Arithmetical definability and computational complexity

Arithmetical Knowledge and Arithmetical Definability: Four Studies

by Sean Walsh The subject of this dissertation is arithmetical knowledge and arithmetical definability. The first two chapters contain respectively a critique of a logicist account of a preferred means by which we may legitimately infer to arithmetical truths and a tentative defense of an empiricist account. According to the logicist account, one may infer from quasi-logical truths to patently ...

Kolmogorov complexity of initial segments of sequences and arithmetical definability

The structure of the K-degrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the K-degrees of infinite binary sequences, X is below Y if the prefix-free Kolmogorov complexity of the first n bits of X is less than the complexity of the first n bits of Y , for each n. Identifying infinite binar...

Arithmetical definability over finite structures

Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform and . We prove finite analogs of three classic results in arithmetical definability, namely that and TIMES can first-order define PLUS, that and DIVIDES can first-order define TIM...

Logical Definability Versus Computational Complexity: Another Equivalence

Sequences of low arithmetical complexity

Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity. Mathematics Subject Classification. 68R15.