complexity of equivalence problems for concurrent systems of finite agents abstract a concurrent system of synchronous communicating agents is assembled from simpler sequential agents by parallel composition and hiding for example hide a1 alin p1 p2 pn describes the system of communicating agentsp1 pnin which the communication eventsa1 alare hidden consider descriptions of two systemspandqof synchronously communicating finite state agents assume that one wants to check whetherp qfor one of the commonly used equivalence we show that this question is pspace hard for all equivalences that lie between strong bisimulation and trace equivalences for some equivalences exponential lower and upper bounds are proven we also show that this problem is np hard and co np hard even for a class of very simple finite agents

the complexity of word problems this time with interleaving abstract we consider regular expressions extended with the interleaving operator and investigate the complexity of membership and inequivalence problems for these expressions for expressions using the operators union concatenation kleene star and interleaving we show that the inequivalence problem deciding whether two given expressions do not describe the same set of words is complete for exponential space without kleene star we show that the inequivalence problem is complete for the class p 2 at the second level of the polynomial time hierarchy certain cases of the membership problem deciding whether a given word is in the language described by a given expression are shown to be np complete it is also shown that certain languages can be described exponentially more succinctly by using interleaving

improved model checking of hierarchical systems we present a unified game based approach for branching time model checking of hierarchical systems such systems are exponentially more succinct than standard state transition graphs as repeated sub systems are described only once early work on model checking of hierarchical systems shows that one can do better than a naive algorithm that flattens the system and removes the hierarchy given a hierarchical system s and a branching time specification j for it we reduce the model checking problem does s satisfy j to the problem of solving a hierarchical game obtained by taking the product of s with an alternating tree automaton a j for j our approach leads to clean uniform and improved model checking algorithms for a variety of branching time temporal logics in particular by improving the algorithm for solving hierarchical parity games we are able to solve the model checking problem for the m calculus in pspace and time complexity that is only polynomial in the depth of the hierarchy our approach also leads to an abstraction refinement paradigm for hierarchical systems the abstraction maintains the hierarchy and is obtained by merging both states and sub systems into abstract states

distributed rate distortion with common components we describe a scheme for rate distortion with distributed encoding in which the sources to be compressed contain a common component we show that this scheme is optimal in some situations and that it strictly improves upon existing schemes which do not make full use of common components this establishes that independent quantization followed by independent binning is not optimal for the two encoder problem with a distortion constraint on one source we also show that independent quantization and binning is suboptimal for the three encoder problem in which the goal is to reproduce one of the sources losslessly this provides a counterexample that is fundamentally different from one provided earlier by korner and marton the proofs rely on the binary analogue of the entropy power inequality and the existence of a rate loss for the binary symmetric wyner ziv problem

on the optimality of binning for distributed hypothesis testing we study a hypothesis testing problem in which data are compressed distributively and sent to a detector that seeks to decide between two possible distributions for the data the aim is to characterize all achievable encoding rates and exponents of the type 2 error probability when the type 1 error probability is at most a fixed value for related problems in distributed source coding schemes based on random binning perform well and are often optimal for distributed hypothesis testing however the use of binning is hindered by the fact that the overall error probability may be dominated by errors in the binning process we show that despite this complication binning is optimal for a class of problems in which the goal is to test against conditional independence we then use this optimality result to give an outer bound for a more general class of instances of the problem

ccs expressions finite state processes and three problems of equivalence abstract we examine the computational complexity of testing finite state processes for equivalence in milner s calculus of communicating systems ccs the equivalence problems in ccs are presented as refinements of the familiar problem of testing whether two nondeterministic finite automata nfa are equivalent i e accept the same language three notions of equivalence proposed for ccs are investigated namely observational equivalence strong observational equivalence and failure equivalence we show that observational equivalence can be tested in polynomial time as defined in ccs observational equivalence is the limit of a sequence of successively finer equivalence relations k where 1 is nondeterministic finite automaton equivalence we prove that for each fixed k deciding k is pspace complete we show that strong observational equivalence can be decided in polynomial time by reducing it to generalized partitioning a new combinatorial problem of independent interest finally we demonstrate that testing for failure equivalence is pspace complete even for a very restricted type of process

rate region of the quadratic gaussian two encoder source coding problem we determine the rate region of the quadratic gaussian two encoder source coding problem this rate region is achieved by a simple architecture that separates the analog and digital aspects of the compression furthermore this architecture requires higher rates to send a gaussian source than it does to send any other source with the same covariance our techniques can also be used to determine the sum rate of some generalizations of this classical problem our approach involves coupling the problem to a quadratic gaussian ldquoceo problem rdquo

an achievable rate region for the multiple access channel with feedback an achievable rate region r 1 leq i x 1 y x 2 u r 2 leq i x 2 y x 1 u r 1 r 2 leq i x 1 x 2 y where p u x l x 2 y p u p x l u p x 2 u p y x l x 2 is established for the multiple access channel with feedback time sharing of these achievable rates yields the rate region of this paper this region generally exceeds the achievable rate region without feedback and exceeds the rate point found by gaarder and wolf for the binary erasure multiple access channel with feedback the presence of feedback allows the independent transmitters to understand each other s intended transmissions before the receiver has sufficient information to achieve the desired decoding this allows the transmitters to cooperate in the transmission of information that resolves the residual uncertainty of the receiver at the same time independent information from the transmitters is superimposed on the cooperative correction information the proof involves list codes and block markov encoding

multiple access channels with states causally known at transmitters it has been recently shown by lapidoth and steinberg that strictly causal state information can be beneficial in multiple access channels macs specifically it was proved that the capacity region of a two user mac with independent states each known strictly causally to one encoder can be enlarged by letting the encoders send compressed past state information to the decoder in this study a generalization of the said strategy is proposed whereby the encoders compress also the past transmitted codewords along with the past state sequences the proposed scheme uses a combination of long message encoding compression of the past state sequences and codewords without binning and joint decoding over all transmission blocks the proposed strategy has been recently shown by lapidoth and steinberg to strictly improve upon the original one capacity results are then derived for a class of channels that include two user modulo additive state dependent macs moreover the proposed scheme is extended to state dependent macs with an arbitrary number of users finally output feedback is introduced and an example is provided to illustrate the interplay between feedback and availability of strictly causal state information in enlarging the capacity region

the multiple access channel with causal side information common state we show that if a memoryless multiple access channel mac is governed by an independent and identically distributed state sequence then unlike the single user case the capacity region is typically increased if the state is revealed to the encoders in a strictly causal way for this scenario we derive inner and outer bounds on the capacity region for the gaussian mac whose state sequence comprises the channel noise we compute the capacity region and propose a variation on the schalkwijk kailath scheme that achieves capacity with a double exponential decay of the maximal probability of error we also study the causal case for which we derive an achievable region which is typically strictly larger than the region achievable with naive shannon strategies

on the scaling of three dimensional homogeneous and isotropic turbulence abstract in this paper we investigate the scaling properties of three dimensional isotropic and homogeneous turbulence we analyze a new form of scaling extended self similarity recently introduced in the literature we found that anomalous scaling of the velocity structure functions is clearly detectable even at a moderate and low reynolds number and it extends over a much wider range of scales with respect to the inertial range

multiterminal source encoding with one distortion criterion the authors unify earlier investigations concerning the encoding of two correlated sources x sub k y sub k by means of separate encoders decoding is done by a single decoder which receives the outputs from both encoders the reconstruction of x sub k is required to be perfect in the usual shannon sense the authors determine the admissible rate region r d where d is the distortion of the reconstruction of y sub k the binary hamming case is investigated explicitly

mac with action dependent state information at one encoder the growing interest in action dependent channels motivates us to extend the study of action dependent settings which until now focused on point to point models to multiple access channels macs in this paper we consider a two user state dependent mac in which one of the encoders called the informed cognitive encoder is allowed to take an action that affects the formation of the channel states two independent messages are to be sent through the channel 1 a common message known to both encoders and 2 a private message known only to the informed encoder in addition the informed encoder has access to the sequence of channel states in a noncausal manner our framework generalizes the previously evaluated settings of state dependent point to point channels with actions and macs with common messages we derive a single letter characterization of the capacity region for this setting using this general result we obtain and compute the capacity region for the gaussian action dependent mac the special methods used in solving the gaussian case are then applied to obtain the capacity of the gaussian action dependent point to point channel a problem left open and solved now finally we establish some dualities between action dependent channel coding and source coding problems in particular we obtain a duality equivalence between the considered mac setting and the rate distortion model known as successive refinement with actions this is done by developing a set of simple duality principles that enables us to successfully evaluate the outcome of one problem given the outcome of the other

on optimal causal coding of partially observed markov sources in single and multiterminal settings the optimal causal zero delay coding of a partially observed markov process is studied where the cost to be minimized is a bounded nonnegative additive measurable single letter function of the source and the receiver output a structural result is obtained extending witsenhausen s and walrand varaiya s structural results on optimal causal coders to more general state spaces and to a partially observed setting the decentralized multiterminal setup is also considered for the case where the source is an i i d process it is shown that an optimal solution to the decentralized causal coding of correlated observations problem is memoryless for markov sources a counterexample to a natural separation conjecture is presented

an improved outer bound for multiterminal source coding we prove a new outer bound on the rate distortion region for the multiterminal source coding problem this bound subsumes the best outer bound in the literature and improves upon it strictly in some cases the improved bound enables us to obtain a new conclusive result for the binary erasure version of the ceo problem the bound recovers many of the converse results that have been established for special cases of the problem including the recent one for the gaussian two encoder problem

lattices for distributed source coding jointly gaussian sources and reconstruction of a linear function consider a pair of correlated gaussian sources x 1 x 2 two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder the decoder is interested in reconstructing a linear combination of x 1 and x 2 to within a mean square distortion of d we obtain an inner bound to the optimal rate distortion region for this problem a portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components x 1 and x 2 first this results in a better rate region for certain parameter values our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory we then consider the case of linear reconstruction of k sources and provide an inner bound to the optimal rate distortion region some parts of the inner bound are achieved using the following coding structure lattice vector quantization followed by ldquocorrelatedrdquo lattice structured binning

on the rate loss of multiple sources codes in the wyner ziv setting in this paper we consider the rate loss and achievable region of multiple sources codes also named multiterminal source codes in the wyner ziv setting wz msc in which two stationary memoryless sources x y are compressed separately from each other at respective rates r1 and r2 and the joint decoder having access to the side information s which is correlated with sources x y reconstructs x and y with distortion d1 and d2 respectively we define the rate loss l1 r1 rx y s d1 l2 r2 ry x s d2 and l0 r1 r2 rx y s d1 d2 for general real valued stationary memoryless sources and mean squared error distortion measure we show that the rate loss can be bounded by li ki 1 i 1 2 and l0 1 where k1 rx y s d1 d2 ry s d2 rx y s d1 and k2 rx y s d1 d2 rx s d1 ry x s d2 when the sources and side information have a jointly gaussian distribution we give upper bounds of k1 and k2 by a function of conditional correlation coefficient xy s for a special case when there is no side information our result tightens feng s rate loss bounds using our upper bounds of the rate loss we give good approximations of the achievable region

on the loss of single letter characterization the dirty multiple access channel for general memoryless systems the existing information theoretic solutions have a ldquosingle letterrdquo form this reflects the fact that optimum performance can be approached by a random code or a random binning scheme generated using independent and identically distributed copies of some scalar distribution is that the form of the solution of any information theoretic problem in fact some counter examples are known the most famous one is the ldquotwo help onerdquo problem korner and marton showed that if we want to decode the modulo two sum of two correlated binary sources from their independent encodings then linear coding is better than random coding in this paper we provide another counter example the ldquodoubly dirtyrdquo multiple access channel mac like the korner marton problem this is a multiterminal scenario where side information is distributed among several terminals each transmitter knows part of the channel interference while the receiver only observes the channel output we give an explicit solution for the capacity region of the binary doubly dirty mac demonstrate how this region can be approached using a linear coding scheme and prove that the ldquobest known single letter regionrdquo is strictly contained in it we also state a conjecture regarding the capacity loss of single letter characterization in the gaussian case

the wyner ziv problem with multiple sources this paper provides bounds on the rate distortion region for the distributed compression scenario where two or more sources are compressed separately for a decoder that has access to side information conclusive rate distortion results are found for the case where the sources are conditionally independent given the side information

the rate distortion function for the quadratic gaussian ceo problem a new multiterminal source coding problem called the ceo problem was presented and investigated by berger zhang and viswanathan recently viswanathan and berger have investigated an extension of the ceo problem to gaussian sources and call it the quadratic gaussian ceo problem they considered this problem from a statistical viewpoint deriving some interesting results in this paper we consider the quadratic gaussian ceo problem from a standpoint of multiterminal rate distortion theory we regard the ceo problem as a certain multiterminal remote source coding problem with infinitely many separate encoders whose observations are conditionally independent if the remote source is given this viewpoint leads us to a complete solution to the problem we determine the tradeoff between the total amount of rate and squared distortion deriving an explicit formula of the rate distortion function the derived function has the form of a sum of two nonnegative functions one is a classical rate distortion function for single gaussian source and the other is a new rate distortion function which dominates the performance of the system for a relatively small distortion it follows immediately from our result that the conjecture of viswanathan and berger on the asymptotic behavior of minimum squared distortion for large rates is true

three partition refinement algorithms we present improved partition refinement algorithms for three problems lexicographic sorting relational coarsest partition and double lexical ordering our double lexical ordering algorithm uses a new efficient method for unmerging two sorted sets

memoryless multiple access channel with asymmetric noisy state information at the encoders the problem of reliable communication over the memoryless state dependent multiple access channel mac is considered where the encoders and the decoder are provided with various degrees of asymmetric noisy channel state information csi for the case where the encoders observe causal asymmetric noisy csi and the decoder observes complete csi inner and outer bounds to the capacity region which are tight for the sum rate capacity are provided next single letter characterizations for the channel capacity regions under each of the following system settings are established 1 the csi at the encoders are asymmetric deterministic functions of the csi at the decoder and the encoders have noncausal noisy csi 2 the encoders observe asymmetric noisy csi with asymmetric delays and the decoder observes complete csi 3 a degraded message set scenario with asymmetric noisy csi at the encoders and complete and or noisy csi at the decoder the main component in these results is a generalization of a recently introduced converse coding approach for the mac with asymmetric quantized csi at the encoders and herein considerably extended and adapted for the noisy csi setup

scaling properties of the streamwise component of velocity in a turbulent boundary layer abstract experimental results concerning the statistics of the streamwise component of velocity in a turbulent boundary layer near a solid wall are presented energy spectra probability density functions and velocity structure functions are investigated as a function of the distance to the wall energy spectra and structure functions display power law scalings which depend on this distance on the other hand the relative behavior of structure functions remains constant in the logarithmic sublayer this behavior exhibits two distinct power laws at small scales relative scaling exponents are relatively close to those usually reported in isotropic and homogeneous turbulence at larger scales where shear effects are dominant relative exponents display larger intermittency corrections

predecessor existence problems for finite discrete dynamical systems we study the predecessor existence problem for finite discrete dynamical systems given a finite discrete dynamical system s and a configuration c the predecessor existence or pre problem is to determine whether there is a configuration c such that s has a transition from c to c in addition to the decision version we also study the following variants the predecessor existence or pre problem counting the number of predecessors the unique predecessor existence or upre problem deciding whether there is a unique predecessor and the ambiguous predecessor existence or apre problem given a configuration c and a predecessor c of c deciding whether there is a different predecessor c of c general techniques are presented for simultaneously characterizing the computational complexity of the pre problem and its three variants our hardness results are based on the concept of simultaneous reductions single transformations that can be used to simultaneously prove the hardness of the different variants of the pre problem for their respective complexity classes our easiness results are based on dynamic programming and they extend the previous results on pre problem for one dimensional cellular automata the hardness results together with the easiness results provide a tight separation between easy and hard instances further the results imply similar bounds for other classes of finite discrete dynamical systems including discrete hopfield and recurrent neural networks concurrent state machines systolic networks and one and two dimensional cellular automata our results extend the earlier results of green sutner and orponen on the complexity of the predecessor existence problem and its variants

the distributed karhunen loeve transform the karhunen loeve transform klt is a key element of many signal processing and communication tasks many recent applications involve distributed signal processing where it is not generally possible to apply the klt to the entire signal rather the klt must be approximated in a distributed fashion this paper investigates such distributed approaches to the klt where several distributed terminals observe disjoint subsets of a random vector we introduce several versions of the distributed klt first a local klt is introduced which is the optimal solution for a given terminal assuming all else is fixed this local klt is different and in general improves upon the marginal klt which simply ignores other terminals both optimal approximation and compression using this local klt are derived two important special cases are studied in detail namely the partial observation klt which has access to a subset of variables but aims at reconstructing them all and the conditional klt which has access to side information at the decoder we focus on the jointly gaussian case with known correlation structure and on approximation and compression problems then the distributed klt is addressed by considering local klts in turn at the various terminals leading to an iterative algorithm which is locally convergent sometimes reaching a global optimum depending on the overall correlation structure for compression it is shown that the classical distributed source coding techniques admit a natural transform coding interpretation the transform being the distributed klt examples throughout illustrate the performance of the proposed distributed klt this distributed transform has potential applications in sensor networks distributed image databases hyper spectral imagery and data fusion

symbolic model checking for m calculus requires exponential time we investigate the complexity of symbolic model checking for calculus the exponential time lower bound is proved moreover this lower bound holds even for a fixed alternation free formula

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

double scaling and intermittency in shear dominated flows the re ned kolmogorov similarity hypothesis is a valuable tool for the description of intermittency in isotropicconditions for ows in presence of a substantial mean shear the nature of intermittency changes since the process ofenergy transfer is a ected by the turbulent kinetic energy production associated with the reynolds stresses in theseconditions a new form of re ned similarity law has been found able to describe the increased level of intermittencywhich characterizes shear dominated ows ideally a length scale associated with the mean shear separates the tworanges i e the classical kolmogorov like inertial range below and the shear dominated range above however the data analyzed in previous papers correspond to conditions where the two scaling regimes can only be observedindividually in the present letter we give evidence of the coexistence of the two regimes and support the conjecture thatthe statistical properties of the dissipation eld are practically insensible to the mean shear this allows for atheoretical prediction of the scaling exponents of structure functions in the shear dominated range based on theknown intermittency corrections for isotropic ows the prediction is found to closely match the available numericaland experimental data

rate distortion for correlated sources with partially separated encoders we resolve some open problems concerning the encoding of a pair of correlated sources with respect to a fidelity criterion two encoders are used one encoder observes only the output of the first source while the other is supplied both with the second source output and with partial information about the first source at some predetermined rate a general coding theorem is proved which established that cal s ast a certain region defined in terms of single letter information theoretic quantities is an inner bound to the region of all attainable vectors of rates and distortions for certain cases of interest the converse is proved too thereby establishing the rate distortion region for these cases

lattice strategies for the dirty multiple access channel in costa s dirty paper channel gaussian random binning is able to eliminate the effect of interference which is known at the transmitter and thus achieve capacity we examine a generalization of the dirty paper problem to a multiple access channel mac setup where structured lattice based binning seems to be necessary to achieve capacity in the dirty mac two additive interference signals are present one known to each transmitter but none to the receiver the achievable rates using costa s gaussian binning vanish if both interference signals are strong in contrast it is shown that lattice strategies lattice precoding can achieve positive rates independent of the interference power furthermore in some cases which depend on the noise variance and power constraints high dimensional lattice strategies are in fact optimal in particular they are optimal in the limit of high snr where the capacity region of the dirty mac with strong interference approaches that of a clean mac whose power is governed by the minimum of the users powers rather than their sum the rate gap at high snr between lattice strategies and optimum rather than gaussian random binning is conjectured to be 1 2 log2 e 6 0 254 bit thus the doubly dirty mac is another instance of a network setting like the korner marton problem where linear structured coding is potentially better than random binning

dichotomy results for fixed point counting in boolean dynamical systems we present dichotomy theorems regarding the computational complexity of counting fixed points in boolean discrete dynamical systems i e finite discrete dynamical systems over the domain 0 1 for a class f of boolean functions and a class g of graphs an f g system is a boolean dynamical system with local transitions functions lying in f and graphs in g we show that if local transition functions are given by lookup tables then the following complexity classification holds let f be a class of boolean functions closed under superposition and let g be a graph class closed under taking minors if f contains all min functions all max functions or all self dual and monotone functions and g contains all planar graphs then it is p complete to compute the number of fixed points in an f g system otherwise it is computable in polynomial time we also prove a dichotomy theorem for the case that local transition functions are given by formulas over logical bases this theorem has a significantly more complicated structure than the theorem for lookup tables a corresponding theorem for boolean circuits coincides with the theorem for formulas

fluctuations of a passive scalar in a turbulent mixing layer the turbulent flow originating downstream of the kelvin helmholtz instability in a mixing layer has great relevance in many applications ranging from atmospheric physics to combustion in technical devices the mixing of a substance by the turbulent velocity field is usually involved in this paper a detailed statistical analysis of fluctuations of a passive scalar in the fully developed region of a turbulent mixing layer from a direct numerical simulation is presented passive scalar spectra show inertial ranges characterized by scaling exponents ensuremath 4 3 and ensuremath 3 2 in the streamwise and spanwise directions in agreement with a recent theoretical analysis of passive scalar scaling in shear flows celani et al j fluid mech 523 99 2005 scaling exponents of high order structure functions in the streamwise direction show saturation of intermittency with an asymptotic exponent ensuremath zeta ensuremath infty 0 4 at large orders saturation of intermittency is confirmed by the self similarity of the tails of the probability density functions of the scalar increments at different scales r with the scaling factor r ensuremath ensuremath zeta ensuremath infty and by the analysis of the cumulative probability of large fluctuations conversely intermittency saturation is not observed for the spanwise increments and the relative scaling exponents agree with recent results for homogeneous isotropic turbulence with mean scalar gradient probability density functions of the scalar increments in the three directions are compared to assess anisotropy

fast iterative coding techniques for feedback channels a class of capacity achieving low complexity high reliability variable rate coding schemes is developed for communication over discrete memoryless channels with noiseless feedback algorithms for encoding and decoding that require computations growing linearly with the number of channel inputs used are developed the error exponent associated with the scheme is shown to be optimal and implies that capacity is achievable simulations are performed and support the analytically predicted high performance and low complexity

generalized scaling in fully developed turbulence in this paper we report numerical and experimental results on the scaling properties of the velocity turbulent fields in several flows the limits of a new form of scaling named extended self similarity ess are discussed we show that when a mean shear is absent the self scaling exponents are universal and they do not depend on the specific flow 3d homogeneous turbulence thermal convection mhd in contrast ess is not observed when a strong shear is present we propose a generalized version of self scaling which extends down to the smallest resolvable scales even in cases where ess is not present this new scaling is checked in several laboratory and numerical experiments a possible theoretical interpretation is also proposed a synthetic turbulent signal having most of the properties of a real one has been generated

the ceo problem multiterminal source coding we consider a new problem in multiterminal source coding motivated by the following decentralized communication estimation task a firm s chief executive officer ceo is interested in the data sequence x t sub t 1 sup spl infin which cannot be observed directly perhaps because it represents tactical decisions by a competing firm the ceo deploys a team of l agents who observe independently corrupted versions of x t sub t 1 sup spl infin because x t is only one among many pressing matters to which the ceo must attend the combined data rate at which the agents may communicate information about their observations to the ceo is limited to say r bits per second if the agents were permitted to confer and pool their data then in the limit as l spl rarr spl infin they usually would be able to smooth out their independent observation noises entirely then they could use their r bits per second to provide the ceo with a representation of x t with fidelity d r where d spl middot is the distortion rate function of x t in particular with such data pooling d can be made arbitrarily small if r exceeds the entropy rate h of x t suppose however that the agents are not permitted to convene agent i having to send data based solely on his own noisy observations y sub i t we show that then there does not exist a finite value of r for which even infinitely many agents can make d arbitrarily small furthermore in this isolated agents case we determine the asymptotic behavior of the minimal error frequency in the limit as l and then r tend to infinity

on the capacity of the discrete memoryless broadcast channel with feedback a coding scheme for the discrete memoryless broadcast channel with noiseless noisy generalized feedback is proposed and the associated achievable region derived the scheme is based on a block markov strategy combining the marton scheme and a lossy version of the gray wyner scheme with side information in each block the transmitter sends fresh data and update information that allows the receivers to improve the channel outputs observed in the previous block for a generalization of dueck s broadcast channel our scheme achieves the noiseless feedback capacity which is strictly larger than the no feedback capacity for a generalization of blackwell s channel and when the feedback is noiseless our new scheme achieves rate points that are outside the no feedback capacity region it follows by a simple continuity argument that for both these channels and when the feedback noise is sufficiently low our scheme improves on the no feedback capacity even when the feedback is noisy