Hiro-o Tokunaga

Inferability of closed set systems from positive data

In this paper, we generalize previous results showing connections between inductive inference from positive data and algebraic structures by using tools from universal algebra. In particular, we investigate the inferability from positive data of language classes defined by closure operators. We show that some important properties of language classes used in inductive inference correspond closely to commonly used properties of closed set systems. We also investigate the inferability of algebraic closed set systems, and show that these types of systems are inferable from positive data if and only if they contain no infinite ascending chain of closed sets. This generalizes previous results concerning the inferability of various algebraic classes such as the class of ideals of a ring. We also show the relationship with algebraic closed set systems and approximate identifiability as introduced by Kobayashi and Yokomori [11]. We propose that closure operators offer a unifying framework for various approaches to inductive inference from positive data.

Pencils and infinite dihedral covers of ℙ

In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.

Dihedral coverings of algebraic surfaces and their application

In this article, we study dihedral coverings of algebraic surfaces branched along curves with at most simple singularities. A criterion for a reduced curve to be the branch locus of some dihedral covering is given. As an application we have the following: Let B be a reduced plane curve of even degree d having only a nodes and b cusps. If 2a + 6b > 2d2 − 6d+ 6, then π1(P \B) is non-abelian. Note that Nori’s result implies that π1(P \ B) is abelian, provided that 2a + 6b < d2.

Computing characteristic sets of bounded unions of polynomial ideals

The surprising fact that Hilberts basis theorem in algebra shows identifiabilty of ideals of polynomials in the limit from positive data is derived by the correspondence between ideals and languages in the context of machine learning. This correspondence also reveals the difference between the two and raises new problems to be solved in both of algebra and machine learning. In this article we solve the problem of providing a concrete form of the characteristic set of a union of two polynomial ideals. Our previous work showed that the finite basis of every polynomial ideal is its characteristic set, which ensures that the class of ideals of polynomials is identifiable from positive data. Union or settheoretic sum is a basic set operation, and it could be conjectured that there is some effective method which produces a characteristic set of a union of two polynomial ideals if both of the basis of ideals are given. Unfortunately, we cannot find a previous work which gives a general method for how to find characteristic sets of unions of languages even though the languages are in a class identifiable from positive data. We give methods for computing a characteristic set of the union of two polynomial ideals.

Nodal degenerations of plane curves and galois covers

Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois covers.

Learning Bounded Unions of Noetherian Closed Set Systems Via Characteristic Sets

In this paper, we study a learning procedure from positive data for bounded unions of certain class of languages. Our key tools are the notion of characteristic sets and hypergraphs. We generate hypergraphs from given positive data and exploit them in order to find characteristic sets.

On local trigonal fibrations

We show that certain degenerating hyperelliptic curves, called horizontal types, of genus g appear as the central fiber of a local family of trigonal curves of genus g by constructing such a family explicitly.

Inferability of unbounded unions of certain closed set systems

In this article, we study inferability from positive data for the unbounded union of certain class of languages. In order to show inferability, we put an emphasis on a characteristic set of a given language. We consider a class of closed set systems such that there exists an algorithm for generating a characteristic set consisting of one element. Two concrete examples of closed set systems with such algorithms are given. Furthermore, we consider applications of these examples to the study of transaction databases.