47 research outputs found
A Note on Encodings of Phylogenetic Networks of Bounded Level
Driven by the need for better models that allow one to shed light into the
question how life's diversity has evolved, phylogenetic networks have now
joined phylogenetic trees in the center of phylogenetics research. Like
phylogenetic trees, such networks canonically induce collections of
phylogenetic trees, clusters, and triplets, respectively. Thus it is not
surprising that many network approaches aim to reconstruct a phylogenetic
network from such collections. Related to the well-studied perfect phylogeny
problem, the following question is of fundamental importance in this context:
When does one of the above collections encode (i.e. uniquely describe) the
network that induces it? In this note, we present a complete answer to this
question for the special case of a level-1 (phylogenetic) network by
characterizing those level-1 networks for which an encoding in terms of one (or
equivalently all) of the above collections exists. Given that this type of
network forms the first layer of the rich hierarchy of level-k networks, k a
non-negative integer, it is natural to wonder whether our arguments could be
extended to members of that hierarchy for higher values for k. By giving
examples, we show that this is not the case
The algebra of functions with antidomain and range
We give complete, finite quasiequational axiomatisations for algebras of unary partial functions under the operations of composition, domain, antidomain, range and intersection. This completes the extensive programme of classifying algebras of unary partial functions under combinations of these operations. We look at the complexity of the equational theories and provide a nondeterministic polynomial upper bound. Finally we look at the problem of finite representability and show that finite algebras can be represented as a collection of unary functions over a finite base set provided that intersection is not in the signature