36 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
Short Definitions in Constraint Languages
A first-order formula is called primitive positive (pp) if it only admits the use of existential quantifiers and conjunction. Pp-formulas are a central concept in (fixed-template) constraint satisfaction since CSP(?) can be viewed as the problem of deciding the primitive positive theory of ?, and pp-definability captures gadget reductions between CSPs.
An important class of tractable constraint languages ? is characterized by having few subpowers, that is, the number of n-ary relations pp-definable from ? is bounded by 2^p(n) for some polynomial p(n). In this paper we study a restriction of this property, stating that every pp-definable relation is definable by a pp-formula of polynomial length. We conjecture that the existence of such short definitions is actually equivalent to ? having few subpowers, and verify this conjecture for a large subclass that, in particular, includes all constraint languages on three-element domains. We furthermore discuss how our conjecture imposes an upper complexity bound of co-NP on the subpower membership problem of algebras with few subpowers
Short definitions in constraint languages
A first-order formula is called primitive positive (pp) if it only admits the
use of existential quantifiers and conjunction. Pp-formulas are a central
concept in (fixed-template) constraint satisfaction since CSP() can be
viewed as the problem of deciding the primitive positive theory of ,
and pp-definability captures gadget reductions between CSPs.
An important class of tractable constraint languages is
characterized by having few subpowers, that is, the number of -ary relations
pp-definable from is bounded by for some polynomial .
In this paper we study a restriction of this property, stating that every
pp-definable relation is definable by a pp-formula of polynomial length. We
conjecture that the existence of such short definitions is actually equivalent
to having few subpowers, and verify this conjecture for a large
subclass that, in particular, includes all constraint languages on
three-element domains. We furthermore discuss how our conjecture imposes an
upper complexity bound of co-NP on the subpower membership problem of algebras
with few subpowers
The Subpower Membership Problem for Finite Algebras with Cube Terms
The subalgebra membership problem is the problem of deciding if a given
element belongs to an algebra given by a set of generators. This is one of the
best established computational problems in algebra. We consider a variant of
this problem, which is motivated by recent progress in the Constraint
Satisfaction Problem, and is often referred to as the Subpower Membership
Problem (SMP). In the SMP we are given a set of tuples in a direct product of
algebras from a fixed finite set of finite algebras, and are
asked whether or not a given tuple belongs to the subalgebra of the direct
product generated by a given set.
Our main result is that the subpower membership problem SMP() is
in P if is a finite set of finite algebras with a cube term,
provided is contained in a residually small variety. We also
prove that for any finite set of finite algebras in a variety
with a cube term, each one of the problems SMP(), SMP(), and finding compact representations for subpowers in
, is polynomial time reducible to any of the others, and the first
two lie in NP