57 research outputs found
Generic Expression Hardness Results for Primitive Positive Formula Comparison
We study the expression complexity of two basic problems involving the
comparison of primitive positive formulas: equivalence and containment. In
particular, we study the complexity of these problems relative to finite
relational structures. We present two generic hardness results for the studied
problems, and discuss evidence that they are optimal and yield, for each of the
problems, a complexity trichotomy
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 of 2-nilpotent algebras
The subpower membership problem SMP(A) of a finite algebraic structure A asks
whether a given partial function from A^k to A can be interpolated by a term
operation of A, or not. While this problem can be EXPTIME-complete in general,
Willard asked whether it is always solvable in polynomial time if A is a
Mal'tsev algebras. In particular, this includes many important structures
studied in abstract algebra, such as groups, quasigroups, rings, Boolean
algebras. In this paper we give an affirmative answer to Willard's question for
a big class of 2-nilpotent Mal'tsev algebras. We furthermore develop tools that
might be essential in answering the question for general nilpotent Mal'tsev
algebras in the future.Comment: 17 pages (including appendix
On the reduction of the CSP dichotomy conjecture to digraphs
It is well known that the constraint satisfaction problem over general
relational structures can be reduced in polynomial time to digraphs. We present
a simple variant of such a reduction and use it to show that the algebraic
dichotomy conjecture is equivalent to its restriction to digraphs and that the
polynomial reduction can be made in logspace. We also show that our reduction
preserves the bounded width property, i.e., solvability by local consistency
methods. We discuss further algorithmic properties that are preserved and
related open problems.Comment: 34 pages. Article is to appear in CP2013. This version includes two
appendices with proofs of claims omitted from the main articl
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