Termination Detection of Local Computations

Contrary to the sequential world, the processes involved in a distributed system do not necessarily know when a computation is globally finished. This paper investigates the problem of the detection of the termination of local computations. We define four types of termination detection: no detection, detection of the local termination, detection by a distributed observer, detection of the global termination. We give a complete characterisation (except in the local termination detection case where a partial one is given) for each of this termination detection and show that they define a strict hierarchy. These results emphasise the difference between computability of a distributed task and termination detection. Furthermore, these characterisations encompass all standard criteria that are usually formulated : topological restriction (tree, rings, or triangu- lated networks ...), topological knowledge (size, diameter ...), and local knowledge to distinguish nodes (identities, sense of direction). These results are now presented as corollaries of generalising theorems. As a very special and important case, the techniques are also applied to the election problem. Though given in the model of local computations, these results can give qualitative insight for similar results in other standard models. The necessary conditions involve graphs covering and quasi-covering; the sufficient conditions (constructive local computations) are based upon an enumeration algorithm of Mazurkiewicz and a stable properties detection algorithm of Szymanski, Shi and Prywes

Polynomial functors and combinatorial Dyson-Schwinger equations

We present a general abstract framework for combinatorial Dyson-Schwinger equations, in which combinatorial identities are lifted to explicit bijections of sets, and more generally equivalences of groupoids. Key features of combinatorial Dyson-Schwinger equations are revealed to follow from general categorical constructions and universal properties. Rather than beginning with an equation inside a given Hopf algebra and referring to given Hochschild $1$-cocycles, our starting point is an abstract fixpoint equation in groupoids, shown canonically to generate all the algebraic structure. Precisely, for any finitary polynomial endofunctor $P$ defined over groupoids, the system of combinatorial Dyson-Schwinger equations $X=1+P(X)$ has a universal solution, namely the groupoid of $P$-trees. The isoclasses of $P$-trees generate naturally a Connes-Kreimer-like bialgebra, in which the abstract Dyson-Schwinger equation can be internalised in terms of canonical $B_+$-operators. The solution to this equation is a series (the Green function) which always enjoys a Fa\`a di Bruno formula, and hence generates a sub-bialgebra isomorphic to the Fa\`a di Bruno bialgebra. Varying $P$ yields different bialgebras, and cartesian natural transformations between various $P$ yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to truncation of Dyson-Schwinger equations. Finally, all constructions can be pushed inside the classical Connes-Kreimer Hopf algebra of trees by the operation of taking core of $P$-trees. A byproduct of the theory is an interpretation of combinatorial Green functions as inductive data types in the sense of Martin-L\"of Type Theory (expounded elsewhere).Comment: v4: minor adjustments, 49pp, final version to appear in J. Math. Phy

Shared Rough and Quasi-Isometries of Groups

We present a variation of quasi-isometry to approach the problem of defining a geometric notion equivalent to commensurability. In short, this variation can be summarized as "quasi-isometry with uniform parameters for a large enough family of generating systems". Two similar notions (using isometries and rough isometries instead, respectively) are presented alongside. This article is based mainly on a chapter of the author's doctoral thesis (\cite{Lochmann_dissertation}).Comment: 16 page

Classification of unital simple Leavitt path algebras of infinite graphs

We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if L_K(E) and L_K(F) are simple Leavitt path algebras, then L_K(E) is Morita equivalent to L_K(F) if and only if K_0^\textnormal{alg} (L_K(E)) \cong K_0^\textnormal{alg} (L_K(F)) and the graphs $E$ and $F$ have the same number of singular vertices, and moreover, in this case one may transform the graph E into the graph F using basic moves that preserve the Morita equivalence class of the associated Leavitt path algebra. We also show that when K is a field with no free quotients, the condition that E and F have the same number of singular vertices may be replaced by K_1^\textnormal{alg} (L_K(E)) \cong K_1^\textnormal{alg} (L_K(F)), and we produce examples showing this cannot be done in general. We describe how we can combine our results with a classification result of Abrams, Louly, Pardo, and Smith to get a nearly complete classification of unital simple Leavitt path algebras - the only missing part is determining whether the "sign of the determinant condition" is necessary in the finite graph case. We also consider the Cuntz splice move on a graph and its effect on the associated Leavitt path algebra.Comment: Version IV Comments: We have made some substantial revisions, which include extending our classification results to Leavitt path algebras over arbitrary fields. This is the version that will be published. Version III Comments: Some typos and errors corrected. New section (Section 10) has been added. Some references added. Version II Comments: Some typos correcte