4,863 research outputs found

    An insertion operator preserving infinite reduction sequences

    Get PDF
    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

    Full text link
    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 SLREL\mathbf{SLREL} of linear canonical relations and the result WW(SLREL){\rm WW}(\mathbf{SLREL}) of our version of the WW construction, identifying the morphisms in the latter with pairs (L,k)(L,k) 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 SLREL\mathbf{SLREL} 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

    Full text link
    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

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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
    corecore