scheduling problems form one of the most important class of combinatorial optimization problems .they arise in situations where a set of operations ( tasks ) have to be performed on a set of resources ( machines ) , optimizing a given quality criterion .flowshop problems constitute a special case of scheduling problems in which an operation must pass through all the set of resources before being completed .such scheduling problems are often difficult to solve , because of the large search space they induce , and then represent a great challenge for combinatorial optimization .therefore many optimization methods have been proposed so far and experimented on a set of widely - used benchmark instances .regarding , the minimization of makespan in flowshop problems , iterated local search ( ils ) approaches seem to achieve very good performance .in particular , sttzle s ils stays one of the references of the literature .it has been listed as one of the best performing metaheuristics on a review of heuristic approaches for the flowshop problem investigated in the paper  .more recently , ruiz and sttzle have proposed an iterated greedy algorithm to solve the flowshop problem , based on similar mechanisms , and they have shown that is outperforms the classical metaheuristics for this problem .the aim of the paper is to analyze characteristics of the flowshop problems in order to understand and to explain why sttzle s method achieves such good performance .a quick analysis shows that the neutrality is high in those problems and we want to explain how this neutrality influences the behavior of heuristic methods .it will then become possible to propose mechanisms that are able to exploit this neutrality .  the method proposed by sttzle consists of an iterated local search ( ils ) approach based on the _ insertion _ neighborhood operator .this operator is argued to be the best one by the original author , as it produces better results than the _ transpose _ operator , for example , while allowing a faster evaluation compared to the _ exchange _ operator .the method starts from a solution constructed using a greedy heuristic ( the neh heuristic ) , initially proposed by nawaz et al .next , the local search algorithm , based on a _ first improvement _ exploration of the neighborhood , is iterated until a local minimum is reached .then , between each local search , a small perturbation is applied on the current solution using random applications of the _ transpose _ and _ exchange _ neighborhood operators .an important characteristic of this approach is the acceptance criterion of the ils algorithm , which is based on the metropolis condition ( as in simulated annealing ) .indeed , such a condition allows to accept a solution with a same or worse fitness value than the current one .hence , the contributions of this work are the following ones . on the one hand ,the specific problem of flowshop scheduling is deeply studied in terms of landscape analysis and neutrality . on the other hand ,some propositions are drawn in order to exploit neutrality in the design of a local search algorithm .of course , these considerations are still valid for other combinatorial optimization problems with a neutrality .the paper is organized as follows .section [ sec : background ] is dedicated to the presentation of the flowshop scheduling problem investigated in this paper , and of the required notions about neutrality analysis in fitness landscapes .section [ sec : analysis ] presents the neutral networks analysis for the permutation flowshop problem under study , whereas section [ sec : heuristic ] gives some hints on how to exploit the neutrality property in order to solve such problems efficiently by means of local search algorithms .finally , the last section is devoted to discussion and future works ..notations used in the paper . [ cols=">,<",options="header " , ]   [ tab : param ]   the flowshop scheduling problem ( fsp ) is one of the most investigated scheduling problem from the literature .the problem consists in scheduling  jobs on machines .machines are critical resources , _ i.e. _ two jobs can not be assigned to the same machine at the same time .a job is composed of  tasks , where  is the task of  , requiring machine  . a processing time  is associated with each task  .we here focus on a permutation fsp , where the operating sequences of the jobs are identical and unidirectional for every machine . as consequence, a feasible solution can be represented by a permutation of size ( the ordered sequence of scheduled jobs ) , and the size of the search space is then .  in this study, we will consider that the makespan , _i.e. _ the total completion time , is the objective function to be minimized .let be the completion date of task  , the makespan ( ) can be computed as follows : according to graham et al . , the problem under study can be denoted by .the fsp can be solved in polynomial time by the johnson s algorithm for two machines  . however , in the general case , minimizing the makespan has been proven to be np - hard for three machines and more  . as a consequence, large - size problem instances can generally not be solved to optimality , and then metaheuristics may appear to be good candidates to obtain well - performing solutions .  [ [ benchmark - instances . ] ] benchmark instances .+ + + + + + + + + + + + + + + + + + + +  experiments will be driven using a set of benchmark instances originally proposed by taillard and widely used in the literature .we investigate different values of the number of jobs and of the number of machines .the processing time of job and machine is generated randomly , according to a uniform distribution ) ] , is a neighbor of and .  a way to describe neutral networksnn is given by the _ autocorrelation of neutral degree _ along a neutral random walk .from neutral degrees collected along this neutral walk , we computed its autocorrelation function  , that is the correlation coefficient of the neutral degree between the solutions and for all possible .the autocorrelation measures the correlation structure of a nn . if the first correlation coefficient is close to 1 , the variation of neutral degree is low ; and so , there are some areas in nn of solutions which have close neutral degrees , which shows that nn are not random graphs .  another interesting information to determine if a local search could find a better solution on a neutral network , is the position of portals .the number of steps before finding a portal during a neutral random walk is a good indicator of the probability to find better solution(s ) according to the computational cost to find it , _i.e. _ the number of evaluations .  moreover , to design a local search which explores the neutral networks in an efficient way , we need to find some information around the nn where , _ a priori _ , there is a lack of information ._ evolvability _ is defined by altenberg as `` the ability of random variations to sometimes produce improvement '' .the concept of evolvability could be difficult to define in combinatorial optimization .for example , the evolvability could be the minimum fitness which can be reached in the neighborhood .in this work , we choose to define the evolvability of a solution as the average fitness in its neighborhood .it gives the expectation of fitness reachable after a random move .the _ autocorrelation of evolvability _ allows to measure the information around neutral networks .this autocorrelation is the autocorrelation function of a evolvability measure collected during a neutral random walk . when this correlation is large , the solutions which are close from each other on a neutral network have evolvabilities which are close tooso , the evolvability could guide the search on neutral networks such as the fitness guides the search in the landscape where the autocorrelation of fitness values is large  .to analyze neutral networks , for each instance of taillard s benchmarks , 30 different neutral walks were performed .the neutral walks all start from a local optimum .it has been obtained by a steepest descent algorithm initialized with a random solution .the length of each neutral walk depends on the length of the descents which lead to local optima .we consider 10 times the maximal length found on the 30 descents . in the following ,the results are presented according to the number of jobs ( ) and the number of machines ( ) . for each problem size , an average value and the corresponding standard deviation are represented . by the term size , we mean both the number of jobs ( ) and the number of machines ( ) .this average value is computed from the means obtained from the 10 instances of the same size , themselves calculated from the values given by the 30 neutral walks .computed between and of the neutral degree according to the number of jobs . ]  in this section , we first measure the neutral degree of the fsp .then , we describe the structure of the neutral networks ( nn ) .figure [ neutraldegtonghsizeplot ] shows the average neutral degree to the size of the neighborhood , collected along the 30 neutral walks .whatever the number of machines , the neutral degree ratio increases when the number of jobs increases .this ratio is higher for small number of machines . for -machine , and for - or -job and -machine instances , the neutral degree is huge , higher than . for - or -job and -machine instances , the ratio seems to be very low ( ) , but the number of neighbors with same fitness value is significant ( about and neutral neighbors for and jobs , respectively ) .there is no local optimum without a neighbor with the same fitness value , which means that each local optimum belongs to a local optima neutral network .the neutral degree is high enough to describe the fitness landscape with neutral networks .a neutral walk corresponds to a sequence of neighbor solutions on a nn of the fitness landscape , where all solutions share the same fitness value . during those neutral walks ,we compute the autocorrelation of the neutral degree ( see section [ measureneutralfit ] ) . figure [ neutraldegreeplot ] shows the first autocorrelation coefficient for , and machines with respect to the number of jobs . in order to prove that those correlations are significative , we compare them to a null model .it consists of shuffling the same values of neutral degrees collected during the neutral walks .then , the autocorrelation of this model is compared to the original one .for all sizes , the first autocorrelation coefficient of the null model is below .therefore , we can conclude that the autocorrelation is a consequence of the succession of solutions encountered during the walk .obviously , for , and jobs , the neutral degree is highly correlated ( higher than ) .moreover , the standard deviations are very low , which indicates that the average values reflect properly this property on instances of same size. for -job and - or -machine instances , the standard deviation gets higher .this can possibly be explained by a higher correlation .nevertheless , these values allow us to conclude that the neutral degree of a solution is partially linked to the one of its neighbor solutions .let us remark that the correlation for -job -machine instances is very low , due to the small average value of the neutral degree for this size .the first conclusions of this analysis is that ( ) there exists a high neutrality over the fitness landscape , particularly for large - size instances ( ) the neutral networks , defined as the graphs of neighbor solutions with the same fitness value , are not random . as a consequence, we should not expect to explore the neutral networks efficiently with a random walk .hence , heuristic methods should exploit the information available in the neighborhood of the solutions .a metaheuristic such as ils visits several local optima . in the previous section , we have seen that the local optima often belong to a nn .a natural question arises when the metaheuristic reaches a nn : is it possible to escape from this nn ? in this section , we classify the local optima nn in three different types , and we analyze their size .three types of nn typologies may exist ( see figure [ plateaugeometry ] ) :  1 .the local optimum is the single solution on the nn ( type t1 ) , _ i.e. _ it has no neighbor with the same fitness value , we call it a degenerated nn .the neutral walk from the local optimum did not show any neighbors with a better fitness values for all the solutions encountered along the neutral walk ( type t2 ) .of course , as the whole nn has not been enumerated , we can not decide if it is possible to escape from them .at least one solution having a neighbor with better fitness value than the local optimum fitness is found along the neutral walk ( type t3 ) .+      figures [ percentt1 ] , [ percentt23 ] and [ percentt45 ] show the proportion of nn of each type ( t1 , t2 or t3 ) counted along the neutral walks . for - , - and -job instances ,the neutral walks show only nn of types t2 and t3 .no local optimum solution is alone on the nn . for -job instances ,the number of type ( t1 ) is also small , except for 20 machines ( of type t1 ) .hence , the neutrality is important to keep in mind while solving such instances .the number of nn without any escaping solutions found ( t2 ) is significative only for -machine instances ( higher than ) and stays very low for - an -machine instances ( lower than ) .the -machine instances , which are known to be the hardest to solve optimally , are the ones where the probability to escape from local optimum by neutral exploration of the nn is close to one .when the neutral networks size is very small , the number of visited solutions is very small .indeed , a nn of type t2 or t3 could contain very few solutions and , the neutral walk could loop on some solutions .these situations have to be considered with attention .figure [ dist0 ] shows the average percentage of solutions visited more than once during the neutral walk . for the - , - and -job instances, there is no re - visited solutions during the neutral walks . for -job and -machine instances ,the number of re - visited solutions is approximatively during neutral walks on nn of type t2 or t3 .this result points out two remarks .first , the nn of local optima seems to be large for most instances .second , the number of re - visited solutions is low , which means that the probability to escape the nn of type t2 is below the inverse of the size of the neutral walk .  in conclusion , for most instances, a metaheuristic could escape the local optimum by exploring the nn .the next section will show some hints on how to guide a metaheuristic on neutral networks .in the previous section , we proposed to use neutral exploration to escape from local optimum , as there exists solutions having neighbor(s ) with a better fitness value around neutral networks .we called those solutions , portals .an efficient metaheuristic has to find such portal with a minimum number of evaluations .first , we study the number of steps to reach a portal , and then we propose an insight to get information to find them quickly .     as shown on figure [ percentt45 ] at least of neutral random walks for fsp with 50 , 100 and 200 jobs can reach a portal ( more than for 10 and 20 machines ) .the performance of a metaheuristic which explores neutral networks highly depends on the probability to find a portal .indeed , it could become more time consuming to consider a neutral walk than applying a smart restart .figure [ portestepnb ] gives the average number of steps to reach the first portal during the 30 neutral walks .the larger the number of machines , the less the number of steps is required by the neutral walk to reach a portal . for -machine instances ,the neutral random walks need around steps to reach a portal , which is very small compared to the length of the descents ( 19 , 40 , 64 , 101 respectively for 20 , 50 , 100 , and 200 jobs ) . for -machine instances , the length of the neutral walksis around the length of the descents .hence , it is probably more advantageous to perform a neutral random exploration than a random restart .moreover , the fitness value obtained after the neutral walk is better than after the descent . consequently , if an _ a priori _ study highlights that a portal is supposed to be encountered quickly , a metaheuristic that takes into account information on the neutral walk should move on the nn , and then finally find an improving solution .   in the previous section ,the role of neutrality was demonstrated by the correlation of the neutral degree between the neutral walk neighbors and the high frequency of neutral networks .neutral networks lead , with very few steps , to a portal .the neutrality could give interesting information about the landscape in order to guide the search .however , since the neutral network is large , the search has to be guided to find quickly a portal and not to stagnate on the nn .thus , proper information has to be collected and interpreted along the neutral walk to help the metaheuristic to take good decision : is it more interesting to continue the neutral walk until a portal is reached or to restart ? as suggested in section [ measureneutralfit ] , we compute the evolvability of a solution as the average fitness values of its neighbors for all visited solutions .we analyze the evolvability of solutions on neutral networks and we give some results about the correlation of evolvability and portals on a neutral network .this allows us to propose new ideas for the design of a metaheuristic .   of the average fitness values of neighbors solutions between and according to the number of jobs .]  during those neutral walks , we compute the evolvability of each solution along the neutral walk , and then its autocorrelation ( see section [ measureneutralfit ] ) . figure [ correlmeanfitnghplot ] shows the first autocorrelation coefficient for , and machines with respect to the number of jobs . in order to show that those correlations are significative , as in section [ sectionneutraldegree ] , we compare them to a null model .for all sizes , the first autocorrelation coefficient of the null model is below .therefore , we can conclude that the autocorrelation is a consequence of the succession of solutions encountered during the walk .the average fitness values of the neighbors are not distributed randomly : they can then be exploited by a metaheuristic .the neutral networks present evolvability and portals .so , we can wonder if the evolvability would be able to guide a metaheuristic quickly to a portal . to test this hypothesis , along the neutral walks ,we compute the correlation between the average fitness values in the neighborhood and the number of steps required to reach the closer portal of the walk .this is presented in figure [ correlmeanfitnbimpngh ] .the larger the number of machines , the higher ( in absolute value ) the negative correlation . for - or -machine instances, this correlation belongs to  ] .this is a specific choice which could have an impact on the difficulty of instances .future works will consider other instance generators , and study the neutrality according to the instance parameters .this work also helps to understand some experimental results on the efficiency of metaheuristics . in a study of iterated local search to solve the fsp , sttzle designs several efficient ils , called ils - s - pfsp and compares them to local search algorithms .he writes : `` experimentally , we found that rather small modifications [ of the solution ] are sufficient to yield very good performance '' . in section [ reachportals ] , we show that improving solutions can be reached very quickly applying _ insertion _ operator on a neutral network .so , sttzle s remark can be explained by the neutrality and the high probability on the neutral networks to move on a solution with an improving neighbor .moreover , this works supports the experimentations on ils design for -machine instances .the study of neutral walks highlights features that explain the efficient design of the ils - s - pfsp .indeed , remember that the ils - s - pfsp , initialized with a random solution , applies a local search based on _insertion_-neighborhood mapping to get a local optimum , and then applies iteratively the steps ( ) perturbation , ( ) local search , and ( ) acceptance criterion , until a termination condition is met .all acceptance criteria tested in ils - s - pfsp are based on the metropolis condition : they always accept a solution with equal fitness value .so the neutral moves are always accepted .besides , sttzle work shows that the perturbation based on the application of several _ swap _ operators ( also called _ transpose _ operators ) is efficient . and , the _ swap _ neighborhood is included in the _ insertion _ neighborhood as the job can be inserted at the positions or .so , applying the _ swap _ operator several times could correspond to a walk on a neutral network defined by _insertion_-neighborhood relation .thus , steps ( ) and ( ) allow the ils - s - pfsp to move on the neutral network that could be frequent for those fsp instances .moreover , we show that the distance is small between a local optimum and a portal .so , such an ils - s - pfsp is able to quickly improve the current best solution , which could explain its performances .furthermore , our work proposes to consider the neutrality to guide a metaheuristic on the search space .the fsp instances shows neutrality , it is easy to encounter portals along a neutral walk and the evolvability leads quickly to them . with such information , a metaheuristic is proposed : first a local search is performed from a random solution , and then iteratively ( ) the evolvability on the neutral network is optimized until a portal is found and ( ) the local search is applied to move to an other local optimum .the metaheuristic finishes when the termination criterion is met .similar ideas have been ever tested on other problems with neutrality such as max - sat and nk - landscapes with neutrality  . a first attempt for developing such a strategy leads to the proposition of nils that has been successfully tested on flowshop problems .erel , s. , collard , p. , clergue , m. : easuring the evolvability landscape to study neutrality . in keijzer ,m. , et al .: poster at genetic and evolutionary computation gecco-2006 genetic and evolutionary computation gecco-2006 , seattle , wa united states , acm press ( 07 2006 ) 613614 tea team .erel , s. , collard , p. , clergue , m. : cuba search : when selection meets innovation . in : evolutionary computation , 2004 .cec2004 evolutionary computation , 2004 .cec2004 , portland ( oregon ) united states , ieee press ( 06 2004 ) 924 931  marmion , m.e . ,dhaenens , c. , jourdan , l. , liefooghe , a. , verel , s. : : a neutrality - based iterated local search and its application to flowshop scheduling . in : 11th european conference on evolutionary computation in combinatorial optimisation(evocop11 ) .lncs , springer - verlag ( 2011 )

Abstract: solving efficiently complex problems using metaheuristics , and in particular local search algorithms , requires incorporating knowledge about the problem to solve . in this paper , the permutation flowshop problem is studied . it is well known that in such problems , several solutions may have the same fitness value . as this neutrality property is an important issue , it should be taken into account during the design of search methods . then , in the context of the permutation flowshop , a deep landscape analysis focused on the neutrality property is driven and propositions on the way to use this neutrality in order to guide the search efficiently are given .


memory - based learning of classification tasks is a branch of supervised machine learning in which the learning phase consists simply of storing all encountered instances from a training set in memory .memory - based learning algorithms do not invest effort during learning in abstracting from the training data , such as eager - learning ( e.g. , decision - tree algorithms , rule - induction , or connectionist - learning algorithms , ) do .rather , they defer investing effort until new instances are presented .on being presented with an instance , a memory - based learning algorithm searches for a best - matching instance , or , more generically , a set of the best - matching instances in memory .having found such a set of best - matching instances , the algorithm takes the ( majority ) class with which the instances in the set are labeled to be the class of the new instance .pure memory - based learning algorithms implement the classic -nearest neighbour algorithm ; in different contexts , memory - based learning algorithms have also been named lazy , instance - based , exemplar - based , memory - based , case - based learning or reasoning )  memory - based learning has been demonstrated to yield accurate models of various natural language tasks such as grapheme - phoneme conversion , word stress assignment , part - of - speech tagging , and pp - attachment .for example , the memory - based learning algorithm ib1-ig , which extends the well - known ib1 algorithm with an information - gain weighted similarity metric , has been demonstrated to perform adequately and , moreover , consistently and significantly better than _ eager - learning _ algorithms which do invest effort in abstraction during learning ( e.g. , decision - tree learning , and connectionist learning ) when trained and tested on a range of morpho - phonological tasks ( e.g. , morphological segmentation , grapheme - phoneme conversion , syllabification , and word stress assignment ) .thus , when learning nlp tasks , the abstraction occurring in decision trees ( i.e. , the explicit _ forgetting _ of information considered to be redundant ) and in connectionist networks ( i.e. , a non - symbolic encoding and decoding in relatively small numbers of connection weights ) both hamper accurate generalisation of the learned knowledge to new material .these findings appear to contrast with the general assumption behind eager learning , that data representing real - world classification tasks tends to contains ( i ) redundancy and ( ii ) exceptions : redundant data can be removed , yielding smaller descriptions of the original data ; some exceptions ( e.g. , low - frequency exceptions ) can ( or should ) be discarded since they are expected to be bad predictors for classifying new ( test ) material .however , both redundancy and exceptionality can not be computed trivially ; heuristic functions are generally used to estimate them ( e.g. , functions from information theory ) .the lower generalisation accuracies of both decision - tree and connectionist learning , compared to memory - based learning , on the above - mentioned nlp tasks , suggest that these heuristic estimates may not hold for data representing nlp tasks .it appears that in order to learn such tasks successfully , a learning algorithm should not forget ( i.e. , explicitly remove from memory ) any information contained in the learning material : it should not abstract from the individual instances.  an obvious type of abstraction that is not harmful for generalisation accuracy ( but that is not always acknowledged in implementations of memory - based learning ) is be the straightforward abstraction from tokens to types with frequency information .in general , data sets representing natural language tasks , when large enough , tend to contain considerable numbers of duplicate sequences mapping to the same output or class .for example , in data representing word pronunciations , some sequences of letters , such as ing at the end of english words , occur hundreds of times , while each of the sequences is pronounced identically , viz ./i/. instead of storing all individual sequence tokens in memory , each set of identical tokens can be safely stored in memory as a single sequence type with frequency information , without loss of generalisation accuracy .thus , forgetting instance tokens and replacing them by instance types may lead to considerable computational optimisations of memory - based learning , since the memory that needs to be searched may become considerably smaller .given the safe , performance - preserving optimisation of replacing sets of instance tokens by instance types with frequency information , a next step of investigation into optimising memory - based learning is to measure the effects of _ forgetting instance types _ on grounds of their exceptionality , the underlying idea being that the more exceptional a task instance type is , the more likely it is that it is a bad predictor for new instances .thus , exceptionality should in some way express the unsuitability of a task instance type to be a best match ( nearest neighbour ) to new instances : it would be unwise to copy its associated classification to best - matching new instances . in this paper, we investigate three criteria for estimating an instance type s exceptionality , and removing instance types estimated to be the most exceptional by each of these criteria .the criteria investigated are  1 .typicality of instance types ; 2 .  class prediction strength of instance types ; 3 .friendly - neighbourhood size of instance types ; 4 .random ( to provide a baseline experiment ) .we base our experiments on a large data set of english word pronunciation .we briefly describe this data set , and the way it is converted into an instance base fit for memory - based learning , in section [ data ] . in section [ experimental ] we describe the settings of our experiments and the memory - based learning algorithm ib1-ig with which the experiments are performed .we then turn to describing the notions of typicality , class - prediction strength , and friendly - neighbourhood size , and the functions to estimate them , in section [ criteria ] .section [ results ] provides the experimental results . in section [ discussion ] , we discuss the obtained results and formulate our conclusions .converting written words to stressed phonemic transcription , i.e. , word pronunciation , is a well - known benchmark task in machine learning .we define the task as the conversion of fixed - sized instances representing parts of words to a class representing the phoneme and the stress marker of the instance s middle letter . to generate the instances ,windowing is used .table [ overall - windows - ex ] displays example instances and their classifications generated on the basis of the sample word booking .classifications , i.e. , phonemes with stress markers ( henceforth pss ) , are denoted by composite labels .for example , the first instance in table [ overall - windows - ex ] , _ _ _ book , maps to class label /b/1 , denoting a /b/ which is the first phoneme of a syllable receiving primary stress .in this study , we chose a fixed window width of seven letters , which offers sufficient context information for adequate performance , though extension of the window decreases ambiguity within the data set .  [ cols="^,^,^,^,^,^,^,^,^ " , ]figure [ edited - gs ] displays the generalisation accuracies in terms of incorrectly classified test instances obtained with all performed experiments . the leftmost point in the figure ,from which all lines originate , indicates the performance of ib1-ig when trained on the full data set of 222,601 types , viz .6.42% incorrectly classified test instances ( when computed in terms of incorrectly pronounced test words , ib1-ig pronounces 64.61 of all test words flawlessly ) .the line graph representing the four experiments in which instance types are removed randomly can be seen as the baseline graph .it can be expected that removing instances randomly leads to a degradation of generalisation performance .the upward curve of the line graph denoting the experiments with random selection indeed shows degrading performance with increasing numbers of left - out instance types .the relative decrease in generalisation accuracy is 2.0% when 1% of the training material is removed randomly , 3.8% with 2% random removal , 10.7% with 5% random removal , and 20.7% with 10% random removal .=  surprisingly , the only experiments showing lower performance degradation than removal by random selection are those with class - prediction strength ; the other criteria for removing exceptional instances lead to worse degradations . it does not matter whether instance types are removed on grounds of their typicality : apparently , a markedly low , neutral , or high typicality value indicates that the instance type is ( on average ) important , rather than removable .the same applies to friendly - neighbourhood size : instances with small neighbourhood sizes appear to contribute significantly to performance on test material .it is remarkable that the largest errors with 1% and 2% removal are obtained with the friendly - neighbourhood size criterion : it appears that on average , the instances with few or no nearest neighbours are important in the classification of test material .when using class - prediction strength as removal criterion , performance does not degrade until about 5% of the instance types with the lowest strength are removed from memory .the reason is that class - prediction strength is the only criterion that detects minority ambiguities , i.e. , instance types with prediction strength 0.0 , that can not contribute to classification since they are always overshadowed by their counterpart instance types with majority classes , even for their own classification . in the trainingset , 9,443 instance types are minority ambiguities , i.e. , 4.2% of the instance types ( accounting for 3.8% of the instance tokens in the original token set ) .thus , among the tested methods for reducing the memory needed for storing an instance base in memory - based learning , only two are performance - preserving while accounting for a substantial reduction in the amount of memory needed by ib1-ig :  1 .replacing instance tokens by instance types accounts for a reduction of about 63% of memory needed to store instances , excluding the memory needed to store frequency information .when frequency information is stored in two bytes per instance type , the memory reduction is about 54% .removing instance types that are minority ambiguities on top of the type / token - reduction accounts only for an additional memory reduction of 2% , i.e. , for a total memory reduction of 65% ; 56% with two - byte frequency information stored per instance .as previous research has suggested , keeping full memory in memory - based learning of word pronunciation strongly appears to yield optimal generalisation accuracy .the experiments in this paper show that optimisation of memory use in memory - based learning while preserving generalisation accuracy can only be performed by ( i ) replacing instance tokens by instance types with frequency information , and ( ii ) removing minority ambiguities .both optimisations can be performed straightforwardly ; minority ambiguities can be traced with less effort than by using class - prediction strength .our implementation of ib1-ig described in already makes use of this knowledge , albeit partially ( it stores class distributions with letter - window types ) .we note that with ( in our tested implementation of ib1-ig , ) , removing minority ambiguities may distort performance , since taking more than single nearest neighbours into account may allow for minority ambiguities to play a constructive role in classification .our results also show that atypicality , non - typicality , and typicality , and friendly - neighbourhood size are all estimates of exceptionality that indicate the importance of instance types for classification , rather than their removability . as far as these estimates of exceptionalityare viable , our results suggest that exceptions should be kept in memory and not be thrown away .   *the tested criteria can be employed as instance weights as in each and pebls , rather than as criteria for instance removal .instance weighting may add relevant information to similarity matching , and may improve ib1-ig s performance rather than just preserving it .* the tested implementation of ib1-ig performs a -nn search through instance space with .when , ib1-ig s performance may change , as well as the effect of applying the instance - removal techniques tested here . removing instances from memory may have a less drastic effect when more instance types at more distances are allowed to match a new instance .daelemans , w. 1996 .abstraction considered harmful : lazy learning of language processing . in h. j. van den herik and a. weijters , editors , _ proceedings of the sixth belgian dutch conference on machine learning _, pages 312 , maastricht , the netherlands . matriks .daelemans , w. and a. van den boschgeneralisation performance of backpropagation learning on a syllabification task . in m.f. j. drossaers and a. nijholt , editors , _ twlt3 : connectionism and natural language processing _ , pages 2737 , enschede .twente university .daelemans , w. , a. weijters , and a. van den bosch .empirical learning of natural language processing tasks .lecture notes in artificial intelligence , , number 1224 , pages 337344 .berlin : springer - verlag .rumelhart , d. e. , g. e. hinton , and r. j. williams .learning internal representations by error propagation . in d.e. rumelhart and j. l. mcclelland , editors , _ parallel distributed processing : explorations in the microstructure of cognition _, volume 1 : foundations .cambridge , ma : the mit press , pages 318362 .      , a. , w. daelemans , and a. weijters .morphological analysis as classification : an inductive - learning approach . in k.oflazer and h. somers , editors , _ proceedings of the second international conference on new methods in natural language processing , nemlap-2 , ankara , turkey _ , pages 7989 .

Abstract: memory - based learning , keeping full memory of learning material , appears a viable approach to learning nlp tasks , and is often superior in generalisation accuracy to eager learning approaches that abstract from learning material . here we investigate three _ partial _ memory - based learning approaches which remove from memory specific task instance types estimated to be exceptional . the three approaches each implement one heuristic function for estimating exceptionality of instance types : ( i ) typicality , ( ii ) class prediction strength , and ( iii ) friendly - neighbourhood size . experiments are performed with the memory - based learning algorithm ib1-ig trained on english word pronunciation . we find that removing instance types with low prediction strength ( ii ) is the only tested method which does not seriously harm generalisation accuracy . we conclude that keeping full memory of types rather than tokens , and excluding minority ambiguities appear to be the only performance - preserving optimisations of memory - based learning .


one of the most important branches of classical and quantum cryptography is secure multi - party computation ( smc ) ( and references therein ) .smc is a primitive for distributed computation .it enables the distributed computing of correct output of a function in a situation , where the inputs are given by a group of mutually distrustful users .a smc is required to be fair , and secure .specifically , it should not leak the secret inputs of the individual players .efforts have been made to achieve this in various ways , both classically and quantum mechanically .however , in those efforts , it has been assumed that some of the users follow the protocol honestly ( which implies that some of the users are semi - honest ) . among the variants of smc schemes ,a specific variant having particular importance is `` socialist millionaire problem '' which was first introduced by yao in 1982 as a computing task where two millionaires ( alice and bob ) wish to know who is richer without disclosing the amount of their wealth to each other .subsequently , boudot et al . , modified it to a task where the millionaires are only interested to know whether they are equally rich or not .thus , they wish to compute a function , where the subscripts and represent inputs from alice and bob , respectively .thus , and are essentially classical information corresponding to the assets of alice and bob , and a scheme for boudot s version of the socialist millionaire problem is actually a scheme for private comparison of equality of information .a lot of work has been done on classical private comparison , but as the security of all classical cryptographic schemes are conditional , it can never lead to an unconditionally secure scheme of private comparison .in contrast , we can achieve unconditional security in the quantum world .this fact led to many proposals for private comparison of equality of information using quantum resources , we may refer to all such protocols as protocols for `` quantum private comparison ( qpc ) '' .before we proceed further , it would be apt to note that qpc problem , the socialist millionaire problem and tierce problem are equivalent , and in what follows we will mostly refer to all such problems as qpc problem .  in the original definition of socialist millionaire problem , it was a two - party computation task , but a pioneering work of lo established that two - party secure qpc is not possible .this implies that to implement a secure qpc , we must have a third party ( tp ) , who would assist the users to compare the equality of their secrets .interestingly , the tp may be semi - honest , semi - honest having an intelligent robot , dishonest , or almost - dishonest . despite , the strong proof of lo, some efforts have been made to achieve qpc without tp , but they have been found to be insecure and/or unfair .thus , in what follows , we will concentrate on three party protocols of qpc , where a tp helps alice and bob to compare the equality of their information . such protocols for qpc have been proposed using different types of entangled states .for example , in ref . , a scheme for qpc has been proposed using -type state , state was used in ref . , bell state was used in , ghz state was used in .further , in ref . , a group theoretic structure of the protocols of quantum dialogue was proposed using a large number of different types of entangled states ( e.g. , and brown states ) , and it was shown that the quantum dialogue scheme proposed there can be converted to a protocol of the socialist millionaire problem , which is equivalent to qpc .thus , in , several options for realization of protocols for qpc were provided .further , in the similar line , in , a set of new options ( e.g. , 4-qubit state , 4-qubit cat state , etc . ) for realization of qd , and thus , qpc have been proposed .it s already established that the schemes for qpc have useful applications in private bidding and auctions , secret ballot elections , e - commerce , etc .( and references therein ) . due to the fact that a scheme for qpc has applications in many fields ,many variants of qpc have been studied in the recent past .for example , huang et al ., have recently proposed a ghz - state - based qpc scheme for users .huang et al.s scheme considers an almost - dishonest tp and allows him to compare the equality of the secrets of a subset of users , too .  inwhat follows , we plan to propose two protocols for qpc in the line of modified tseng - lin - hwang ( tlh ) protocol , which was proposed in its original form in 2012 . in the original tlh protocola scheme for quantum private comparison was proposed using bell states , but almost immediately after the publication of tlh scheme , yang et al ., had shown that there exist a security flaw in the original tlh scheme and other similar schemes , which assume tp to be semi - honest .yang et al ., also proposed a modified scheme which they claimed to be free from the limitations of the previous protocols as their scheme considered a dishonest tp .subsequently , in an interesting work on cryptanalysis of yang et al.s scheme , it was observed by zhang et al ., that yang et al.s scheme was also not completely secure . keeping these facts in mind , here we aim to propose two schemes of qpc that are free from the attacks described earlier and are fundamentally different from all the existing schemes as one of the proposed schemes is semi - quantum in nature and the other one is a completely orthogonal - state - based scheme .  in a practical situation ,all communication schemes are affected by the noise present in the communication channel .so it s extremely important to know how qpc schemes behave in the presence of noise .recently , some efforts have been made to investigate the effect of noise on qpc schemes .for example , the effects of pauli noise , and collective amplitude damping have been studied very recently .however , to the best of our knowledge , no serious effort has yet been made to rigorously investigate the effect of a complete set of possible noise models on the schemes of qpc .this paper aims to do that for the protocols proposed in this paper .remaining part of the paper is organized as follows . in section [ sec : protocols - for - quantum ] , we have proposed two schemes of qpc that are fundamentally different from the existing schemes of qpc . in section [ sec : analysis - of - security - efficiency ] , the security and efficiency of the proposed schemes are analyzed .the effect of various types of noise models on the proposed schemes is are investigated in section [ sec : effect - of - different - noise ] .finally , the paper is concluded in section [ sec : conclusion ] .in this section , we introduce two protocols for qpc .the first one is an orthogonal - state - based protocol , and the second one is a semi - quantum protocol . here, we assume that alice and bob wish to compare their assets with the help of a tp .   in quantum cryptography , orthogonal - state - based protocols are of fundamental interest as in contrast to bb84 type conjugate coding based protocol , where the security arises from the conjugate coding or noncommutivity , here ( i.e. , in case of orthogonal - state - based schemes ), the security arises from the duality ( monogamy of entanglement ) for single particle ( multipartite entangled ) states .the first ever orthogonal - state - based scheme of qkd was introduced in 1995 , and is now known as gv protocol . since then several quantum cryptographic schemes based on orthogonal states have been proposed for various cryptographic tasks .for example , orthogonal - state - based schemes have been proposed for qka , qsdc , dsqc , cdsqc , etc .a few of these schemes have also been experimentally realized .however , to the best of our knowledge , no effort has yet been made to design an orthogonal - state - based protocol for quantum private comparison . inwhat follows , the same is proposed in 8 steps , which are denoted by where osb stands for orthogonal - state - based .this notation is adopted to distinguish steps of this protocol from the steps of our second protocol , which is a semi - quantum scheme , and whose steps are denoted by where stands for semi - quantum .the proposed orthogonal - state - based scheme works as follows :  osb1 : : :  tp prepares -epr pairs randomly chosen from the set of  four bell states   .tp prepares two quantum sequences , namely and   , which contain all the first and second particles  of the epr pairs , respectively .+  thus , both and , contain   qubits , and if a sequence is transmitted via quantum  channel , we would require to check half of the travel qubits for  eavesdropping .the same can be done by adding decoy  qubits in each sequence .this is so because that would provide an  upper bound on the errors in the qubits not checked for eavesdropping ,  and the error would become asymptotically small for large values of   .osb2 : : :  tp prepares two copies of   .he uses the first  ( second ) copy as a set of decoy qubits   ( ) to be randomly inserted in the sequence   to perform eavesdropping check using  gv - subroutine ( detail of gv - subroutine can be found in ( * ? ? ?* ; * ? ? ?* ;  * ? ? ?specifically , tp randomly inserts in   ( in ) to  obtain a new enlarged sequence   .later , he sends the quantum  sequence to alice and   to bob .osb3 : : :  after receiving quantum sequences   alice / bob perform gv subroutine for eavesdropping check with the help  of tp . they compute the error rate , if this error rate is more than a  predetermined allowed threshold value , then they abort the protocol  and restarts from obs1 considering that an eavesdropper is present .  otherwise , they proceed to the next step .+  as the eavesdropping is checked by measuring the decoy qubits , after  this step , alice and bob are left with and   , respectively , because they remove the qubits  measured for eavesdropping check .osb4 : : :  qubits of and are now  measured in computational basis by alice and bob , respectively . both  would individually obtain -bits key strings of bit  values and corresponding to the  measurement outcomes and   , respectively .the bit string with alice ( bob )  is denoted as ( ) .osb5 : : :  alice and bob prepare a shared key   of -bits using an orthogonal - state - based scheme of qkd   or quantum key agreement ( qka ) .osb6 : : :  alice and bob have the private information regarding their assets  which they wish to compare .consider that the private information of  alice and bob is represented by the bit strings   and , respectively .alice encrypts her secret   with her key string and shared  key by using an exclusive - or  operation to obtain   .meanwhile , bob also encrypts his message with keys   and by using  the same operation to obtain   .  here, the superscript denotes the   bit of the -bits string .alice and bob send the  calculated strings and   separately to tp via a public channel .osb7 : : :  tp generates a classical bit string of  -bits corresponding to the choice of the initial bell  states in obs1 , such that for the bell state being     he generates   bit value in as 0 ( 1 ) .osb8 : : :  tp computes , which is exclusive - or result of   , , and as   .the  bit value 0 ( 1 ) of corresponds to the same  ( different ) values of and   .thus , iff   both  alice and bob have the same amount of assets .checking values of   , tp announces whether alice and bob have equal  amount of assets .interestingly , following similar argument , the existing controlled qd scheme proposed by some of the present authors in ref . can also be modified to design a completely orthogonal - state - based qpc protocol .however , we are not interested to elaborately discuss another orthogonal - state - based scheme . instead , here , we prefer to propose another foundationally relevant scheme which can work with limited quantum resources .now , we would try to design a semi - quantum protocol for qpc .in which , tp would be considered as the only quantum party ( i.e. , possess quantum resources ) while both alice and bob are classical parties . in convention ,a classical party can only access and process classical piece of information , i.e. , can only work in the computational basis .thus , being classical user , alice and bob can only measure and prepare qubits in computational basis .additionally , they can reflect the qubits without disturbing them .the first semi - quantum protocol of key distribution ( sqkd ) was introduced in 2007 .in which an unconditionally secure quantum key was shared between a classical and quantum party by using measure and resend and permutation of particles techniques to avoid eavesdropping . in what follows, we refer to these two equivalent techniques ( i.e. , the schemes discussed in refs . ) for eavesdropping checking as semi - quantum subroutine .the protocol proposed in ref . was widely discussed and was followed by a set of sqkd protocols ( and references therein ) , semi - quantum dsqc , etc . a protocol for semi - quantum private comparison ( sqpc )is described below . in the protocol proposed below, we will assume that alice and bob share a secure key , which has been distributed / produced using sqkd protocol proposed in ref . , in which alice and bob restricted to classical resources can share a quantum key .in other words , we assume that alice and bob know before they start following protocol for semi - quantum private comparison .sq1 : : :  same as osb1 , with the only difference that the number of epr states  prepared is sq2 : : :   and the classical users alice  and bob , respectively . as mentioned beforehand, the classical users  can only perform two operations , i.e. , either `` measure and resend ''  * * in the computational basis or * * `` reflect '' the qubit without any  change .* * sq3 : : :  both the users measure half of the received qubits ( i.e. ,   qubits ) randomly in computational basis and  replace them with the freshly prepared qubits in the same basis in  accordance with the measurement outcome .they also store the bit  values obtained in their measurement outcome corresponding to each  measured qubit .sq4 : : :  after receiving all the qubits from alice and bob , tp performs bell  measurements on the received pairs of qubits ( i.e. ,   and ) .a particular  measurement would yield one of the bell states . if tp s measurement  yields the same bell state as was initially prepared by him , he  announces 0 , otherwise he announces 1 .+  +  .[tab : sq - pc ] all possible cases that may appear in a semi - quantum  private comparison scheme are summarized in this table . here , both the  classical users can either choose to measure or reflect half of the  qubits , randomly and independently .we have explicitly shown all the  cases when the initial state was or   .it is only case 4 , which is of  interest , because only this case leads to successful key sharing  between alice and bob .case 1 is used for eavesdropping checking ,  i.e. , tp s announcement `` 1 '' for the qubits corresponding to case 1  may be considered as the signature of disturbance caused by eve .[ cols="^,^,^,^ , < " , ]  |=======================================================================  | |alice s |bob s | |outcome   | |choice |choice |initial state |initial state    |case 1 |r |r |  |   |case 2 |m |r |  |   |case 3 |r |m |  |   |case 4 |m |m |  |  |=======================================================================  +   +  out of initial bell states , in sq3 , both alice  and bob randomly measure one half of the received qubits .therefore ,  for a particular pair of qubits ( bell state , which  was prepared initially ) , there are four possible cases as listed below  and summarized in table [ tab : sq - pc ] .+  * case 1 : * both alice and bob reflect the incoming qubits undisturbed.  this case may not be useful in sharing a key between alice and bob ,  but it can be used for eavesdropping checking . as in the absence of an  eavesdropper and/or noise the initial state prepared by tp and the  final state obtained by him as the outcome of his bell measurement  should be the same .however , the presence of noise and eavesdropper  can lead to other bell states .thus , if tp prepares   and obtain any  other state at the end , he can conclude that there is an eavesdropper  or noise , but the converse is not true .however , since   any effort of eavesdropping will always be  detected .+  * case 2 and case 3 : * either alice or bob measures the qubit while the  other party reflects it .all these cases will be discarded as these  cases are neither useful in sharing key nor for eavesdropping  checking .+  * case 4 : * both alice and bob perform measurements in computational  basis on the qubits received by them , and store the measurement  outcome ( 0 or 1 ) to be used in the future .this case would lead to a  key .sq5 : : :  after the announcement of tp , both alice and bob disclose the  coordinates of the qubits they have measured .subsequently , both of  them make two separate lists corresponding to qubits falling in cases  1 and 4 .then , alice and bob check eavesdropping using semi - quantum  subroutine using qubits of case 1 . in which , both the classical  parties ensure that tp s measurement outcome is 0 when both have  reflected the qubits ( case 1 ) . in this case , about one - fourth of the  initial states ( i.e. , bell states ) will be  used for eavesdropping checking .if the errors are found to be lower  than a tolerable limit , they proceed to the next step , otherwise they  discard the scheme , and restart from sq1 .+  on the successful completion of this step ( i.e. , if the error rate is  found to be less than the tolerable limit ) , alice and bob obtain   and , respectively ,  where and are  correlated strings of length -bits .sq6 : : :  on request of alice and bob , tp discloses his initial choices of bell  states for cases , that are selected randomly .  subsequently , using the values of and   ,alice and bob compare the parity values  for the bell states prepared by tp and the bit values obtained by them  for all cases .if it matches , they proceed , otherwise  they abort .+  this would check whether tp had genuinely prepared a bell state or  not , which in turn would ensure the correlation in the reduced  -bit strings and   .this disclosure will also be relevant in some of  the intercept and resend attacks of dishonest alice or bob .this  particular aspect will be discussed in detail in the next section .sq7 : : :  same as obs6 .sq8 : : :  same as obs7 .sq9 : : :  same as obs8 .before we analyze the security of the proposed protocols , it would be beneficial to summarize the information known ( or unknown ) to each party . to be specific , in both the protocols proposed in the last section , tp has the information of the initial bell states , which is unknown to both alice and bob .this ensures 1 bit of ignorance for both alice and bob .this is so because the initial bell states can be distinguished as parity 0 ( ) state and parity 1 ( ) state . for a parity ( 1 )state the final key bits obtained by alice and bob would be the same ( complementary of each other ) .hence , the ignorance due to the lack of knowledge about the initial bell state is 1 bit .interestingly , the initial choice of bell states also ensure 1 bit of ignorance for tp as he has knowledge about the parity of the initial bell state , but he does not know whether alice s ( or bob s ) measurement outcome was 0 or 1 .further , alice ( bob ) has the shared quantum key and correlated key ( ) , which are not known to tp .it is evident that anyone interested to know the secrets of alice and/or bob ( and/or ) would require all three parts of information .now , we can consider various possible attacks by each party to gain the information not accessible to them .for example , tp can try to take benefit of state preparation by preparing a state other than bell state to gain advantage by entanglement swapping attack .in fact , he can prepare an arbitrary quantum state to exploit the state preparation , and try to extract correlated keys and .however , the unconditional security of the quantum key would maintain security against any such attempt as both alice and bob encrypt their messages regarding asset information and , respectively using the quantum key .in other words , it can be viewed as alice and bob performing communication in a secure manner by using a quantum key and the security of message can be attributed completely to the security of the quantum key .additionally , alice ( bob ) may wish to extract bob s ( alice s ) key ( ) by designing an intercept and resend attack . the eavesdropping checking mechanism adopted in both the protocols ( by gv and semi - quantum subroutines ,respectively ) ensures security against this kind of an attack .it may be noted that the eavesdropping checking procedure adopted here would provide security against both insider and outsider attackers .a modified intercept and resend attack strategy by a classical alice ( bob ) in sqpc scheme could be to intercept all the qubits sent to bob ( alice ) and keep them in a memory .though , possession of a quantum memory by a classical party is beyond his / her limits , but it is an interesting scenario to be investigated .suppose alice measures half of the qubits which tp has sent to her , and follows the same for the corresponding bob s qubits in her possession .depending upon the outcomes of her measurement , alice prepares new qubits and sends ( the same number of qubits tp would have sent ) them to bob after adding the remaining auxiliary qubits .bob proceeds with the protocol as expected , then alice intercepts all the qubits bob returns to tp and replaces them with the qubits ( initially sent by tp to bob ) in her possession .using this approach , she will get completely as it will be prepared using a subset of the string of qubits , she had measured and sent to bob . though this kind of an attack can be circumvented by bob s critical analysis of the final key ( ideally ( i.e. , when alice has not made any attempt to attack the protocol ) , in all those cases , where bob has measured and resend his qubit while alice has not measured her qubit , tp s announcement should have almost equal probabilities for 0 and 1 . however ,if alice has performed this attack , then tp would broadcast 0s for all these cases . ) .further , a standard solution would be a strict time checking .in which , tp announces the sending time of the qubits . using this fact , and the time of receipt of the qubits both alice and bob can check the travel time and discard the protocol if they find any anomaly .the qubit efficiency of the proposed qpc protocols can be calculated using the quantitative measure proposed in refs .the efficiency where number of classical bits are transmitted using the total number of qubits , and -bits classical communication is involved .it should be noted that only the classical communication used for decoding the message is considered , not the classical communication required for eavesdropping checking . in both the orthogonal - state - based qpc and sqpc protocols , alice andbob share their -bit secrets , contributing equally in .the orthogonal - state - based qpc involves qubits , which are shared in a secure manner using number of additional decoy qubits .finally , alice and bob announce of -bits each which is followed by 1 bit of classical communication by tp ( i.e. , the announcement that discloses whether the assets are equal ( say , 0 ) or not ( say , 1 ) ) .additionally , alice and bob also share a quantum key and the qubits and bits used to obtain this shared key should also be counted in the computation of efficiency . here , we have chosen orthogonal - state - based qka scheme for considering the resources involved in sharing the quantum key . using shuklaet al.s scheme , and exploiting the dense coding capacity of the quantum channel used , an -bit quantum key can be shared by two parties using qubits ( as quantum channel and decoy qubits ) , which involves -bit classical communication .therefore , total number of qubits used are * * with additional classical communication * *. thus , the efficiency of the orthogonal - state - based qpc protocol would be , which becomes 15.38% for large .similarly , in case of sqpc protocol , qubits are initially prepared by tp , subsequently qubits are prepared by alice and bob ( by each ) as a replacement of the qubits measured by them .tp announces bit measurement outcomes which alice and bob use to generate their keys with the help of bit of classical messages available with each of them .finally , alice and bob both share an -bit encrypted messages which helps tp to obtain and announce 1 bit final result .additionally , both classical parties also share a secure quantum key for which they use bits ( i.e. , tp initially prepares qubits , and alice ( bob ) prepares additional ( ) qubits ) .similar to the sqpc protocol , tp also announces -bit classical communication which is followed by -bit announcement by both alice and bob .it is important to note that alice and bob keep the bit values only for specific announcements by tp . here, we have considered the asymptotic case where equal probabilities for favorable and unfavorable measurement outcomes of tp has been considered . using all these valuesthe efficiency of the sqpc protocol is . for large values of , . thus, the protocol 1 proposed here is more efficient than the protocol 2 , which is not surprising as we have considered two classical users in the second protocol .  here, it would be worth mentioning that in analogy with the classical communication involved in eavesdropping checking process ( which is not counted in computation of , we have not counted the classical bits that are used to check correlations between and .in this section , we aim to study the effect of a set of noise models on the feasibility of both the qpc schemes proposed here . for this we will use the operator sum representation of a transformed quantum state ( ) ,initially prepared in , which is {n}{\sum}k_{i}\rho k_{i}^{\dagger}.\label{eq : noise - affected}\ ] ] here , the operators satisfying {n}{\sum}k_{i}^{\dagger}k_{i}=\mathbb{i}$ ] ensure the trace - preserving nature of the quantum channel . inwhat follows , first we will describe the kraus operators of the noise channels and mathematical formulation used in the present paper .subsequently , we will analyze both of the proposed schemes under the effect of various noise models .specifically , we will consider the amplitude damping ( ad ) channel , bit flip channels , phase flip channels and depolarizing channels .let us first describe an ad channel , which causes the loss of energy from the system to its surrounding considered as a reservoir at vacuum .  1 .: :  * amplitude damping : * the kraus operators for ad channels are   and   . here , andin what follows , is  the probability of error due to the specific noise discussed .we will also study the effect of pauli noise , which affects the qubit independent of its initial state with a certain probability and leaves it unaffected with the remaining probability .the error caused due to this kind of noise can be studied as bit flip ( bf ) , phase flip ( pf ) , and depolarizing channel ( dc ) noise .  2 .: :  * bit flip channels : * the bf channel , which flips the qubit with  probability and leaves it unchanged with remaining  probability , can be given by the kraus operators   and   .3 . : :  * phase flip channels : * similar to the bf channels , the kraus operators  for a phase flip channel are given by   and   , which flips the phase of the qubits with   probability while identity acts with the remaining  probability .it is interesting that the effect of phase damping noise  on certain scheme can be reduced from the obtained results for phase  flip channels ( and references therein ) , so we have not  explicitly studied the phase damping kind of noise .4 . : :  * depolarizing channels : * if the noisy channels leaves the state  unchanged with a certain probability and completely mixed with the  remaining probability , the kraus operators for such a channel would be   ,   ,   , and    in both the qpc schemes proposed here , tp initially prepares one of the bell states and shares that with two distant parties .therefore , will be a two qubit density matrix evolving under two independent quantum channels .mathematically , the transformed density matrix can be written as where corresponds to the probability of decoherence in the qubit and are the kraus operators of the noisy channel acting on the particular qubit , with . to quantify the effect of noisewe will use a distance based measure , fidelity , between the quantum state received by the receivers ( alice and bob here ) after the effect of a specific noise channel and the quantum state expected in the ideal condition , when noisy channel do not alter the quantum state .in fact , in the ideal condition , the state at the receivers end should be the same prepared by tp .therefore , the fidelity between the initial ( ) and final ( ) state is here , it is also worth mentioning that the fidelity expression used here is square of the conventional fidelity expression but is also used widely as a measure ( and references therein ) .now , we will discuss the effect of all these noise models on the proposed schemes .it is noteworthy that the qubits used by alice and bob to prepare their respective keys undergo the same fate in both the schemes .therefore , the discussion what follows is applicable to both orthogonal - state - based qpc and sqpc protocols . specifically , when both the qubits of the initial state are subjected to ad noise the compact analytic expression of the obtained fidelity is    where and are the probabilities of errors in the first and second qubits , respectively .the same calculated for the different choice of initial state , i.e. ,  in what follows , the same fidelity expressions are obtained irrespective of the initial choice of bell state under different noisy environment .  similar to the earlier case , when the first qubit evolves under the effect of ad channel , whereas the second qubit is affected by bf channel the calculated fidelity turns out to be    similarly , when the first qubit is subjected to ad noise and the second qubit evolves under the effect of pf and dc noise models , we obtain the fidelity expressions as    and    respectively .  before we consider the effect of other kind of noise models ( other than ad ) on the first qubit , it is worth noting that the scheme is symmetric with respect to alice and bob ( due to the symmetry of the bell states used ) , and the similar fidelity expressions are expected whether the first ( second ) qubit is affected by ad ( bf ) or the first ( second ) qubit is affected by bf ( ad ) noise .in fact , if we interchange the values of and we can obtain the fidelity expression of one from the other . due to this fact, we are enlisting here only the expressions for fidelity , for those cases which lack this kind of symmetry ( i.e. , which can not be obtained from another expression by simply using a symmetry argument ) .if the first qubit is subjected to bf noise , while the second qubit evolves under bf , pf and dc noises , then a systematic computation would yield the fidelity expressions as      and    respectively .it is interesting to report that the fidelity obtained while both the qubits evolve under pf environment is exactly the same as that under bf noise ( [ eq : bf - bf ] ) .if we consider pf noise on the alice s qubit and dc noise on the bob s qubit we obtain a closed form analytic expression of fidelity as    finally , when both the qubits are subjected to dc noise , the obtained fidelity between the quantum state affected by noise and the initial state is    before we proceed with the analysis of the various fidelity expressions obtained under different combinations of noisy channels , it is worth establishing the motivation of this study .specifically , we have already mentioned the fidelity between the initial and final state in the ideal conditions is expected to be unity . for the sake of argument , consider that one of the channel ( either tp to alice or tp to bob ) is noiseless , then this unit fidelity falls considerably ( cf .[ fig : qubit1or2 ] ) .specially , for the higher values of decoherence rate , it even becomes null for bf or pf channels .more realistic scenario would be where both the qubits undergo decoherence .for simplicity of discussion , if we consider that the rate of decoherence in both the channels is the same , then the fidelity of the obtained state evolving under bf or pf noise turned upside down to become unity for higher probability of errors .though , a similar benefit appears when the qubits evolve under ad noise .however , no such advantage is seen in presence of the dc noise .the fidelity variation of the quantum states for the qpc protocols in specific cases of noisy environment is shown here .the smooth ( blue ) , small - dashed ( red ) and dot dashed ( cyan ) curves correspond to the fidelity of the obtained states when only alice s ( or bob s ) qubit is subjected to ad , bf and dc noises , respectively .similarly , the dotted ( black ) , large - dashed ( orange ) and large - dot dashed ( magenta ) lines show variation in the fidelity of the obtained states with probability of errors , when both alice s and bob s qubits are subjected to the same kind of noise with equal probabilities of errors , i.e. , ad , bf and dc noises , respectively . in both the cases of pf channels , the same fidelity as that under bf noiseis obtained . as discussed in the text the choice of initial bell states only matter when both the qubits are subjected to ad noise .here , the dotted ( black ) line shows the fidelity variation for the initial state while the small - dashed ( red ) line corresponds to the initial state . ]we have already reported ten fidelity expressions ( eqs .( [ eq : ad - ad])-([eq : dn - dn ] ) ) calculated for the quantum states shared between all the parties and evolving under different noise channels .all the analytic expressions depend on two independent variables ( probability of error ). therefore , variation of fidelity with these variables can be explicitly illustrated using contour plots .specifically , in figs .[ fig : ad]-[fig : pf - dc ] , we have shown the contour variation of eqs .( [ eq : ad - ad])-([eq : dn - dn ] ) . in fig .[ fig : ad ] , the fidelity as a function of both the probabilities is shown when the initial choice of bell state by tp was and in ( a ) and ( b ) , respectively .a gradual decay in fidelity with each independent probability appears in ( a ) , and when both the decoherence rates are higher , relatively better results in terms of fidelity are visible .whereas ( b ) gives a contrasting picture , and fidelity is found to fall continuously with both the probabilities .the comparative analysis reveals that gets less affected than .it may be easily observed in both of these plots ( i.e. , figs .[ fig : ad ] ( a ) and ( b ) ) that the contour variation is symmetric along a diagonal from lower left to top right corner due to the fact that the same noisy channel is affecting both the qubits .it will remain valid in all such plots to follow ( cf . figs .[ fig : bf1-pauli2 ] ( a ) and [ fig : pf - dc ] ( b ) ) where the same noise acts on both qubits and symmetric nature is visible in the contour plots .qpc protocols subjected to ad channels , i.e. , both the qubits evolve under ad noise . in ( a ) and ( b ), the choice of the initial bell state by tp is and , respectively . ]if we consider that quantum channel for tp to alice is characterized as ad while bob s qubit is affected by one of the bf , pf or dc noise , then the symmetry observed in fig .[ fig : ad ] is lost ( cf . fig .[ fig : ad1-pauli2 ] ( a ) , ( b ) , and ( c ) , respectively ) .the variations of fidelity in all these cases show that it is not sufficient to characterize one quantum channel , as the type of noise applied to the second qubit is also relevant .if we look closely at / in these three contour plots ( i.e. , fig .[ fig : ad1-pauli2 ] ( a ) , ( b ) , and ( c ) ) , we can see the same seven contour lines at the starting point at / on the -axis undergoing different variation due to the effect of the noisy channel of the second qubit .among all these noise models , dc noise is found to minimally affect the fidelity , while pf noise is found to have most devastating effects .qpc protocols subjected to a noisy environment , when the first qubit of the bell state ( alice s qubit ) is subjected to ad , while the second ( bob s ) qubit evolves under different noisy channels . in ( a ) , ( b ) , and ( c ) , it evolves under bf , pf , and dc , respectively . ]another possibility is that the first qubit is subjected to bf noise . in fig .[ fig : bf1-pauli2 ] ( a ) , another qubit is also traveling through the bf channel .interestingly , the degrading of fidelity with an increase in probability of error in one of the channels can be controlled by increasing the decoherence rate for the other channel .the same intercepts on the -axis for all three plots correspond to the same type of noise acting on the first qubit . unlike the second qubit evolving under bf channel, pf noise does not check the decay in fidelity ( cf .[ fig : bf1-pauli2 ] ( b ) ) .in fact , least fidelity is obtained in fig .[ fig : bf1-pauli2 ] ( b ) when compared among the second qubit evolving under bf , pf , and dc noise . while the second qubit is subjected to dc noise ( fig .[ fig : bf1-pauli2 ] ( c ) ) , the fidelity can be further maintained , but the variation is quite different from fig . [fig : bf1-pauli2 ] ( a ) .it should be noted that when the first ( second ) qubit is subjected to bf ( ad ) noise contour variation of the fidelity can be obtained by interchanging the and axes in fig .[ fig : ad1-pauli2 ] ( a ) . due to this reason, these cases will not be further discussed .similarly , fig .[ fig : bf1-pauli2 ] ( a ) also shows the variation of fidelity when both qubits travel through the pf channels .qpc protocols subjected to noisy environment , when the first qubit of the bell state ( alice s qubit ) is subjected to bf , while the second ( bob s ) qubit evolves under different noisy channels . in ( a ) ,( b ) and ( c ) bob s qubits is affected by bf , pf , and dc , respectively . as mentioned in the text ( a ) also corresponds to the pf noise on both the qubits . ]another interesting possibility would be to consider the first and second qubits evolving under pf and dc noise , respectively .it can be observed from fig .[ fig : pf - dc ] ( a ) that dc noise affects the fidelity comparatively less than other noises .finally , we consider both the qubits subjected to dc noise in fig .[ fig : pf - dc ] ( b ) .the symmetric decay in fidelity shown in this case has a specific characteristic that even the lowest value of the obtained fidelity is an appreciable amount when compared with the remaining cases .( a ) qpc protocol subjected to a noisy environment , when the first qubit of the bell state ( alice s qubit ) is subjected to pf while the second ( bob s ) qubit evolves under dc . ( b ) similarly , when both the qubits of the bell state ( i.e. , alice s and bob s qubits ) evolve under the effect of dc noisy environment . ]it is also important to observe a comparative analysis of the effect of different kinds of noisy channels .specifically , if we consider the first qubit travels through a specific channel with the known rate of error than the effect of noisy environment due to the second qubit is studied here in two dimensional variation .specifically , in fig .[ fig:2dad ] , we consider two cases ( namely , decoherence rate and 0.8 ) of variation of fidelity with / in error rate for the second channel .the effects of different values of are shown in the figure , as for its higher value , the fidelity remains the same for initial state , while falls sharply for initial .it can be seen that for a higher rate of damping the effect of noise on the other qubit is hardly relevant ( cf .[ fig:2dad ] ( b ) ) .  the effect of noise on the fidelity obtained in the qpc protocol when the first qubit is subjected to ad noise , while the second qubit is subjected to different noisy channels .the smooth ( blue ) , large - dashed ( orange ) lines correspond to the effect of ad noise on the second qubit for the initial bell state and , respectively .similarly , the dotted ( red ) , small - dashed ( magenta ) , large - dot dashed ( cyan ) curves correspond to the fidelity of the obtained states when bob s qubit is subjected to bf , pf and dc noises , respectively . in ( a ) and ( b ), the decoherence rate for the alice s channel is 0.2 and 0.8 , respectively . in the plots , ( also in the following figures )the probability of error is written as . ]  similarly , when the first qubit evolves under the effect of bf channel with low error rate , the second qubit subjected to dc noise performed best as shown in fig .[ fig:2dbf ] ( a ) .however , in fig .[ fig:2dbf ] ( b ) the second qubit under bf channel outperformed it for higher probability of error .interestingly , the two dimensional cut shown in fig . [ fig:2dbf ] ( b ) the fidelity is seen to increase with increasing probabilities of errors in the second channel .  the effect of noise on the fidelity obtained in the qpc protocol when the first qubit is subjected to bf noise , while the second qubit is subjected to different noisy channels .the smooth ( blue ) , dotted ( red ) , dashed ( magenta ) , dot dashed ( cyan ) curves correspond to the fidelity of the obtained states when bob s qubit is subjected to ad , bf , pf and dc noises , respectively . in ( a ) and( b ) , the probability of error in alice s channel is 0.2 and 0.8 , respectively . ]  when the first qubit is fixed to be traveling through a pf channel with intermediate error rate , the second qubit evolving under dc noise suffers most ( cf . fig .[ fig:2dpf ] ) . as the fidelity in this case is initially highest for lower values of error in the second channel , becoming second to the pf channel for higher errors .the obtained fidelity in the same case , i.e. , second qubit under dc noise , falls comprehensively even below ad noise for sufficiently large values of error rates for the second channel .the effect of noise on the fidelity obtained in the qpc protocol when the first qubit is subjected to pf noise , while the second qubit is subjected to different noisy channels .the smooth ( blue ) , dotted ( red ) , dashed ( magenta ) , dot dashed ( cyan ) curves correspond to the fidelity of the obtained states when bob s qubit is subjected to ad , bf , pf and dc noises , respectively . in ( a ) and ( b ), the probability of error in alice s channel is 0.4 and 0.6 , respectively . ]  finally , we will consider when the first qubit subjected to dc noise in fig .[ fig:2ddc ] . here , we can observe that for higher values of error in the first channel though fidelity in two of the cases appears increasing , but it is consistent with corresponding contour plots ( cf . figs .[ fig : ad]-[fig : pf - dc ] ) .it is worth commenting here that fidelity obtained when the second qubit is evolving under the bf channel improves considerably to be better than in damping and dc noise .the effect of noise on the fidelity obtained in the qpc protocol when the first qubit is subjected to dc noise , while the second qubit is subjected to different noisy channels .the smooth ( blue ) , dotted ( red ) , dashed ( magenta ) , dot dashed ( cyan ) curves correspond to the fidelity of the obtained states when bob s qubit is subjected to ad , bf , pf and dc noises , respectively . in ( a ) and ( b ) , the probability of error in alice s channel is 0.2 and 0.6 , respectively . ]so far , we have been discussing the effect of noise on the qubits which are used by alice and bob to obtain random key strings . however , it is also desirable that during eavesdropping checking stage the presence of eve and noise can be discriminated .this was the motivation of our last work , where we have reported that different suitable choices of decoy qubits and eavesdropping checking subroutines suitable over different noisy channels . as the gv subroutine was explicitly discussed in ref . , we will only discuss the effect of noise on the semi - quantum subroutine .it is worth noting that unlike bb84 or gv subroutine , where the principle of security varies from each other , in a semi - quantum subroutine , the security can be inherently achieved by using one of them .specifically , in a semi - quantum subroutine , a quantum party prepares the decoy qubits and classical party reflects them to quantum party , who later calculates the error rate for that . depending upon the choice of decoy qubits bb84 or gv subroutine can be used .however , from the point of view of the effect of noise on this subroutine , to and fro communication of decoy qubits should be considered while studying the effect of noisy environments . here ,for the simplicity of discussion , we will consider the same kind of noise with the same error rate acting on certain qubit while its to and fro travel .  specifically , when both the qubits undergo damping during both rounds of travels , the obtained fidelity expressions are    and    for the initial choice of bell state as and , respectively . in the remaining cases , when the first qubit is subjected to damping and the second qubit is considered under bf , pf , and dc noises , the obtained fidelity is calculated as      and    respectively . in another case , when the first qubit is subjected to bf channel and second qubit evolves under the effect of bf , pf , and dc noise the analytic expressions of fidelity are      and  ,\label{eq : bfp1-dnp2 } \end{array}\ ] ]  respectively .similarly , when the first ( second ) qubit is affected by pf ( dc ) noise , then the fidelity can be calculated as    finally , when both the qubits are subjected to dc noise , we obtain the expression for fidelity as  .\label{eq : dnp1-dnp2 } \end{array}\ ] ]  the fidelity expressions for the remaining cases are not explicitly written here , as those expressions can be derived from the above expressions by symmetry argument . the analysis of the obtained fidelity expression is simplified here by considering the case when both the qubits are affected by the same kind of noise with the same error rate . in this particular case , as shown in fig .[ fig : semi - quantum ] , it is observed that the fidelity for the damping effect , for as the initial choice of tp , gradually decays to zero .further , for the state evolving over the bf or pf channel , a revival of with an increasing error rate is observed . for the remaining cases ( i.e., for state under ad channels and an arbitrary bell state subjected to dc noise ) , fidelity falls gradually and becomes constant around .  the effect of noise on the fidelity of the quantum states used by tp for eavesdropping checking in the sqpc protocol when the first both the qubits are subjected to ad noise for initial choice of bell states as and in the smooth ( blue ) and dashed ( magenta ) lines , respectively .the dotted ( red ) and dot dashed ( cyan ) curves correspond to the fidelity of the obtained states when both qubits are affected by bf ( equivalently pf ) and dc noises , respectively . ]two protocols for qpc have been proposed .the essential beauty of the present work underlies in the fact that both of the proposed protocols for qpc are fundamentally different from all the existing protocols of qpc . specifically , the first protocol is designed solely using orthogonal states and the security of the protocol does not rely on conjugate coding .this is interesting because of the fact that bb84 protocol for qkd and early protocols of secure quantum communication ( see chapter 8 of ref . for a review ) , whose security arise from the fact that one can not perform simultaneous measurements in mutually unbiased bases ( conjugate coding ) , led to a perception that unconditional security of qkd and other quantum tasks arises from the conjugate coding ( i.e. , through the use of non - orthogonal states as the states prepared in mutually unbiased bases are not orthogonal to each other ) .interestingly , in refs . it has been firmly established that qkd is possible without using non - orthogonal states , i.e. , by using orthogonal states only .subsequently , a set of orthogonal - state - based protocols for various secure quantum communication tasks has been proposed by some of the present authors , and those protocols have established that most of the quantum cryptographic tasks that can be performed using non - orthogonal states can also be performed using orthogonal states . specifically , it has been established that one can perform quantum secure direct communication , qka , etc . , solely using orthogonal states .these orthogonal - state - based protocols are fundamentally different from the conjugate coding based schemes as their security does not arise from our inability to perform simultaneous measurement using two mutually unbiased bases .however , until now , no effort has been made to design an orthogonal - state - based protocol for qpc .thus , the first protocol of the present paper is fundamentally different from all the existing protocols for qpc , and it is unique in that sense .interestingly , the second protocol proposed here is also an orthogonal - state - based , but in the case of the second protocol we did not stress much on this characteristic , because it has another unique characteristic , which has not been explored in any of the existing schemes for qpc : the protocol is semi - quantum in nature , i.e. , it allows some of the users to be classical .these two protocols answer couple of questions ( in context of qpc ) that are foundationally important . for example , both the protocol answer : which quantum properties are essential for the implementation of schemes of qpc ?it establishes that conjugate coding is not required and ensuring simultaneous non - availability of all the pieces of information to eve after splitting it into several pieces is sufficient to ensure security of schemes for qpc .specifically , conjugate - coding - based schemes achieve security by hiding the basis information of two mutually unbiased bases , whereas an orthogonal - state - based employs geographical information splitting using pop technique .the second protocol answers another question : how much quantumness is needed ( or in other words , how many quantum users are needed ) to implement a scheme for qpc ?it is established that only tp needs to be quantum and all other users can be classical .this is in sharp contrast to the existing protocols where all the users are required to be quantum .thus , the second protocol of ours clearly uses reduced quantum resources , as two parties comparing their assets are completely classical in nature .however , there exists a trade - off between the amount of quantum resources used and the qubit efficiency achieved .specifically , the qubit efficiency of the sqpc protocol ( i.e. , our second protocol ) is found to be much lower compared to the first protocol where all the users are quantum in nature .clearly , this happens because in sqpc , a lesser number of parties possess quantum resources and that increases the requirement of and for accomplishing the same task ( i.e. , to communicate the same amount of classical information by following the restrictions of the same cryptographic task ) .the feasibility of both of the proposed schemes is analyzed under well - known noise models , such as ad , bf , pf , and dc .the study has led to many interesting conclusions . in general ,the effect of any specific noise model is independent of the choice of the initial bell state .however , it is observed that when both the qubits are subjected to ad noise , the fidelity expressions depend on the parity of the bell state .further , the study has established that the quantum state is least affected due to dc illustrated through a higher value of fidelity .interestingly , it has also been observed that the higher error rates in one of the quantum channel can also lead to positive effects .specifically , a few cases have been observed where the higher decoherence rate in one of the quantum channel had resulted in higher fidelity compared to the situation having lower error rates in the same channel .another interesting result has been obtained when both the qubits were traveling over a bf channel , as in this case a revival in fidelity with increasing error rate in a quantum channel has been observed .the effect of noise in the eavesdropping checking for sqpc protocol is also considered , as it is necessary to differentiate between noise and eve .it has been observed that the parity 1 bell states have least fidelity when subjected to ad noise .  keeping the above in mind, we conclude the paper by noting that a qpc scheme neither requires user other than tp to be quantum , nor it requires to use the quantum states prepared in mutually unbiased bases . however , to implement a scheme for qpc in a realistic situation , characterization of the channel ( knowledge of noise(s ) present in the quantum channel ) would play a very important role , as in the absence of this knowledge , tp may reach to an incorrect conclusion .

Abstract: private comparison is a primitive for many cryptographic tasks , and recently several schemes for the quantum private comparison have been proposed , where two users can compare the equality of their secrets with the help of a semi - honest third party ( tp ) without knowing each other s secret and without disclosing the same to the tp . all the schemes proposed so far achieve the secrecy by using conjugate coding , and all the users involved in the process are required to be quantum in the sense that they can perform measurement(s ) and/or create states in basis other than computational basis . in contrast to these existing schemes , here we propose two new protocols for quantum private comparison , which are fundamentally different from the existing schemes . the first one does not use conjugate coding ( uses orthogonal states only ) and the second one allows the users other than tp to be classical whose activities are restricted to either reflecting a quantum state or measuring it in computational basis . further , the performance of the protocols is evaluated under various noise models ( e.g. , amplitude damping , bit flip , phase flip , depolarizing channel , etc . ) , and thus , the feasibility of performing quantum private comparison in a realistic situation has been established .