Tree structures, Entropy, and Action Principles for Neighbourhood Topologies

We recast the idea of decision trees as they emerge in Information theory and Complexity theory into a set theoretical language; the result we call tree structures over a given set. We identify all main structural elements of tree structures, the most important of which is the tree function, defined as a sum over certain quantities at every nod in the tree. We show in detail that the minimization of the tree function on, possibly constrained, sets of tree structures over a given set renders the functional form of entropy, or of Wiener-Shannon information, depending on the context. We suggest three natural axioms defining tree structures, which are valid also when the underlying set is infinite; in this case the resulting trees are fractal-like objects. These axioms turn out to be related to the neighbourhood axioms describing neighbourhoods on a topological space. In fact we will show that the paths in a tree structure, which are totally ordered subsets of a tree, can be regarded as a countable neighbourhood basis, which in turn defines a topology on the underlying set. The tree function then assigns nonnegative numbers to every neighbourhood topology on the underlying set that arises from a tree structure. On the preferred topology, this number becomes minimal, in which case it represents an entropy-like quantity. This has the distinct flavour of an action principle, distinguishing certain topologies by means of minimization of the tree function. This result hints at a deep relation between entropy-like quantities and preferred neighbourhood topologies on sets.