Testing from semi-independent communicating finite state machines with a slow environment

Some systems may be modelled as a set of communicating finite state machines with a slow environment. These machines communicate through the exchange of values. While it is possible to convert such a model into one finite state machine, from which test cases can be derived, this process may lead to an explosion in the number of states. Alternatively it is possible to utilise any independence that exists. The problem of producing a minimal test set, in the presence of certain types of independence and unique input/output sequences, can be represented as a variant of the vehicle routing problem. Possible heuristics for solving this problem are outlined and the method is applied to an example

A Linear-time Independence Criterion Based on a Finite Basis Approximation

Detection of statistical dependence between random variables is an essential component in many machine learning algorithms. We propose a novel independence criterion for two random variables with linear-time complexity. We establish that our independence criterion is an upper bound of the Hirschfeld-Gebelein-Rényi maximum correlation coefficient between tested variables. A finite set of basis functions is employed to approximate the mapping functions that can achieve the maximal correlation. Using classic benchmark experiments based on independent component analysis, we demonstrate that our independence criterion performs comparably with the state-of-the-art quadratic-time kernel dependence measures like the Hilbert-Schmidt Independence Criterion, while being more efficient in computation. The experimental results also show that our independence criterion outperforms another contemporary linear-time kernel dependence measure, the Finite Set Independence Criterion. The potential application of our criterion in deep neural networks is validated experimentally

Extremal Choice Equilibrium: Existence and Purification with Infinite-Dimensional Externalities

We prove existence and purification results for equilibria in which players choose extreme points of their feasible actions in a class of strategic environments exhibiting a product structure. We assume finite-dimensional action sets and allow for infinite-dimensional externalities. Applied to large games, we obtain existence of Nash equilibrium in pure strategies while allowing a continuum of groups and general dependence of payoffs on average actions across groups, without resorting to saturated measure spaces. Applied to games of incomplete information, we obtain a new purification result for Bayes-Nash equilibria that permits substantial correlation across types, without assuming conditional independence given the realization of a finite environmental state. We highlight our results in examples of industrial organization, auctions, and voting.

Curious properties of free hypergraph C*-algebras

A finite hypergraph $H$ consists of a finite set of vertices $V(H)$ and a collection of subsets $E(H) \subseteq 2^{V(H)}$ which we consider as partition of unity relations between projection operators. These partition of unity relations freely generate a universal C*-algebra, which we call the "free hypergraph C*-algebra" $C^*(H)$. General free hypergraph C*-algebras were first studied in the context of quantum contextuality. As special cases, the class of free hypergraph C*-algebras comprises quantum permutation groups, maximal group C*-algebras of graph products of finite cyclic groups, and the C*-algebras associated to quantum graph homomorphism, isomorphism, and colouring. Here, we conduct the first systematic study of aspects of free hypergraph C*-algebras. We show that they coincide with the class of finite colimits of finite-dimensional commutative C*-algebras, and also with the class of C*-algebras associated to synchronous nonlocal games. We had previously shown that it is undecidable to determine whether $C^*(H)$ is nonzero for given $H$. We now show that it is also undecidable to determine whether a given $C^*(H)$ is residually finite-dimensional, and similarly whether it only has infinite-dimensional representations, and whether it has a tracial state. It follows that for each one of these properties, there is $H$ such that the question whether $C^*(H)$ has this property is independent of the ZFC axioms, assuming that these are consistent. We clarify some of the subtleties associated with such independence results in an appendix.Comment: 19 pages. v2: minor clarifications. v3: terminology 'free hypergraph C*-algebra', added Remark 2.2

Using matrix graph grammars for the analysis of behavioural specifications: Sequential and parallel independence

Proceedings of the Seventh Spanish Conference on Programming and Computer Languages (PROLE 2007)In this paper we present a new approach for the analysis of rule-based specification of system dynamics. We model system states as simple digraphs, which can be represented with boolean matrices. Rules modelling the different state changes of the system can also be represented with boolean matrices, and therefore the rewriting is expressed using boolean operations only. The conditions for sequential independence between pair of rules are well-known in the categorical approaches to graph transformation (e.g. single and double pushout). These conditions state when two rules can be applied in any order yielding the same result. In this paper, we study the concept of sequential independence in our framework, and extend it in order to consider derivations of arbitrary finite length. Instead of studying one-step rule advances, we study independence of rule permutations in sequences of arbitrary finite length. We also analyse the conditions under which a sequence is applicable to a given host graph. We introduce rule composition and give some preliminary results regarding parallel independence. Moreover, we improve our framework making explicit the elements which, if present, disable the application of a rule or a sequence.Work sponsored by Spanish Ministry of Science and Education, project MOSAIC (TSI2005-08225-C07-06)

Connections on the State-Space over Conformal Field Theories

Motivated by the problem of background independence of closed string field theory we study geometry on the infinite vector bundle of local fields over the space of conformal field theories (CFT's). With any connection we can associate an excluded domain $D$ for the integral of marginal operators, and an operator one-form $\omega_\mu$. The pair $(D, \omega_\mu)$ determines the covariant derivative of any correlator of local fields. We obtain interesting classes of connections in which $\omega_\mu$'s can be written in terms of CFT data. For these connections we compute their curvatures in terms of four-point correlators, $D$, and $\omega_\mu$. Among these connections three are of particular interest. A flat, metric compatible connection \HG, and connections $c$ and $\bar c$ having non-vanishing curvature, with $\bar c$ being metric compatible. The flat connection cannot be used to do parallel transport over a finite distance. Parallel transport with either $c$ or $\bar c$, however, allows us to construct a CFT in the state space of another CFT a finite distance away. The construction is given in the form of perturbation theory manifestly free of divergences.Comment: 54pp. MIT-CTP-219