39 research outputs found
Taylor's modularity conjecture and related problems for idempotent varieties
We provide a partial result on Taylor's modularity conjecture, and several
related problems. Namely, we show that the interpretability join of two
idempotent varieties that are not congruence modular is not congruence modular
either, and we prove an analogue for idempotent varieties with a cube term.
Also, similar results are proved for linear varieties and the properties of
congruence modularity, having a cube term, congruence -permutability for a
fixed , and satisfying a non-trivial congruence identity.Comment: 27 page
The wonderland of reflections
A fundamental fact for the algebraic theory of constraint satisfaction
problems (CSPs) over a fixed template is that pp-interpretations between at
most countable \omega-categorical relational structures have two algebraic
counterparts for their polymorphism clones: a semantic one via the standard
algebraic operators H, S, P, and a syntactic one via clone homomorphisms
(capturing identities). We provide a similar characterization which
incorporates all relational constructions relevant for CSPs, that is,
homomorphic equivalence and adding singletons to cores in addition to
pp-interpretations. For the semantic part we introduce a new construction,
called reflection, and for the syntactic part we find an appropriate weakening
of clone homomorphisms, called h1 clone homomorphisms (capturing identities of
height 1).
As a consequence, the complexity of the CSP of an at most countable
-categorical structure depends only on the identities of height 1
satisfied in its polymorphism clone as well as the the natural uniformity
thereon. This allows us in turn to formulate a new elegant dichotomy conjecture
for the CSPs of reducts of finitely bounded homogeneous structures.
Finally, we reveal a close connection between h1 clone homomorphisms and the
notion of compatibility with projections used in the study of the lattice of
interpretability types of varieties.Comment: 24 page
Complexity of term representations of finitary functions
The clone of term operations of an algebraic structure consists of all
operations that can be expressed by a term in the language of the structure. We
consider bounds for the length and the height of the terms expressing these
functions, and we show that these bounds are often robust against the change of
the basic operations of the structure
Local consistency as a reduction between constraint satisfaction problems
We study the use of local consistency methods as reductions between
constraint satisfaction problems (CSPs), and promise version thereof, with the
aim to classify these reductions in a similar way as the algebraic approach
classifies gadget reductions between CSPs. This research is motivated by the
requirement of more expressive reductions in the scope of promise CSPs. While
gadget reductions are enough to provide all necessary hardness in the scope of
(finite domain) non-promise CSP, in promise CSPs a wider class of reductions
needs to be used.
We provide a general framework of reductions, which we call consistency
reductions, that covers most (if not all) reductions recently used for proving
NP-hardness of promise CSPs. We prove some basic properties of these
reductions, and provide the first steps towards understanding the power of
consistency reductions by characterizing a fragment associated to
arc-consistency in terms of polymorphisms of the template. In addition to
showing hardness, consistency reductions can also be used to provide feasible
algorithms by reducing to a fixed tractable (promise) CSP, for example, to
solving systems of affine equations. In this direction, among other results, we
describe the well-known Sherali-Adams hierarchy for CSP in terms of a
consistency reduction to linear programming
Mechanical behaviour of soils in cyclic loading and its effect on simulations of geotechnical problems
Institute of Hydrogeology, Engineering Geology and Applied GeophysicsÚstav hydrogeologie, inž. geologie a užité geofyzikyPřírodovědecká fakultaFaculty of Scienc
Functors on relational structures which admit both left and right adjoints
This paper describes several cases of adjunction in the homomorphism preorder
of relational structures. We say that two functors and
between thin categories of relational structures are adjoint if for all
structures and , we have that maps
homomorphically to if and only if maps homomorphically
to . If this is the case is called the left
adjoint to and the right adjoint to . In 2015,
Foniok and Tardif described some functors on the category of digraphs that
allow both left and right adjoints. The main contribution of Foniok and Tardif
is a construction of right adjoints to some of the functors identified as right
adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to
arbitrary relational structures, and coincidently, we also provide more right
adjoints on digraphs, and since these constructions are connected to finite
duality, we also provide a new construction of duals to trees. Our results are
inspired by an application in promise constraint satisfaction -- it has been
shown that such functors can be used as efficient reductions between these
problems
Revisiting alphabet reduction in Dinur’s PCP.
Dinur’s celebrated proof of the PCP theorem alternates two main steps in several iterations: gap amplification to increase the soundness gap by a large constant factor (at the expense of much larger alphabet size), and a composition step that brings back the alphabet size to an absolute constant (at the expense of a fixed constant factor loss in the soundness gap). We note that the gap amplification can produce a Label Cover CSP. This allows us to reduce the alphabet size via a direct long-code based reduction from Label Cover to a Boolean CSP. Our composition step thus bypasses the concept of Assignment Testers from Dinur’s proof, and we believe it is more intuitive - it is just a gadget reduction. The analysis also uses only elementary facts (Parseval’s identity) about Fourier Transforms over the hypercube
Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs
A linearly ordered (LO) -colouring of a hypergraph is a colouring of its vertices with colours such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO -colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO -colourable, and the case that it is not even LO -colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opr\v{s}al, Wrochna, and \v{Z}ivn\'y (2023)
Distributive and anti-distributive Mendelsohn triple systems
We prove that the existence spectrum of Mendelsohn triple systems whose
associated quasigroups satisfy distributivity corresponds to the Loeschian
numbers, and provide some enumeration results. We do this by considering a
description of the quasigroups in terms of commutative Moufang loops.
In addition we provide constructions of Mendelsohn quasigroups that fail
distributivity for as many combinations of elements as possible.
These systems are analogues of Hall triple systems and anti-mitre Steiner
triple systems respectively