This paper focuses on the convergence of information in distributed systems of agents communicating over a network.

The information on which the convergence is sought is not represented by real numbers, as often in the literature, rather by sets. The dynamics of the evolution of information across the network is accordingly described by set-valued iterative maps. While the study of convergence of set-valued iterative maps is highly complex in general, this paper focuses on Boolean maps, which are comprised of arbitrary combinations of unions, intersections, and complements of sets. For these important class of systems, we provide tools to study both global and local convergence. A distributed geographic information system, leading to successful information reconstruction from partial and corrupted data, is used to illustrate the applications of the proposed methods.

10aEmbedded Control10aRobotics1 aFagiolini, A1 aDubbini, N1 aMartini, S1 aBicchi, A. uhttp://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=727207501404nas a2200157 4500008004100000022001400041245012700055210006900182300001000251490000800261520088500269653001301154100001501167700001301182856005101195 2012 eng d a0065-103600aAn equivalent of Kronecker's Theorem for powers of an Algebraic Number and Structure of Linear Recurrences of fixed length0 aequivalent of Kroneckers Theorem for powers of an Algebraic Numb a15-330 v1533 aAfter defining a notion of epsilon-density, we provide for any integer m>1 and real algebraic number alpha an estimate of the smallest epsilon such that the set of vectors of the form (t,t^alpha,...,t alpha^{m-1}) for tR is epsilon-dense modulo 1 in terms of the multiplicative Mahler measure M(A(x)) of the minimal integral polynomial A(x) of alpha, which is independent of m. In particular, we show that if alpha has degree d it is possible to take epsilon = 2^{[d/2]}/M(A(x)). On the other side we show using asymptotic estimates for Toeplitz determinants that we cannot have epsilon$-density for sufficiently large m whenever epsilon is strictly smaller than 1/M(A(x)). In the process of proving this we obtain a result of independent interest about the structure of the Z-module of integral linear recurrences of fixed length determined by a non-monic polynomial.

10aRobotics1 aDubbini, N1 aMonge, M uhttp://journals.impan.pl/cgi-bin/doi?aa153-1-200557nas a2200133 4500008004100000245009500041210006900136100001500205700001700220700001800237700001300255700001600268856013900284 2011 eng d00aCausality as a unifying approach between activation and connectivity analysis of fMRI data0 aCausality as a unifying approach between activation and connecti1 aDubbini, N1 aRicciardi, E1 aGaglianese, A1 aMarmi, S1 aPietrini, P uhttp://www.centropiaggio.unipi.it/publications/causality-unifying-approach-between-activation-and-connectivity-analysis-fmri-data.html01685nas a2200169 4500008004100000245010100041210006900142260001700211300001400228490000700242520107300249653001301322100001501335700001501350700001501365856013501380 2011 eng d00aLeft invertibility of discrete-time output-quantized systems: the linear case with finite inputs0 aLeft invertibility of discretetime outputquantized systems the l cNovember, 16 a117 - 1390 v233 aThis paper studies left invertibility of discrete-time linear outputquantized systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a nite alphabet. Left invertibility, i.e. injectivity of I/O map, is reduced to left D-invertibility, under suitable conditions. While left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set, and is much easier to detect. The condition under which left invertibility and left D-invertibility are equivalent is that the elements of the dynamic matrix of the system form an algebraically independent set. Our main result is a method to compute left D-invertibility (so also left invertibility for a full measure matrix set) for all linear systems with no eigenvalue of modulus one. Therefore we are able to check left invertibility of output-quantized linear systems for a full measure matrices set. Some examples are presented to show the application of the proposed method.

10aRobotics1 aDubbini, N1 aPiccoli, B1 aBicchi, A. uhttp://www.centropiaggio.unipi.it/publications/left-invertibility-discrete-time-output-quantized-systems-linear-case-finite-inputs01674nas a2200169 4500008004100000245006300041210006200104260003600166300001800202520109500220653002101315653001301336100001501349700001301364700001501377856011201392 2011 eng d00aLeft invertibility of output-quantized MISO linear systems0 aLeft invertibility of outputquantized MISO linear systems aMilano, italycAugust 28 - Sept a11278 - 112833 aThis paper studies left invertibility of single-output discrete-time quantized linear systems. Quantized outputs are generated according to a given partition of the state-space, while inputs are sequences on a finite alphabet. Left invertibility deals with the possibility of recovering unknown inputs from the only knowledge of the outputs. It is reduced, under suitable conditions, to left D-invertibility: while left invertibility takes into account membership to sets of a given partition, left D-invertibility considers only membership to a single set, and is easily (and algorithmically) detectable. Our main result is a sufficient condition for the equivalence between left invertibility and left D-invertibility in MISO system. In unidimensional systems the equivalence is valid except at most a finite (and computable) number of cases. These results allows the effective detection of left invertibility by means of left Dinvertibility, which is algorithmically detectable. An example with effective computations is presented to show the application of the proposed method.

10aEmbedded Control10aRobotics1 aDubbini, N1 aMonge, M1 aBicchi, A. uhttp://www.centropiaggio.unipi.it/publications/left-invertibility-output-quantized-miso-linear-systems.html01790nas a2200169 4500008004100000245008300041210006900124300001400193490000700207520120000214653002101414653001301435100001501448700001501463700001501478856012701493 2010 eng d00aLeft invertibility of discrete systems with finite inputs and quantized output0 aLeft invertibility of discrete systems with finite inputs and qu a798 - 8090 v833 aThe aim of this paper is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems which evolve in discrete time within a continuous state-space; quantized outputs are generated by the system according to a given partition of the state-space, while inputs are arbitrary sequences of symbols in a finite alphabet, which are associated to specific actions on the system. Our main results are obtained under some contractivity hypotheses. The problem of left invertibility, i.e. recovering an unknown input sequence from the knowledge of the corresponding output string, is addressed using the theory of Iterated Function Systems (IFS), a tool developed for the study of fractals. We show how the IFS naturally associated to a system and the geometric properties of its attractor are linked to the invertibility property of the system. Our main result is a necessary and sufficient condition for left invertibility and uniform left invertibility for joint contractive systems. In addition, an algorithm is proposed to recover inputs from output strings. A few examples are presented to illustrate the application of the proposed method.

10aEmbedded Control10aRobotics1 aDubbini, N1 aPiccoli, B1 aBicchi, A. uhttp://www.centropiaggio.unipi.it/publications/left-invertibility-discrete-systems-finite-inputs-and-quantized-output.html01002nas a2200169 4500008004100000024002000041245008300061210006900144260002700213300001400240520039300254653001300647100001500660700001700675700001500692856012500707 2010 eng d app. 4078 - 408300aLeft invertibility of output-quantized systems: an application to cryptography0 aLeft invertibility of outputquantized systems an application to aAtlanta, USAcDecember a4078-40833 aIn this paper a secure communication method is proposed, based on left invertibility of output-quantized dynamical systems. The sender uses an output-quantized linear system with a feedback function to encode messages, which are sequences of inputs of the system. So left invertibility property enables the receiver to recover the messages. The secret key is formed by the system

10aRobotics1 aDubbini, N1 aCarluccio, A1 aBicchi, A. uhttp://www.centropiaggio.unipi.it/publications/left-invertibility-output-quantized-systems-application-cryptography.html01562nas a2200181 4500008004100000245004900041210004900090260003600139300001200175520099900187653002101186653001301207100001701220700001501237700001501252700001501267856009801282 2009 eng d00aDistributed Consensus on Boolean Information0 aDistributed Consensus on Boolean Information aVenice, ItalycSeptember, 24 - a72 - 773 aIn this paper we study the convergence towards consensus on information in a distributed system of agents communicating over a network. The particularity of this study is that the information on which the consensus is seeked is not represented by real numbers, rather by logical values or compact sets. Whereas the problems of allowing a network of agents to reach a consensus on logical functions of input events, and that of agreeing on set-valued information, have been separately addressed in previous work, in this paper we show that these problems can indeed be attacked in a unified way in the framework of Boolean distributed information systems. Based on a notion of contractivity for Boolean dynamical systems, a necessary and suficient condition ensuring the global convergence toward a unique equilibrium point is presented. This result can be seen as a first step toward the denition of a unified framework to uniformly address all consensus problems on Boolean algebras.

10aEmbedded Control10aRobotics1 aFagiolini, A1 aMartini, S1 aDubbini, N1 aBicchi, A. uhttp://www.centropiaggio.unipi.it/publications/distributed-consensus-boolean-information.html01875nas a2200157 4500008004100000245008300041210006900124300001600193520130000209653002101509653001301530100001501543700001501558700001501573856012901588 2008 eng d00aLeft invertibility of discrete systems with finite inputs and quantized output0 aLeft invertibility of discrete systems with finite inputs and qu a4687 - 46923 aThe aim of this paper is to address left invertibility for dynamical systems with inputs and outputs in discrete sets. We study systems that evolve in discrete time within a continuous state-space. Quantized outputs are generated by the system according to a given partition of the state-space, while inputs are arbitrary sequences of symbols in a finite alphabet, which are associated to specific actions on the system. We restrict to the case of contractive dynamics for fixed inputs. The problem of left invertibility, i.e. recovering an unknown input sequence from the knowledge of the corresponding output string, is addressed using the theory of Iterated Function Systems (IFS), a tool developed for the study of fractals. We show how the IFS naturally associated to a system and the geometric properties of its attractor are linked to the left invertibility property of the system. Our main results are a necessary and sufficient condition for a given system to be left invertible with probability one on the space of inputs (i.e. for almost all input sequences), and necessary and sufficient conditions for left invertibility and uniform left invertibility under some weak additional hypotheses. A few examples are presented to illustrate the application of the proposed method.

10aEmbedded Control10aRobotics1 aDubbini, N1 aPiccoli, B1 aBicchi, A. uhttp://www.centropiaggio.unipi.it/publications/left-invertibility-discrete-systems-finite-inputs-and-quantized-output.html-0