28 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
Confluence Algebras and Acyclicity of the Koszul Complex
International audienceThe -Koszul algebras are -homogeneous algebras satisfying a homological property. These algebras are characterised by their Koszul complex: an -homogeneous algebra is -Koszul if and only if its Koszul complex is acyclic. Methods based on computational approaches were used to prove -Koszulness: an algebra admitting a side-confluent presentation is -Koszul if and only if the extra-condition holds. However, in general, these methods do not provide an explicit contracting homotopy for the Koszul complex. In this article we present a way to construct such a contracting homotopy. The property of side-confluence enables us to define specific representations of confluence algebras. These representations provide a candidate for the contracting homotopy. When the extra-condition holds, it turns out that this candidate works. We make explicit our construction on several examples
The Diamond Lemma for non-terminating rewriting systems using deterministic reduction strategies
International audienc
Topological rewriting systems applied to standard bases and syntactic algebras
International audienceWe introduce topological rewriting systems as a generalisation of abstract rewriting systems, where we replace the set of terms by a topological space. Abstract rewriting systems correspond to topological rewriting systems for the discrete topology. We introduce the topological confluence property as an approximation of the confluence property. Using a representation of linear topological rewriting systems with continuous reduction operators, we show that the topological confluence property is characterised by lattice operations. Using this characterisation, we show that standard bases induce topologically confluent rewriting systems on formal power series. Finally, we investigate duality for reduction operators that we relate to series representations and syntactic algebras. In particular, we use duality for proving that an algebra is syntactic or not
The Diamond Lemma for non-terminating rewriting systems using deterministic reduction strategies
International audienc