out for several other structures, including the structure of c.e. many one degrees, the structure of c.e. weak truth table degrees and the lattice of c.e. sets under inclusion. In all cases we will obtain undecidability of, or even an interpretation of true arithmetic in the theory of the structure. For the c.e. many one-degrees, we also obtain definability results and restrictions on automorphisms. This work appeared first as the author's habilitation thesis at the Universitaet Heidelberg, 1998.

math.LO↗

arXiv2013

Logic Blog 2012

The 2012 logic blog has focussed on the following: Randomness and computable analysis/ergodic theory; Systematizing algorithmic randomness notions; Traceability; Higher randomness; Calibrating the complexity of equivalence relations from computability theory and algebra.

math.LO↗

arXiv2013

Calibrating the complexity of Delta 2 sets via their changes

The computational complexity of a Delta 2 set will be calibrated by the amount of changes needed for any of its computable approximations. Firstly, we study Martin-Loef random sets, where we quantify the changes of initial segments. Secondly, we look at c.e. sets, where we quantify the overall amount of changes by obedience to cost functions. Finally, we combine the two settings. The discussions lead to three basic principles on how complexity and changes relate.

math.LO↗

arXiv2011

Characterizing the strongly jump-traceable sets via randomness

We show that if a set $A$ is computable from every superlow 1-random set, then $A$ is strongly jump-traceable. This theorem shows that the computably enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets computable from every superlow 1-random set. We also prove the analogous result for superhighness: a c.e.\ set is strongly jump-traceable if and only if it is computable from every superhigh 1-random set. Finally, we show that for each cost function $c$ with the limit condition there is a 1-random $Δ^0_2$ set $Y$ such that every c.e.\ set $A \le_T Y$ obeys $c$. To do so, we connect cost function strength and the strength of randomness notions. This result gives a full correspondence between obedience of cost functions and being computable from $Δ^0_2$ 1-random sets.

math.LO↗

arXiv2007

Automatic Structures: Richness and Limitations

We study the existence of automatic presentations for various algebraic structures. An automatic presentation of a structure is a description of the universe of the structure by a regular set of words, and the interpretation of the relations by synchronised automata. Our first topic concerns characterising classes of automatic structures. We supply a characterisation of the automatic Boolean algebras, and it is proven that the free Abelian group of infinite rank, as well as certain Fraisse limits, do not have automatic presentations. In particular, the countably infinite random graph and the random partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. Our second topic is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is Σ_1^1-complete.

cs.DM↗