Finite isomorphically complete systems

AbstractIn the theory of finite automata it is an important problem to characterize such systems of automata from which any automaton can be built under a given composition and representation. Such systems are called complete with respect to the fixed composition and representation. From practical point of view, it is useful to determine those compositions and representations for which there are finite complete systems. In this paper we show that the existence of finite complete systems implies the unboundedness of the feedback dependency of the composition

Algorithmic decidability of Engel's property for automaton groups

We consider decidability problems associated with Engel's identity ($[\cdots[[x,y],y],\dots,y]=1$ for a long enough commutator sequence) in groups generated by an automaton. We give a partial algorithm that decides, given $x,y$, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk's $2$-group is not Engel. We consider next the problem of recognizing Engel elements, namely elements $y$ such that the map $x\mapsto[x,y]$ attracts to $\{1\}$. Although this problem seems intractable in general, we prove that it is decidable for Grigorchuk's group: Engel elements are precisely those of order at most $2$. Our computations were implemented using the package FR within the computer algebra system GAP

XML Publishing: Bridging Theory and Practice

Abstract. Transforming relational data into XML, as known as XML publishing, is often necessary when one wants to exchange data residing in databases or to create an XML interface of a traditional database. This paper aims to provide an overview of recent advances in XML publish-ing. We present a notion of publishing transducers recently developed for studying the expressive power and complexity of XML publishing languages. In terms of publishing transducers we then characterize XML publishing languages being used in practice. In addition, we address dy-namic aspects of XML publishing, namely, incremental maintenance and update management of XML views published from relational data.