66 research outputs found
Bounded time computation on metric spaces and Banach spaces
We extend the framework by Kawamura and Cook for investigating computational
complexity for operators occurring in analysis. This model is based on
second-order complexity theory for functions on the Baire space, which is
lifted to metric spaces by means of representations. Time is measured in terms
of the length of the input encodings and the required output precision. We
propose the notions of a complete representation and of a regular
representation. We show that complete representations ensure that any
computable function has a time bound. Regular representations generalize
Kawamura and Cook's more restrictive notion of a second-order representation,
while still guaranteeing fast computability of the length of the encodings.
Applying these notions, we investigate the relationship between purely metric
properties of a metric space and the existence of a representation such that
the metric is computable within bounded time. We show that a bound on the
running time of the metric can be straightforwardly translated into size bounds
of compact subsets of the metric space. Conversely, for compact spaces and for
Banach spaces we construct a family of admissible, complete, regular
representations that allow for fast computation of the metric and provide short
encodings. Here it is necessary to trade the time bound off against the length
of encodings
Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes
We survey diverse approaches to the notion of information: from Shannon
entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov
complexity are presented: randomness and classification. The survey is divided
in two parts in the same volume. Part I is dedicated to information theory and
the mathematical formalization of randomness based on Kolmogorov complexity.
This last application goes back to the 60's and 70's with the work of
Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last
years.Comment: 40 page
Classification from a computable viewpoint
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory.
Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.
Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work
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