Our aim is to see which practices of Greek geometry can be expressed in various logics. Thus we refine Detlefsen’s notion of descriptive complexity by providing a scheme of increasing more descriptive formalizations of geometry Following Hilbert we argue that defining a field structure on a line in ‘Euclidean geometry’ provides a foundation for both geometry and algebra. In particular we prove ...

The descriptional complexity of semi-conditional grammars is studied. It is proved that every recursively enumerable language is generated by a semi-conditional grammar of degree (2, 1) with no more than seven conditional productions and eight nonterminals.

In this paper we investigate the descriptional complexity of knot theoretic problems and show upper bounds for planarity problem of signed and unsigned knot diagrams represented by Gauss words. Since a topological equivalence of knots can involve knot diagrams with arbitrarily many crossings then Gauss words will be considered as strings over an infinite (unbounded) alphabet. For establishing t...

We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce the class of (hyper)graph polynomials definable in second order logic, and outline a research progr...

v Acknowledgements vii Chapter

This paper proves that every recursively enumerable language is generated by a scattered context grammar with no more than four nonterminals and three non-context-free productions. In its conclusion, it gives an overview of the results and open problems concerning scattered context grammars and languages.

This paper proposes a new method to rank the cases classified by a decision tree. The method applies a posteriori without modification of the tree and doesn’t use additional training cases. It consists in computing the distance of the cases to the decision boundary induced by the decision tree, and to rank them according to this geometric score. When the data are numeric it is very easy to impl...