Observable Graphs

An edge-colored directed graph is \emph{observable} if an agent that moves along its edges is able to determine his position in the graph after a sufficiently long observation of the edge colors. When the agent is able to determine his position only from time to time, the graph is said to be \emph{partly observable}. Observability in graphs is desirable in situations where autonomous agents are moving on a network and one wants to localize them (or the agent wants to localize himself) with limited information. In this paper, we completely characterize observable and partly observable graphs and show how these concepts relate to observable discrete event systems and to local automata. Based on these characterizations, we provide polynomial time algorithms to decide observability, to decide partial observability, and to compute the minimal number of observations necessary for finding the position of an agent. In particular we prove that in the worst case this minimal number of observations increases quadratically with the number of nodes in the graph. From this it follows that it may be necessary for an agent to pass through the same node several times before he is finally able to determine his position in the graph. We then consider the more difficult question of assigning colors to a graph so as to make it observable and we prove that two different versions of this problem are NP-complete.Comment: 15 pages, 8 figure

On the Finiteness Property for Rational Matrices

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This conjecture, known as the ``finiteness conjecture", is now known to be false but no explicit counterexample to the conjecture is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that finite sets of nonnegative rational matrices have the finiteness property if and only if \emph{pairs} of \emph{binary} matrices do. We also show that all {pairs} of $2 \times 2$ binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for sets of nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.Comment: 12 pages, 1 figur

On Primitivity of Sets of Matrices

A nonnegative matrix $A$ is called primitive if $A^k$ is positive for some integer $k>0$. A generalization of this concept to finite sets of matrices is as follows: a set of matrices $\mathcal M = \{A_1, A_2, \ldots, A_m \}$ is primitive if $A_{i_1} A_{i_2} \ldots A_{i_k}$ is positive for some indices $i_1, i_2, ..., i_k$. The concept of primitive sets of matrices comes up in a number of problems within the study of discrete-time switched systems. In this paper, we analyze the computational complexity of deciding if a given set of matrices is primitive and we derive bounds on the length of the shortest positive product. We show that while primitivity is algorithmically decidable, unless $P=NP$ it is not possible to decide primitivity of a matrix set in polynomial time. Moreover, we show that the length of the shortest positive sequence can be superpolynomial in the dimension of the matrices. On the other hand, defining ${\mathcal P}$ to be the set of matrices with no zero rows or columns, we give a simple combinatorial proof of a previously-known characterization of primitivity for matrices in ${\mathcal P}$ which can be tested in polynomial time. This latter observation is related to the well-known 1964 conjecture of Cerny on synchronizing automata; in fact, any bound on the minimal length of a synchronizing word for synchronizing automata immediately translates into a bound on the length of the shortest positive product of a primitive set of matrices in ${\mathcal P}$. In particular, any primitive set of $n \times n$ matrices in ${\mathcal P}$ has a positive product of length $O(n^3)$

Continuous-time average-preserving opinion dynamics with opinion-dependent communications

We study a simple continuous-time multi-agent system related to Krause's model of opinion dynamics: each agent holds a real value, and this value is continuously attracted by every other value differing from it by less than 1, with an intensity proportional to the difference. We prove convergence to a set of clusters, with the agents in each cluster sharing a common value, and provide a lower bound on the distance between clusters at a stable equilibrium, under a suitable notion of multi-agent system stability. To better understand the behavior of the system for a large number of agents, we introduce a variant involving a continuum of agents. We prove, under some conditions, the existence of a solution to the system dynamics, convergence to clusters, and a non-trivial lower bound on the distance between clusters. Finally, we establish that the continuum model accurately represents the asymptotic behavior of a system with a finite but large number of agents.Comment: 25 pages, 2 figures, 11 tex files and 2 eps file

Searches for Clean Anomalous Gauge Couplings effects at present and future e+e−e^+e^- colliders

We consider the virtual effects of a general type of Anomalous (triple) Gauge Couplings on various experimental observables in the process of electron-positron annihilation into a final fermion-antifermion state. We show that the use of a recently proposed "$Z$-peak subtracted" theoretical description of the process allows to reduce substantially the number of relevant parameters of the model, so that a calculation of observability limits can be performed in a rather simple way. As an illustration of our approach, we discuss the cases of future measurements at LEP2 and at a new 500 GeV linear collider.Comment: 23 pages incl. 5 figures (e-mail [email protected]

Physics opportunities with future proton accelerators at CERN

We analyze the physics opportunities that would be made possible by upgrades of CERN's proton accelerator complex. These include the new physics possible with luminosity or energy upgrades of the LHC, options for a possible future neutrino complex at CERN, and opportunities in other physics including rare kaon decays, other fixed-target experiments, nuclear physics and antiproton physics, among other possibilities. We stress the importance of inputs from initial LHC running and planned neutrino experiments, and summarize the principal detector R&D issues.Comment: 39 page, word document, full resolution version available from http://cern.ch/pofpa/POFPA-arXive.pd

The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations

We study decidability and complexity questions related to a continuous analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of a linear recurrent sequence. In particular, we show that the continuous version of the nonnegativity problem is NP-hard in general and we show that the presence of a zero is decidable for several subcases, including instances of depth two or less, although the decidability in general is left open. The problems may also be stated as reachability problems related to real zeros of exponential polynomials or solutions to initial value problems of linear differential equations, which are interesting problems in their own right.Comment: 14 pages, no figur

Reachability problems for PAMs

Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.Comment: 16 page