We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields. 1. Existentially closed real-closed difference fields. In the first part of this paper we will consider on one hand difference totally ordered fields, namely totally ordered fields with a distinguished automorphism σ and on the other ...

Datalog∃ is the extension of Datalog, allowing existentially quantified variables in rule heads. This language is highly expressive and enables easy and powerful knowledge-modeling, but the presence of existentially quantified variables makes reasoning over Datalog∃ undecidable, in the general case. The results in this paper enable powerful, yet decidable and efficient reasoning (query answerin...

Online learning algorithms such as Winnow have received much attention in Machine Learning. Their performance degrades only logarithmically with the input dimension, making them useful in large spaces such as relational theories. However, online first-order learners are intrinsically limited by a computational barrier: even in the finite, function-free case, the number of possible features grow...

A subset $A$ of a semigroup $S$ is called $chain$ ($antichain$) if $xy\in\{x,y\}$ ($xy\notin\{x,y\}$) for any (distinct) elements $x,y\in S$. ($anti$)$chain$-$finite$ contains no infinite (anti)chains. We prove that each antichain-finite periodic and every idempotent $e$ the set $\sqrt[\infty]{e}=\{x\in S:\exists n\in\mathbb N\;\;(x^n=e)\}$ finite. This property semigroups used to finite only i...

We investigate the conditions under which a Steiner triple system can have a 2or 3-existentially closed block intersection graph.