Abstract In this article, we define and study the notion of a $(c,c_{1})$-cylinder, which turns out to be very useful instrument for investigating the relationships between conjunctive reducibility ($c$-reducibility) and its injective version $c_{1}$-reducibility. Using this notion, we prove the following results: (i) Neither hypersimple sets nor hemimaximal sets can be $(c,c_{1})$-cylinders; (ii) The $c$-degree of a noncomputable c.e. set contains either only one or infinitely many noncomputable $c_{1}$-degrees; (iii) the $c$-degree of either a hemimaximal set or a hypersimple set contains infinitely many noncomputable $c_{1}$-degrees.

Artificial Intelligence is developing using statistical and logical approaches. Considerable efforts have been devoted to combining logical and probabilistic methods in a single framework, which influenced the development of several formalisms and programming tools. Such formalisms allow representation and reasoning about uncertain knowledge. Uncertainty happens in many areas, like medicine, manufacturing, weather forecasts, prediction (of e.g. voting intentions, natural disaster), etc. Ontologies are machine-processable formalisms for knowledge representation. Their purpose is to describe objects according to domain of interests. This knowledge is used by (automated) reasoning systems for query answering. Probabilistic ontologies are obtained by adding a probabilistic interpretation to the constraints forming the ontology, and adapting corresponding reasoning methods to handle these probabilities. The RFEPO is an interdisciplinary project and aims at formulating unification and matching problems used in probabilistic ontology reasoning, and to search and compare algorithms for their solution. Additionally, when there is no algorithm for solving them, our project aims to study the algebraic structures of degrees induced by Turing and other algorithmic reducibilities.

Abstract We consider three strong reducibilities, $s_{1}, s_{2}, Q_{1}$ (where we identify a reducibility $\leqslant _r$ with its index $r$). The first two reducibilities can be viewed as injective versions of $s$-reducibility, whereas $Q_1$-reducibility can be viewed as an injective version of $Q$-reducibility. We have, with proper inclusions, $s_{1} \subset s_{2} \subset s$. It is well known that there is no minimal $s$-degree, and there is no minimal $Q$-degree. We show on the contrary that there exist minimal $\varDelta ^{0}_{2}$$s_{2}$-degrees and minimal $\varDelta ^{0}_{2}$$s_{1}$-degrees. On the other hand, both the $\varPi ^{0}_{1}$$s_{2}$-degrees and the $\varPi ^{0}_{1}$$s_{1}$-degrees are downwards dense. By the isomorphism of the $s_1$-degrees with the $Q_1$-degrees induced by complementation of sets, it follows that there exist minimal $\varDelta ^0_2$$Q_1$-degrees, but the c.e. $Q_{1}$-degrees are downwards dense.

Abstract In this article we prove the following results: (i) Every hemimaximal set has minimal $c_{1}$-degree, i.e. if $B$ is hemimaximal and $A$ is a c.e. set such that $A \le _{c_{1}} B$ then either $B \leq _{{c}_{1}} A$ or $A$ is computable. (ii) The $sQ$-degree of a c.e. set contains either only one or infinitely many c.e. $c$-degrees. (iii) If $A,B$ are c.e. cylinders in the same $sQ_{1}$-degree and $A<_{c_{1}} B$, then this $sQ_{1}$-degree contains infinitely many c.e. $c_{1}$-degrees.