The principle of pointfree continuity

In the setting of constructive pointfree topology, we introduce a notion of continuous operation between pointfree topologies and the corresponding principle of pointfree continuity. An operation between points of pointfree topologies is continuous if it is induced by a relation between the bases of the topologies; this gives a rigorous condition for Brouwer's continuity principle to hold. The principle of pointfree continuity for pointfree topologies $\mathcal{S}$ and $\mathcal{T}$ says that any relation which induces a continuous operation between points is a morphism from $\mathcal{S}$ to $\mathcal{T}$. The principle holds under the assumption of bi-spatiality of $\mathcal{S}$. When $\mathcal{S}$ is the formal Baire space or the formal unit interval and $\mathcal{T}$ is the formal topology of natural numbers, the principle is equivalent to spatiality of the formal Baire space and formal unit interval, respectively. Some of the well-known connections between spatiality, bar induction, and compactness of the unit interval are recast in terms of our principle of continuity. We adopt the Minimalist Foundation as our constructive foundation, and positive topology as the notion of pointfree topology. This allows us to distinguish ideal objects from constructive ones, and in particular, to interpret choice sequences as points of the formal Baire space

Localic completion of uniform spaces

We extend the notion of localic completion of generalised metric spaces by Steven Vickers to the setting of generalised uniform spaces. A generalised uniform space (gus) is a set X equipped with a family of generalised metrics on X, where a generalised metric on X is a map from the product of X to the upper reals satisfying zero self-distance law and triangle inequality. For a symmetric generalised uniform space, the localic completion lifts its generalised uniform structure to a point-free generalised uniform structure. This point-free structure induces a complete generalised uniform structure on the set of formal points of the localic completion that gives the standard completion of the original gus with Cauchy filters. We extend the localic completion to a full and faithful functor from the category of locally compact uniform spaces into that of overt locally compact completely regular formal topologies. Moreover, we give an elementary characterisation of the cover of the localic completion of a locally compact uniform space that simplifies the existing characterisation for metric spaces. These results generalise the corresponding results for metric spaces by Erik Palmgren. Furthermore, we show that the localic completion of a symmetric gus is equivalent to the point-free completion of the uniform formal topology associated with the gus. We work in Aczel's constructive set theory CZF with the Regular Extension Axiom. Some of our results also require Countable Choice.Comment: 39 page

A quotient of the Lubin-Tate tower II

In this article we construct the quotient M_1/P(K) of the infinite-level Lubin-Tate space M_1 by the parabolic subgroup P(K) of GL(n,K) of block form (n-1,1) as a perfectoid space, generalizing results of one of the authors (JL) to arbitrary n and K/Q_p finite. For this we prove some perfectoidness results for certain Harris-Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze's candidate for the mod p Jacquet-Langlands and the mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of M_1/P(K) when n = 2, and shows that M_1/Q(K) is not perfectoid for maximal parabolics Q not conjugate to P.Comment: with an appendix by David Hanse

Adaptive Lock-Free Data Structures in Haskell: A General Method for Concurrent Implementation Swapping

A key part of implementing high-level languages is providing built-in and default data structures. Yet selecting good defaults is hard. A mutable data structure's workload is not known in advance, and it may shift over its lifetime - e.g., between read-heavy and write-heavy, or from heavy contention by multiple threads to single-threaded or low-frequency use. One idea is to switch implementations adaptively, but it is nontrivial to switch the implementation of a concurrent data structure at runtime. Performing the transition requires a concurrent snapshot of data structure contents, which normally demands special engineering in the data structure's design. However, in this paper we identify and formalize an relevant property of lock-free algorithms. Namely, lock-freedom is sufficient to guarantee that freezing memory locations in an arbitrary order will result in a valid snapshot. Several functional languages have data structures that freeze and thaw, transitioning between mutable and immutable, such as Haskell vectors and Clojure transients, but these enable only single-threaded writers. We generalize this approach to augment an arbitrary lock-free data structure with the ability to gradually freeze and optionally transition to a new representation. This augmentation doesn't require changing the algorithm or code for the data structure, only replacing its datatype for mutable references with a freezable variant. In this paper, we present an algorithm for lifting plain to adaptive data and prove that the resulting hybrid data structure is itself lock-free, linearizable, and simulates the original. We also perform an empirical case study in the context of heating up and cooling down concurrent maps.Comment: To be published in ACM SIGPLAN Haskell Symposium 201

Formalized linear algebra over Elementary Divisor Rings in Coq

This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms computing this normal form on a variety of coefficient structures including Euclidean domains and constructive principal ideal domains. We also study different ways to extend B\'ezout domains in order to be able to compute the Smith normal form of matrices. The extensions we consider are: adequacy (i.e. the existence of a gdco operation), Krull dimension $\leq 1$ and well-founded strict divisibility