|
Logic & Semantics |
An X-tree is a finite tree T=(V,E), together with a map for
which
Such trees are equivalent to ``pairwise compatible" system of bipartitions of X, and are widely used in classification, particularly in biology, but also in linguistics, philology etc.
In such applications we are often given a list where
is a
-tree, for a (small) subset
of X, and we wish to
determine if these trees can be consistently combined into one
(or more) unknown ``parent" X-tree(s). This tree reconstruction problem is
closely related to another which takes as its input a collection of
partitions of subsets of X.
We will discuss some of the resulting combinatorial and computational features
of this problem. In particular we consider the class of all possible
``inference rules" for enlarging any consistent collection of trees.
We show that (as conjectured) this class is infinite, but under certain restrictions
a small finite subset always suffices to reconstruct a tree. We also consider the question of whether
or not this system of inference rules suffices to determine the consistency of .