4,863 research outputs found
An insertion operator preserving infinite reduction sequences
International audienceA common way to show the termination of the union of two abstract reduction systems, provided both systems terminate, is to prove that they enjoy a specific property (some sort of 'commutation' for instance). This specific property is actually used to show that, for the union not to terminate, one of the systems must itself be non-terminating, which leads to a contradiction. Unfortunately, the property may be impossible to prove because some of the objects that are reduced do not enjoy an adequate form. Hence the purpose of this paper is threefold: - First, it introduces an operator enabling us to insert a reduction step on such an object, and therefore to change its shape, while still preserving the ability to use the property. Of course, some new properties will need to be verified. - Second, as an instance of our technique, the operator is applied to relax a well-known lemma stating the termination of the union of two termination abstract reduction systems. - Finally, this lemma is applied in a peculiar and then in a more general way to show the termination of some lambda calculi with inductive types augmented with specific reductions dealing with: (i) copies of inductive types; (ii) the representation of symmetric groups
Selective Categories and Linear Canonical Relations
A construction of Wehrheim and Woodward circumvents the problem that
compositions of smooth canonical relations are not always smooth, building a
category suitable for functorial quantization. To apply their construction to
more examples, we introduce a notion of highly selective category, in which
only certain morphisms and certain pairs of these morphisms are "good". We then
apply this notion to the category of linear canonical
relations and the result of our version of the WW
construction, identifying the morphisms in the latter with pairs
consisting of a linear canonical relation and a nonnegative integer. We put a
topology on this category of indexed linear canonical relations for which
composition is continuous, unlike the composition in itself.
Subsequent papers will consider this category from the viewpoint of derived
geometry and will concern quantum counterparts
Generalizing the Paige-Tarjan Algorithm by Abstract Interpretation
The Paige and Tarjan algorithm (PT) for computing the coarsest refinement of
a state partition which is a bisimulation on some Kripke structure is well
known. It is also well known in model checking that bisimulation is equivalent
to strong preservation of CTL, or, equivalently, of Hennessy-Milner logic.
Drawing on these observations, we analyze the basic steps of the PT algorithm
from an abstract interpretation perspective, which allows us to reason on
strong preservation in the context of generic inductively defined (temporal)
languages and of possibly non-partitioning abstract models specified by
abstract interpretation. This leads us to design a generalized Paige-Tarjan
algorithm, called GPT, for computing the minimal refinement of an abstract
interpretation-based model that strongly preserves some given language. It
turns out that PT is a straight instance of GPT on the domain of state
partitions for the case of strong preservation of Hennessy-Milner logic. We
provide a number of examples showing that GPT is of general use. We first show
how a well-known efficient algorithm for computing stuttering equivalence can
be viewed as a simple instance of GPT. We then instantiate GPT in order to
design a new efficient algorithm for computing simulation equivalence that is
competitive with the best available algorithms. Finally, we show how GPT allows
to compute new strongly preserving abstract models by providing an efficient
algorithm that computes the coarsest refinement of a given partition that
strongly preserves the language generated by the reachability operator.Comment: Keywords: Abstract interpretation, abstract model checking, strong
preservation, Paige-Tarjan algorithm, refinement algorith
On Termination for Faulty Channel Machines
A channel machine consists of a finite controller together with several fifo
channels; the controller can read messages from the head of a channel and write
messages to the tail of a channel. In this paper, we focus on channel machines
with insertion errors, i.e., machines in whose channels messages can
spontaneously appear. Such devices have been previously introduced in the study
of Metric Temporal Logic. We consider the termination problem: are all the
computations of a given insertion channel machine finite? We show that this
problem has non-elementary, yet primitive recursive complexity
Effective Scalar Products for D-finite Symmetric Functions
Many combinatorial generating functions can be expressed as combinations of
symmetric functions, or extracted as sub-series and specializations from such
combinations. Gessel has outlined a large class of symmetric functions for
which the resulting generating functions are D-finite. We extend Gessel's work
by providing algorithms that compute differential equations these generating
functions satisfy in the case they are given as a scalar product of symmetric
functions in Gessel's class. Examples of applications to k-regular graphs and
Young tableaux with repeated entries are given. Asymptotic estimates are a
natural application of our method, which we illustrate on the same model of
Young tableaux. We also derive a seemingly new formula for the Kronecker
product of the sum of Schur functions with itself.Comment: 51 pages, full paper version of FPSAC 02 extended abstract; v2:
corrections from original submission, improved clarity; now formatted for
journal + bibliograph
Spectral triples from Mumford curves
We construct spectral triples associated to Schottky--Mumford curves, in such
a way that the local Euler factor can be recovered from the zeta functions of
such spectral triples. We propose a way of extending this construction to the
case where the curve is not k-split degenerate.Comment: 25 pages, LaTeX, 4 eps figures (v4: to appear in IMRN
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