Ultrametric automata have properties similar to the properties of probabilistic automata but the descriptional power of these types of automata can differ very much. In this paper, we compare ultrametric automata with deterministic, nondeterministic, probabilistic and alternating automata with various state complexities. We also show that two-way ultrametric automata can have a smaller state co...

A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is ultrametric. We show that every ultrametric minimum cost spanning tree game is a cost allocation game on a rooted tree. It follows that there exist optimal algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametric minimum cost s...

Following a review of metric, ultrametric and generalized ultrametric, we review their application in data analysis. We show how they allow us to explore both geometry and topology of information, starting with measured data. Some themes are then developed based on the use of metric, ultrametric and generalized ultrametric in logic. In particular we study approximation chains in an ultrametric ...

We develop an analysis of wavelets and pseudodifferential operators on multidimen-sional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and Lizorkin generalized functions on multidimensional ultrametric spaces. We also consider some family of pseudodifferential operators on multidimensional ...

The quotient space theory based on fuzzy tolerance relation is put forward to solve the problem of clustering in this paper. The similarity matrix does not always satisfy ultrametric inequality, theoretically and practically. We give the method to construct the hierarchical quotient space chain if the similarity matrix is only reflexive and symmetric. We consider not only the subset of data (cl...

The category of 1-bounded compact ultrametric spaces and non-distance increasing functions (KUM's) have been extensively used in the semantics of concurrent programming languages. In this paper a universal space U for KUM's is introduced, such that each KUM can be isomet-rically embedded in it. U consists of a suitable subset of the space of functions from 0; 1) to IN, endowed with a \preex-bas...

We investigate the interrelations between labeled trees and ultrametric spaces generated by these trees. The trees, which generate complete ultrametrics, totally bounded discrete ones, are characterized up to isomorphism. As corollary, we obtain a characterization of generating compact ultrametrics ultrametrics. It is also shown that every space tree contains dense subspace.

Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces (Lawvere 1973, Rutten 1995). Combining Lawvere's (1973) enriched-categorical and Smyth' (1987, 1991) topological view on generalized (ultra)metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized ultrametric spaces. Restricted to the special...

Ultrametric algorithms are similar to probabilistic algorithms but they describe the degree of indeterminism by p-adic numbers instead of real numbers. No wonder that only very few examples of advantages for ultrametric algorithms over probabilistic ones have been published up to now, and all they are slightly artificial. This paper considers ultrametric and probabilistic one-counter automata w...

A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is an ultrametric. We show that every ultrametric minimum cost spanning tree game is reduced to a cost allocation game on a rooted tree. It follows that there exist O(n) time algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametri...