measuring the flow velocity of liquid metals is a challenging task because of their opacity , chemical reactivity and - in most cases - elevated ambient temperature .fortunately , the high electrical conductivity of liquid metals often allows to use magnetic inductive measurement techniques .these techniques generally rely on applying magnetic fields to the fluid and measuring appropriate features , e.g. amplitudes , phases , or forces , of the flow induced magnetic fields .a local embodiment of this technique is the eddy current flow meter ( ecfm ) as patented by lehde and lang in 1948 , which consists of two primary coils excited by an ac generator , and one secondary coil located midway between them .modifications of this method , using one primary coil and two secondary coils , were described in .another version of this local sensor , which measures the flow induced change of the amplitude in the vicinity of a small permanent magnet , is the magnetic - distortion probe described by miralles et al . .a global embodiment of the same principle , the contactless inductive flow tomography ( cift ) , is able to reconstruct entire two or three - dimensional flow fields from induced field amplitudes that are measured at many position around the fluid when it is exposed to one or a few external magnetic fields .another inductive measurement concept relies on the determination of magnetic phase shifts due to the flow .further , the lorentz force velocimetry ( lfv ) determines the force acting on a permanent magnet close to the flow , which results as a direct consequence of newton s third law applied to the braking force acting by the magnet on the flow . with this technique , it is even possible to measure velocities of fluids with remarkably low conductivities , such as salt water .a common drawback of ( nearly ) all those methods is that they require extensive calibration since the flow induced magnetic field perturbations depend both on geometric details of the measuring system and on the conductivity of the fluid , which is , in turn , temperature - dependent . actually , the signals are proportional to the magnetic reynolds number , where is the magnetic permeability constant , the conductivity of the liquid , and and denote typical velocity and length scales of the relevant fluid volume .further to this , the use of permanent magnets , as necessary for the magnetic distortion probe and for lfv , or of magnetic yoke materials , as for the phase - shift method , set serious limitations to the ambient temperature at the position of the respective sensors .transient eddy current flow metering ( tecfm ) aims at overcoming both drawbacks . building upon earlier work of zheigur and sermons , this is accomplished by impressing a traceable eddy current system into the liquid metal and detecting its movement with appropriately positioned magnetic field sensors . since the eddy current moves with the velocity of the liquid, there is no need for a calibration of the sensor .the non - invasive tecfm sensor for measuring the liquid metal velocity close to the fluid boundary from outside , as described in , represents a specific external realization of tecfm .  here, we present a modified variant of tecfm , an invasive sensor that can be placed within a liquid metal pool or a pipe to measure the local velocity in the surrounding metal . after describing the main functioning principle of this _ immersed transient eddy current flow metering _ ( itecfm ) , we will illustrate the method by numerical simulations .then , first flow measurements in the eutectic alloy gainsn will be presented .the paper closes with some conclusions and a discussion of the prospects to use the method under high temperature conditions .itecfm is intended to measure the local velocity or the flow rate around the sensor in liquid metal pools or large pipes ( for small pipes there will be some distortion of the results , when the penetration depth of the magnetic field into the liquid metal is larger than the radius of the pipe ) . basically , the itecfm sensor is an invasive tube - shape sensor which is put inside the liquid metal , parallel to the flow direction .there is no direct contact between the liquid metal and the pick - up coils because the latter are protected by a cladding , made of stainless steel for example .in contrast to the external variant of tecfm , this configuration traces the zero crossing of the magnetic field of the eddy current system instead of the position of a magnetic pole . for this purpose ,the coils are arranged differently in order to allow a velocity measurement of the surrounding liquid .   and are displayed on the right hand side.,width=529 ]  the eddy currents within the liquid metal , which are to be used for inferring the fluid velocity , are induced by the excitation coils e1 and e2 ( see figure [ fig : image1 ] ) .figure [ fig : image1a ] shows a simplified scheme of these eddy currents . both magnetic fields and are generated by current steps which occur at the same time , but in opposite directions .the result are two oppositely directed magnetic fields with the same amplitude , which will induce opposing eddy currents during switching on or off of the excitation currents .because of the symmetric arrangement of the coils ( and , therefore , the magnetic fields ) , the zero crossing of the total magnetic field is located exactly in the middle between the receiver coils r1 and r2 , when is zero and/or immediately after the current step for .the excitation currents are assumed to be switched off at ., when the excitation currents are switched off at ( left ) and for ( right ) when the eddy currents have started to move in flow direction and begin to dissipate.,width=529 ]  although the eddy currents ec1 , ec2 and their magnetic fields , are dissipating , for the zero crossing remains exactly in their middle , regardless of their magnitude .this changes , however , if the fluid is moving .then , the eddy currents are transported in flow direction with the velocity of the fluid . under the reasonable assumption that the electrical conductivity of the liquid metal is homogeneous around the sensor, both eddy currents will dissipate with the same rate and the zero crossing of the magnetic field will also move with the fluid velocity .the position of the zero crossing can be tracked by means of the receiver coils r1 and r2 .  just as in ,the position of the zero crossing can be calculated according to where and are the positions of the receiver coils r1 and r2 , and and are the respective voltages measured there .although the arrangement of the coils for the external tecfm is different , this simplified formula can be used to approximate the liquid metal velocity in the case of itecfm , too .this will be validated by numerical simulations in the next section .the simulations of itecfm were implemented in comsol multiphysics 5.0 , using a time dependent 2d axisymmetric model and the magnetic fields ( mf ) physics environment .   for the simulation , some simplifications have been made .the flow velocity of the liquid metal is assumed constant and homogeneous around the sensor thimble .the liquid metal does not contain foreign particles or gas bubbles .furthermore , any lorentz forces exerted by the excitation coils on the liquid metal are neglected .  for an optimal operation of the sensor ,the receiver coils should be arranged symmetrically , with the centre of symmetry exactly in the middle between the two excitation coils ( see dashed horizontal line in figure [ p : simmodel ] ) .reasonable positions of the receiver and excitation coils have been determined by multiple simulations with variations in arrangement , size and spacing between the coils . in principle, there are two possibilities for the arrangement of the coils : the receiver coils can be placed between the excitation coils or vice versa . however , since the initial position of the zero crossing of the magnetic field is located exactly in the middle between the excitation coils , the receiver coils should be placed as close as possible to this point in order to achieve maximum sensitivity of the sensor . placing the excitation coils between the receiver coils is also possible but would result in a much lower signal amplitude and sensitivity because of the increased distance from .indicates the flow direction of the liquid metal , the dashed vertical line represents the symmetry axis for the 2d axisymmetric model , the dashed horizontal line shows the position of the centre of symmetry regarding the coils ( mm , mm , mm , mm ) .b ) excitation current pulses for the top and bottom excitation coil , width=415 ]  the size of the excitation coils turn out to have only a minor influence on the functionality of the sensor , as long as the absolute values of the excitation current pulses are the same , since is always in the middle between them .their actual size was chosen to accommodate a reasonable number of turns for the coil wires in the actual prototype .however , the axial extension of the receiver coils should be as small as possible because this will increase their sensitivity for detecting the zero crossing of the magnetic field , and also minimize the dependence on the conductivity of the liquid .further to this , since the distance between the coils and the boundary of the liquid metal has a significant influence on the signal strength , the air gap between coils and inner wall of the sensor , as well as the wall thickness of the sensor thimble , should be as small as possible .the actual sizes , turn numbers , and wire thicknesses of the coils for a low temperature ( lt ) and a high temperature ( ht ) prototype of the itecfm sensor are shown in table 1 ( see also figure [ p : simmodel ] a ) . a photography of the high temperature sensor is shown in figure [ fig : foto ] ..properties of the low temperature(lt ) and high temperature(ht ) prototypes [ cols="<,^,^",options="header " , ]   c.,width=340 ]   the edge steepness of the excitation current plays an important role for itecfm because the fluid velocity can only be extracted from the magnetic fields when the excitation currents have reached their final value , i.e. . from this instant on , r1 and r2 detect exclusively the magnetic fields of the eddy currents within the liquid .  in figure[ fig : image2 ] we see how the time derivatives of the fields and at the two receiver coils r1 and r2 change with increasing fluid velocity .for this example , the three top curves show the values for and the three bottom curves the values for . while they are symmetrical for , there is a growing asymmetry between and for increasing .   and as a function of time , computed for three fluid velocities and the conductivity of gainsn / m .the excitation current is supposed to be switched off at and to reach zero at .,width=340 ]  when plotting on a line parallel to the flow direction at different instants in time after the current steps , the movement of the zero crossing can clearly be seen ( figure [ fig : image8 ] ) . although the magnetic field is significantly dissipating with time, the zero crossing of keeps moving with .this is shown in the right panel of figure [ fig : image8 ] , where marks the time - dependent position of the zero crossing .m / s in a range of mm around the centre of the sensor ( left ) , and a zoom to mm ( right ) after 0.1 ms , 0.2 ms and 0.3 ms . mm is located exactly in the middle between the coils r1 and r2 . shows the time - dependent position of the zero crossing of .,width=529 ]  figure [ fig : image9 ] shows the movement of over time .the fluid velocity is then inferred from the slope of the respective line in the data set . since , in this particular simulation , the excitation currents reach zero only at , before that instant appears to move slower than the fluid velocity .   for three different velocities .the current steps reach their final value after .,width=340 ]  the reason for this effect is the superimposition with the remaining magnetic fields of the excitation coils . for laboratory experimentsit is therefore advisable to use a current source with a high edge steepness .otherwise and would have already dissipated too much for an accurate measurement .while we have considered the case of switching off the excitation current , almost the same results are obtained for switching on the currents .although the magnetic fields of the excitation coils are not zero , they are constant after they reach their final value and would not induce currents within the excitation coils or the liquid metal .yet , the difference between both methods would become larger for increasing .until now , the simulations have been calculated only for one electrical conductivity of the liquid metal ( / m for gainsn ) . with view on the strong influence of the ( temperature dependent ) conductivity for conventional ecfm s , we present in figure [ fig : image10 ] the calculated velocity for different conductivities and a variety of hypothetical ( and real ) coil geometries .m / s when switching off ( on ) the excitation current.,width=529 ]  in general , despite some slight dependence on the electrical conductivity of the fluid , the deviation from the ideal velocity is relatively weak for a wide range of conductivities . hence , itecfm can essentially be considered as calibration - free . in the case of extremely thin receiver coils ( a ) , the overall deviation would remain less than 5% for / m / m . for the standard geometry ( as embodied in the prototypes for which the coil size and distance between the coils have to be larger in order to accommodate a suitable number of turns and to facilitate the construction of the core )it can be seen ( b and c ) that the results for switching on or switching off the excitation currents are almost the same .the further simulation ( c , d , e ) show also that the deviation from the ideal velocity is smaller , when the coils are positioned as close to each other as possible , especially at high conductivities . at low conductivities of the liquid metal ,the positioning of the coils has only a small influence on the results .the increasing deviations for higher fluid conductivities are related to the size of and the distance between the coils as well as the dissipation of the eddy currents .m / s when switching off the excitation current .the coils are arranged like in the standard case b).,width=377 ]  as can be seen in figure [ corewall ] , the electrical conductivity of the sensor core , which holds the coils , and of the sensor thimble have a significant influence on the measurement results for the velocity , especially at low of the liquid metal .the eddy currents which are induced within the conductive components of the sensor , are stronger for higher electrical conductivities . unlike the eddy currents within the liquid metal, they are not moving with but are stationary at all times .because the magnetic fields of the eddy currents from liquid metal are superimposed with the fields of the sensor components , the measured velocity appears to be lower .this effect is stronger at low of the liquid metal because the eddy currents within the sensor components are in the same order of magnitude or even larger than the ones in the liquid metal .another aspect to consider is the volume of the respective sensor components .because the core has a considerable larger volume than the thimble wall , its conductivity has a larger influence on the velocity measurement . at higher of the liquid metalthe effect gets more and more negligible because of the stronger eddy currents and the larger volume of the liquid metal .a first test of a itecfm sensor was carried out with the low - temperature prototype in a liquid metal loop with the eutectic alloy gainsn .this sensor has a plastic coil holder , the excitation coils have 100 turns , the receiver coils have 120 turns , and conventional copper wire of diameter 0.25 mm was used ( see table 1 ) .rectangular voltage pulses of with a frequency of 1 khz , a duty cycle of 50 % and a fall time of have been used to generate the excitation currents .the sensor was put inside a stainless steel tube to prevent direct contact with the liquid metal .the receiver voltages were measured with a memory oscilloscope and was calculated with equation ( [ f1 ] ) .figure [ fig : image3 ] shows the measurement results for four different fluid velocities and a linear fit of each dataset .the displayed results represent the mean value of 2500 measurement sweeps with one measurement taken every millisecond for . as can be seen in the previous section in figure [ fig : image9 ] ,the results for are expected to have a linear rise .for itecfm in a pipe flow of gainsn .the fluid velocity is inferred from the slope of the respective linear fit for each pre - adjusted velocity .the inferred velocity is indicated above each curve.,width=415 ]  there are some deviations from the expected results for , especially the disturbances around and .the overall slope of the linear fit however , is very close to the pre - adjusted flow velocity in the gainsn - loop ( which is , as a matter of fact , also not exactly known ) .the disturbances appear at the same times for each measurement and are most likely caused by the resonant frequency of the receiver coils .future experiments using tailored current sources instead of the presently used voltage source are expected to improve this situation .in this paper , we have presented the principle of itecfm and some promising results obtained both in simulations as well as in an experiment with a first prototype in gainsn . while its calibration - free character makes the method a promising candidate for a number of laboratory and industrial applications , it certainly needs further tests and optimization .although both the external and immersed configurations of tecfm are based on the same principle , there are some differences with view on the different arrangement of the excitation and receiver coils , which have to be addressed in detail .future work will be devoted to more experiments with optimized excitation schemes and different liquid metals to validate the simulation results and the calibration free - free character of the sensor . another advantage of itecfm is the avoidance of any magnetic materials which makes it particularly suited for high temperature applications .tests with the high temperature prototype consisting of heat resistant materials are planned for ambient temperatures of up to 650  as they are typical , e.g. , for sodium fast reactors .this work was supported by cea in the framework of the ardeco programme .20     poornapushpakala s , gomathy c , sylvia ji , babu b 2014 design , development and performance testing of fast response electronics for eddy current flow meter in monitoring sodium flow _ flow meas .instrum . _* 38 * ( 2014)98 - 107

Abstract: eddy current flow meters ( ecfm ) are widely used for measuring the flow velocity of electrically conducting fluids . since the flow induced perturbations of a magnetic field depend both on the geometry and the conductivity of the fluid , extensive calibration is needed to get accurate results . transient eddy current flow metering ( tecfm ) has been developed to overcome this problem . it relies on tracking the position of an impressed eddy current system which is moving with the same velocity as the conductive fluid . we present an immersed version of this measurement technique and demonstrate its viability by numerical simulations and a first experimental validation .   _ keywords _ : flow measurement , inductive methods , calibration - free +


in entanglement - based protocols for quantum key distribution ( qkd ) two honest parties ( alice and bob ) can obtain a secure secret key by performing measurements on shared epr pairs .they can also certify that they have epr pairs by observing sufficiently strong violations of bell inequalities .when the epr pairs are noisy , measurements lead to noisy and partially secret correlations . in order to obtain perfect secret bits , error correction and privacy amplification have to be performed , with the assistance of local operations and public communication ( lopc ) . before implementing this procedure , however , an estimate of the quality of the correlations needs to be performed . formulated in a different way ,an estimate of the maximal amount of information that an eavesdropper ( eve ) has about alice s and bob s bits has to be performed .this is done by exploiting the monogamy of entanglement , which imposes trade - offs between the entanglement between alice and bob , and eve s correlations with them .  a way of estimating the degree of entanglement that alice and bob share is to perform quantum tomography .in order to do so , they have to assume that the quantum systems they measure live on a state space of a particular dimension ( usually two ) .this assumption , though strong , is usually not mentioned in the presentations of qkd . in particular, it implies that alice and bob must trust their apparatuses ( see for a detailed discussion ) .  a framework in which one can analyze quantum correlations without knowledge of the dimension is to consider them in the larger set of no - signaling correlations .no - signaling correlations are characterized by the assumption that _ no measuring process can be used to send information between distant locations_. in this framework , the origin of the correlations , the kind of system that has been measured , and in particular , the dimension of the underlying quantum system , do not matter .it is shown in that , if the obtained correlations violate some bell inequality then there is some degree of privacy in them in the sense that secret key is needed to create these correlations by lopc .the first protocol proved secure against a no - signaling eavesdropper is the bhk - protocol , introduced in .however , the security analysis provided was limited , it only applies to the noiseless regime and has a vanishing secret key rate . in was shown that the bhk - protocol with a positive key rate is secure against individual attacks , even in the noisy regime . in the present paperwe generalize this result to completely general attacks .the security definition that we use is the strongest one , the so called _ universally - composable security_. one calls a cryptographic primitive ( for instance key distribution ) universally composable if it is secure in any arbitrary context ( for instance one - time pad encryption ) .the secret key rate that we obtain is comparable to the one obtained when the eavesdropper is constrained by quantum mechanics , and when the devices are fully specified and trusted .  in order to do so, we introduce an exponentially - accurate scheme for estimating symmetric properties of arbitrary multipartite probability distributions . also , we prove the security of privacy amplification in a similar way as in .our proof has the advantage that can accommodate any error - correction scheme .the paper is structured in the following way . in sectionii we introduce some preliminaries : nonsignaling correlations , nonlocality , and their relation to privacy . in section iiiwe describe the protocol , and explain how to implement it with quantum devices . in sectioniv we explain the security criterion . in sectionv we compute the secret key rate in a practical scenario . in sectionvi we provide the complete security proof , distributed in several subsections .section vii contains the conclusions .we use upper - case to denote the random variable whose particular outcome is the corresponding lower - case .we use bold letters to denote strings of variables or random variables .alice and bob share pairs of physical systems , labeled by .alice measures her system with one of the observables , obtaining the outcome .analogously , bob measures his system with one of the observables and obtains the outcome .the chosen observables and their corresponding outcomes for the pairs of systems are represented by the random variables , , , , which are correlated according to the joint conditional probability distribution .the number is the probability of obtaining the strings of outcomes when measuring and .the only assumption about this distribution is the following .  * the no - signaling assumption : * _ the choice of observable for one system can not modify the marginal distribution for the rest of systems ._  more formally , we impose the following condition among any two sets of subsystems with input , and output , : for all .although the two sets of subsystems are arbitrary , the above constrains turn out to be equivalent to the ones where corresponds to a single subsystem or .it is important to note that if these equalities were not satisfied , arbitrarily fast signaling between separated subsystems could be achieved .also , if not for this assumption , the notion of subsystem would have no sense .general properties for nonsignaling correlations are shown in .  in the cryptographic scenarioone assumes that the only information accessible to eve ( apart from the public messages exchanged by alice and bob ) is the outcome obtained when measuring a physical system with an observable . without loss of generalitywe assume that eve has only one system .the sole assumption that we use in the security proof is that the global -partite distribution is a nonsignaling one .apart from this , this distribution is completely arbitrary .it is important to stress that systems inside alice s laboratory must not signal each other , and the same for bob . this may be quite difficult to implement in practice , but it is , in principle , possible .a bipartite distribution is said to be local if it can be written as local distributions can be generated by shared randomness ( denoted above ) between the parties , plus local operations .a distribution which can not be written as ( [ local co ] ) is said to be nonlocal .  by definition ,bell inequalities are satisfied by all local distributions ( [ local co ] ) . in this paper, we concentrate on the braunstein - caves ( bc ) bell inequality . for any distribution with and ,let be uniform on the set and define the random variable = \frac{1}{2}+ m\big ( a\oplus b \oplus    \mathrm{i}\{x = m-1\ } \mathrm{i}\{y=0\ } \big)\ ] ] where the indicator function is defined as , .the bc - inequality for observables can be written as the bipartite distribution can be generated between two noncommunicating parties in the following way : ( i ) are independently generated with uniform distribution over , ( ii ) after the measurements , once communication is allowed , the two parties post - select the evens where is in the set . as mentioned above , any local distribution ( [ local co ] ) satisfies ( [ bc inequality ] ) .the bc - inequality for is equivalent to the chsh - inequality  where here , the random variables are independent and uniform on .suppose that eve is correlated to alice through the global distribution .if alice measures , we can quantify the knowledge that eve has about with the ( optimal ) correct - guessing probability if then eve knows with certainty . if then eve is completely ignorant about the value of . in this paperit is shown that the knowledge that eve has about can be upper - bounded by the amount of nonlocality that alice and bob share if the marginal for the honest parties violates the bc - inequality ( [ bc inequality ] ) , then according to ( [ aaa ] ) , the probability that eve guesses correctly is smaller than one .this is the reason why the bell inequality ( [ bc inequality ] ) is unconventionally written as a lower bound : the more nonlocality the honest parties share , the lower is , and the less knowledge eve has ( [ aaa ] ) .this is one manifestation of the monogamy of nonlocal correlations .here we explain how to implement the protocol with quantum - mechanical devices .this is not necessary for defining the protocol , or prove its security .however it helps to understand the reasons behind its particular design .suppose alice and bob share many copies of the noisy epr state where is the purity , is the projector onto , and the four - dimensional identity matrix .they perform the measurements in the following orthogonal basis .the observable for alice is the observable for bob is and the observable for bob is the same as alice s . in the bloch sphere ,these observables correspond to the directions represented in fig . 1 ..,title="fig:",width=226 ] +  the observables are the ones used to obtain large violations of the bc - inequality . for ,the settings ( [ a basis ] , [ b basis ] ) are the ones which maximize the violation of the chsh - inequality for the state ( [ noise ] ) .the observables maximize the correlation between alice and bob , and hence , are used to generate the raw key .   recall that for each value of we have a different protocol .* 1 . distribution and measurements . * alice and bob are given pairs of systems .alice generates the random bits independently and with identical distribution : , for a small .analogously , bob generates the random bits independently and with identical distribution .pairs such that are used to generate the raw key , and pairs such that are used to estimate how much nonlocality alice and bob share .for each , if alice measures her system with , if she measures it with chosen uniformly on , if bob measures his system with , if he measures it with chosen uniformly on .estimation of nonlocality .* they publish and for the pairs such that they publish the outcomes . the subset of those pairs such that is denoted by . with those pairsthey compute the average value for the bc - inequality where  ] and is defined by ( [ var b ] ) .  _proof : _ using the no - signaling condition we can write let us show that for any and any .first , expand each side of this inequality according to definitions and ; second , note that ; and finally , use this to show that each term in the left is component - wise bounded by the corresponding term in the right . let us show for the case . in the following chain of equalities and inequalities we use , respectively: the definition of in ( [ pmax ] ) ; lemma [ lemma 1 ] ; inequality and positivity of the vectors ; the linearity of the scalar product ; decomposition ( [ eq : marginal ] ) ; and the identity . in order to extend this inequality to all values of , consider the relabeling . for any  relabeling corresponds to a permutation of the entries of the vectors such that this relabeling leaves the vector invariant .hence , performing the relabeling to each pair with , the above inequality for is generalized to any value of .   this privacy amplification scheme is similar to the one introduced in .it has the advantage that one can hash out any information about the raw key , that is , the function is arbitrary .contrary , the scheme introduced in only works when the function is generic .our privacy amplification scheme has the disadvantage that it needs a random hash function , in particular a two - universal one , while the one in works with a deterministic hash function .[ 2universal ] a random function is called two - universal if for any pair such that we have   [ gros ] if is a two - universal random function , then for any subset we have where runs over .  _proof _ in what follows we take the square of the left - hand side of ( [ gros f ] ) ; use the convexity of the square function ; sum over ; partially sum over ; use the two - universality of ; and a trivial bound .   [ lluis c2 ] let be a -partite nonsignaling distribution , let where is a given function , and let where a two - universal random function , then where $ ] and is defined in ( [ var b ] ) ._ proof _ for any subset we have the following chain of component - wise inequalities . in the first step we use the expansion and the component - wise triangular inequality . in the second step we use the following triangular inequality for any  lemma [ gros ] , and the concavity of the square - root function for the last inequality all termsare summed up by using .  in the rest of this proofthe following notation is used .we denote by the vector with entries for all values of and fixed values of . following this notationwe can write . for any subsets and any set of coefficients we have the following chain of equalities and inequalities , where we have respectively used : lemma [ lemma 1 ] , the cauchy - schwarz inequality , the linearity of the scalar product , and the definition of the conditional distribution. the following establishes ( [ ll ] ) . in the above we have respectively used : the definition of conditional distribution ; equality ; inequality ( [ mezzanine ] ) with ; the component - wise inequality ( [ ffinal ] ) together with the fact that the components of the vector are positive ; and the last inequality follows from ( [ concavity ] ) and .   according to the previous theorem , the security of the secret key can be bounded in terms of the quantity , which does not depend on at all !this is a particular manifestation of the monogamy of nonlocal correlations . in the unconditional - security scenario alice and bobdo not know the distribution , hence , how can they estimate ?the only thing they know is the estimated information , defined in ( [ est nonlocality ] ) .the following result establishes the security of the secret key in terms of .[ uc pa ] let be a -partite nonsignaling distribution whose marginal is symmetric with respect to the 4-component variables .suppose the first systems of alice are measured with , obtaining the outcomes .suppose the last pairs are measured with chosen uniformly on , and let    \ .\ ] ] let where is a given function , and where is a two - universal random function with output size the inequality holds with probability larger than ._ proof _ applying lemma [ clau ] to and we conclude that holds with probability larger than } \\ & \geq &     1- 3n \exp{\![-(3m)^{-2 } \sqrt{n\e } ] } \ . \end{aligned}\ ] ] for the last , note that the maximum value the variable can achieve is . using theorem [ lluis c2 ] , inequality ( [ bon ] ) , and the assignation ( [ assign ns uc ] ) we obtain ^{n\k }   \\ & \leq & \sqrt{2}^{-\sqrt{n\e}}\ , \end{aligned}\ ] ] which concludes the security proof .  now , a few comments are in order .  1 .note that the distribution considered in theorem [ uc pa ] does not represent all pairs of systems that alice and bob share at the beginning of the protocol .it does not include the pairs such that but do not satisfy condition ( [ condition ] ) .however , this is irrelevant in establishing the security of the secret key ( see comments below ) .there is no reason to believe that the honest parties marginal distribution is symmetric .however , it is measured and processed in a completely symmetric way . for example, the pairs used in the estimation of are chosen at random .this is equivalent to the situation considered in theorem [ uc pa ] , where the distribution is assumed to be symmetric and the pairs used in the estimation constitute a fixed subset .3 .  theorem [ uc pa ] limits the knowledge that eve has about the secret key , even if she hears the messages published in the error correction step .however , the messages published in the estimation of , denoted by in ( [ uc ] ) , are not considered .the information is not a function of , is generated by measuring other systems .therefore , we can consider those systems ( the ones used in the estimation ) , as well as the rest of the universe , as part of eve s power .summarizing , the situation considered in theorem [ uc pa ] is as complete as required in ( [ uc ] ) .we show that it is possible to generate secret key from correlations that violate the braunstein - caves inequality by a sufficient amount .we prove this according to the strongest notion of security , the so - called universally - composable security .the only assumption used in the security proof is the impossibility of arbitrarily - fast signaling between subsystems by performing local measurements .we introduce an exponentially - accurate scheme for estimating symmetric properties of general distributions .this allows alice and bob to treat any unknown given correlations as if they where generated by independent and identically - distributed samples .this can be useful in order to quantify bell - inequality violations without the i.i.d .assumption .our approach to qkd goes beyond the philosophy of in which there is still quantum mechanics , in particular , the validity of tsirelson s bound is assumed .in contrast , our approach is conceptually simpler in that _ all _ we assume is no - signaling .it is remarkable that , although our security is based on weaker assumptions , the secret key rates that we obtain are comparable to the ones where the adversary s attack is constrained by no - signaling plus quantum mechanics . in particular , we obtain the optimal rate of one secret bit per singlet consumed .our results also contribute to the understanding of quantum cryptography where the honest users do not have a complete control of their quantum apparatuses , or distrust them .qkd is a present - day technology .entanglement - based protocols are usually implemented with a source that sequentially sends entangled pairs of systems to alice and bob .each pair is measured in alice and bob s locations with the same two apparatuses .those measuring apparatuses could generate outcomes depending on previous inputs .if this is the case , our assumptions for the security proof do not hold , because there is signaling between measuring events within the same lab. it would be desirable to have a security proof which accommodates this situation .therefore , an important open problem is to obtain a security proof from weaker no - signaling assumptions .llm is supported by caixa manresa , the spanish mec projects ( fis2008 - 00784 toqata " , fis2007 - 60182 , consolider ingenio 2010 qoit " ) , and the eu - ip programme scala " .rr is supported by the swiss national science foundation , grant no .200021 - 119868 .mc is supported by the excellence network of bavaria ( tmp , qccc ) and the dfg grants ch 843/1 - 1 and ch 843/2 - 1 .aw is supported by the u.k .epsrc through the qip irc " and an advanced fellowship , by a royal society wolfson merit award , a philip leverhulme prize , by the european commission through ip qap " , and by the singapore ministry of education .jb is supported by an epsrc career acceleration fellowship .m. ben - or , m. horodecki , d. w. leung , d. mayers , j. oppenheim ; theory of cryptography : second theory of cryptography conference , tcc 2005 , j.kilian ( ed . ) springer verlag 2005 , vol .3378 of lecture notes in computer science , pp .386 - 406 .j. l. carter , m. n. wegman ; journal of computer and system sciences , 18:143 - 154 , 1979 . m. n. wegman , j. l. carter ; journal of computer and system sciences , 22:265279 , 1981 . b. s. tsirelson ; hadronic j. suppl .* 8 * , 329 ( 1993 ) .

Abstract: we analyze a protocol which generates secret key from correlations that violate a bell inequality by a sufficient amount , and prove its security against eavesdroppers which are only constrained by the fact that any information accessible to them must be compatible with the impossibility of arbitrarily fast signaling . we prove unconditional security according to the strongest notion , the so called universally - composable security . the no - signaling assumption is imposed at the level of the outcome probabilities given the choice of the observable , therefore , the protocol remains secure in situations where the honest parties do not have a complete control over their quantum apparatuses , or distrust them . the techniques developed are very general and can be applied to other bell inequality - based protocols . in particular , we provide a scheme for estimating bell - inequality violations when the samples are not independent and identically distributed .


boolean networks ( bns ) are dynamical models originally introduced by s. kauffman in the late 60s . since kauffman s seminal work , they have been used as abstract modeling schemes in many different fields , including cell differentiation , immune response , evolution , and gene - regulatory networks ( for an introductory review see and references therein ) . in recent days , bns have received a renewed attention as a powerful scheme of data analysis and modeling of high - throughput genomics and proteomics experiments .  the bottom - line of previous research has been the description and classification of different attractor types present in bns under deterministic parallel update dynamics .special attention was dedicated to so - called critical bns situated at the transition between ordered and chaotic regimes .recently , a new point of view has been introduced that casts the original dynamical problem into a _ constraint satisfaction problem _ , that can be studied with theoretical and algorithmic tools inspired from the statistical mechanics of disordered systems .  following this new approach presented in it is possible to study the organization of fixed points in the thermodynamic limit in random boolean networks .this leads to the identification of the sudden emergence of a computational core , whose existence is a necessary ( but not sufficient ) condition for a globally complex phase where all fixed points are organized in an exponential number of macroscopically separated clusters .this phenomenon is found to be robust with respect to the choice of the boolean functions , and missing only in networks where all boolean functions are of and or or type .in addition , the size of the complex regulatory phase is found to grow with the number of inputs to the boolean functions .the main motivation for this work is to check the robustness of the above - reported results in networks of finite size , both from the theoretical and algorithmic point of view .we aim at understanding how reliable the predictions of message passing algorithms such as belief propagation ( bp ) are on graphs of finite - size .we performed an extensive numerical check on the number of solutions in small samples , comparing bp predictions to exact exhaustive enumeration algorithms results .results of bp on directed bns are found to be in very good agreement with exact ones when counting the expected number of fixed points . a worse performance of bp is found in its prediction of local quantities such as the single nodes marginal probabilities .nevertheless , the performance seems to increase with system sizes for already moderate size values .  inwe define the model , in we describe the belief propagation algorithm and the strategy we adopted to obtain exhaustively all solutions of our bn s model . inwe quantify numerically the agreement between bp and the exhaustive search results .furthermore , a brief analysis on the distribution of the overlap of the solutions is presented .finally in we present the analytical computation of the first and second moment of the average number of solutions of our model of bns .numerical solutions of the bp equations strongly suggest that the annealed calculation of the entropy ( logarithm of the number of bn fixed points ) is exact in the large size random case .fluctuations over the annealed value are computed numerically and successfully confronted with analytical estimates of the second moment of the number of fixed points .the large size limiting value of the fluctuations around the annealed result is also computed , and some considerations are discussed .in the following we will consider a bn of interacting variables with .in general , each variable can be regulated by other _ parent variables _ , and can enter in the regulation of an arbitrary number of _ child variables_. we then consider boolean functions , represented by with , depending on inputs and having a single output .the truth value of a given output variable is then fixed by the truth values of the regulating variables via the relation : with running over all regulated genes . as shown in fig .[ fig : network ] , not all variables need to be controlled by a boolean function , _i.e. _ in general we have with .are external inputs ( non - regulated variables ) . ][ fig : network ]  on the other hand , each regulated variable is the output of one and only one function .the whole set of boolean constraints completely specifies the network topology .we aim at computing , _i.e. _ the number of stationary points of the network .we introduce a hamiltonian , that it is equal to the number of unsatisfied boolean constraints : where the symbol stands for the logical operation xor .we will consider _ random factor graphs _ characterized ( among all possible random graphs ) by :  * function nodes have fixed in - degree and out - degree one . *variables have in - degree at most one .this means that all regulating variables are collected in one single constraint ( see eq .( [ eq : function ] ) ) .  setting , the degree distribution of variable nodes approaches asymptotically   we now have to specify the functions in the factor nodes present on the random factor graph defined so far , i.e we have to specify not only the topology of the graph , but also the content .there are possible boolean functions with 2 inputs and 1 output .following , we group them into 4 classes :  * the two constant functions equal to 0 and 1 . *four functions depending only on one of the two inputs ,_ i.e. _ . * and - or class: there are eight functions , which are given by the logical and or or of the two input variables , or of their negations .these functions are _canalizing_. if , e.g. , in the case the value of is set to zero , the output is fixed to zero independently of the value of .it is said that is a _ canalizing variable _ of with the _ canalizing value _ zero . * xor class : the last two functions are the xor of the two inputs , and its negation .these two functions are not canalizing , whatever input is changed , the output changes too .  for the sake of clarity we concentrate only on classes depending effectively on two inputs , _i.e. _ on those in the and - or class and the xor class .as we will see in the following sections , the average number of fixed points does not depend on the relative appearance of the functions within each class , but only on the relative appearance of the classes .we therefore require functions to be in the xor class , and the remaining functions to be of the and - or type , with being a free model parameter . in this simple casethe network ensemble is then completely defined by and .belief propagation is a marginalization algorithm used with success in many different fields .it is exact on a tree , _i.e. _ on a graph with no loops , but it is also known to give good estimates of the marginals in other cases .  in this section we will first introduce the bp algorithm , and then we will explain the strategy adopted for the exhaustive search .the bp algorithm is an iterative strategy for computing the fraction of zero - energy configurations having , say , .more generally , bp calculates marginal probability distributions of solutions of problems defined on factor graphs .a more detailed introduction on the bp equations in the case of bns can be found in .let us define one of the clauses where appears .one can introduce the following quantities :  * : the probability that variable takes value in the absence of clause .* : are a function proportional to the probability that clause is satisfied when variable takes value .the above - defined quantities satisfy the following set of equations : % gp f iisuppose  \prod_{b \in a \setminus i } \mu_{b\rightarrow i } ( x_i )  \nonumber\\ \mu_{i\rightarrow a}(x_i ) & = & c_{i\rightarrow a } \prod _ { b\in i  \setminus a } m_{b\rightarrow i}(x_i)\,\,\ , , \end{aligned}\ ] ] where represents the set of all variable indexes belonging to function _ except _ variable index , the set of all function indexes to which variable belongs , _ except _ index , and is a constant enforcing the normalization of the probability distribution .this set of equations can be numerically iterated up to their fixed point , if present .it turns out that , in the case of our model , the iterative procedure always converges .marginals can be computed in terms of the messages in eq .( [ eq : bp ] ) via the following set of relations : \prod_{i \in a }  \mu_{i\rightarrow a}(x_i)\nonumber \\p_i ( x_i )  & = & c_i \prod_{a\in i }  m_{a\rightarrow i}\,\,\ , , \end{aligned}\ ] ] where and are again normalization constants . from the marginals we can eventually compute the entropy , _i.e. _ the logarithm of the number ( ) of zero energy configurations :    no known polynomial algorithm is generally able to exhaustively find all the solutions of a boolean constraint satisfaction problem like this one .there is however a number of efficient implementations of exhaustive search strategies - still exponential in the running times - that allow to explore problems of reasonable size .  in our casewe have mapped the random bn instances onto _ conjunctive normal form _ ( cnf ) formulas .such instances are made of several clauses put in and disjunction , each clause being made of literals in or conjunction .this is the natural form of the well - known satisfiability problem .the mapping is done writing in the cnf formula all the configurations which violate a clause in the rbn instance , and negating the literals for the true variables . as an example, it is simple to verify that a and node involving e as : can be written in cnf in the following way :   once the problem is cast in this form , we can exploit the vast number of very well performing algorithms existing for solving satisfiability instances .we are interested now in _ exhaustive search _ programs , among which we choose ` relsat 2.00 ` .this award - winning program can perform a complete search of the solution space for instances up to variables ( in our model ) in accessible time using a common pc .in this section we will check the predictive precision of the bp algorithm measuring the entropy and the magnetization vector .   in has been pointed out using an heuristic argument that , for this model of random bns , the average number of solutions is always equal to _ i.e. _ to the number of all possible external input configurations ( note that is exactly the number of non - regulated sites ) . a direct numerical check of eq .( [ eq : bp ] ) on single samples shows that the bp predictions are always in agreement with the above - mentioned heuristic prediction , so that for any sample , the entropy density , independently from the sample realization and the percentage of xor functions , apart from terms that vanish when .samples of ( red boxes ) and ( blue boxes ) for four different choices of and expressed as a percentage .the black arrow is the numerical estimate using bp , that agrees perfectly with the theoretical value .,title="fig : " ]  samples of ( red boxes ) and ( blue boxes ) for four different choices of and expressed as a percentage .the black arrow is the numerical estimate using bp , that agrees perfectly with the theoretical value .,title="fig : " ]  samples of ( red boxes ) and ( blue boxes ) for four different choices of and expressed as a percentage . the black arrow is the numerical estimate using bp , that agrees perfectly with the theoretical value .,title="fig : " ]  samples of ( red boxes ) and ( blue boxes ) for four different choices of and expressed as a percentage .the black arrow is the numerical estimate using bp , that agrees perfectly with the theoretical value .,title="fig : " ]  in fig .( [ fig : entropy - n50 - 100 ] ) we display the frequency distribution of for samples at different values of and . comparing these histograms one can observe that  * the most probable value of depends strongly on and only mildly on . *the distributions at increasing seem to peak around the value .    compared withthe reference histogram of ( left panel ) . in the right panelwe display the histogram of both and .measurement are done at  and a percentage of xor functions of over an ensemble of samples.,title="fig : " ]  compared with the reference histogram of ( left panel ) . in the right panelwe display the histogram of both and .measurement are done at  and a percentage of xor functions of over an ensemble of samples.,title="fig : " ]  so far we have analyzed the behavior of the entropy alone .although the entropy is very well predicted by bp , this is not always the case of the single variables marginal probabilities these quantities are of extreme importance in any decimation procedure that is expected to find a specific fixed point via bp .necessary condition for any decimation procedure to be effective is , for each non pathological sample , a strong positive correlation between magnetization vectors and , -dimensional vectors whose -th component represents the magnetization of each variable defined as where . in order to assess this point ,we define two global parameters testing respectively :  * the probability distribution of the relative angular overlap of the two magnetization vectors in the dimensional space , and * the probability distribution of relative magnetization euclidean distances .  for each sample , the first overlap parameter is defined , _i.e _ as the cosine of the angle between the exhaustive and bp magnetization vectors : the second is defined as the euclidean distance between non normalized magnetizations : in both cases the predictions have been tested against a null hypothesis . in the null hypothesis ,random magnetization vectors are extracted in the following way : for each sample is a random permutation of the components of , so that where is a random permutation of the ordered set .the quantities and are then calculated substituting to in eqs.([overlap1 ] ) and ( [ overlap2 ] ) .distributions of the overlaps over the sample populations are plotted in .the results show a strong correlation between the true and the predicted magnetization vector distributions .it has been indicated in that in this model the entropy is analytic in all the phase diagram , while the organization of the fixed points undergoes a sudden reorganization at some :  * at all solutions are in a single cluster , _i.e. _ any pair of solution is connected by a path via other solutions , where in each step only a finite number of variables can be changed .* at the space of solutions spontaneously breaks into an exponential number of macroscopically separated clusters of fixed points .their number , or more precisely its normalized logarithm , is called complexity .it is a first - order phase transition .this behavior is characterized by the appearance of a non - trivial structure of the space of solutions . in other wordsthe fixed points , rather than being uniformly scattered over the -dimensional hypercube , start to organize themselves in clusters , with a well defined _intra - cluster _ and _ inter - cluster _ overlap . nevertheless , by means of the exhaustive enumeration technique introduced above, one can easily write down all the solutions for a given sample and hence calculate all the overlaps , defined in a standard way as .note that we are using spin variables here , and indicates two distinct solutions .the distribution of the overlaps for two sizes and three different choices of are shown in .we choose a value far apart form the clustering transition line ( no xor functions ) , a value in the vicinity of the transition line ( of xor functions ) and a value deep inside the clustered phase ( of xor functions ) .  . ( left panel ) , ( central panel ) , ( right panel ) , for ( thin line ) and ( thick line).,title="fig : " ] . ( left panel ) , ( central panel ) , ( right panel ) , for ( thin line ) and ( thick line).,title="fig : " ] . ( left panel ) , ( central panel ) , ( right panel ) , for ( thin line ) and ( thick line).,title="fig : " ]  it is clear that as one moves from the unclustered to the clustered phase ( which at fixed is equivalent to increasing ) , the distribution changes from a broad shape to a two - peaked one .however , this two - peaked pattern is just a finite size effect , as it seems to be indicated by the reduction of the weight of the right peak of the from to shown in the left panel of . as the number of clusters is exponential in the system size, the probability of extracting two random solutions from the same cluster is negligibly small even for moderate sizes .this phenomenon can be understood by the following qualitative argument : let us suppose that the number of clusters as well as the size of each single cluster are exponential in the system size .let us further suppose that the distribution of cluster sizes is strongly peaked around a single value , ( less stringent conditions can be found , together with pathological cases where those conditions do not hold , but a complete treatment would go beyond the scope of present work ) . in a completely clustered phase the weight contribution of the intra - cluster overlap will then be proportional to where is the number of clusters .on the other hand , the contribution to the inter - cluster overlap is proportional to ( _ i.e. _ all couple of solutions except those in the same clusters ) .if is exponential in , the intra - cluster contribution will be negligible with respect to the inter - cluster one in the thermodynamic limit .[ nsol_dist ]  in this section we will show that the logarithm of the number of solutions ( _ i.e. _ the entropy ) of our model of bns computed within the annealed approximation scheme agrees with the numerical estimate for the entropy we performed by exhaustive enumeration and by the numerical solution of the bp equations on single sample .  this feature allows us to make some conjectures , and check whether the annealed approximation is exact in the limit .we will show that the variance of as well as all higher moments are proportional to in the large limit , with a proportionality constant depending only on and .this implies that a non zero contribution to the entropy is given at the leading order only by the external regulating variables , in accordance with results obtained in .however , we will also show how , in the general case of , the proportionality constants are different from one and greater than it , not ensuring the exactness of the annealed result , albeit replica results given in are a strong hint in that direction .moreover , we will see that in the thermodynamic limit : where is a constant independent of . only in the case of pure xor classes ( ) , it is possible to show that the proportionality constant is exactly the large limit for any order moment and any value of , implying that the annealed approximation is exact .let s consider a distribution of positive random variables .as we are interested in the relation between the quenched entropy and the annealed approximation , we want to investigate the relation between and , where identifies here the average over the distribution of .we can write : where we have expressed the quenched entropy in term of the annealed one plus a series of the moments of the distribution . in our casewe take . in cases where the sum of the series of moments is finite or for any and as a function of the size of the graph such that , then the quenched and the annealed entropies coincide .we will show that this is the case at least for the second order moment .it is important to state , however , that the previous series expansion is valid only if , and that averaging the resulting series term by term is possible only if the sum can be taken out of the integral which defines the average .these conditions are not always met in our model , and in particular we expect the existence of a certain range of and values beyond which the conditions are not necessarily satisfied . in particular we will also show that in the thermodynamic limit however this will not necessarily undermine the validity of the annealed calculation .  under the hypothesis of equation ( [ diverge ] ) , andthanks to a straightforward implementation of the tschebichev inequality , one can easily show that , ( see [ app : cheby ] ) where is a positive constant .this implies that , in the thermodynamic limit , the probability distribution of the number of solutions has support in the interval . unfortunately in order to take under control also the left tail of the distribution one should compute moments of the type with , which is , as indicated in [ app : high_mom ] , a rather complicated task .given a probability distribution of classes of boolean functions , drawn independently , where is a generic boolean function of inputs , and given a value of  ] .the variables . averaging over the values of and is therefore equivalent of averaging over the . in particular , in the case of flat classes distribution , which is the object of the present work , one has .therefore : let us now define regulated variables and external inputs ( including the isolated nodes ) . assumingthe input variables are extracted randomly and independently on each of the clauses ; one can write ^{n^+_o } \nonumber \\   & & \left [ { \cal p}(++|-)g(++- ) +  { \cal p}(+-|-)g(+-- ) + \right.\nonumber \\ & & \left .{ \cal p}(-+|-)g(-+- ) + { \cal p}(--|-)g(--- )  \right]^{n_o - n^+_o }  \label{eeq2}\end{aligned}\ ] ] where is the probability of drawing two inputs of sign and given the fact that one is looking at a clause with output variable sign , and is the value of the function node times the probability of extracting a certain boolean function type . in the case of uniform , is trivially  signs triplet , leading to identically .  for a different distributionthis is in general _ not _ the case . asa title of example , in the case of a mixture of pure xor and and functions , without any literal negation , and eq.([eeq2 ] ) reads ^{n^+_o } \cdot \nonumber \\   & & \left [ x\frac{n^+(n^+-1)}{(n-1)(n-2 ) } + \frac{(n - n^+-1)(n - n^+-2)}{(n-1)(n-2 ) } \right .\nonumber \\ & & + \left .  2 ( 1-x)\frac{n^+(n - n^+-1)}{(n-1)(n-2 ) } \right]^{n_o - n^+_o }  \label{eeq2b}\end{aligned}\ ] ] leading order terms can be computed following a calculation identical to the one shown below for the second moment , and will be omitted here .   for the second moment : averaging uniformly over the function types one obtains with averages over different function types distributions are also possible , but beyond the scope of this work .along the same line of arguments of the previous section , averaging over the poissonian graph structure , one obtains ^{n^+_o } \nonumber \\   & & \left [ { \cal p}(++|-)g(x;++- ) +  { \cal p}(+-|-)g(x;+-- ) + \right.\nonumber \\ & & \left .  { \cal p}(-+|-)g(x;-+- ) + { \cal p}(--|-)g(x;--- )  \right]^{n_o - n^+_o }  \label{eeq6}\end{aligned}\ ] ] where is now the value of given and the sign of the inputs . equations ( [ eeq2 ] ) and ( [ eeq6 ] ) have an identical structure .this observation can be generalized and holds for any higher order momentum , changing the structure of the functions .the details of the computation of the second moment are reported in [ sec : app ] . at the leading order it turns out that : the value of the multiplicative constant as a function of for increasing values of plotted in fig.([figura - m1 ] ) .for the second moment of the distribution of the number of solutions in the large limit .the red uppermost line is the function , diverging for ,width=453 ]  note that , unless in the pure xor case , .this means that the analysis of the second order momentum is not enough to assess the theoretical validity of the annealed calculation of the entropy in the sense that the convergence of the logarithmic correction series is not ensured a priori. moreover , whenever , .   at simplifications in the computation of the second moment hold . both the entropic contribution due to the external regulators , and the kullback - leibler terms are identically zero at the saddle point , as one can check from the saddle point equation for presented in [ sec : app ] .it turns out that the contribution to the saddle point includes also the term corresponding to the non - zero boundary term of integration . for the second moment in particular , one has to take into account the contribution of all terms around .going back to eq.([eeq6 ] ) and taking explicitly into account those terms , one obtains in the large limit where .the values of both contributions diverge for .analytic estimates suggest that the divergence goes as . in fig .[ n_square ] we display the numerical estimate of obtained via exhaustive enumeration and the analytic estimate presented in eq .( [ eeq6 ] ) .the agreement is good , as it should be , since the computation of the second moment is exact , also for finite .in this work we have investigated several aspects of the fixed points solutions set of a particular class of _ finite size _ diluted random boolean networks with inputs per function .a search algorithm based on a state - of - the - art satisfiability solver was used to exhaustively enumerate fixed points up to moderate system sizes . for fixed system size ensemblesthe average number of solutions were plotted together with average fluctuations and with two additionally ad hoc defined order parameters indicating average distance between solutions within the fixed points set .a throughout comparison of the exact results with those given by the heuristic belief propagation algorithm was done in order to assess the performance of bp for small size samples whose network structure significantly deviates from tree - like .bp was shown to perform significantly well in the prediction of the correct number of solutions even in small sizes cases .bp seems to loose its predicting power in the calculation of single boolean variables marginals at the fixed points ; still it was shown to retain a significant correlation with exact results in the global spatial arrangements of the solutions .furthermore , the analytical results closing the paper , together with their agreement with exact enumeration results , give a strong hint on the exactness of the annealed calculation of the entropy as well as the high order moments of the probability distribution of the number of solutions .we are happy to acknowledge martin weigt for his suggestions about the computation of the first and second moment .we want also to thank matteo marsili , and riccardo zecchina , for many interesting discussions .we first present the details of the computations of the second moment and we will then give some hints on the structure of the generic moment .   with respect to , if we keep only the leading order terms in eq.([eeq6 ] ) , the sums can be approximated by the following integral , where the exponent is given by the entropy contribution of the external variables plus a kullback - leibler distance term vanishing at the saddle point : } \nonumber \\ k(\alpha , x , b_o^+,b_i^+ ) & = & \frac{\sqrt{\alpha ( 1-\alpha)}}{\sqrt{b_o^+ b_i^+(1-b_o^+)(1-b_i^+ ) } } e^{\alpha b(\alpha , x , b_o^+,b_i^+ ) } \nonumber \\ { \cal f}[\alpha , x , b_o^+,b_i^+ ] & = & ( 1-\alpha)h(b_i^+ ) + \alpha d_{kl } ( b_o^+|g(\alpha , x , b_o^+,b_i^+ ) ) \nonumber \\ b(\alpha , x , b_o^+,b_i^+ ) & = & 3 - \frac{3 b^+ b_o^+ + ( 1-x)(1-b^+)b_o^+ + ( 1+x)(1-b^+)b_o^+/2}{(b^+)^2+(1-x)(1-b^+)b_o^+ + ( 1+x)(1-b^+)^2/2 } \nonumber \\ & - & \frac{(1+x)(b^+(1-b_o^+ ) + 3(1-x)(1-b^+)(1-b_o^+)/2}{1 - ( b^+)^2+(1-x)(1-b^+)b_o^+ - ( 1+x)(1-b^+)^2/2 } \nonumber \\h(x ) & = & -x \log(x ) - ( 1-x ) \log(1-x ) \nonumber \\ d_{kl } ( x|y ) & = & x\log\left ( \frac{y}{x } \right ) + ( 1-x)\log\left ( \frac{1-y}{1-x}\right ) \nonumber \\ g(\alpha , x , b_o^+,b_i^+ ) & = & ( b^+)^2 + ( 1-x)b^+(1-b^+ ) + \frac{1+x}{2}(1-b^+)^2 \nonumber\\ b^+(\alpha , b_o^+,b_i^+ ) & = & \alpha b_o^+ + ( 1-\alpha)b_i^+ \nonumber \label{eeq8}\end{aligned}\ ] ] where and .the value of the integral can be calculated at the saddle point finite corrections can be in principle computed extending the calculation of to higher orders in , and performing an asymptotic expansion around the saddle point . whenever and it can be seen that the condition ( [ eqsaddle1 ] ) finds the only maximum , which lies within the integration interval .   in this subsectionwe will show that in the thermodynamic limit the support of is contained in the interval , using , as we have already shown in the previous section , that : where is a constant independent from .the one - tailed chebyshev inequality states that : under the condition we can express where . inserting eqs .( [ eq:2mom ] ) into eq .( [ eq : cheby ] ) we get : where is a positive constant making the right tail of the distribution ( _ i.e. _ for values of ) exponentially small in , as we wanted to show .indeed , this simple result is enough to imply that no contribution to the entropy is given by instances whose number of solutions is larger than the annealed value .the support of the probability distribution of entropy values must be therefore $ ] . in order to prove that also smaller values do not take part of the support, one would need to calculate fractional order moments , as explained in the end of next section .for the general order momentum one can write , along the same line of reasoning : with ^{n^{\sigma_1 ...\sigma_{n-1}}_o } \nonumber \label{eeq7b}\end{aligned}\ ] ] where we are summing over all configurations overlaps with output / input variables of signs , the represent the probability of finding real replicas of an input variables couple in a given function , and the value of the averaged product of the replicated boolean function .as before , from the leading terms of eq.([eeq7 ] ) , one gets } \nonumber\end{aligned}\ ] ] with & = & ( 1-\alpha ) h^{(n ) } +  \alpha d^{(n)}_{kl } ( \{b_o \ } | g^{(n ) } ) \nonumber \\ h^{(n)}(\alpha , x,\ { b_o , b_i \ } ) & = & \sum_{{\vec \sigma } }  b_i^{\sigma_1 ... \sigma_{n-1 } } \log(b_i^{\sigma_1 ... \sigma_{n-1 } } ) \nonumber \\ d^{(n)}_{kl } ( \{b_o \ } | g^{(n ) } ) & = & \sum_{{\vec \sigma } } b_o^{\sigma_1 ...\sigma{n-1 } } \log\left(\frac{g^{(n)}}{b_0^{\sigma_1 ... \sigma{n-1 } } } \right )  \label{eeq9}\end{aligned}\ ] ] with and . the explicit form of functions and not given here for brevity .due to the general form of the exponent , the unique saddle point can always be computed as the one fulfilling conditions : such that unfortunately , the explicit calculation of the function for all seems not to be easily accessible .let us just finally note that one could prove that the distribution of the logarithm of the number of solutions tends to in the large one by showing that : which seems reasonable given the structure of eqs .( [ eeq9 ] , [ eqsaddle2 ] , [ eq : in ] ) , and the numerical result displayed in section [ sec : entropy ] , but , again , too difficult to be shown explicitly . +                   r. j. bayardo jr . and r. c. schrag . using csp look - back techniques to solve real world sat instances . _ in proc . of the 14th national conf . on artificial intelligence _ , 203 - 208 ( 1997 ) .

Abstract: since their introduction , boolean networks have been traditionally studied in view of their rich dynamical behavior under different update protocols and for their qualitative analogy with cell regulatory networks . more recently , tools borrowed from statistical physics of disordered systems and from computer science have provided a more complete characterization of their equilibrium behavior . however , the largest part of the results have been obtained in the thermodynamic limit , which is often far from being reached when dealing with realistic instances of the problem . the numerical analysis presented here aims at comparing - for a specific family of models - the outcomes given by the heuristic belief propagation algorithm with those given by exhaustive enumeration . in the second part of the paper some analytical considerations on the validity of the annealed approximation are discussed .   _ keywords _ : message - passing algorithms , random graphs , boolean networks


bag - of - words ( bow ) was originally proposed for document retrieval . in recent years , the method has been successfully applied to image retrieval tasks in computer vision community .the method is attractive because of its efficient image representation and retrieval .bow represents an image as a sparse vector of visual words , and thus images can be searched efficiently using an inverted index file system .other major application areas of the bow method are appearance - based mobile robot localization and slam problems .the fundamental issue involved with the appearance - based approach to both visual slam and global localization is the place recognition .robot should be able to recognize the places it has visited before to localise itself or refine the map of the environment .this task is performed by matching the current view of the robot to the existing map that contains the images of the previously visited locations . in this paperwe consider the problem of appearance - based localization in which the map is known _ a priori_.  in large - scale environments maps contain too many images to match .the image search in such a large map is still a challenging and open problem .matching images by comparing the local features of each image directly to the local features of all other images in the map is not practical .bag - of - words proposes a more efficient approach ; first , rather than matching with a pool of million visual features extracted from many thousands of images , the local features are mapped to a smaller number of vocabulary words that are built in an offline phase . this process is called vector - quantization .once the visual words are identified in the query image , they are used as indices into the image database , to directly retrieve the images that share the same words .when the vocabulary is large , the vector - quantization process can be a computationally expensive task in real - time localization .considerable research has been done to speed up the search ; some papers do approximate nearest - neighbor search using structures like vocabulary trees ; some methods reduce the number of local image features by selecting only a fraction of features that are highly discriminative ; another group makes use of more compact feature descriptors like . in this paper , we first show that some image retrieval tasks are _ easier _ for bow method .the hardness criteria , defined later , concerns how distinctive the image query is among all images in the dataset . given this criteria ,we show that the bow search can be terminated earlier for easier queries .this means , in such queries , mapping only a portion of features can be sufficient to yield a relatively good result .the stopping rule saves considerable amount of computational resources .the intuition behind this is that when there are many similar images to the query image , in terms of the number of common words they share , the search becomes more difficult as more processing is required to find the closest match candidate .whereas when the query image has only a few match candidates , i.e. , it shares its visual words with only a few images , the search becomes easy as vector - quantizing only a small number of features is sufficient to find the closest match to the query . by exploiting this fact, we can stop the vector - quantization when the search is easy and the nearest neighbor to the query is easy to find .our method acts as an approximate image search algorithm .our experimental results show that the accuracy decreases only slightly while the computational cost decreases dramatically.  our approach can be best compared with the approach of cummins and newman who use concentration inequalities ( bennett s inequality in their case ) for early bail - out in multi - hypothesis testing that excludes unlikely location hypotheses from further evaluation .however , we use a different bail - out strategy for the process of vector - quantization . in the next section ,we briefly review the image representation and the inverted - index search algorithm used in bow framework .this is followed by a review of the localization algorithms that employ bow for the image search .our proposed method to improve the efficiency of bow is described in section [ sec : method ] .section [ sec : results ] presents the experimental results and the evaluation criteria , and the result of our comparisons .finally , we conclude the paper in section [ sec : conclud ] .bag - of - words is a popular model that has been used in image classification , objection recognition , and appearance - based navigation . because of its simplicity and search efficiencyit has been used as a successful method in web search engines for large - scale image and document retrieval .bag - of - words model represents an image by a sparse vector of visual words .image features , e.g. , sifts , are sampled and clustered ( e.g. , using k - means ) in order to quantize the space into a discrete set of visual words .the centroids of clusters are then considered as visual words which form the visual vocabulary .once a new image arrives , its local features are extracted and vector - quantized into the visual words .each word might be weighed by some score which is either the word frequency in the image ( i.e. , _ tf _ ) or the `` term frequency - inverse document frequency '' or _tf - idf _ .a histogram of weighted visual words , which is typically sparse , is then built and used to represent the image .an inverted index file , used in the bow framework , is an efficient image search tool in which the visual words are mapped to the database images .each visual word serves as a table index and points to the indices of the database images in which the word occurs .since not every image contains every word and also each word does not occur in every image , the retrieval through inverted - index file is fast .bag - of - words model has been extensively used as the basis of the image search in appearance - based localization or slam algorithms .cummins and newman propose a probabilistic framework over the bag - of - words representation of locations , for the appearance - based place recognition . along with the visual vocabulary they also learn the chow liu tree to capture the co - occurrences of visual words .similarly , angeli _ et al . _ develop a probabilistic approach for place recognition in slam .they build two visual vocabularies incrementally and use two bow representations as an input of a bayesian filtering framework to estimate the likelihood of loop closures .  assuming each image has hundreds of sift features , mapping the features to the visual words , using a linear search method , is computationally expensive and not practical for real - time localization .researchers have tackled this problem with different approaches that speeds up the search but at the expense of accuracy .a number of papers have employed compact feature descriptors that speeds up the search .glvez - lpez and tards in propose to use fast and breif binary features and introduce a bow model that descritizes a binary space .similarly , use surf to have a more compact feature descriptor .another approach is to use approximate nearest neighbor search algorithms , like hierarchical k - means , kd - trees or graph - based search methods and , to speed up the quantization process .achar _ et al ._ and zhang propose reducing the number of features in each image , thereby reducing ( removing ) the vector - quantization process .they keep track of the features that are repeatable over time . however , these approaches are more specific to the navigation problems where the data is sequential and there exists considerable overlaps between consecutive images .our approach is more similar to this group as we also reduce the amount of feature mapping .however rather than only selecting a small set of features , we use all features but we stop the mapping process when necessary . our approach is also more general as it does not depend on the sequential property of the data .vector - quantization ( vq ) is an expensive process when the bow - based image retrieval is performed in large - scale environments , in which hundreds of features extracted from an image need to be matched against hundreds of thousands of visual words .the question is if we really need to vector - quantize all features ?depending on the difficulty level of the search , the number of features to be converted to visual words may vary .we call a search difficult when there are many similar images to the query and thus finding the nearest neighbor among all those candidates requires more computations. whereas in an easy search , the query image is similar to only a few images in the database and it can find its true match after processing only a small percentage of features .figure [ diff_search ] and figure [ easy_search ] show examples of easy and difficult searches .the histograms show the similarity of each database image to the query based on the _ tf - idf _ score .the difficult search needs to process at least of features to find the closest match to the query ( indicated by the peak of the histogram ) , however the easy search can stop the search after processing only of features as the peak does not change until the end of the search . comparing the distance between the peak and average of the histograms , it can be seen that in difficult search this distance is smaller than that in easy search , which is expected .initially each image starts with no vote . in original bow, image features are converted to visual words one by one and the histogram ( of images scores ) is built incrementally .each bin of the histogram corresponds to one of the database images and indicates the score of that image .each visual word will cast a distance - weighted vote for multiple images , i.e. , histogram bins .this process continues until all words cast their votes and then the peak of the histogram ( the bin with the highest score ) determines the nearest neighbor to the query .  [cols="^,^ " , ]   the experimental results shown in tables [ exp1:city1 ] to [ exp1:newcol2 ] have been performed by rule 1 .figures [ fig : rules ] and [ fig : rules2 ] show the result of applying different rules to the bow search on the new college ( right - side sequence ) and city center ( right - side sequence ) datasets , respectively .the graphs have been generated by varying the stopping thresholds . as can be seen ,stopping rules # 1 and # 3 generate better results . by reducing the number of processed features to half the accuracy only decreases slightly .  figure [ fig : feat_im ] shows another experiment that validates our proposed stopping method .figure shows the relation between the number of images and the percentage of features they need to process to obtain the same result as that of the original bow .the experiment has been done on city center dataset ( left - side sequence ) with the stopping rule # 1 used .we set the threshold to to get exactly the same accuracy of original bow .as can be seen , only images need to process of their features , another only need to vector - quantize smaller percentages of features .we observed that the computational requirements of the bow method can be significantly reduced by allocating less computations to easier search queries .deciding when to terminate the search is treated as a stopping problem .we proposed several stopping rules and showed their effectiveness on an appearance - based localization problem .a. angeli , d. filliat , s. doncieux , and j .- a .meyer , `` a fast and incremental method for loop - closure detection using bags of visual words '' , _ ieee transactions on robotics , special issue on visual slam _ , pp .1027 - 1037 , 2008 .k. hajebi , y. abbasi - yadkori , h. shahbazi , and h. zhang , `` fast approximate nearest - neighbor search with k - nearest neighbor graph '' , in _ proc .of the twenty - second international joint conference on artificial intelligence ( ijcai ) _ , vol .2 , pp . 1312 - 1317 , 2011 .r. i. hartley and a. zisserman , _ multiple view geometry in computer vision ._ , cambridge university press , second edition , 2004. w. hoeffding , `` probability inequalities for sums of bounded random variables '' , _ journal of the american statistical association _ , vol .13 - 30 , 1963 .f. li and j. kosecka , `` probabilistic location recognition using reduced feature set '' , in _ proc . of the ieee international conference on robotics and automation ( icra ) _ , 2006 .o. maron and a. w. moore , `` hoeffding races : accelerating model selection search for classification and function approximation '' , in _ advances in neural information processing systems ( nips ) _ , vol .59 - 66 , 1993 .d. nister and h. stewenius , `` scalable recognition with a vocabulary tree '' , in _ proc . of the ieee international conference on computer vision and pattern recognition ( cvpr )2 , pp . 2161 - 2168 , 2006 .j. philbin , o. chum , m. isard , j. sivic , and a. zisserman , `` object retrieval with large vocabularies and fast spatial matching '' , in _ proc . of the ieee international conference on computer vision and pattern recognition ( cvpr ) _ , pp . 1 - 8 , 2007 .h. shahbazi and h. zhang , `` application of locality sensitive hashing to real - time loop closure detection '' , in _ proc . of the ieee / rsj international conference on intelligent robots and systems ( iros ) _ , 2011 .j. sivic and a. zisserman , `` video google : efficient visual search of videos : toward category - level object recognition '' , in _ proc . of the ieee international conference on computer vision and pattern recognition ( iccv )_ , pp . 1470 - 1477 , 2003 .

Abstract: we propose a technique to improve the search efficiency of the bag - of - words method for image retrieval . we introduce a notion of difficulty for the image matching problems and propose methods that reduce the amount of computations required for the feature vector - quantization task in bow by exploiting the fact that easier queries need less computational resources . measuring the difficulty of a query and stopping the search accordingly is formulated as a stopping problem . we introduce stopping rules that terminate the image search depending on the difficulty of each query , thereby significantly reducing the computational cost . our experimental results show the effectiveness of our approach when it is applied to appearance - based localization problem .