298 research outputs found
Algebra and the Complexity of Digraph CSPs: a Survey
We present a brief survey of some of the key results on the interplay between algebraic and graph-theoretic methods in the study of the complexity of digraph-based constraint satisfaction problems
Triangulations, orientals, and skew monoidal categories
A concrete model of the free skew-monoidal category Fsk on a single
generating object is obtained. The situation is clubbable in the sense of G.M.
Kelly, so this allows a description of the free skew-monoidal category on any
category. As the objects of Fsk are meaningfully bracketed words in the skew
unit I and the generating object X, it is necessary to examine bracketings and
to find the appropriate kinds of morphisms between them. This leads us to
relationships between triangulations of polygons, the Tamari lattice, left and
right bracketing functions, and the orientals. A consequence of our description
of Fsk is a coherence theorem asserting the existence of a strictly
structure-preserving faithful functor from Fsk to the skew-monoidal category of
finite non-empty ordinals and first-element-and-order-preserving functions.
This in turn provides a complete solution to the word problem for skew monoidal
categories.Comment: 48 page
The number of clones determined by disjunctions of unary relations
We consider finitary relations (also known as crosses) that are definable via
finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite
parameter set . We prove that whenever contains at least one
non-empty relation distinct from the full carrier set, there is a countably
infinite number of polymorphism clones determined by relations that are
disjunctively definable from . Finally, we extend our result to
finitely related polymorphism clones and countably infinite sets .Comment: manuscript to be published in Theory of Computing System
Quantified Constraints in Twenty Seventeen
I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions
A Categorical Construction of the Real Unit Interval
The real unit interval is the fundamental building block for many branches of
mathematics like probability theory, measure theory, convex sets and homotopy
theory. However, a priori the unit interval could be considered an arbitrary
choice and one can wonder if there is some more canonical way in which the unit
interval can be constructed. In this paper we find such a construction by using
the theory of effect algebras. We show that the real unit interval is the
unique non-initial, non-final irreducible algebra of a particular monad on the
category of bounded posets. The algebras of this monad carry an order,
multiplication, addition and complement, and as such model much of the
operations we need to do on probabilities. On a technical level, we show that
both the categories of omega-complete effect algebras as well as that of
omega-complete effect monoids are monadic over the category of bounded posets
using Beck's monadicity theorem. The characterisation of the real unit interval
then follows easily using a recent representation theorem for omega-complete
effect monoids.Comment: 13 pages + 2 page appendi
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