Objective
The project aims to a systematic investigation of computable numberings in the arithmetical hierarchy and the hierarchy of Ershov, with emphasis both on the theoretical aspects as well as on applications to computer science.
The relative complexity of numberings is measured by the notion of a Rogers's semilattice. For families of sets in the arithmetical and Ershov hierarchies, the present project aims
(1) to investigate the algebraic properties of the Rogers semilattices of the computable numberings (cardinality, density, description of segments and ideals, existence of minimal and least elements);
(2) to investigate the spectrum of minimal numberings for a given family, and describe conditions for families to have minimal numberings of a given type;
(3) to investigate global properties of Rogers semilattices (embedding questions, isomorphism types, elementary types, elementary theories, decidability questions, decidable fragments, automorphisms);
(4) to investigate automorphism types of computable objects;
(5) to investigate continuous numberings of effectively given topological spaces.
It is expected :
- to show that any nontrivial Rogers semilattice of a family of sets in the Ershov hierarchy is not lattice and consists of infinitely many elements, or construct counter-examples;
- to describe the initial segments of Rogers semilattices in terms of Lachlan semilattices;
- to show that every Lachlan semilattice can be embedded above any non-greatest element of the Rogers semilattice of a family of arithmetical sets;
- to find examples of families in the hierarchy of Ershov whose Rogers semilatticies contain ideals without minimal elements;
- to find algorithmic criteria and sufficient structural conditions for the existence of positive and Friedberg numberings;
- to determine the number of computable minimal numberings of various types (positive, Friedberg, strongly effectively minimal, effectively minimal) for a given family;
- to show differences in the elementary types of nontrivial Rogers semilattices, depending on whether we consider finite or infinite families;
- to provide examples of different elementary types of Rogers semilattices of infinite families;
- to determine the levels of undecidability of the elementary theories of Rogers semilattices;
- to describe the spectrum of the isomorphism types of the Rogers semilattices and the elementary types depending on the level of the arithmetical hierarchy to which the families of sets belong;
- to find characterizations of permutation groups of sets lying in some level of the arithmetical hierarchy;
- to extend the class of topological spaces with computable numberings;
- to find the relation between continuous maps and computable maps;
- to solve the problem of which partial numberings can be totalised.
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Programme(s)
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Coordinator
53100 Siena
Italy
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