D-completions and the d-topology

In this article we give a general categorical construction via reflection functors for various completions of T0-spaces subordinate to sobrification, with a particular emphasis on what we call the D-completion, a type of directed completion introduced by Wyler [O. Wyler, Dedekind complete posets and Scott topologies, in: B. Banaschewski, R.-E. Hoffmann (Eds.), Continuous Lattices Proceedings, Bremen 1979, in: Lecture Notes in Mathematics, vol. 871, Springer Verlag, 1981, pp. 384-389]. A key result is that all completions of a certain type are universal, hence unique (up to homeomorphism). We give a direct definition of the D-completion and develop its theory by introducing a variant of the Scott topology, which we call the d-topology. For partially ordered sets the D-completion turns out to be a natural dcpo-completion that generalizes the rounded ideal completion. In the latter part of the paper we consider settings in which the D-completion agrees with the sobrification respectively the closed ideal completion. © 2008 Elsevier B.V. All rights reserved

Detecting Mutations by eBWT

In this paper we develop a theory describing how the extended Burrows-Wheeler Transform (EBWT) of a collection of DNA fragments tends to cluster together the copies of nucleotides sequenced from a genome G. Our theory accurately predicts how many copies of any nucleotide are expected inside each such cluster, and how an elegant and precise LCP array based procedure can locate these clusters in the EBWT. Our findings are very general and can be applied to a wide range of different problems. In this paper, we consider the case of alignment-free and reference-free SNPs discovery in multiple collections of reads. We note that, in accordance with our theoretical results, SNPs are clustered in the EBWT of the reads collection, and we develop a tool finding SNPs with a simple scan of the EBWT and LCP arrays. Preliminary results show that our method requires much less coverage than state-of-the-art tools while drastically improving precision and sensitivity

Extending algebraic operations to D -completions

In this article, we show how separately continuous algebraic operations on T0-spaces and the laws that they satisfy, both identities and inequalities, can be extended to theD-completion, that is, the universal monotone convergence space completion. Indeed we show that the operations can be extended to the lattice of closed sets, but in this case it is only the linear identities that admit extension. Via the Scott topology, the theory is shown to be applicable to dcpo-completions of posets. We also explore connections with the construction of free algebras in the context of monotone convergence spaces. © 2011 Elsevier B.V. All rights reserved

Representing Guardedness in Call-By-Value

Like the notion of computation via (strong) monads serves to classify various flavours of impurity, including exceptions, non-determinism, probability, local and global store, the notion of guardedness classifies well-behavedness of cycles in various settings. In its most general form, the guardedness discipline applies to general symmetric monoidal categories and further specializes to Cartesian and co-Cartesian categories, where it governs guarded recursion and guarded iteration respectively. Here, even more specifically, we deal with the semantics of call-by-value guarded iteration. It was shown by Levy, Power and Thielecke that call-by-value languages can be generally interpreted in Freyd categories, but in order to represent effectful function spaces, such a category must canonically arise from a strong monad. We generalize this fact by showing that representing guarded effectful function spaces calls for certain parametrized monads (in the sense of Uustalu). This provides a description of guardedness as an intrinsic categorical property of programs, complementing the existing description of guardedness as a predicate on a category