1,559 research outputs found
Constraint Satisfaction and Semilinear Expansions of Addition over the Rationals and the Reals
A semilinear relation is a finite union of finite intersections of open and
closed half-spaces over, for instance, the reals, the rationals, or the
integers. Semilinear relations have been studied in connection with algebraic
geometry, automata theory, and spatiotemporal reasoning. We consider semilinear
relations over the rationals and the reals. Under this assumption, the
computational complexity of the constraint satisfaction problem (CSP) is known
for all finite sets containing R+={(x,y,z) | x+y=z}, <=, and {1}. These
problems correspond to expansions of the linear programming feasibility
problem. We generalise this result and fully determine the complexity for all
finite sets of semilinear relations containing R+. This is accomplished in part
by introducing an algorithm, based on computing affine hulls, which solves a
new class of semilinear CSPs in polynomial time. We further analyse the
complexity of linear optimisation over the solution set and the existence of
integer solutions.Comment: 22 pages, 1 figur
The Reducts of the Homogeneous Binary Branching C-relation
Let (L;C) be the (up to isomorphism unique) countable homogeneous structure
carrying a binary branching C-relation. We study the reducts of (L;C), i.e.,
the structures with domain L that are first-order definable in (L;C). We show
that up to existential interdefinability, there are finitely many such reducts.
This implies that there are finitely many reducts up to first-order
interdefinability, thus confirming a conjecture of Simon Thomas for the special
case of (L;C). We also study the endomorphism monoids of such reducts and show
that they fall into four categories.Comment: 39 pages, 4 figure
- …