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

Fast evaluation of solid harmonic Gaussian integrals for local resolution-of-the-identity methods and range-separated hybrid functionals

An integral scheme for the efficient evaluation of two-center integrals over contracted solid harmonic Gaussian functions is presented. Integral expressions are derived for local operators that depend on the position vector of one of the two Gaussian centers. These expressions are then used to derive the formula for three-index overlap integrals where two of the three Gaussians are located at the same center. The efficient evaluation of the latter is essential for local resolution-of-the-identity techniques that employ an overlap metric. We compare the performance of our integral scheme to the widely used Cartesian Gaussian-based method of Obara and Saika (OS). Non-local interaction potentials such as standard Coulomb, modified Coulomb and Gaussian-type operators, that occur in range-separated hybrid functionals, are also included in the performance tests. The speed-up with respect to the OS scheme is up to three orders of magnitude for both, integrals and their derivatives. In particular, our method is increasingly efficient for large angular momenta and highly contracted basis sets.Comment: 18 pages, 2 figures; accepted manuscript. v2: supplementary material include

An interpretation of the Sigma-2 fragment of classical Analysis in System T

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0 and 1 by showing how to avoid using bar recursion altogether. Our result is proved via a conservative extension of System T with an operator for composable continuations from the theory of programming languages due to Danvy and Filinski. The fragment of Analysis is therefore essentially constructive, even in presence of the full Axiom of Choice schema: Weak Church's Rule holds of it in spite of the fact that it is strong enough to refute the formal arithmetical version of Church's Thesis

Some applications of logic to feasibility in higher types

In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalize the corresponding properties of the class of feasible functions. We also improve the Kapron - Cook result on mashine representation of basic feasible functionals. Our proofs are based on essential applications of logic. We introduce a weak fragment of second order arithmetic with second order variables ranging over functions from N into N which suitably characterizes basic feasible functionals, and show that it is a useful tool for investigating the properties of basic feasible functionals. In particular, we provide an example how one can extract feasible "programs" from mathematical proofs which use non-feasible functionals (like second order polynomials)