12 research outputs found
Computation with Advice
Computation with advice is suggested as generalization of both computation
with discrete advice and Type-2 Nondeterminism. Several embodiments of the
generic concept are discussed, and the close connection to Weihrauch
reducibility is pointed out. As a novel concept, computability with random
advice is studied; which corresponds to correct solutions being guessable with
positive probability. In the framework of computation with advice, it is
possible to define computational complexity for certain concepts of
hypercomputation. Finally, some examples are given which illuminate the
interplay of uniform and non-uniform techniques in order to investigate both
computability with advice and the Weihrauch lattice
The computational content of Nonstandard Analysis
Kohlenbach's proof mining program deals with the extraction of effective
information from typically ineffective proofs. Proof mining has its roots in
Kreisel's pioneering work on the so-called unwinding of proofs. The proof
mining of classical mathematics is rather restricted in scope due to the
existence of sentences without computational content which are provable from
the law of excluded middle and which involve only two quantifier alternations.
By contrast, we show that the proof mining of classical Nonstandard Analysis
has a very large scope. In particular, we will observe that this scope includes
any theorem of pure Nonstandard Analysis, where `pure' means that only
nonstandard definitions (and not the epsilon-delta kind) are used. In this
note, we survey results in analysis, computability theory, and Reverse
Mathematics.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
Algorithmic randomness, reverse mathematics, and the dominated convergence theorem
We analyze the pointwise convergence of a sequence of computable elements of
L^1(2^omega) in terms of algorithmic randomness. We consider two ways of
expressing the dominated convergence theorem and show that, over the base
theory RCA_0, each is equivalent to the assertion that every G_delta subset of
Cantor space with positive measure has an element. This last statement is, in
turn, equivalent to weak weak K\"onig's lemma relativized to the Turing jump of
any set. It is also equivalent to the conjunction of the statement asserting
the existence of a 2-random relative to any given set and the principle of
Sigma_2 collection
Metastable convergence theorems
The dominated convergence theorem implies that if (f_n) is a sequence of
functions on a probability space taking values in the interval [0,1], and (f_n)
converges pointwise a.e., then the sequence of integrals converges to the
integral of the pointwise limit. Tao has proved a quantitative version of this
theorem: given a uniform bound on the rates of metastable convergence in the
hypothesis, there is a bound on the rate of metastable convergence in the
conclusion that is independent of the sequence (f_n) and the underlying space.
We prove a slight strengthening of Tao's theorem which, moreover, provides an
explicit description of the second bound in terms of the first. Specifically,
we show that when the first bound is given by a continuous functional, the
bound in the conclusion can be computed by a recursion along the tree of
unsecured sequences. We also establish a quantitative version of Egorov's
theorem, and introduce a new mode of convergence related to these notions
The computational content of Nonstandard Analysis
Kohlenbach's proof mining program deals with the extraction of effective information from typically ineffective proofs. Proof mining has its roots in Kreisel's pioneering work on the so-called unwinding of proofs. The proof mining of classical mathematics is rather restricted in scope due to the existence of sentences without computational content which are provable from the law of excluded middle and which involve only two quantifier alternations. By contrast, we show that the proof mining of classical Nonstandard Analysis has a very large scope. In particular, we will observe that this scope includes any theorem of pure Nonstandard Analysis, where 'pure' means that only nonstandard definitions ( and not the epsilon-delta kind) are used. In this note, we survey results in analysis, computability theory, and Reverse Mathematics
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page