How functional programming mattered

In 1989 when functional programming was still considered a niche topic, Hughes wrote a visionary paper arguing convincingly ‘why functional programming matters’. More than two decades have passed. Has functional programming really mattered? Our answer is a resounding ‘Yes!’. Functional programming is now at the forefront of a new generation of programming technologies, and enjoying increasing popularity and influence. In this paper, we review the impact of functional programming, focusing on how it has changed the way we may construct programs, the way we may verify programs, and fundamentally the way we may think about programs

On framings of knots in 3-manifolds

We show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented $3$-manifold is when the manifold has a properly embedded non-separating $S^2$. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough's work on the mapping class groups of $3$-manifolds. We also relate our results to the theory of skein modules.Comment: 12 pages, 9 figure

Closed Choice and a Uniform Low Basis Theorem

We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, of the natural numbers, of Cantor space and of Baire space correspond to the class of computable functions, of functions computable with finitely many mind changes, of weakly computable functions and of effectively Borel measurable functions, respectively. We also prove that all these classes correspond to classes of non-deterministically computable functions with the respective spaces as advice spaces. Moreover, we prove that closed choice on Euclidean space can be considered as "locally compact choice" and it is obtained as product of closed choice on the natural numbers and on Cantor space. We also prove a Quotient Theorem for compact choice which shows that single-valued functions can be "divided" by compact choice in a certain sense. Another result is the Independent Choice Theorem, which provides a uniform proof that many choice principles are closed under composition. Finally, we also study the related class of low computable functions, which contains the class of weakly computable functions as well as the class of functions computable with finitely many mind changes. As one main result we prove a uniform version of the Low Basis Theorem that states that closed choice on Cantor space (and the Euclidean space) is low computable. We close with some related observations on the Turing jump operation and its initial topology

Learning-assisted Theorem Proving with Millions of Lemmas

Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and re-used in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL Light lemmas for proving further theorems. We use these criteria to mine the large inference graph of the lemmas in the HOL Light and Flyspeck libraries, adding up to millions of the best lemmas to the pool of statements that can be re-used in later proofs. We show that in combination with learning-based relevance filtering, such methods significantly strengthen automated theorem proving of new conjectures over large formal mathematical libraries such as Flyspeck.Comment: journal version of arXiv:1310.2797 (which was submitted to LPAR conference

An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations

These introductory lectures show how to define finite type invariants of links and 3-manifolds by counting graph configurations in 3-manifolds, following ideas of Witten and Kontsevich. The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first section. As an important example, we present it as the algebraic intersection of a torus and a 4-chain called a propagator in a configuration space. In the second section, we introduce the simplest finite type 3-manifold invariant, which is the Casson invariant (or the Theta-invariant) of integer homology 3-spheres. It is defined as the algebraic intersection of three propagators in the same two-point configuration space. In the third section, we explain the general notion of finite type invariants and introduce relevant spaces of Feynman Jacobi diagrams. In Sections 4 and 5, we sketch an original construction based on configuration space integrals of universal finite type invariants for links in rational homology 3-spheres and we state open problems. Our construction generalizes the known constructions for links in the ambient space, and it makes them more flexible. In Section 6, we present the needed properties of parallelizations of 3-manifolds and associated Pontrjagin classes, in details.Comment: 68 pages. Change of title, updates and minor reorganization of notes of five lectures presented in the ICPAM-ICTP research school of Mekn{\`e}s in May 2012. To appear in the Proceedings of the conference "Quantum topology" organized by Chelyabinsk State University in July 2014 (Vestnik ChelGU