Positive undecidable numberings in the Ershov hierarchy

We give a su cient condition for an in nite computable family of 1 a sets, to have computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved by Talasbaeva for the nite levels of the Ershov hierarchy. In par- ticular the family of all 1 a sets has positive undecidable numberings: this veri es for all levels of the Ershov hierarchy a conjecture due to Badaev and Goncharov. We point out also that for every ordinal notation a of a nonzero ordinal, there are families of 1 a sets having positive numberings, but no Friedberg numberings: this answers for all levels (whether nite or in nite) of the Ershov hierarchy, a question originally raised, only for the nite levels over level 1, by Badaev and Goncharov

Joins and meets in the structure of Ceers

We study computably enumerable equivalence relations (abbreviated as ceers) under computable reducibility, and we investigate the resulting degree structure Ceers, which is a poset with a smallest and a greatest element. We point out a partition of the ceers into three classes: the finite ceers, the light ceers, and the dark ceers. These classes yield a partition of the degree structure as well, and in the language of posets the corresponding classes of degrees are first order definable within Ceers. There is no least, no maximal, no greatest dark degree, but there are infinitely many minimal dark degrees. We study joins and meets in Ceers, addressing the cases when two incomparable degrees of ceers X,Y have or do not have join or meet according to where X,Y are located in the classes of the aforementioned partition: in particular no pair of dark ceers has join, and no pair in which at least one ceer is dark has meet. We also exhibit examples of ceers X,Y having join which coincides with their uniform join, but also examples when their join is strictly less than the uniform join. We study join-irreducibility and meet-irreducibility. In particular we characterize the property of being meet-irreducible for a ceer E, by showing that it coincides with the property of E being self-full, i.e. every reducibility from E to itself is in fact surjective on its equivalence classes (this property properly extends darkness). We then study the quotient structure obtained by dividing the poset Ceers by the degrees of the finite ceers, and study joins and meets in this quotient structure. We look at automorphisms of Ceers, and show that there are continuum many automorphisms fixing the dark ceers, and continuum many automorphisms fixing the light ceers. Finally, we compute the complexity of the index sets of the classes of ceers studied in the paper

Intuitionistic logic and Muchnik degrees

We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now classic result of Skvortsova for the Medvedev lattice

Friedberg numberings in the Ershov hierarchy

We show that for every n 1, there exists a 1n -computable family which up to equivalence has exactly one Friedberg numbering which does not induce the least element of the corresponding Rogers semilattice

Classifying word problems of finitely generated algebras via computable reducibility

We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence relations (ceers), which has considerably grown in recent times. To pursue our analysis, we rely on the most popular way of assessing the complexity of ceers, that is via computable reducibility on equivalence relations, and its corresponding degree structure (the c-degrees). On the negative side, building on previous work of Kasymov and Khoussainov, we individuate a collection of c-degrees of ceers which cannot be realized by the word problem of any finitely generated algebra of finite type. On the positive side, we show that word problems of finitely generated semigroups realize a collection of c-degrees which embeds rich structures and is large in several reasonable ways