874 research outputs found
Layer by layer - Combining Monads
We develop a method to incrementally construct programming languages. Our
approach is categorical: each layer of the language is described as a monad.
Our method either (i) concretely builds a distributive law between two monads,
i.e. layers of the language, which then provides a monad structure to the
composition of layers, or (ii) identifies precisely the algebraic obstacles to
the existence of a distributive law and gives a best approximant language. The
running example will involve three layers: a basic imperative language enriched
first by adding non-determinism and then probabilistic choice. The first
extension works seamlessly, but the second encounters an obstacle, which
results in a best approximant language structurally very similar to the
probabilistic network specification language ProbNetKAT
Knots and distributive homology: from arc colorings to Yang-Baxter homology
This paper is a sequel to my essay "Distributivity versus associativity in
the homology theory of algebraic structures" Demonstratio Math., 44(4), 2011,
821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arc
colorings and survey associative and distributive magmas and their homology
with relation to knot theory. We outline potential relations to Khovanov
homology and categorification, via Yang-Baxter operators. We use here the fact
that Yang-Baxter equation can be thought of as a generalization of
self-distributivity. We show how to define and visualize Yang-Baxter homology,
in particular giving a simple description of homology of biquandles.Comment: 64 pages, 29 figures; to be published as a Chapter in: "New Ideas in
Low Dimensional Topology", World Scientific, Vol. 5
Applications of self-distributivity to Yang-Baxter operators and their cohomology
Self-distributive (SD) structures form an important class of solutions to the
Yang--Baxter equation, which underlie spectacular knot-theoretic applications
of self-distributivity. It is less known that one go the other way round, and
construct an SD structure out of any left non-degenerate (LND) set-theoretic
YBE solution. This structure captures important properties of the solution:
invertibility, involutivity, biquandle-ness, the associated braid group
actions. Surprisingly, the tools used to study these associated SD structures
also apply to the cohomology of LND solutions, which generalizes SD cohomology.
Namely, they yield an explicit isomorphism between two cohomology theories for
these solutions, which until recently were studied independently. The whole
story leaves numerous open questions. One of them is the relation between the
cohomologies of a YBE solution and its associated SD structure. These and
related questions are covered in the present survey
Consistency, Amplitudes and Probabilities in Quantum Theory
Quantum theory is formulated as the only consistent way to manipulate
probability amplitudes. The crucial ingredient is a consistency constraint: if
there are two different ways to compute an amplitude the two answers must
agree. This constraint is expressed in the form of functional equations the
solution of which leads to the usual sum and product rules for amplitudes. A
consequence is that the Schrodinger equation must be linear: non-linear
variants of quantum mechanics are inconsistent. The physical interpretation of
the theory is given in terms of a single natural rule. This rule, which does
not itself involve probabilities, is used to obtain a proof of Born's
statistical postulate. Thus, consistency leads to indeterminism.
PACS: 03.65.Bz, 03.65.Ca.Comment: 23 pages, 3 figures (old version did not include the figures
Convolution, Separation and Concurrency
A notion of convolution is presented in the context of formal power series
together with lifting constructions characterising algebras of such series,
which usually are quantales. A number of examples underpin the universality of
these constructions, the most prominent ones being separation logics, where
convolution is separating conjunction in an assertion quantale; interval
logics, where convolution is the chop operation; and stream interval functions,
where convolution is used for analysing the trajectories of dynamical or
real-time systems. A Hoare logic is constructed in a generic fashion on the
power series quantale, which applies to each of these examples. In many cases,
commutative notions of convolution have natural interpretations as concurrency
operations.Comment: 39 page
- …