99 research outputs found
Reduction Operators and Completion of Rewriting Systems
We propose a functional description of rewriting systems where reduction
rules are represented by linear maps called reduction operators. We show that
reduction operators admit a lattice structure. Using this structure we define
the notion of confluence and we show that this notion is equivalent to the
Church-Rosser property of reduction operators. In this paper we give an
algebraic formulation of completion using the lattice structure. We relate
reduction operators and Gr\"obner bases. Finally, we introduce generalised
reduction operators relative to non total ordered sets
Syzygies among reduction operators
We introduce the notion of syzygy for a set of reduction operators and relate
it to the notion of syzygy for presentations of algebras. We give a method for
constructing a linear basis of the space of syzygies for a set of reduction
operators. We interpret these syzygies in terms of the confluence property from
rewriting theory. This enables us to optimise the completion procedure for
reduction operators based on a criterion for detecting useless reductions. We
illustrate this criterion with an example of construction of commutative
Gr{\"o}bner basis
A lattice formulation of the F4 completion procedure
We write a procedure for constructing noncommutative Groebner bases.
Reductions are done by particular linear projectors, called reduction
operators. The operators enable us to use a lattice construction to reduce
simultaneously each S-polynomial into a unique normal form. We write an
implementation as well as an example to illustrate our procedure. Moreover, the
lattice construction is done by Gaussian elimination, which relates our
procedure to the F4 algorithm for constructing commutative Groebner bases
Extremes for the inradius in the Poisson line tessellation
A Poisson line tessellation is observed within a window. With each cell of
the tessellation, we associate the inradius, which is the radius of the largest
ball contained in the cell. Using Poisson approximation, we compute the limit
distributions of the largest and smallest order statistics for the inradii of
all cells whose nuclei are contained in the window in the limit as the window
is scaled to infinity. We additionally prove that the limit shape of the cells
minimising the inradius is a triangle
- …