the concept of natural selection , proposed by charles darwin and alfred russell wallace , is underlyed by a fundamental driving force that explains how the different types of species are in a permanent flux .several models apply this principle to modelate different kinds of populations using simple concepts , namely mutation ( i.e. changes in the genotype and phenotype of the individuals ) and selection ( according to the fitness differences between the elements of a population ) .additionally , a random drift in the whole population must be included in order to account for fluctuations and effects of noise in the population . in the recent years, some works have been analyzed fundamental mathematical aspects of co - evolutive dynamics in populations assuming frequency dependent selection ( a good review is given by nowak et . ; an estandart reference is the book of hofbauer and sigmund ) .however , this concept of evolution does not only concerns only dynamics of populations , but also systems in economy and even molecular biology .general abilities are usually assigned to competing populations of individuals of two different classes and , in order to model mutation - selection processes .mathematically , such processes are usually modeled by game theory , where two individuals playing a game increase or decrease their scores according to the actions they adopt . in the context of biological sciences ,the score of the game is equivalent to the fitness of an individual .furthermore , if the individual plays in a rational way , the probability to obtain a higher fitness increases .therefore , the selection of a rational action depends on whether selfishness or altruism may improve the final fitness of the individual .  based on this principle, a kind of phase behavior can be defined for infinite populations . if the fitness ( defined by a pay - off matrix ) is used as a control parameter , then it is possible to control the number of individuals of both classes and into the population ( see for example , where the interaction matrix is rescaled ; lee et al . make an analysis of games with different pay - off functions ) .however , population dynamics can not be simply reduced to this phenotypic point of view .internal characteristics ( which can be more or less simple ) , related with the way in which individuals try to find out in an even more rational way the best action , must also be included .  within the present work, the phenotype will be related to the action each individual adopts ; this phenotype is a dynamical variable that depends on the perception and memory of individuals , which is of variable size .hence , the selection of a new action is based on the information stored from past moves , i.e. is a rational choice based on the direct information of the game ( and indirectly on the score each individual obtains ) .the individual s action evolves in order to adapt itself to the changes in the fitness .simultaneously , the size of the storage device evolves depending on the variation of the individual s fitness . from a biological point of view, the memory size is related to the individual s genetic pool , because it influences the strategies that each individual adopts as response to the pressure of the evolution dynamics .hence , depending on the actions , the genetic pool can mutate in order to increase the fitness .thus , individual s identity is changing .several works make an analysis of the effect of learning schemas in the co - evolutive dynamics and equilibrium of games . c. hauert also analyze the effect of the pay off matrix in the evolution of strategies in agents with large memories ; an alternative schema in economics is proposed by r. selten and r. stoecker ) . in theseworks the storage device is implemented using a linear schema , as a kind of genetic code .other models have used notions of learning implemented into the dynamics of the whole population and not into the dynamics of the individual itself .such schemas are not suitable to implement large memories .additionally , they do not allow the implementation of an analysis of the system under different cognitive scenarios , in particular under the influence of _ external information sources_.  the present model combines the notion of an ising learning schema and game theory , joining the stochastic of the population with the stochastic of a primitive perception model .such a model allows us to implement individuals with larger memories , implying a more complex dynamics of the memory distribution .furthermore , with this schema it is possible to model primitive learning aspects .we are also able to analyze the effect that external information may have onto the system ( similar to an _ external field _ ) .such external information source induces a kind of _ normative imitation _ into the individuals .the selection of strategies is therefore not only influenced by the dynamics of the population , but also by the dynamics related to the individual learning schema .the actions as well as the storage size co - evolve and are two observables in this model .the consideration of this dynamical memory as an observable of the population is a way to define diversity in the population accurately . from a biological point of view , the observation of the distribution of actions is a measure of the phenotypic diversity ; at the same time the observation of the distribution of memories is a measure of the genetic diversity .the basis of both distributions is a source of energy provided by the environment . in his book published in 1944, schrdinger explains that life maintains order by degradation of free energy and producing high entropy waste .however , this argument seems to be not enough to explain why diversity is not uniformly distributed . in the present modelthe _ diversity _ is measured as a function of the _ number of microstates with a given memory_. this observable has been overlooked , ignoring its wide - ranging effects on community structure and ecosystem process . given that this distribution depends on the pay - off and the internal energy of the individuals , as well as the external information , we are able to explore the _ shannon s distribution index _ , equivalent to the negative entropy of the system , as a function of the learning characteristics of the individuals as well as the influence of external information sources ( for example genetic material transported by water , wind or insects ) .an important question to solve is if the evolutive process takes place at the individual s or at the population s level .if the dynamics of the population is stochastic , then local variations of the processes may introduce qualitative changes in the dynamics . in the present approach individual s information processingintroduces an additional element into the system .the whole population affects the fitness of each individual .but , if the individual is able to process information in order to produce an action , then the coevolutive process does not have only origin at the population level , i.e. individual characteristics influence the co - evolution of the population .  in the next sectionthe fundamental definitions and the mathematical basis of the model are introduced . in the first subsectionthe mathematical basis of game theory is explained ; in the second and third subsections the dynamics of the actions and evolution of the memory size is explained .the last subsection contains the implementation of the model . in section number threeare two big parts : one for the analysis of the distribution of actions and a second for the analysis of the distribution of memories . in the last section a discussion of the resultsis presented .we describe a population of individuals .each individual adopts an individual action that influences its fitness . according to this action ,each individual can be of two types , either ( for individuals that cooperate ) or ( for individuals that defect ) .additionally the individuals can observe and store its own as well as the opponent s actions . with this informationthe agent can decides the new action .the size of the storage device ( memory size ) is also an individual characteristic and is defined as an observable .the action of individual is represented by a unitary vector , that can be either or for the two classes of actions and respectively .the pay - off in the system depends on these actions and is given by the following matrix    where is the opponent s action , is the fitness associated to the action of both individuals and . in the present model only pair interactions are allowed .the fitness matrix is represented by the scenario implemented in this work is for a prisoner s dilemma game , i.e. the matrix has the values , where is for temptation , for reward , for punishment and for sucker . in this model , , and . using eq .( [ util_i ] ) , the outcome for the individual can be computed as where is the number of neighbours .the utility is a fundamental quantity that should determine the co - evolutionary process .   _memory _ is defined as an individual device where the information of previous actions of the individual and its neighbours is stored . using this informationa new action can be adopted .there are several techniques that simulate how this storage of information and implementation of new moves takes place ; usually such techniques are based on linear methods . in the present work , the memory works like a field that influences the dynamics of the new action ( that means the memory is associated to a kind of internal energy ; see fig .[ figissing ] ) . using this schemathere is a relation between the perception of the individual ( observation of the actions in the game ) and the perturbations and errors implemented in the game ( noise ) .is adopted if it minimizes the energy asociated to the chain of spins in the memory.,scaledwidth=40.0% ]  an ising schema is adopted . for simplicity , the perception lattice in the mind of the individual is one dimensional .each of the actions in the past are bits stored in the memory of the individual and are coupled to the new action .this new action is adopted if the internal energy is minimized .the individual s actions depend off course on the appropriate selection of the internal coupling parameters .the hamilton function for the ising perception of the individual depends on the individual s past actions and its opponents  where and are respectively the past actions of individual and individual stored in the memory of individual ; and are the coupling constant between the new action and the internal stored states of respectively the individual and its opponent . and are the number of bits that the individual can store from respectively own and opponents past actions . the total storage size is defined as   additionally an external field can also be introduced . in this context , this external field represents the influence that external information may have in the individual s decision .this should induce a normative imitation by means of an external influence .an external influence is for example a kind of signal trying to orient the actions of the individuals to either of both states or .the individual s internal energy function adopts the following form where is the intensity of the external field .when the individuals have a normative tendency to cooperate .otherwise , the individuals tend to defect .the state is the probability that the action of the individual is in a given state is described by a master equation defined on the ensemble of the storage element .\label{master_i}\ ] ] the transition probabilities in the equation eq .( [ master_i ] ) can be defined such that it reaches an equilibrium state , implying a balance between the entropic and the energetic coefficients .this balance is reached when the individual optimizes its stored information .hence , a transition probability of metropolis form is adopted .the action of the individual can flip with a transition probability where , the learning parameter , equivalent to the level of noise in the environment , is the period of time and is the difference between two internal states .the mechanism for the definition of new actions is based on the perception of the information of the eigen actions of the reference individual and the actions of its opponents .when the system reaches an equilibrium state , i.e. the fluctuation of the action of each individual is equilibrated with the dissipation of the system , the probability that the action is part of the class of elements is given by where is the partition function on the storage distribution of the individual given by the dynamics of the actions depends on the size of the storage , which simultaneously determines the size of the internal energy associated to each individual .in the present context , it represents the change of the storage size of the individual .the evolutionary dynamics is represented by a mutation process of the individuals .each individual in the lattice is characterized by its storage size .if the fitness is not large enough , the storage size changes ; this change in the memory size automatically implies a change in the identity of the individual .the fundamental supposition is that the mutation of the memory size can increase the fitness of the individual .this mutation takes place only some times in a random way .such a process can be described by a moran process ( see traulsen et .al. ) : ( i ) _ selection _an individual is selected for reproduction with a probability related to its fitness ; ( ii ) _ reproduction _ the individual produces one offspring ; ( iii ) the offspring replaces a randomly selected individual . the microscopic state represents the probability for a reproduction in a given time , and can be either 1 if a reproduction takes place or 0 if nothence , the storage size of is given by .the transition probability of is given by where is a heaviside step function , is the utility given by eq .( [ utility ] ) and is the number of neighbour individuals .the probability is the probability that the individual has this mutation process .the dynamics of is given by this is the must simplest model of growth .hence , equation ( [ master_i ] ) and ( [ master_ii ] ) are two coupled equations describing the population s dynamics .the first describes how the information is handled by each of the individuals in the lattice ; the second describes how the internal characteristics of the individuals evolves ( storage capacity ) depending on the individual s fitness .both equations depend on the action each individual adopts .the first equation eventually reaches an equilibrium state after infinite times .therefore , the single dynamical equation that remains is the equation for the mutation of memories .however , the fluctuation on equilibrium of the first equation produces a stochastic drive for the second one .the system is defined on a square lattice . in each position of the latticethere is an individual doted with a storage size interacting with its four nearest individuals .each individual is in a cell at the position of the lattice .the whole system has periodic boundary conditions .two lattice sizes are considered in this work , with 4600 and 8100 individuals .   if individual s fitness is not large enough .this last step only takes place in a random way.,scaledwidth=40.0% ]  the system is based on a von neumann cellular automata , i.e. each individual interacts with its four nearest neighbors . as the system is defined on pair interactions , each individual plays parallel games with the neighbors . for each actionthe individual uses a particular memory .this implies , each individual adopts different game strategies for different opponents . in each time unitthe following process takes place ( see fig . [ schema ] ) : each individual store the last own action and the action of its opponent in its memory ( in the figure represented as a semicircle ) ; with this information the individual decides its new action . in the next step the score of the gameis computed ; this information is the individual utility in each lattice position .if this utility is lower than the utility on the neighbors , and the individual is allowed to do a mutation , then a change of the memory size from to takes place .this process is repeated in the next time step .the system is initialized with random initial interactions and random initial memories .however , we want to observe how the evolution in the system takes place .therefore , the first random memories in the system are small ones . given that the system reaches an equilibrium state in the actions , the simulations are stopped until this state is reached .the dynamics of the system depends on the evolution of the memory size , as well as the learning parameters . in a real modelsuch parameters should be defined in an individual way . however , for simplicity we assume that learning parameters are common for the whole individuals , such that it can be used as a control parameter .   as a function of the time . in the insetthe utility function is plotted as a function of the time .the dependence on the external field is shown.,scaledwidth=40.0% ]the actions are discrete states for and .after the system relaxes , the distribution of the number of each one of these states depends on the learning parameters .this dynamics is shown in fig .( [ fig_coop_t ] ) , where the number of is computed for different external fields .the distribution function of elements of class is given by (t ) = \frac { \sum_{i=1}^{n } \rho^{i}(t)}{n},\ ] ] where is the probability that the individual is in the state , defined by eq .( [ master_i ] ) , and is the total number of elements into the cell . the number of cooperators can be tuned by means of the external field . for the first time stepsthere is an equilibrium state in the number of cooperators .but for a split in the function of number of cooperators takes place , depending on the external field . for number of cooperators is larger than for .this is equivalent to say that a set of individuals react to a cooperative attitude if the external influence is also cooperative ; other wise , an external influence tending to defect may convince the individuals also to act as defect .this is equivalent to say that the individuals collectively adhere to a norm which is in this case imposed from outside .  the distribution of and into the lattice is shown in fig .( [ fig_distr_c ] ) .this plot is done for a fix learning parameter and different external fields .the system shows a correlation between different individuals .however , no ordered domains are evident . for are large clusters of . by tuning the field ,the clusters of leaves place for clusters of .the distribution of the clusters of cooperators for a positive external field should be symmetric to the distribution of defectors in a negative field .however , the snapshots show qualitatively a different cluster sizes of and at both extreme values of .hence , the distribution of either of both classes of individuals probably depends on the game , in particular on the fact that defect is more rewarded than cooperate .   for fixed .in the inset plot the number of cooperators is drawn as a function of the learning parameter for fixed .for there is no dependence on the learning parameter.,scaledwidth=40.0% ]  the dependence of the number of cooperators on the external field is shown in fig .( [ fig_coop_h ] ) .the behavior of the system with respect to the external field is mathematically related to the behavior of a 1d ising model . in such case the distribution of individuals is equivalent to the _ magnetization _ of the system . in the following analysiswe denote  ] , where is the shift on the field , is a scaling factor according to the scale of and is a scaling factor on .if the memory size is constant , this dependence should be exactly equal to the fit function .however , the variation of the memory size , and the definition of the game in a square lattice , affects the smoothness of the magnetization curve .therefore , the dynamics of the game has influence on the equilibrium distribution of cooperators and defectors and its dependence on external learning parameters .an interesting aspect is the dependence on the learning parameter .given that the learning schema is one dimensional , there is no possibility to observe a critical behavior as a function of .however the dependence on this parameter is only observed for , i.e. for a population under strong influence to defect an increase in the number of cooperators is related to the increase of .the memory is an important individual feature that may improve its response to evolution pressure .this distribution is defined as where is the total number of elements into the system ( please note that denotes the distribution of cooperators , whereas denotes the distribution of memories ) .this density is off course different to the density of cooperators .a snapshot shows a distribution of memories in the square lattice .the colours represent the variation of different memory sizes , from 1 to 20 , the maximal storage capacity defined for the present computations .the storage size distribution also strongly depends on the external field of the system . for is a relative uniform distribution of individual memories . by increasing the field ,large memories are dominant .the rich structure for in fig . ( [ fig_distr_m ] ) indicates a rapid evolution of large memories .this happens when the external field induces the system to be cooperative .a less rich but stable distribution for is related to a more diversity , i.e. to the coexistence of different individuals with different storage sizes .( left ) and ( right).,title="fig:",scaledwidth=30.0% ]  ( left ) and ( right).,title="fig:",scaledwidth=30.0% ]  here , the structure of the game is also related to the clusters with different size .the gradient in the memory size is relative small .therefore , a phase separation between memories of different sizes takes place .this effect is similar to the formation of domains composed by individuals with similar memory .the complexity and diversity of distribution of memories can be analyzed in the time . in fig .( [ fig_m_distr_t ] ) the distribution of number of individuals with a memory , for few memories , as a function of the time is shown .the distribution in the time is an asymmetric function .each individual has a growing rate , related to a high fitness , and a subsequent decay rate , related to the decrease of the fitness because new individuals appear into the population . at the same time individuals with less memory sizesface extinction from the system .eventually , this individual with memory faces also extinction when the fitness function is not large enough .the relaxation process is related to the growth of a new individual with a different memory size .this result shows the rivality between individuals and the dependence of the memory size on the fitness function of the system . in the extinction processthe distribution functions have a long tail , while the memory still preserves a high fitness .   as a function of time ( left plot ) . in the second plot ( right ) we show the total density of the number of individuals as a function of the memory size . here , the slope of depends on the external field .,title="fig:",scaledwidth=40.0% ]  as a function of time ( left plot ) . in the second plot ( right )we show the total density of the number of individuals as a function of the memory size . here, the slope of depends on the external field .,title="fig:",scaledwidth=40.0% ]  for two external fields the distribution of memories in the time shows a change in the distributions . as is shown in fig .( [ fig_m_distr_t ] ) , for ( defect ) the tails of the distributions are fat in comparison to ( cooperate ) .hence , the extinction rate can be controlled using the external field .  in general the distribution of memories in the timeshows how the effect of the change of is , where (t)=\frac{\sum_{m } \rho_{m}(t)}{n},\ ] ] the number of memories defined into the system . for short and intermediary timesthere is no dependence on the field because the system is still plenty of relative small memories ( fig .[ fig_m_t ] ) . at a critical time new individuals with larger memories co - evolve .after this time , i.e. for long time regimes , a dependence on the external field introduces a change into the tails of the curves .the variation as a function of the external field is more dramatic as the dependence on the learning parameter ( see the inset of fig . ( [ fig_m_t ] ) ) .it is also interesting to note that the total density of , defined as =\frac{\int_{o}^{\infty } \rho_{m}(t)dt}{n \times t},\ ] ] is proportional to the memory size , i.e . in every casethe slope depends on and . according to the result that we obtain , if .the number of individuals in a population is proportional to the storage size . in a extinction process of the population ,the different decays in the number of individuals of each species depends on the size of the window where this process takes place .hence , the size of the extinction channel is equivalent to the memory ( the size of the genetic code ) of each individual .  a very important additional aspect related to the distribution of densities is the measurement of the diversity of the system .there are several measures for diversity ( see for example the simpson s index ) .we adopt a shannon s index , which provides information not only about the population richness , but also the composition of the community .this index is given by  \log \pi[m ] , \label{shannon}\ ] ]  where is the number of memories .the shannon s equitability , a measure of diversity quantifying how equal the population are numerically , can be calculated by dividing the index by the maximal shannon s index defined by   the diversity is therefore a function of and ; in particular the diversity is a function of the information introduced into the system . for the present work for , whereas for , implying that an external influence for defect increases the diversity of the system .therefore , the diversity can be tuned by means of the external field .this is a very important result , because it implies that diversity is not only related to the amount of energy but also is related to the amount and kind of information it is introduced into the system . in both cases , the evenness is equal to the diversity , i.e. for the evenness is larger than for . the relation between diversity and information flow into the system is interesting .if this prediction is right , then the diversity can not only be argument on energy flux but also on information process by the elements of the system ., indicating a critical state in the evolution of memories.,scaledwidth=40.0% ]  the evolution of memories is a non equilibrium process .this fact is corroborated in fig .( [ fig_distr_m ] ) , where different domains are distributed in the lattice .this is a process with different scales that may also appear at different time scales . in this workwe use the function as an input signal in order to compute the power spectra of the distribution of memories .the power spectra of ( see fig .( [ t_fig_m_t ] ) ) can be fitted with a curve of the form , where .the results obtained in the present investigation fit with .this implies that there are long correlations in the time in the aggregate number of memories .in particular , given the definition of >$ ] is possible to conclude that this spectra arises from the superposition of individual evolutionary processes .therefore , the co - evolutive process is in this aspect equivalent a system with self organized criticality .however , the power spectra does not depend on the learning parameters .the obtained results should be analysed from two points of view . _ first _ , the present model reproduces basic characteristics of the ising learning in the lattice of players .however the variation of the memory size , related to the fitness of each individual , introduces variations in the actions of the individuals .the coupling of the system to the co - evolutive dynamics does not introduce phase transitions , but this dynamics , together with the dimension of the lattice where the simulations were performed , affect the dependence on the learning parameters .  _second _ , the dynamics of the game influences the memory distribution into the system .the variation of the external field influence the complexity of the distribution of memories and , therefore , the diversity of the distribution of individuals characterized by .therefore , an important conclusion is that the _ learning characteristics _ of the individuals may influence the distribution of .although the co - evolutive process is far from equilibrium , it may be influenced by external information imprinted into the system .the dimensionality of the storage device of the individuals is an important factor .a one dimensional storage device does not allows that may introduce a change in the phase behavior of the actions of the individuals . like more dimensional ising models , increasing of the dimensionmay introduce interesting effects that may have influence in the phase behavior of the distribution of .the final relevant result in this investigation is the _ role of the external information _ has in the _ entropy _ of the system . using this model we are able to analyse the genetic diversity of a population , which is the neg - entropy of the system .an external information source , changing the phenotype of the population , has an influence on the diversity of the system .in particular , the diversity increases if a large part of the population has a normative tendency to defect .this fact implies a reduction of the entropy of the system . in other words , _ diversity _ is not only a function of the size of the population , but also _ depends on cognitive characteristics and external sources of information_. this result can be a good approach to explain , why in some regions there are more diversity than in their neighbors , taking into account that genetic diversity is correlated to the community structure .a possible answer in such case is that animals , water or wind transport genetic material or signals that may influence the information exchange of the individuals .this phenomenon is not only exclusive in biology .intuitively , it is equivalent , for instance , to the effect of background music and advesting on consumption behavior of individuals in shopping malls . in this last case ,the external information , and not only budget , is the essential factor that affects individual behavior .the model implemented is a very simple approach that joints fundamental aspects from the theory of ferromagnetism with population dynamics in the frame of learning theory .this model can be extended to different directions . from a physical point of viewit is interesting to make an analysis of the dimensionality of the memory in the individual s behavior . for high dimensional memoriesis possible to expect a critical behavior in the distribution of cooperators . in biologyis interesting to make an analysis of the effect that an inhomogeneous and fluctuating external information source may have in the distribution of the population .finally , an extension from the lattice to a different topology in the interaction of agents may have interesting effects in the population s distribution .several scholars have in the last time discussed , if natural selection is a causal process at the population s level or if it takes place at individual s level . by accepting frequency dependent selection this causalityis placed at the population level . however, if the individuals express a kind of cognition , related to the reaction to noise and information sources , then this causality must also account for the individual .in resume , in the present model we propose middle point into this dispute . by tuning external information as well as noise sourceswe are able to observe how same individuals may affect the population . by observing a mutation processwe observe how the population influences the distribution of individuals .we have shown that this coupling have an important connection in the measure of diversity of the population . in particular, we showed that avoiding a cooperative behavior is possible to increase this diversity .

Abstract: the dynamics of populations is rich , taking into account that both , the individual s actions and the population s fitness are coupled . the way in which an individual chooses a strategy depends off course on the interaction with other individuals and the relation between selection and mutation within the population . the present model considers individuals with _ memory_. this memory is represented by a device where information of past actions is stored as bits in a 1d ising chain . the selection of a new individual action depends on the individual s memory . if the selection of a strategy does not improve the individual s fitness , a new individual with different memory size replaces it . both , actions and memory are observables that characterize the population . they can change in time , and both depend on the fitness of the population . this model allows the implementation of _ learning _ parameters as well as an _ external information source _ , acting as an _ external field _ which drives individuals to select one preferred action . in particular we show that the _ diversity _ of the population , measured as a _ shannon s diversity index _ ( equivalent to a neg - entropy ) , is not only related to the _ energy consumption and size _ of the system , but is also related to the way in which the individuals are influenced by the _ external field_.


the evaluation of fuzzy quantified expressions is a topic that has been widely dealt with in literature  since the use of quantified expressions in fields such as fuzzy control , temporal reasoning in robotics , , complex fuzzy queries in databases , information retrieval , data fusion , etc .can take advantage of using vague and interpretable quantification models .moreover , the definition of adequate models to evaluate quantified expressions is fundamental to perform computing with words , topic that was suggested by zadeh to express the ability of programming systems in a linguistic way . in this paperwe analyze the theoretical behavior and some practical consequences of the model defined on .furthermore , we show that the underlying probabilistic interpretation of this model hints the utility of the model for a number of applications .  in general , approaches to fuzzy quantification in the literature use the concept of _ fuzzy linguistic quantifier _ to represent absolute or proportional fuzzy quantities .zadeh defines _ quantifiers of the first kind _ as quantifiers used for representing absolute quantities ( defined by using fuzzy numbers on ) , and _ quantifiers of the second kind _ as quantifiers used for representing relative quantities ( defined by using fuzzy numbers on  ] .moreover , most of the existing approaches for dealing with fuzzy quantification are based on the evaluation of the compatibility between the linguistic quantifier and a scalar , possibilistic or probabilistic cardinality measure for the involved fuzzy sets .scalar approaches , usually consist of a simple evaluation of the quantifier on the cardinality value . for possibilistic approaches , an overlapping measure _ sup - min _is generally used whilst for probabilistic approaches , a weighted mean of all the compatibility values is computed .owa approaches can also be related to the probabilistic interpretation .a different approach is used in , where families of models that are based on a three valued interpretation of fuzzy sets are defined .  for analyzing the behavior of fuzzy quantification models different properties of convenient or necessary fulfillmenthave been defined .most of the approaches in literature fail to exhibit a plausible behavior , and only a few seem to exhibit an adequate behavior in the general case .  in this work we will follow the glckner approximation to fuzzy quantification . in his approach ,the author generalizes the concept of _ generalized classic quantifier _ ( second order predicates or set relationships ) to the fuzzy case ; that is , a _ fuzzy quantifier _ is a fuzzy relationship between fuzzy sets . and then rewrites the fuzzy quantification problem as the problem of looking for mechanism to transform _ semi - fuzzy quantifiers _ ( quantifiers between generalized classic quantifiers and fuzzy quantifiers that are adequate to specify the meaning of quantified expressions ) to fuzzy quantifiers .moreover , glckner has also defined a rigorous axiomatic framework to assure the good behavior of qfms .models fulfilling this framework are called _ determiner fuzzification schemes ( dfss ) _ and they fulfill an important set of appropriate behavior properties .  the main goal of this work is to analyze the behavior of the model .this model has a very solid theoretical behavior , superior to most of the models defined in the literature .moreover , we show that the underlying probabilistic interpretation based on likelihood functions has very interesting consequences , that assure its utility for a number of applications . for example , in the application of the model in a information retrieval task was shown , with competitive results . in model has been used in a summarization application for the evaluation of quantified temporal expressions . from a theoretical point of viewthe model is a dfs , although is only defined in finite domains .the fulfillment of the dfs axioms guarantees a very good theoretical behavior . as an important point ,the fuzzy operators induced by the model are the product t - norm and the probabilistic sum t - conorm .this fact makes the model essentially different of the models defined in because all those standard models induce the min tnorm and the max tconorm . to our knowledge, the model is the unique known non standard dfss .the paper is organized as follows . in the first section ,we resume the glckner s approach to fuzzy quantification , based on quantifier fuzzification mechanisms . in the second section we explain some of the properties that let us to analyze the behavior of the quantification model .most of them are a compilation of the properties defined on ( * ? ? ?* ; * ? ? ?* and 4 ) , but we have added to these properties two very interesting properties fulfilled by the model and by the probabilistic models defined in . in section threethe qfm is defined .we also explores the behavior of the model when the cardinality of the base set tends to infinite , with a surprising relation with original zadeh s model .next section is devoted to some interesting consequences of the probabilistic interpretation of the qfm , with relation with a number of application fields .proofs of the properties and efficient algorithm solutions are collected in two apendixes .a bibliographic analysis of quantification models has not been included as it can be found in .to overcome the zadeh s framework to fuzzy quantification glckner rewrites the problem of fuzzy quantification as the problem of looking for adequate means to convert the specification means ( semi - fuzzy quantifiers ) into the operational means ( fuzzy quantifiers ) . in this sectionwe explain in some detail the framework proposed by glckner to achieve that result .fuzzy quantifiers are just a fuzzy generalization of crisp or classic quantifiers . before giving the definition of fuzzy quantifiers, we will show the definition of classic quantifiers and some examples :  [ classic quantifier . ]57 ) a two valued ( generalized ) quantifier on a base set is a mapping , where is the arity ( number of arguments ) of , denotes the set of crisp truth values , and is the powerset of .  in this workwe assume the base set is finite as the model is only defined on finite base sets .examples of some definitions of classic quantifiers are:{cc}% \frac{\left\vert y_{1}\cap y_{2}\right\vert } { \left\vert y_{1}\right\vert } \geq0.80 & x_{1}\neq\varnothing\\ 1 & x_{1}=\varnothing \end{array } \right .\nonumber\end{aligned}\ ] ]  let us consider the evaluation of the sentence at least eighty percent of the members are lawyers where the properties members and lawyers are respectively defined as , and at least eighty percent is defined in expression [ tfgq_1 ]. then .  in a fuzzy quantifier arguments andresult can be fuzzy .the definition of a fuzzy quantifier is :  [ fuzzy quantifier ] ( * ? ? ?66 ) an n - ary fuzzy quantifier on a base set is a mapping  ] for .let a succession and be a fuzzy set constructed on such sucession .let the crisp set constructed with the crisp elements of . if the following limit there exists: then , when the size of the base set tends to infinite , tends to: for .the assumption of existence of the limit is very weak and irrelevant from a practical point of view . we simply are asking the proportional cardinality of the succession does not oscillate as tends to infinite .the interpretation underlying the model assumes that each represents and independent bernoulli process of probability .let us denote this bernoulli process .the probability represents the probability of the random variable .first , let us suppose that is divergent .using previous theorem , for an enough big , we can approximate this distribution for a normal distribution of parameters and be a random variable following a normal distribution of parameters . if we define then follows a normal distribution of parameters . ] .  as the fuzzy number associated to the fuzzy quantifieris defined on   ] interval .the probability distribution of is a normal distribution of parameters .let us note that the normal distribution fulfills that the probability in standard deviations of the mean is identical for all the normal distributions .as tends to infinite , the probability mass in tends to .that is , we always can find a such that the probability mass in would be as close to as we wanted .let us compute the limit of the variance when tends to infinite   that is , the probability distribution of is more and more concentrated as tends to infinite . andthen , for every we can find a sufficiently large such that .as the fuzzy number is continuous , then it is clear that when tends to infinite .let us consider now that is finite . in this case, we can not apply the central limit theorem .in this situation there are an infinite number of such that or .moreover , when tends to infinite , the proportion of such that with respect to tends to :   then ,  tends to a set in which only a finite number of elements are fuzzy .let and let the finite numbers of such that .it should be noted that by supposition the following limit there exists:  for a sufficiently large , all such that are in .let us consider the shape of the probability distribution of .as the such that are crisp , the probability distribution of will consist on points with and points with . moreover , by construction of all these points are consecutive .when we normalize to apply the fuzzy number that define the quantifier ( by means ot the transformation  ) we are in the same case that when is divergent . the probability distribution will be more and more concentrated around and the approximation would be valid .  although we have not developed a similar approximation for other kinds of quantifiers , it seems easy to extend previous proof to more complex cases .for example , in the case of proportional quantifiers we will have to consider two bernoulli successions and  .as we are assuming independence , we can built a third bernoulli succession and the previous results can be applied for approximating the probability distribution of the cardinality of ) .when tends to infinite , the probability distribution of tends to . for the same reason , the probability distribution of tends to .and then , the proportional cardinality of in tends to .the limit case approximation of the opens the possibility of applying the model to continuous fuzzy signals , fundamental for the application of the model for _ fuzzy quantified temporal reasoning_  .let us consider a continuous fuzzy signal where represents time in an interval   ] on where defined by means of a continuous fuzzy number . for example , could be defined as: where represents the lebesgue measure .  asthe model is finite , it can not be directly applied to continuous quantifiers .a reasonable possibility to apply the model on a continuous set is to discretize the interval   ] then is between the inferior integral and the superior integral of ) . andthen,   let be a probability distribution and a fuzzy label defined on the referential universe of .for example , could be a normal distribution of parameters representing the probability of _ heights for male adults _ , and the fuzzy label _being tall_.  let a random sample of , and let  .then and by using the limit approximation of the model :   that is , the probability of the fuzzy event , or the probability of the label given the population distribution .in this report we described and studied the theoretical behavior of the qfm .the analysis have proved the model is a finite dfs essentially different of the standard dfss proposed by this author .moreover the underlying probabilistic semantics makes the model particularly interesting for applications .other interesting results are the limit case approximation of the model , that allows its application to continuous domains , and the study of the application of the model to populations described by means of a probability distributions .in this section we analyze the most relevant properties of the qfm .a sligthly more detailed discussion can be consulted in .first of all , we will proof some preliminary results .[ lemafaoperinduc]it holds that1 ) where is the bivalued identity and is the fuzzy identity.2 ) where is the standard negation.3 ) ; that is , the product tnorm. ) ; that is , the probabilistic tconorm , the dual of the product.) , in this case the rechenbach fuzzy implication.  we only are going to show the proof of .the rest of the proofs can be consulted in ( * ? ? ?* appendix a ) .first , note that the definicin of  is: then  moreover , in the proofs of the properties of the model we need the following lemmas too .[ lemacasonitido]let be crisp sets .it holds that {lll}% & & \\ & & % \end{tabular } \\right.\ ] ]  the definition of is and as is crisp if and if .[ lemaproyeccion]let be a fuzzy set ( that is ) .let be a crisp set .then where is a fuzzy set we represent the restriction of to the reference universe .] ,   by definition of , the probability of is the product of the probabilities on their projections:  [ lematconormaprobabilistica]let be the probabilistic tconorm . by denote its -ary version ; that is , it is fulfilled it also is fulfilled that . moreover , if we define and previous relationship is also fulfilled . ]  proof is by induction .case : and then  induction supposition : case   case    [ [ axiom - z-1 ] ] axiom z-1 + + + + + + + + +  it holds that   let be a semi - fuzzy quantifier , a crisp set . using the lemma [ lemacasonitido ] we have and then for all .[ [ axiom - z-2 ] ] axiom z-2 + + + + + + + + +  it holds that   using the lemma [ lemaproyeccion]  [ [ axiom - z-3 ] ] axiom z-3 + + + + + + + + +  it holds that  for the proof of this axiom we will need the proofs of the properties of internal and external negation .* proof of the property of external negation *  we have to prove that  in the lemma [ lemafaoperinduc ] we have established that the induced negation operation is the standard negation .  * proof of the property of internal negation *  we have to prove that  [ lemanegacioninterna]let be a fuzzy set .then    proof of the property of internal negation :    and using the lemma [ lemanegacioninterna ] the above expression is equal to  using the properties of internal and external negation duality is trivial.  [ [ axiom - z-4 ] ] axiom z-4 + + + + + + + + +  it holds that  in the proof of [ eqaxiomaz4modelofa_1 ] we will use the following results :  [ lemaencuentrosinternos ] let be given , a crisp set .then where is defined by means of the induced tnorm .let .we will use [ lemaproyeccion ] .  it fulfills the following :   as the sum [ eqlemaencuentrosinternos_1 ] is restricted to the such that then, and using lemma [ lemaproyeccion]  and the equality [ eqlemaencuentrosinternos_2 ] will be fulfilled if: proof is by induction in the cardinality of .we will denote by a referential set with .  * case * ( ( ):  * induction hypothesis : case * ( , ( ) .we suppose that  * case * : , ( )  it should be noted that if then   whilst if then we have   by computation  as the situation in which and is not possible .let we evaluate the three sums in [ eqlemaencuentrosinternos_5 ] . in the computation we use the induction hypothesis .first sum:  second sum:  third sum :   and using expressions [ eqlemaencuentrosinternos_6 ] , [ eqlemaencuentrosinternos_7 ] and [ eqlemaencuentrosinternos_8 ] :   in this way we have proved [ eqlemaencuentrosinternos_4 ] , and the lemma is satisfied .[ lemaencuentrosinternos2]let be a unary semi - fuzzy quantifier .then it is fulfilled:  using lemma [ lemaencuentrosinternos]:  and finally we prove the fulfillment of expression [ eqaxiomaz4modelofa_1 ] :    where we have using [ lemaencuentrosinternos2 ] . and then,  now we prove the fulfillment of the z-4 axiom:   for a semi - fuzzy quantifier it is fulfilled ( represents the trasposition of the and element ): and then  in this way , we can use the properties of external negation , internal negation and trasposition of arguments ( trivially fulfilled ) and the expression [ eqaxiomaz4modelofa_1 ] to obtain: where in the last step we use that and are constructed by means of dual operators .[ [ axiom - z-5 ] ] axiom z-5 + + + + + + + + +  it holds that  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ if is nonincreasing in the -th arg , then is nonincreasing in the -th arg , ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  we will consider first the unary case .let be a nonincreasing semi - fuzzy quantifier .we will proof that is nonincreasing using induction on the cardinality of the referential .* case * ; that is , the referential contains only one element ( ) .let be fuzzy sets fulfilling .note   as is monotonic nonincreasing for it holds that as a consequence of and because is nonincreasing .let be . by [ eqaxiomaz5modelofa_1] and the propertyis fulfilled for a one referential set  * hypothesis of induction : case * ( ) . for that it holds that .* case * ( ) .  based on define the semi - fuzzy quantifiers and as   and are monotonic nonincreasing on .let .then  and using the induction hypothesis because and are monotonic nonincreasing on a referential of elements .we continue the computation:  let us consider now the general case.let be an -ary semi - fuzzy quantifier non increasing in its . then , in the are constant .the unary semi - fuzzy quantifier is monotonic non increasing , and then is also monotonic non increasing .as this fact is fulfilled for all then the proposition is fulfilled .[ [ axiom - z-6 ] ] axiom z-6 + + + + + + + + +  it holds that  to prove this property we need some previous results .* existential quantifier *  [ [propaxiomaz6modelofa_1]let a fuzzy set .then  let be given . using the lemma [ lematconormaprobabilistica]  * universal quantifier *  [ [propaxiomaz6modelofa_2]let a fuzzy set .then  let be given . then   * induced extension principle .*  to compute the induce extension principle of the we will use the definition [ defpropfuncaplprinextension ] .[ [notacionaxiomaz6generalizacionor]let and  a fuzzy set .by we will denote the generalization of the induced tconorm to fuzzy sets ; that is,{ccc}% \widetilde{\vee}\left (  \mu_{x}\left (  e_{1}\right )  , \widetilde{\vee}\left ( \ldots,\widetilde{\vee}\left (  \mu_{x}\left (  e_{m-1}\right )  , \mu_{x}\left ( e_{m}\right )\right )  \right )  \right )  & : & m>0\\ 0 & : & m=0 \end{array } \right.\ ] ]  [ propaxiomaz6modelofa_3]let ( where ) .let be given and let be the inverse image of .then the induced extension principle of for is {ccc}% \widetilde{\vee}\left (  \left\ {  \mu_{x}\left (  e_{i_{1}}\right )  /e_{i_{1}% } , \ldots,\mu_{x}\left (  e_{i_{k_{i}}}\right )  /e_{i_{k_{i}}}\right\ }  \right ) & : & k_{i}\geq1\\ 0 & : & \text{otherwise}% \end{array } \right .\label{eqaxiomaz6modelofa_1}\\ &  = \mathcal{f}^{a}\left (  \exists_{\widehat{f^{-1}}\left (  e_{i}^{\prime } \right )  } \right )  \left (  x^{\widehat{f^{-1}}\left (  e_{i}^{\prime}\right ) } \right ) \nonumber\end{aligned}\ ] ] where represents the existential quantifier on the base universe and is the proyection of the fuzzy set over .{ccc}% 0 & : & e^{\prime}\notin\widehat{f}\left (  y\right ) \\ 1 & : & e^{\prime}\in\widehat{f}\left (  y\right ) \end{array } \right . \nonumber\\ &  = \sum_{y\in\mathcal{p}\left (  e\right )  |\exists e\in y , f\left (  e\right ) = e^{\prime}}m_{x}\left (  y\right ) \nonumber\end{aligned}\ ] ]  if the previous sum is and we are in the second situation of expression [ eqaxiomaz6modelofa_1 ] .let us suppose .it should be noted that for each  then the intersection of with the inverse image of ( ) is not empty .as for all fulfilling this condition can be decomposed in the part intersecting with , and the part that does not intersect with ( ) expression [ eqaxiomaz6modelofa_2 ] is equal to as contains all the sets of .and then , if we denote using expression [ propaxiomaz6modelofa_1 ] we obtain  now we will prove the fulfillment of {l}% \\ where % \end{tabular } \ \\ ] ]  let , finite sets , a semi - fuzzy quantifier , fuzzy sets and .we point that . by computation  using result [ propaxiomaz6modelofa_3 ]we can rewrite as  rewriting expression [ eqaxiomaz6modelofa_3]  let be .we will compute the probability mass :  and by duality we now that .then,,  as then expression [ eqaxiomaz6modelofa_4 ] is equivalent to  in this way , the probability mass is computed by using the probability masses that are associated to the such that the intersection with the inverse image of the is not empty , and such that the intersection with the is empty . moreover , we should note that all is associated to one ; that is , for some . in this way , if the visit , then the visit , and continuing with the computation of expression [ eqaxiomaz6modelofa_3p5 ] we obtain as we want to prove .the model fulfills the property of argument continuity .  in ( * ? ? ?* appendix a ) a detailed proof can be found .but the next arguments are enough to prove continuity .let us consider the definition of the model: note that for crisp sets  are continuous funtions , because is constant and only involves the use of the product operation .but the sum of continuous functions ( that is , the sum over is continuous . and then the model is continuous in arguments .the model fulfills the property of -continuity :  let semi - fuzzy quantifiers .then,{c}% \sum_{y_{1}\in\mathcal{p}\left (  e\right )  } \ldots\sum_{y_{n}\in \mathcal{p}\left (  e\right )  } m_{x_{1}}\left (  y_{1}\right )  \ldots m_{x_{n}% } \left (  y_{n}\right )  q\left (  y_{1},\ldots , y_{n}\right ) \\-\sum_{y_{1}\in\mathcal{p}\left (  e\right )  } \ldots\sum_{y_{n}\in \mathcal{p}\left (  e\right )  } m_{x_{1}}\left (  y_{1}\right )  \ldots m_{x_{n}% } \left (  y_{n}\right )  q^{\prime}\left (  y_{1},\ldots , y_{n}\right ) \end{array } \right\vert \right .\\ &  \left .  :x_{1},\ldots , x_{n}\in\widetilde{\mathcal{p}}\left (  e\right ) \right\ } \\ &  = \sup\left\ {  \left\vert \begin{array } [ c]{c}% \sum_{y_{1}\in\mathcal{p}\left (  e\right )  } \ldots\sum_{y_{n}\in \mathcal{p}\left (  e\right )  } m_{x_{1}}\left (  y_{1}\right )  \ldots m_{x_{n}% } \left (  y_{n}\right ) \\ \left (  q\left (  y_{1},\ldots , y_{n}\right )  -q^{\prime}\left (  y_{1}% , \ldots , y_{n}\right )  \right ) \end{array } \right\vert : x_{1},\ldots , x_{n}\in\widetilde{\mathcal{p}}\left (  e\right ) \right\ } \\ &  \leq\sup\left\ {  \left\vert \sum_{y_{1}\in\mathcal{p}\left (  e\right ) } \ldots\sum_{y_{n}\in\mathcal{p}\left (  e\right )  } m_{x_{1}}\left ( y_{1}\right )  \ldots m_{x_{n}}\left (  y_{n}\right )  d\left (  q , q^{^{\prime}% } \right )  \right\vert : x_{1},\ldots , x_{n}\in\widetilde{\mathcal{p}}\left ( e\right )  \right\ } \\ &  = \sup\left\ {  \left\vert d\left (  q , q^{^{\prime}}\right )  \sum_{y_{1}% \in\mathcal{p}\left (  e\right )  } \ldots\sum_{y_{n}\in\mathcal{p}\left ( e\right )  } m_{x_{1}}\left (  y_{1}\right )  \ldots m_{x_{n}}\left ( y_{n}\right )  \right\vert : x_{1},\ldots , x_{n}\in\widetilde{\mathcal{p}}\left ( e\right )  \right\ } \\ &  = \sup\left\ {  \left\vert d\left (  q , q^{^{\prime}}\right )  \right\vert : x_{1},\ldots , x_{n}\in\widetilde{\mathcal{p}}\left (  e\right )  \right\ } \\ &  = d\left (  q , q^{^{\prime}}\right)\end{aligned}\ ] ]  and the property is fulfilled for .   the verifies the property of fuzzy argument insertion .let a semi - fuzzy quantifier be given and a fuzzy set .for all crisp we have{c}% f:\left (  k_{1},\ldots , k_{n+1}\right )  \in\widetilde{\mathcal{p}}\left ( e\right )  \rightarrow\sum_{z_{1}\in\mathcal{p}\left (  e\right )  } \ldots \sum_{z_{n}\in\mathcal{p}\left (  e\right )  } \sum_{z_{n+1}\in\mathcal{p}\left ( e\right )  } \\m_{k_{1}}\left (  z_{1}\right )  \ldots m_{k_{n}}\left (  z_{n}\right ) m_{k_{n+1}}\left (  z_{n+1}\right )  q\left (  z_{1},\ldots , z_{n},z_{n+1}\right ) \end{array } \right )  \vartriangleleft a\right ) \\ &  \left (  y_{1},\ldots , y_{n}\right ) \\ &  = \mathcal{u}\left (  \left (  f^{\prime}:\left (  k_{1}^{\prime},\ldots , k_{n}^{\prime}\right )  \in\widetilde{\mathcal{p}}\left (  e\right ) \rightarrow f\left (  k_{1}^{\prime},\ldots , k_{n}^{\prime},a\right )  \right ) \right )  \left (  y_{1},\ldots , y_{n}\right ) \\ &  = \mathcal{u}\left (  \left ( \begin{array } [ c]{c}% f^{\prime}:\left (  k_{1}^{\prime},\ldots , k_{n}^{\prime}\right )  \in \widetilde{\mathcal{p}}\left (  e\right )  \rightarrow\sum_{z_{1}\in \mathcal{p}\left (  e\right )  } \ldots\sum_{z_{n}\in\mathcal{p}\left (  e\right ) } \sum_{z_{n+1}\in\mathcal{p}\left (  e\right )  } \\ m_{k_{1}^{\prime}}\left (  z_{1}\right )  \ldots m_{k_{n}^{\prime}}\left ( z_{n}\right )  m_{a}\left (  z_{n+1}\right )  q\left (  z_{1},\ldots , z_{n}% , z_{n+1}\right ) \end{array } \right )  \right ) \\ &  \left (  y_{1},\ldots , y_{n}\right ) \\ &  = \left ( \begin{array } [ c]{c}% f^{\prime\prime}:\left (  k_{1}^{\prime\prime},\ldots , k_{n}^{\prime\prime } \right )  \in\mathcal{p}\left (  e\right )  \rightarrow\sum_{z_{1}\in \mathcal{p}\left (  e\right )  } \ldots\sum_{z_{n}\in\mathcal{p}\left (  e\right ) } \sum_{z_{n+1}\in\mathcal{p}\left (  e\right )  } \\ m_{k_{1}^{\prime\prime}}\left (  z_{1}\right )  \ldots m_{k_{n}^{\prime\prime}% } \left (  z_{n}\right )  m_{a}\left (  z_{n+1}\right )  q\left (  z_{1}% , \ldots , z_{n},z_{n+1}\right ) \end{array } \right ) \\ &  \left (  y_{1},\ldots , y_{n}\right ) \\ &  = \left (  f^{\prime\prime}:\left (  k_{1}^{\prime\prime},\ldots , k_{n}%^{\prime\prime}\right )  \in\mathcal{p}\left (  e\right )  \rightarrow \sum_{z_{n+1}\in\mathcal{p}\left (  e\right )  } m_{a}\left (  z_{n+1}\right ) q\left (  k_{1}^{\prime\prime},\ldots , k_{n}^{\prime\prime},z_{n+1}\right ) \right ) \\ &  \left (  y_{1},\ldots , y_{n}\right)\end{aligned}\ ] ] because are crisp sets , and then  using the previously obtained result ( expression [ eqmodelofainserargborrosa_0p5 ] ) , then :    this property is easily proved by induction .let us denotelatexmath:[\[\pr\left(  card_{x}=j\right )  = \sum_{y\in\mathcal{p}\left (  e\right )  that the cardinality of a crisp representative of let be .  for have  let us begin the induction proof :  * case * , .evident .  *induction hypothesis . * case ( that is , ) .for it is fulfilled  * casw * ( ) .  for an elements referentialis fulfilled ( using the induction hypothesis ) .   let us suppose now that . and let be the fuzzy set  then,  and using expression [ eqmodelofacuantident_1 ] and the induction hypothesis: model fulfills the property of the probabilistic interpretation of quantifiers :  let a probabilistic covering of the quantification universe .then for all the time to compute the result of evaluatign a quantified expression could seem extremely high , it is possible to develop polynomial algorithms for quantitative quantifiers . in table[ algoritmounariofa ] the algorithm to evaluate unary quantitative quantifiers is shown . a quantitative unary semi - fuzzy quantifier depends on a function ; that is , a function of the possible cardinality values in .the idea of the algorithm is that , if we know the probabilities of the cardinalities in a base set of size ; that is , we know the probabilities then when we add one element to the base set ( ) we have to consider two possibilities to compute the change in the probabilities of the cardinalities .one possibility is that the element fulfills the property represented by ( with probability ) and the other is that the element does not fulfill the property represented by ( with probability ) .the next formula expresses the change in the probabilities : {ccc}% \pr\left (  card_{x}=0\right )  \left (  1-\mu_{x}\left (  e_{k+1}\right )  \right ) & : & j=0\\% \begin{array } [ c]{c}% \pr\left (  card_{x}=j\right )  \left (  1-\mu_{x}\left (  e_{k+1}\right )  \right )\\ + \pr\left (  card_{x}=j-1\right )  \mu_{x}\left (  e_{k+1}\right ) \end{array } & : & 1\leq j\leq k\\ \pr\left (  card_{x}=m\right )  \mu_{x}\left (  e_{i+1}\right )  & : & j = k+1 \end{array } \right.\ ] ]    [ c]linput : x[0, ... ,m-1 ] , m ,q:\{0, ,m}*i * + double pr_aux_i , pr_aux_i_minus_1 ; + double pr[0, .... ,m ] ; + result = 0 ; + pr[0 ] = 1 ; + for ( j = 0;j m;j++ ) \ { +  + pr_aux_i ; +  +  +  + pr_aux_i + x[j ] pr_aux_i_minus_1 ; + + } + [j+1 ] = x[j ] pr_aux_i_minus_1 ; + } + for ( j = 0;j = m;j++ ) + = result + pr[j ] q(j ) ; + return result ;         g. bordogna and g. pasi .modeling vagueness in information retrieval . in m.agosti , f. crestani , and g. pasi , editors , _ lectures on information retrieval ( lncs 1980 ) _ , pages 207241 .springer - verlag berlin heidelberg , 2000 .p. bosc , l. lietard , and o. pivert .quantified statements and database fuzzy querying . in p.bosc and j. kacprzyk ,editors , _ fuzziness in database management systems _ , volume 5 of _ studies in fuzziness _ , pages 275308 .physica - verlag , 1995 .p. cariena , a. bugarn , m. mucientes , f. daz - hermida , and s. barro . ,volume 2 , chapter fuzzy temporal rules : a rule - based approach for fuzzy temporal knowledge representation and reasoning , pages 237250 .springer - verlag , 2002 .m. delgado , d. snchez , and m. a. vila .un enfoque lgico para calcular el grado de cumplimiento de sentencias con cuantificadores lingusticos . in _actas vii congreso espaol sobre tecnologas y lgica fuzzy ( estylf97 ) _ , pages 1520 , 1997 .m. delgado , d. snchez , and m. a. vila. un mtodo para la evaluacin de sentencias con cuantificadores lingusticos . in _actas del viii congreso espaol sobre tecnologas y lgica fuzzy ( estylf98 ) _ , pages 193198 , 1998 .m. delgado , d. snchez , and m. a. vila .a survey of methods for evaluating quantified sentences . in _ proc .first european society for fuzzy logic and technologies conference ( eusflat99 ) _ , pages 279282 , 1999 .f. daz - hermida and a. bugarn .linguistic summarization of data with probabilistic fuzzy quantifiers . in _ actas del xv congreso espaol sobre tecnologas y lgica fuzzy ( estylf 2010 ) _, page accepted , 2010 .f. daz - hermida , a. bugarn , p. cariena , and s. barro .evaluacin probabilstica de proposiciones cuantificadas borrosas . in _ actas del x congreso espaol sobre tecnologas y lgica fuzzy ( estylf 2000 ) _ , pages 477482 , 2000 .f. daz - hermida , d.e .losada , a. bugarn , and s. barro .a novel probabilistic quantifier fuzzification mechanism for information retrieval . in _ proc .ipmu 2004 , the 10th international conference on information prosessing and management of uncertainty in knowledge - based systems _ , pages 1357,1364 , perugia ,italy , july 2004 .i. glckner .evaluation of quantified propositions in generalized models of fuzzy quantification . technical report , universitt bielefeld , january 2003 .preprint submitted to the international journal on approximate reasoning , elsevier science , 15th january 2003 .i. glckner and a. knll .application of fuzzy quantifiers in image processing : a case study . in _ in : proceedings of the third international conference on knowledge - based intelligent information engineering systems kes 99 _ , pages 259262 , 1999 .i. glckner and a. knoll . a formal theory of fuzzy natural language quantification and its role in granular computing . in w.pedrycz , editor , _ granular computing : an emerging paradigm _ , volume 70 of _ studies in fuzziness and soft computing _, pages 215256 .physica - verlag , 2001 .e. l. keenan and d. westersthl .generalized quantifiers in linguistics and logic . in j. van benthem and a. ter meulen , editors , _ handbook of logic and language _ , chapter 15 , pages 837893 .elsevier , 1997 .losada , f. daz - hermida , a. a. bugarn , and s. barro .experiments on using fuzzy quantified sentences in adhoc retrieval . in _ proc .sac-04 , the 19th acm symposium on applied computing - special track on information access and retrieval _ , pages 1059,1066 , nicosia , cyprus , march 2004 .m. mucientes , r. iglesias , c.v .regueiro , a. bugarn , and s. barro .a fuzzy temporal rule - based velocity controller for mobile robotics ., 134(3 ; special issue : fuzzy set techniques for intelligent robotic systems):8399 , 2003 .

Abstract: the main goal of this work is to analyze the behaviour of the quantifier fuzzification mechanism . as we prove in the paper , this model has a very solid theorethical behaviour , superior to most of the models defined in the literature . moreover , we show that the underlying probabilistic interpretation has very interesting consequences .   * keywords : * quantifier fuzzification mechanism , determiner fuzzification schemes , probabilistic quantification models


human vitamin k epoxide reductase ( vkor ) plays a critical role in generating gamma - carboxyglutamate , one of the essential ingredient in blood coagulation cascade .the amino acid sequence of human vkor enzyme ( 163-residues ) was identified .many biochemical experiments have been performed on the study of the human vkor enzyme .however , the x - ray crystal structure of the human vkor enzyme is not known yet . in the absence of the x - ray crystal structure , two models for the topology of the human vkor enzymehave been suggested : 3 transmembrane(tm ) /4 transmembrane(tm ) vkor models . for the human 4tm vkor model ,it is derived from the bacterial vkor homolog .the initial tm domains of the human 4tm vkor are defined as tm1 : r12-a31 , tm2:s81-l97 , tm3:l105-v127 , and tm4:c132-r151 . on the other hand , for the human 3tm vkor model, it is determined based on the biochemical experiments and computer - aided program ` topcons ` .a schematic topology for the 3tm/4tm human vkor models .the topology difference between the two models make the active sites , c132 and c135 ( colored as yellow ) , locate in different positions within the lipid bilayer .the definitions of the tm domains for the human 3tm vkor are determined as tm1 : ( g9-v29 ) , tm2 : ( v104-l124 ) , and tm3 ( v127-w147 ) .the most significant difference in the topology of the human 3tm/4tm vkor models is that the n - terminus of the human 3tm vkor model starts from the lumen while the c - terminus ends at the cytoplasm . on the other hand , for the human 4tm vkor model ,the n - terminus starts from the cytoplasm and the c - terminus ends at the cytoplasm .figure 1 shows a schematic topology for the human 3tm/4tm vkor model .the last two tm domains ( 2nd and 3rd tm domains in the human 3tm vkor model and 3rd and 4th tm domains in the human 4tm vkor model ) show common feature in their membrane topology .the difference between the topology of the human 3tm/4tm vkor models makes the two active sites , c132 and c135 , locate in different positions inside the lipid bilayer .+ normal mode analysis ( nma ) is an effective tool for analyzing the slowest motions of protein .      for the system in the equilibrium, the potential energy v can be expanded as taylor series , where is a displacement from the equilibrium position , =+ . 0means the equilibrium position , in the generalized coordinate . for the oscillating system, the deviation can be expressed as where is the angular frequency and is the phase .and is the component of the amplitude matrix .then , we can solve the equation in the matrix form . * a * is the matrix of the oscillation amplitude.*v * is the matrix of the second derivatives of the potential energy , hessian .* * is the eigenvalue matrix .the eigenvectors of the hessian correspond to normal modes .for accurate normal mode analysis for protein , it requires a full potential for all the atoms of the protein .however , it requires an expensive computational cost . to save the computational time ,coarse - grained normal mode analysis ( dealing with atoms ) was developed .the harmonic pair potential of the coarse - grained model , based on the elastic network model ( enm ) , has a form where is the pair distance vector . and the spring constant k(r ) , function of the atomic pair distance , is obtained from the fitting to the amber94 force field . in this study, we analyze the mobility of tm helices of the two proposed models for the human vkor using the normal mode analysis and molecular dynamics simulation .the translational movement along the axis , the rotational movement around the axis , the slide movement perpendicular to the axis towards / away from the center of the bundle , and the tilt perpendicular to the axis away from the center of bundle are defined in the web - application ` tmm ` ( insets in figure 3 ) .the axis of -helix is defined as the principal axis of inertia of atoms .the projections are defined as  where d is the displacement vector including the rotation , translation , slide and tilt . is the normal mode vector of i. n is the number of atom .cumulative squared overlap calculates how much normal mode i contribute to the displacement .the range of the cumulative squared overlap is 0 to 1 with the relation of  mode numbers are shown in x - axis as vibrational frequencies of movements . in the cumulative squared overlap graph , a curve reaches an asymptotic behavior for fewer vibrational modes means greater mobility of tm domain .we performed molecular dynamics simulation of the _ in - silico _human 3tm/4tm vkor model .we employed the namd 2.8 package with the charmm 27 force field with the protein parameters incorporating the cmap terms .a palmitoyl - oleoyl - phosphatidylethanolamine lipid bilayer ( pope ) was prepared with the dimension of 100 100 with the tip3p water model . for physiological condition, we set the system to be 150 mm in nacl environment .we performed an energy minimization over 20,000 steps by the conjugate gradient method .subsequently , the system was heated to 310 k over 60 ps .the particle mesh ewald ( pme ) method was used .molecular dynamics simulation in the npt ensemble ( 310 k , 1 atm ) was performed for over 2 ns .constant pressure ( 1atm ) was maintained by using the langevin piston nose - hoover method .the md simulation scheme was changed to the npnta ensemble for 60 ns simulation .for the last snapshots of the dynamically generated human 3tm/4tm vkor models through 60 ns md simulation ( figure 2 ) , we investigated the mobility of the tm domains employing normal mode analysis .this approach is effective for understanding the mobility of individual tm helices for a ) translational movement along the axis b ) rotational movement around axis c ) slide movement perpendicular to the axis towards / away from the center of the bundle d ) tilt perpendicular to the axis away from the center of bundle .the four different mobility modes are shown for the 4tm human vkor model ( figure 3 ) and the 3 tm human vkor model ( figure 4 ) .for the human 4tm vkor model ( figure 3 ) , the 4th tm domain including cxxc motif ( including c132 and c135 ) shows large mobility in the translation , rotation , slide and tilt movements in relatively small mode number ( 50 - 100 approximately ) .interestingly , the mobility of the 1st tm domain increases in translation , slide and tilt movements in high frequency mode ( mode number 150 - 200 approximately ) . for the human 3tm vkor model ( figure 4 ) ,the 3rd tm domain including cxxc motif shows large mobility in translation , rotation , slide and tilt movements in low frequency mode .it is a common feature that the tm domain including cxxc motif shows significant mobility for both human 3tm/4tm vkor models .we compare the cumulative squared overlap values for the 1st tm domains of the 3tm/4tm human vkor models ( figure 4 ) , 2nd tm domain of the 3tm human vkor and 3rd tm domain of the 4tm human vkor ( figure 5 ) , and 3rd tm domain of the 3tm human vkor and 4th tm domain of the 4tm human vkor ( figure 6 ) .the mobility of the 1st tm domain of the 4tm human vkor model shows more mobility than that of the 1st domain of the 3tm human vkor model in the tilt , slide , and rotation mode ( figure 4 ) . in a similar way, the mobility of the 4th tm domain of the 4tm human vkor model shows more mobility than that of its corresponding 3rd domain of the 3tm human vkor model in the tilt , rotation and translation mode ( figure 6 ) .significantly , the mobility of the 3rd tm domain of the 4tm human vkor model is greater than that of its corresponding 2nd domain of the 3tm human vkor model in the translation , tilt , slide , and rotation mode ( figure 5 ) .nma analysis shows that the mobility of the individual tm domain of the 4tm human vkor is more than that of the its corresponding tm domain of the 3tm human vkor model .we compare the backbone rmsd values of each domain ( excluding loop regions ) for the 3tm/4tm human vkor models during 60 ns simulation ( figure 7 ) .the backbone rmsd values show large fluctuation up to 20 ns for all the tm domains for both 3tm/4tm human vkor models , implying that the dynamical equilibration process occurs up to 20 ns in the simulation . after 20 ns ,backbone rmsd of all the tm domains reach plateau values , implying that the system approaches to equilibrium state .the less mobility of the 3tm human vkor model is relevant to hydrophobic interactions between the leucine / isoleucine residues on each tm domain .the hydrophobic interactions between the leucine / isoleucine residues contributes to the stability of each tm domain : leu22 ( tm1):leu105(tm2 ) , leu15(tm1):leu128(tm3 ) , and leu108(tm2):ile136(tm3 ) .figure 9 shows the hydrophobic interactions between leucine / isoleucine residues and their bond lengthes measured during 60 ns simulation .up to 20 ns simulation , the bond lengths show dramatic changes , implying the dynamical equilibration process .the nma analysis based on the elastic network model estimates the mobility of the tm helices in the translation , rotation , slide and tilt mode for the last snapshot of the 3tm/4tm vkor models . in an alternative approach to nma , the backbone rmsd value ( excluding loop regions ) from the md simulation trajectory shows the entire time profile of individual tm domain movement during 60 ns .interestingly , the nma analysis results are compatible with the backbone rmsd results after 40 ns of the md simulation . from the analysis of the tm domain mobility based on the nma and md simulation, we conclude that the tm domains of the 4tm human vkor model are more mobile than their corresponding tm domains of the 3tm human vkor model .it implies the 3tm human vkor model is more stable than the 4tm human vkor model .the origin of the stability of the 3tm human vkor model comes from the hydrophobic interactions between the leucine / isoleucine residues on each tm domain .the movement of individual tm domain of the 3tm human vkor model is more suppressed by the hydrophobic interactions between the leucine / isoleucine residues than that of the 4tm human vkor model .b. furie , b. c. furie , blood * 75 * , 1753 ( 1990 ) .j. e. sadler , nature * 427 * , 493 ( 2004 ) .t. li , c. y.chang , d. y. jin , p. j. lin , a. khvorova , and d. w. stafford , nature * 427 * , 541 ( 2004 ) .s. rost , a. fregin , v. ivaskevicius , e. conzelmann , k. hrtnagel , h. j. pelz , k. lappegard , e. seifried , i. scharrer , e. g. d. tuddenham , c. r. muller , t. m. storm , and j. oldenburg , nature * 427 * , 537 ( 2004 ) .s. wu , j. tie , d. w. stafford , and l. g. pedersen , j. thromb .* 12 * , 112 ( 2014 ) .k. j. czogalla , a. biswas , s. rost , m. watzka , and j. oldenburg , blood * 124 * , 1354 ( 2014 ) .d. j. timson , current drug targets * 16 * , 1 ( 2015 ) .w. li , s. schulman , r. j. dutton , d. boyd , j. beckwith , and t. a. rapoport , nature * 463 * , 507 ( 2010 ) . j. tie , s. m. wu , d. jin , c. v. nicchitta , and d. w. stafford , blood * 96 * , 973 ( 2000 ). j. tie , v. p. mutucumarana , g. v. heijne , d. l. straight , k. l. carrick , r. m. pope , and d. w. stafford , j. biol .chem . * 278 * , 45468 ( 2003 ) .j. tie , c. nicchitta , g. v. heijne , and d. w. stafford , j. biol .280 * , 16410 ( 2005 ) .s. wu , s. liu , c. h. davis , d. w. stafford , j. d. kulman , and l. g. pedersen , j. theor .biol . * 279 * , 143 ( 2011 ) .a. bernsel , h. viklund , a. hennerdal , and a. elofsson , nucleic acids res .* 37 * , 1 ( 2009 ) .m. levitt , c. sander , and p. s. stern , j. mol .* 181 * , 423 ( 1985 ) .o. marques , and y. h. sanejouand , proteins * 23 * , 557 ( 1995 ) .k. hinsen , proteins * 33 * , 417 ( 1998 ) .l. skjaerven , s. m. hollup , and n. reuter , n. j. molec .theochem * 898 * , 42 ( 2009 ) .m. tirion , phys .* 77 * , 1905 ( 1996 ) .k. hinsen , a. j. petrescu , s. dellerue , m. c. bellissent - funel , and g. r. kneller , chem . phys . * 261 * , 25 ( 2000 ) . l. skaerven , i. jonassen , and w. reuter , bmc bioinformatics * 8 * , 1 ( 2007 ) . j. c. phillips , r. braun , w. wang , j. gumbart , e. tajkhorshid , e. villa , c. chipot , r. d. skeel , l. kale , and k. schulten , j. comput . chem .* 26 * , 1781 ( 2005 ) .a. d. mackerell , n. banavali , and n. foloppe , biopolymers * 56 * , 257 ( 2001 ) .a. d. mackerell , m. feig , and c. l. brooks iii , j. comput. chem . * 25 * , 1400 ( 2004 ) .w. l. jorgensen , j. chandrasekhar , j. d. madura , r. w. impey , and m. l. klein , j. chem . phys . *79 * , 926 ( 1983 ) .t. darden , d. york , and l. g. pedersen , chem . phys . * 98 * , 10089 ( 1993 ) .s. e. feller , y. zhang , r. w. pastor , and b. r. brooks , j. chem .phys . * 103 * , 4613 ( 1995 ) .

Abstract: 3 transmembrane and 4 transmembrane helices models are suggested for the human vitamin k epoxide reductase ( vkor ) . in this study , we investigate the stability of the human 3 transmembrane/4 transmembrane vkor models employing a coarse - grained normal mode analysis and molecular dynamics simulation . based on the analysis of the mobility of each transmembrane domain , we suggest that the 3 transmembrane human vkor model is  more stable than the 4 transmembrane human vkor model .