20,522 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
The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation
In this paper we study the confluence of two regular singular points of the
hypergeometric equation into an irregular one. We study the consequence of the
divergence of solutions at the irregular singular point for the unfolded
system. Our study covers a full neighborhood of the origin in the confluence
parameter space. In particular, we show how the divergence of solutions at the
irregular singular point explains the presence of logarithmic terms in the
solutions at a regular singular point of the unfolded system. For this study,
we consider values of the confluence parameter taken in two sectors covering
the complex plane. In each sector, we study the monodromy of a first integral
of a Riccati system related to the hypergeometric equation. Then, on each
sector, we include the presence of logarithmic terms into a continuous
phenomenon and view a Stokes multiplier related to a 1-summable solution as the
limit of an obstruction that prevents a pair of eigenvectors of the monodromy
operators, one at each singular point, to coincide.Comment: 22 pages v2: revised versio
The lambda-mu-T-calculus
Calculi with control operators have been studied as extensions of simple type
theory. Real programming languages contain datatypes, so to really understand
control operators, one should also include these in the calculus. As a first
step in that direction, we introduce lambda-mu-T, a combination of Parigot's
lambda-mu-calculus and G\"odel's T, to extend a calculus with control operators
with a datatype of natural numbers with a primitive recursor.
We consider the problem of confluence on raw terms, and that of strong
normalization for the well-typed terms. Observing some problems with extending
the proofs of Baba at al. and Parigot's original confluence proof, we provide
new, and improved, proofs of confluence (by complete developments) and strong
normalization (by reducibility and a postponement argument) for our system.
We conclude with some remarks about extensions, choices, and prospects for an
improved presentation
Quasi-Delay-Insensitive Circuits are Turing-Complete
Quasi-delay-insensitive (QDI) circuits are those whose correct operation does not depend on the delays of operators or wires, except for certain wires that form isochronic forks. In this paper we show that quasi-delay-insensitivity, stability and noninterference, and strong confluence are equivalent properties of a computation. In particular, this shows that QDI computations are deterministic. We show that the class of Turing-computable functions have QDI implementations by constructing a QDI Turing machine
Confluence of singularities of differential equation: a Lie algebra contraction approach
We investigate here the confluence of singularities of Mathieu differential
equation by means of the Lie algebra contraction of the Lie algebra of the
motion group M(2) on the Heisenberg Lie algebra H(3). A similar approach for
the Lam\'e equation in terms of the Lie algebra contraction of on
the Lie algebra of the motion group M(2) is outlined
- …