agent - based tax evasion models have gained much popularity over recent years because they allow for analyzing tax compliance behavior in large populations of heterogeneous agents that interact with each other in a direct manner ( see e.g. ) . moreover , these models allow for a high degree of complexity .for example , by simultaneously incorporating various policy parameters of the government , by endowing each individual agent with a different set of attributes regarding income , risk aversion , etc . or by calibrating individual agent behavior with a diversity of different individual human behavior patterns , which may have been discovered in tax evasion experiments with human subjects , in fields such as economic psychology or by empirical analysis ( see e.g. , , ) . to this extent , and in contrast to traditional models ,agent - based tax evasion models allow for analyzing tax evasion dynamics in a fairly realistic way , which in turn may lead to new insights and policy options for combating tax evasion .  in this paper , we develop and analyze an agent - based tax evasion model that is based on a standard model of statistical mechanics , the ising model of ferromagnetism . among other things ,the model allows for the numerical simulation of tax evasion dynamics in very large populations of heterogeneous agents , where heterogeneity refers to several different behavioral patterns .we show that our model allows for reproducing previously published results obtained from fundamentally different types of agent - based models .we then use the model for analyzing tax evasion dynamics that follow from alternative enforcement scenarios to combat tax evasion .for example , we find that ( i ) for given parameter values there is a certain threshold for efficient audits , ( ii ) real world audit rates may well be too low to effectively curb tax evasion and ( iii ) that at some level direct agent interaction may effectively be a substitute for monetary penalty payments .the paper is organized as follows . in section twowe provide some background on standard income tax evasion theory and existing agent - based tax evasion models . in section three, we develop the econophysics model of tax evasion and apply it to an analysis of audit efficiency and for some replication studies .concluding remarks are provided in the final section .in this section we briefly compare and contrast the standard approach to income tax evasion with essential features of agent - based tax evasion models .next , we review a few of these models and offer some background on the ising model , which we use in section three for constructing a novel agent - based tax evasion model .the neoclassical standard approach to income tax evasion considers a representative , self - reporting agent who declares an income so as to maximize expected utility , , according to , where denotes the true income of the representative agent , is the tax rate on declared income and is the tax rate on undeclared income , where indicates a monetary penalty on income tax evasion and is the audit probability , with [ ] .given this modeling frame , risk neutral tax payers ( i.e. ( [ as1 ] ) is a linear function ) declare their full income if , but declare nothing at all if .in contrast , risk - averse tax payers ( i.e. ( [ as1 ] ) is a concave function ) may declare their income fully , partly or not at all .other things being equal , the more risk - averse a tax payer is , the more compliant a tax - payer will be and both absolute and relative risk - aversion may play a role for the extent of tax evasion .hence , in the standard income tax evasion model risk aversion is the driving force that allows for interior solutions .of course , subsequent literature has developed various extensions and alternatives .for example , all - or - nothing decisions of risk - neutral tax payers may be avoided , if the audit probability is an increasing function of the amount of undeclared income [ ] , or if there are two or more income sources each having a different audit probability .further , the penalty may be proportional to unpaid taxes [ ] , which ensures that the tax rate has no influence on the extent of tax evasion .  in any case, essential features of the neoclassical standard approach to tax evasion are : ( i ) each agent has perfect knowledge about his own expected utility function and the relevant parameter values , which allows him to maximize his own expected utility , ( ii ) the mathematical specification of the utility function leads to situations where the agent maximizes his expected utility either by an all - or - nothing decision or by an interior solution in which the agent may declare just some part of his true overall income , ( iii ) heterogeneity in agent behavior may be introduced by individualizing one or more parameter values that enter an agent s expected utility function , ( iv ) dynamics in the behavior of agents can be due to parameter changes only , ( v ) any kind of direct interaction among the agents is ruled out .agent - based tax evasion models deviate from the neoclassical standard approach in at least three ways .a feature that distinguishes any agent - based tax evasion model from the neoclassical approach is the direct interaction among agents . in particular , the behavior of all or at least some agents depends on the behavior shown by a well specified subgroup of other agents , say neighbors , within the same model . besides this type of interaction is non - market based .another important difference is that some agents , if not all agents , may not possess an utility function .finally , in agent - based models tax evasion dynamics may be triggered by either parameter changes or by stochastic processes or a combination of both .although agent - based tax evasion models are a comparatively new tool for analyzing tax compliance issues , substantial differences already exist between these models or model types .therefore , in the next subsection we briefly review the literature on agent - based tax evasion models .as noted , the essential feature of any agent - based model is the direct non - market based interaction of agents , which is combined with some process that allows for changes in individual behavior patterns . therefore , agent - based tax evasion models may be categorized according to the features of this individual interaction process .in fact , according to this criterion models may either fall into the economics domain or into the econophysics domain . in the latter category , this process is driven by statistical mechanics using a model structure that is well known in physics , the ising model .examples include , , and .in contrast , if the interacting process is driven by parameter changes that induce behavioral changes via an utility function and/or by stochastic processes that are not related to models of statistical mechanics , these models belong to the economics domain .examples include , , , , , , , , and . offers a review of the first three models and , therefore , in the following we just consider the remaining models . set up a model where agents compare expected payoffs from three alternatives : non - compliance , partial - compliance and full - compliance .the key feature of their model is the way in which audits and risk aversion are modeled .agents never know the true audit probability .rather , they are heterogeneous with respect to the perceived chance of being audited and they update their individual rate by observing how many members of their social network were actually audited .likewise , agents are endowed with a different perceived taste for risk which they update based on the last period s interaction with the tax authority and by observing relative payoffs within their social network .another important feature of their model is the social network size of nine members , of which eight are neighbors of the agent under consideration .finally , they run a setting with no social network and three scenarios that differ with respect to the weight agents place on their neighbor s payoffs in updating their own perceived audit and risk parameters . with respect to the results they focus on stability of equilibria under changing enforcement regimes . among other things , they find that even low rates of enforcement may support a highly compliant society of taxpayers , despite strong dynamics at the agent level . model an entire economy with one industry consisting of several competing firms that need to hire employees , if they wish to produce output .firms wish to minimize the expected effective salary , whereas employees want to maximize expected effective net income .moreover , there is a government that provides public goods to the firms and the employees and there is a tax authority that conducts audits . to survive in the competitive environment, employers and employees can agree on various forms of contracts of which some are legal and others represent different forms of black labor , inducing tax evasion .activities of the government , the tax authorities , or the social networks of employers and employees may alter the dynamics of tax compliance in one way or another . consider a number of tax evasion models of which one is agent - based . in this model tax payershave identical utility functions and can make binary decisions , to comply or to evade .relevant for this decision is the level of individual utility , which in turn depends on a combination of internal and external factors .the internal factor is utility that depends on the provision of a public good and the external factor is observed behavior within the social network of an agent .moreover , a stochastic process is added with a fixed probability of random switches between the two states , compliance and evasion .however , no audits or penalties are considered . among other things , they find that efficiency in the provision of public goods reduces tax evasion . and both consider a model with different behavioral types of which one type more or less complies with a rational tax payer of the allingham and sandmo type .they then consider income tax evasion dynamics that results from alternative governmental policies and also include lapse of time effects ( back auditing ) .in fact , the model of may be considered as the agent - based model that is closest to the model . in particular , with 100 percent selfish a - type agents ( cf .below ) the hokamp and pickhardt model represents a numerical version of the model . deal with endogenous norm formation over the life cycle of agents .in particular , their simulations support the view that older people hold stronger moral attitudes ( i.e. evade less taxes ) than younger agents due to an age effect rather than a cohort effect . in their modelboth personal and social norms influence an agent s decision of whether to evade taxes or not .moreover , they allow for heterogenous norm - updating via several social psychology mechanisms ( cognitive dissonance , self - signaling , and conformity with social network preferences ) . otherwise , they follow the segregation model approach of .   consider a model with heterogenous agents who differ with respect to income , risk - aversion , preferences for public expenditure , and believes about the actual level of tax compliance in the population .audits and interactions among the agents are randomized and the government not only sets tax and penalty rates , but also spends all revenue to finance public expenditure . the population is set to 1,000 agents , each having a cobb - douglas utility function with their after tax monetary income , their perceived value of public expenditure and some relevant individual characteristics as arguments .findings include that equilibrium situations still arise even in the heterogeneous case , that the presence of public expenditure may establish a positive relationship between the tax rate and tax evasion without assuming tax morale or the like , and that individual characteristics of the agents matter more for individual compliance than audit policy parameters .however , at least the latter finding may depend on the fact that their model does not allow for auditing past periods ( back auditing ) .agent - based tax evasion models that fall into the econophysics domain differ fundamentally from the aforementioned models of the economics domain .an important difference is that in an econophysics model none of the agents has an utility function of some sort .rather , agents are in one of two possible states evading or compliant and the transition between both is a stochastic process .the latter is influenced by the behavior of neighbors and a global influence denoted as temperature , which has , just like a public good , a nonrival and simultaneous impact on the extent to which all agents are influenced by their respective neighbors .in fact , this difference has a number of implications and consequences .first of all , econophysics models are not based on the individualistic premises on which mainstream neoclassical economics rests .therefore , in these models no agent possesses individual properties such as age , income , risk preferences , etc .a major consequence of this feature is that in econophysics models no monetary penalties can be charged and penalties can not be differentiated with respect to the individual behavior or individual properties of the evader .rather , if caught , the penalty of an evader in an econophysics model consists of an obligation to be compliant for a specified number of periods . to this extent ,penalties in econophysics models differ fundamentally from those charged in economics models .another important consequence is that econophysics models can not consider any past decisions of tax payers .moreover , agents in econophysics models are always limited to a binary decision space , in the present context they can be either compliant tax payers or evaders .hence , the rate of tax evasion is measured in these models as the ratio of evaders over the total number of agents .  yet , despite these fundamental differences an interesting aspect is that on the aggregate level econophysics models may mimic results obtained by economics models and that both model types may lead to stable equilibrium situations ( steady states ) , which are characterized by a prevailing positive rate of tax evasion .essentially , this implies that the results of individual rational behavior patterns may be reconstructed at the macro or aggregate level by means of natural stochastic processes such as statistical mechanics .in addition , econophysics models generate some interesting policy conclusions . for example, find that even very small levels of enforcement are sufficient to establish almost full tax compliance and conclude that regardless of how strong group influence may be , enforcement always works to enhance tax compliance .however , in both cases the adjustment process may take a very long time .the ising model was originally introduced by the german physicist wilhelm lenz in order to study the formation of ferromagnetism .lenz gave it as a problem to his student ernst ising who presented the solution for the one - dimensional system in his phd thesis [ ] .it can be viewed as the simplest model for interacting magnetic moments ( here and in the following referred to as spins ) , which only can take the values . as illustrated in fig .[ figising ] , the spins can thus be represented as vectors which are attached to a lattice point ( fig .[ figising ] shows a one - dimensional example ) .these vectors are only allowed to point in the positive ( ) or negative ( ) z - direction .the interaction energy  drives the orientation of the spins where in the simplest case the interaction constant is only non - zero between adjacent sites and . as can be seen from fig .[ figising ] , the corresponding interaction energy is for parallel and for antiparallel spins .a positive ( negative ) exchange constant thus favors parallel ( antiparallel ) alignment of the spins since the configurations are associated with a lower energy .   on a one - dimensional chain .the spin variables can take the value when the vector points in the positive ( negative ) -direction .the interaction energy between parallel spins is and between antiparallel spins.,width=453 ]  however , the spin configuration of the ising model also depends on temperature , which tries to disorder the spins .the associated characteristic quantity is the thermal energy , where is the boltzmann constant which depends on units ( meter ) , ( kilogramm ) , ( seconds ) , and ( kelvin ) .basically the system is in a ( dis)ordered state when the interaction energy per spin is much ( larger ) lower than the thermal energy .a special case in this regard is the one - dimensional ising model which does not have a phase transition at finite temperatures , but only is in an ordered state at exactly .the situation is different for higher dimensional lattices . analytically demonstrated the occurrence of a finite temperature order - disorder transition on a two - dimensional square lattice at , where denotes the critical temperature and is the exchange constant between nearest - neighbor sites as defined in eq .[ eq : eint ] .in contrast , for dimensions despite its simplicity the model can only be solved numerically .for example , numerical evaluation of the transition temperature for a three - dimensional cubic lattice yields , below which the spins are in an ordered state .note that a spin on a -dimensional hypercubic lattice is attached to nearest neighbors so that the higher the dimension ( or , more precisely , the coordination number ) the thermal energy has to overcome a larger interaction energy to disorder the spins . in order to simplify notation we will measure in the following temperature in units of energy ( i.e. we set ) . in section threewe will also couple the spins to an external magnetic field , which can be different at every lattice site .these fields tend to align the magnetic moments of the spins , which can be described by the energy contribution .thus , a positive field lowers the energy for a positive spin value , whereas a negative field favors negative spins .nowadays , the ising model is one of the standard models in statistical physics .moreover , it has become popular in a large variety of different fields like biology , sociology , economics etc . because it is one of the simplest models that describes the interaction of entities which try to adjust their behavior in order to be conform with that of their neighbors .for example , thomas c. schelling s ( 1971 ) segregation model roughly corresponds to an ising model at as shown by , and .in this section we model tax evasion dynamics by developing an agent - based econophysics model . in section [ sec : modeltwo ] we first introduce the standard two - dimensional ising model and reproduce some selected results of zaklan et al .next , we extend this model by incorporating features of heterogeneous multi - agent simulations .we apply the model to an analysis of different enforcement scenarios and for replicating some results of .all results presented in this section are obtained from a fortran code , which is available upon request .  .the spin degrees of freedom sit on the lattice sites marked by solid dots and each spin interacts with its four adjacent neighbors .( b ) sketch of a ringworld structure as additionally considered in sec .[ sec : hopi ] . here each spinis coupled to two adjacent neighbors only.,width=453 ]   our considerations are based on the two - dimensional ising model eq .( [ eq : eint ] ) , implemented on a square lattice ( see fig .[ fig5 ] ) .in addition to the bare model eq .( [ eq : eint ] ) we now allow for the coupling of each spin to a local magnetic field .the corresponding hamiltonian reads as , the spin degrees of freedom can take the values and denotes the interaction between two spins on sites and .we take to be a constant within a given interaction range ( corresponding to the nearest neighbors , i.e. four agents in the present section ) and otherwise . in the present context interpreted as a compliant tax payer and as a non - compliant one .we use the heat - bath algorithm [ cf . ] in order to evaluate statistical averages of the model .the probability for a spin at lattice site to take the values is given by /t\}}\]]and is the energy change for a spin - flip at site . upon picking a random number spin takes the value when and otherwise .one time step then corresponds to a complete sweep through the lattice . in analogy to we further implement the probability of a tax audit .if tax evasion is detected the agent has to remain compliant over periods of time . and interpret this number of periods of enforced compliance as a penalty or as a consequence of shame and guilt feelings , respectively .but , as noted , this kind of penalty is fundamentally different from monetary penalties applied in tax evasion models of the allingham and sandmo type .rather , periods of after audit compliance are used elsewhere to accommodate extreme subjective audit expectations ( see e.g. ) .   for illustrative purposes and in order to define the limits for our multi - agent generalization below , we consider first the zero - field case ( in eq .[ eq : ising ] ) .this corresponds to the model introduced by zaklan et al .( 2009 , 2008 ) where the system is composed of one type of agent only and spin - flip probability is solely determined by the parameter . in the present context, the ratio can be viewed as a measure for the autonomous behavior of the tax payers . in the high - temperature limit agent at site decides autonomously from his or her social environment ( i.e. ) , whereas for the probability strongly depends on the compliance of his or her neighbors within the social network .note that this is related to the ferromagnetic phase transition of the two - dimensional ising model ( for ) , which occurs at as discussed in the previous section . in the followingwe consider a population of agents , except in sec .[ sec : hopi ] where the population is set to agents .[ fig1a ] ( panels a and b ) displays the corresponding extent of tax evasion ( i.e. the fraction of -spins ) as a function of time for small and large audit probabilities and , respectively . we define as reference energy scale and as initial condition we set all agents to compliant , i.e. . due to the small social temperature ( which is , thus , of the same order than the exchange energy ) only a small number of agents become non - compliant in the first time step and because the audit probability is also small , tax evasion increases continuously up to some saturation value . at this valuethe system of agents has reached equilibrium between the influence of the neighborhood to become a non - compliant tax payer and the tax audits which enforce compliance over time periods . upon increasing the audit probability to ( panel b )the number of tax evaders reaches its maximum already after about three time steps and then starts to decrease due to the high audit probability .after time steps , initially detected tax evaders may again switch from compliant to non - compliant behavior , which leads to the observed small increase of before it reaches its saturation value of .note that the social temperature is below the ordering temperature of the two - dimensional ising model . for zero audit probabilitythe long - term tax evasion would therefore also converge to a small value ( ) , corresponding to the fraction of minority spins at this particular temperature .the effect of a finite audit probability is , therefore , just a further reduction from this small value .  in contrast , above the ordering temperature , the zero audit equilibrium state is of course reached for an equal number of compliant and non - compliant tax payers , i.e. .finite audit probabilities then lead to a further reduction of this equilibrium value .this situation is depicted in panels c and d of fig .[ fig1a ] , which display the extent of tax evasion for a system build up from agents who decide mainly autonomously , due to the large social temperature ( i.e. much larger than the exchange energy ) . as a consequence , already after two time steps approximately half of the agents become non - compliant which in turn induces a reduction of due to the auditing .it is remarkable that even for small audit probability ( fig .[ fig1a]c ) decreases by before it saturates at after more than time steps . upon invoking a larger audit probability , ,the number of autonomous agents decreases rapidly after the first initial time steps and consequently decreases to almost zero within the first time steps .half the fraction of those who have been detected at the first time step will then select the possibility of non - compliance again , which leads to the oscillatory behavior of .a stable situation is only reached at time steps at .note that the results of fig .[ fig1a ] , which we have discussed so far , correspond to those presented by [ fig . 1 and fig . 4 ] .in fact , our results represent the first independent replication of results originally obtained from an agent - based econophysics tax evasion model .  .panels a and b display results for agents with a social temperature of and audit probabilities and , respectively . panelsc and d show the corresponding results for a social temperature of .reference energy scale is .tax audits enforce compliance over time steps.,width=453 ]   we now follow the scheme proposed in and discuss the case of societies where the agents have an endogenous preference with regard to compliance or non - compliance . this feature can be incorporated via the field parameter in eq .[ eq : ising ] .a value of enhances the probability of values and , therefore , corresponds to selfish agents , who try to maximize their payoff by non - compliant behavior . on the other hand ,a positive field favors positive values and , therefore , can be used to describe ethical agents who wish to be compliant , but may make occasional mistakes . with regard to the ising model eq .[ eq : ising ] , a dominant endogenous behavior further requires in both cases .the results in the present section are obtained for a society of identical agents , i.e. with and .  .panels a and b display results for selfish agents , i.e. endogenous non - compliant agents with a bias towards tax evasion and audit probabilities and , respectively .the inset to panel b displays the small time behavior to visualize the oszillations in .panels c and d show the corresponding results for ethical agents with a tax evasion bias and audit probabilities and , respectively . for all results the ratio between interaction constant and temperature is set to .tax audits enforce compliance over time steps.,width=453 ]  the first case of endogenous non - compliant selfish agents ( i.e. ) is shown in panels a and b of fig .[ fig1b ] . at timestep zero we correspondingly set the share of tax evasion to . for zero audit probabilitythe equilibrium state would , therefore , yield a tax evasion rate close to one ( ) due to rare spin flip events .however , a small audit probability reduces tax evasion significantly because at each time step five percent of the remaining non - compliant agents are forced to become compliant .after time steps the first detected agents can become non - compliant again , which after a small increase leads to the stabilization of tax evasion at . a larger audit probability ( fig .[ fig1b]b ) again drives tax evasion rapidly to zero and , in addition , induces weaker peaks at those time steps ( i.e. multiples of ) when forced compliant agents have the possibility to be non - compliant again ( cf . inset to fig .[ fig1b]b ) .when the peak height is below some value ( after time steps ) , then the audit is no longer able to reduce tax evasion to .however , the minimum value starts to increase , leading again to an equilibrium state on very long time scales .finally , the time evolution for ethical agents ( i.e. ) is rather obvious .initial tax evasion is set to and there is only a very small probability that one of the agents becomes non - compliant .this probability is controlled by the ratio and would become zero in the limit . also , since ethical agents wish to minimize tax evasion , the results are also almost independent of the audit probability .hence , any positive audit probability would be inefficient in this case .it is now straightforward to generalize the results of the previous section in order to describe societies which are characterized by behaviorally different types of agents . to achieve this, we define a local temperature at each site so that an ensemble of heterogeneous agents can be classified according to their ratios and .as already outlined in the previous section the parameter describes the exogenous conditioning of agent , whereas his or her endogenous ( non - manipulable ) code of conduct is measured by the ratio .following , we introduce the following four types of agents : ( i ) _ selfish a - type agents _ , which take advantage from non - compliance and , thus , are characterized by and ; ( ii ) _ copying b - type agents _ , which copy tax behavior of their social environment or neighborhood .this can be modelled by and ; ( iii ) _ ethical c - type agents _ , who are practically always compliant and which are parametrized by and ; ( iv ) _ random d - type agents _ , which act by chance , within a certain range , due to some confusion caused by tax law complexity .we implement this behavior by and . here and in the following all parametersare measured with respect to . within these definitions the results presented in the previous section correspond to homogeneous societies of b - type ( fig .[ fig1a]a , b ) , d - type ( fig .[ fig1a]c , d ) , a - type ( fig .[ fig1b]a , b ) , and c - type agents ( fig .[ fig1b]c , d ) .note that by introducing b - type and d - type agents , we extend the model of , who consider just a - type and c - type agents .we now take the number of agents for each behavioral type as fixed , but introduce some heterogeneity for each type .this is achieved by randomizing the parameters and within a given range , which is specified in table [ tab1 ] , and relevant parameters for the following results are given in table [ tab2 ] ..parameter ranges for each group of agents .note that and are measured in units of . [ cols="<,^,^",options="header " , ]   in the results displayed in fig .[ fig2 ] we fix the percentage of b- and d - type agents to and , respectively ( see hokamp and pickhardt , 2010 ) and vary the ratio of a- and c - type agents for two audit probabilities ( panel a ) and ( panel b ) .the initial tax evasion is set to the fraction of a - type agents , i.e. .   percent and percent , respectively .the percentage of a- and c - type agents is varied according to the labels and the order within each label corresponds to the solid lines shown in each panel .audit probability ( a ) , ( b ) and .the horizontal dotted line marks the stationary tax evasion value without audit and the vertical arrows indicate the difference to the stationary value with audit.,title="fig:",width=359 ]  percent and percent , respectively . the percentage of a- and c - type agents is varied according to the labels and the order within each label corresponds to the solid lines shown in each panel .audit probability ( a ) , ( b ) and .the horizontal dotted line marks the stationary tax evasion value without audit and the vertical arrows indicate the difference to the stationary value with audit.,title="fig:",width=359 ]  let us first discuss the case of a small audit probability and ( first curve from below in fig . [ fig2]a ) . herethe society consists of copying b - types ( ) , ethical c - types ( ) and random d - type agents ( ) .the time evolution thus is similar to a mixture of the homogeneous agent systems shown in figs .[ fig1a]a , c and [ fig1b]c , i.e. the stationary state for large time steps is determined by non - compliant b- and d - type agents , the number of which is limited by the audit .note , however , that the difference to the homogeneous cases lies in the fact that now the copying b - type agents are embedded in a matrix of ethical c - type agents , which almost never change their behavior .one observes that the stationary value is not reduced significantly with respect to the limit without audit ( indicated by the dotted horizontal lines ) , since the audit can essentially detect only non - compliant d - type agents .the other limiting case is that for ( first curve from above in fig .[ fig2]a ) , where half of the society consists of tax evading a - type agents . during the first time step most b - type agents copy non - compliant behavior so that rises slightly above .moreover , a comparison of the stationary states with and without audit , , reveals a significant reduction , because a- , b- and d - type agents may be forced to be compliant over a period of time steps .this reduction obviously increases for larger audit probabilities , as can be deduced from fig .naturally , a small audit probability mostly affects a- , and b - type agents because d - type agents essentially act randomly and , thus , the d - type specific reduction is only of the order ( i.e. for and for ) .[ fig3a ] reports the reduction with respect to the stationary state without audit .time steps ( steps ) in the stationary state .] this quantity , thus , measures by which percentage points tax evasion is reduced by the audit as compared to the non - audited society .it is obviously an increasing function of the audit probability and , as anticipated , also increases with the percentage of a - type agents .an interesting feature of the curves shown in fig .[ fig3a ] is the slope , which decreases with increasing .we find that for small audit probabilities the percentage of detected tax evadors increases more strongly than for large .therefore , increasing the audit probability becomes more and more inefficient in increasing the number of detected tax evadors .a natural criterion for an efficient audit probability range is set by the requirement that the slope of the curves in fig .[ fig3a ] is larger than one. this critical value ( indicated by the circles in fig .[ fig3a ] ) is reached at for a - type agents and at for a - type agents .hence , subject to the aforementioned specifications an audit would in general be efficient for audit probabilities .   and finite audit probability normalized to for different percentages of a - type agents .the circles indicate the point where the slope of the curves equals one.,width=453 ]  whereas our previous measure of efficiency is related to the effort which has to be put in an audit , efficiency can also be defined with regard to the success of an audit . in fig .[ fig3 ] we plot again the difference of stationary tax evasion values with and without an audit , now as a function of the percentage of a - type agents .for example , given the percentage of a - type agents as we can read from fig .[ fig3 ] that tax evasion is reduced by . as discussed above this reductiondistributes approximately equal between the a- and b - type agents whereas the reduction of tax evading d - type agents is at most some few percentage points .if we define an audit as efficient when it can at least detect the percentage of a - type agents , then we see from fig .[ fig3 ] that this criterion is almost fulfilled for , but not for small audit probability .in fact , let us consider the specific case of and audit probability .then we can read from fig .[ fig2]b that the percentage of tax evasion is reduced from to , i.e. .this reduction is composed of an audit induced reduction for a - type agents from to and for b - type agents from to , while d - type agents contribute with a reduction from to .efficiency , in the sense defined here , thus means that the audit detects at least the number of tax evadors which corresponds to the fraction of inherently non - compliant tax payers ( i.e. a - type agents ) . yet, it does not mean that all the detected tax evadors are a - type agents because from the example above it is obvious that the reduction involves to approximately equal parts a- and b - type agents .( see fig .[ fig2 ] ) and .,width=453 ]  comparing these results with the results for the previously introduced definition of efficiency it turns out that an audit is efficient with regard to effort for , whereas , on the other hand , with regard to success an audit is efficient for .hence , for the society under consideration the optimal audit probability would be in the range of , corresponding to the boundaries of both , otherwise disjunct , efficiency ranges .of course , other parameter values may lead to different ranges for the optimal audit probability .moreover , in both cases audit efficiency is a pure quantity measure , whereas in the real world costs and revenue aspects may distort the two quantity measures .this notwithstanding , and subject to the model specifications , our results seem to indicate that tax audit rates applied in many countries , e.g. less than one percent in the u.s . according to , may be suboptimally low . to this extent, our results also differ from those of or , who find that even small audit probabilities lead to tax compliance in the very long run .  ;( b ) agent - based model by , but with fixed parameters [ squares ] ; ( c ) ising model on a square lattice with fixed parameters [ triangles ] ; ( d ) ising model on a ringworld with fixed parameters [ diamonds].,width=453 ]   finally , we show in this section how the dynamics of the agent - based model by ( ) can be captured within the present multi - agent ising - model approach .solid dots in fig .[ fig6 ] show the original dynamics in terms of percentage of tax evasion obtained for the first scenario and the set of governmental tax policy parameter changes used by , table 1 , and figure 1 , first line from below .these tax policy parameters are the audit probability , tax rate , penalty rate and a parameter which specifies the complexity of the tax laws .obviously , there is no direct mapping of these parameters to our ising type approach since , for example , a monetary penalty rate presupposes the definition of a tax relevant income , whereas the ising model can only deal with compliant or non - compliant tax payers .however , we will demonstrate in this section that both approaches lead to very similar dynamics when the multi - agent ising model is defined with appropriate field and temperature parameters for individual agents .therefore , we restrict to a constant parameter set , i.e. the , table 1 , tax policy parameters of the first tax period . for this fixed parameter set , the original results of the hokamp and pickhardt approach are displayed in fig .[ fig6 ] by the curve with square symbols .interestingly , these results are by and large in line with recent findings of , who report evidence from a large scale tax audit field experiment in denmark and find a tax evasion rate of for self - employment income .  yet, with respect to the replication it is worth noting that the hokamp and pickhardt ring world model is constructed such that four neighbors to either the left or right of an agent are taken into account .unfortunately , this is not compatible with the symmetry of the ising model which requires , as in eq .[ eq : ising ] .thus , for definiteness we fit the hokamp and pickhardt results with a one - dimensional ising model ( ring world with two neighbors ) and , in addition , show differences to the results obtained with the two - dimensional ising model ( square lattice with four neighbors ) and the same parameter set .  the curves labeled with diamonds and triangles in fig .[ fig6 ] show our replication results .the initial conditions are generated by just percent a - types and percent c - types .percent a - types , percent b - types , no c - types and percent d - types .] this is because in the first time step of the hokamp and pickhardt model b - types must use their default option ( i.e. full compliance ) and d - types are on average also compliant , so that the joint share of b - types and d - types is on average equivalent to a percent c - type share .a - type agents are specified by fields and temperature , whereas c - type agents have and . as already noted ,all units are measured with respect to .equilibrium is reached after time steps so that time step corresponds to the first tax relevant time step or year shown in fig .[ fig6 ] . as of the second tax relevant time step ,we reduce the percentage of c - type agents to percent ( i.e. the d - type share in hokamp and pickhardt ) , because in the hokamp and pickhardt model d - types may also randomly overpay their tax bill , which is not possible in our multi - agent ising model . ] and introduce instead a percent share of b - types , which are characterized by and .hence , the ratio is large for b - types so that they copy the behavior of others in their neighborhood .the number of a - type agents remains unchanged .thus , in the present multi - agent ising - based model , a - type , b - type and c - type behavior patterns are reproduced with different local fields and temperatures , whereas d - type behavior is approximated on average by an equivalent percentage of c - type agents .it turns out that during the first few time steps the ising model multi - agent approach yields a fairly good match with the hokamp and pickhardt results because the lines with triangles , diamonds and squares ( dots ) closely match .however , in the large time limit the latter leads to a saturation at slightly smaller values .this is essentially due to several reasons with minor effects . in particular , differences in the stochastic processes , in the c - type versus d - type modeling , in the individual extent of tax evasion , in the penalties charged and in the lattice structures .moreover , it is worth noting that differences with the square lattice replication ( triangles in fig .[ fig6 ] ) are due to the fact that the b - type agents may copy the a - type behavior much faster in the ringworld . as the ( standard ) ising model is based on the assumption that agents are influenced only by their direct neighbors , the one - dimensional ising model ( ringworld )reduces the size of the social network to two agents , whereas the social network size is four agents in both the two dimensional square lattice ising model and the hokamp and pickhardt ringworld model .hence , given the parameter setting , in the one - dimensional ising ringworld it is more likely that b - type agents are influenced by their a - type neighbors .thus , in each time step , tax evasion is higher in the ising ringworld [ diamonds ] compared to the ising square lattice world [ triangles ] ( see fig .[ fig6 ] ) , except in the long run .regarding the hokamp and pickhardt ringworld , copying of tax evasion behavior is comparatively fast because the average tax evasion in the social network is taken into account in every time step , which almost guarantees that tax evasion behavior quickly spreads .in this paper we have independently reproduced results obtained from an agent - based econophysics model on tax evasion .next , we extended the standard econophysics model of tax evasion in such a way that different behavioral agent types can be included .this represents another novelty because it extends previous work of towards a society which is build up from four different types of agents .we then used the extended model for an analysis of audit efficiency .it turned out that under the given circumstances substantially higher audit probabilities , in comparison to those that are effectively applied in most countries , would be desirable to curb tax evasion . further , overall tax compliance of the extent displayed in fig .[ fig2 ] is achieved by a non - monetary penalty , which consists of periods of enforced after - audit compliance .hence , a non - monetary penalty combined with copying behavior due to interactions among agents may be an effective substitute for the standard approach of charging monetary penalties .moreover , the new feature moves econophysics models of tax evasion closer to agent - based models of tax evasion that are found in the economics domain .therefore , we were also able to use our extended econophysics model for reproducing results originally obtained from an agent - based tax evasion model of the economics domain .again , this is the first time that such a reproduction has been achieved with an econophysics model .an important finding of this reproduction is that a micro foundation of individual behavior , expressed in terms of individual utility or payoff functions in models of the economics domain , is not needed for the aggregate results obtained by these models .put differently , rational individual utility maximizing behavior can be reproduced on the aggregate level by so called zero intelligence agents , some stochastic changes in individual behavior patterns , and a certain degree of direct agent interaction that is controlled by the parameter temperature , which influences all agents in a nonrival manner like a public good ( i.e. tax morale , massmedia , etc . ) .moreover , econophysics tax evasion models are based on a natural stochastic process derived from the ising model of ferromagnetism .this is one of their major advantages in comparison with models from the economics domain in which either simple stochastic processes are used or unrecognized attempts are made that essentially try to reproduce elements of the ising model .hence , if econophysics models of tax evasion would be complemented with a micro foundation they might well make a fundamental contribution to the analysis of tax evasion . by extending the standard econophysics model of tax evasion in a way that various behavioral agent types could be incorporated, we made a first step towards a micro foundation of econophysics models .of course , further steps are required , but these tasks delineate a future research agenda .antunes , luis , joao balsa , ana respicio and helder coelho ( 2007 ) .tactical exploration of tax compliance decisions in multi - agent based simulation , in : luis antunes and keiki takadama ( eds . ) , _ multi - agent - based simulation vii _, international workshop mabs 2006 , lnai 4442 , heidelberg : springer .bloomquist , kim m. ( 2004 ) .modeling taxpayers response to compliance improvement alternatives .paper presented at the annual conference of the north american association for computational social and organizational sciences , pittsburgh , pa .kleven , henrik j. , martin b. knudsen , claus t. kreiner , sren pedersen and emmanuel saez ( 2011 ) .unwilling or unable to cheat ?evidence from a tax audit experiment in denmark , _ econometrica _ * 79*(3 ) , 651 - 692 .lima , francesco w. s. and georg zaklan ( 2008 ) . a multi - agent - based approach to tax morale . _ international journal of modern physics c : computational physics and physical computation _ * 19*(12 ) , 17971808 .mder , zsombor z. , andrs simonovits and jnos vincze ( 2010 ) .tax morale and tax evasion : social preferences and bounded rationality , discussion paper , institute of economics , hungarian academy of siences , hungary .mittone , luigi and paolo patelli ( 2000 ) .imitative behaviour in tax evasion , in : francesco luna and benedikt stefansson ( eds . ) , _ economic simulations in swarm : agent - based modelling and object oriented programming _ , dordrecht , boston , london : kluwer academic publishers , 133158 .szab , attila , lszl gulys and istvn j. tth ( 2008 ) .`` taxsim agent based tax evasion simulator '' , paper presented at the 5th conference of the european social simulation association , university of brescia , italy .szab , attila , lszl gulys and istvn j. tth ( 2009 ) .sensitivity analysis of a tax evasion model applying automated design of experiments , in : luis s. lopes , nuno lau , pedro mariano , and luis m. rocha ( eds . ) , _ progress in artificial intelligence _, 14th portuguese conference on artificial intelligence , epia 2009 , aveiro , portugal , lnai 5816 , berlin heidelberg : springer , 572 - 583 .

Abstract: we analyze income tax evasion dynamics in a standard model of statistical mechanics , the ising model of ferromagnetism . however , in contrast to previous research , we use an inhomogeneous multi - dimensional ising model where the local degrees of freedom ( agents ) are subject to a specific social temperature and coupled to external fields which govern their social behavior . this new modeling frame allows for analyzing large societies of four different and interacting agent types . as a second novelty , our model may reproduce results from agent - based models that incorporate standard allingham and sandmo tax evasion features as well as results from existing two - dimensional ising based tax evasion models . we then use our model for analyzing income tax evasion dynamics under different enforcement scenarios and point to some policy implications .  _ keywords : tax evasion , tax compliance , ising model , econophysics , numerical simulation _  _ jel codes : h26 , o17 , c15 _


during the past four decades , the decrease in transistor size and the increase in integration factor have led to very small , fast , and power efficient chips .as the demand for power efficiency continues , a wide range of new nano - scale technologies is being actively investigated for processing and storage of digital data .although it is difficult to discern which of these approaches will become a technological basis for computers in the future , it is widely recognized that due to their miniature size and variations in technological process , the nano - components will be inherently unreliable . even in more traditional semiconductor technologies , reducing transistor size has already started affecting circuit reliability , and it is widely believed that transistor failures ( both transient and permanent ) will become one of the main technological obstacles as the trend of increasing the integration factor continues . in this paper , we consider storage circuits built from such unreliable ( faulty ) components .we consider an unreliable component ( a logic gate or a memory element ) to be a component that is subject to _ transient faults _, i.e. , faults that manifest themselves at particular time steps but do not necessarily persist for later times .von neumann was the first to study computation using faulty gates . in , he showed that , under certain conditions , increased gate redundancy can lead to increased reliability of a circuit .however , it was shown that , in general , computation by faulty gates with non - zero computational capacity is not possible ( see ) .the study of storage circuits made of unreliable components led to much more optimistic results .taylor in proved that a memory has an associated information storage capacity , , such that arbitrarily reliable information storage is possible for all memory redundancies greater than .the methodology of the proof , however , does not allow one to explicitly calculate the storage capacity .taylor considered two models of component failures and proposed construction of fault - tolerant memories based on low - density parity - check ( ldpc ) codes . in the first model ,the failures of a particular component are assumed to be statistically independent from one use to another and is referred to as the independent failure model . in the second model ,the components fail permanently but bad components are replaced with good ones at regular intervals .the failures in different components are assumed to be independent in both the models .this construction was further studied by kuznetsov in and we will refer to it as the taylor - kuznetsov ( tk ) scheme .hadjicostis was able to generalize taylor s scheme to fault tolerant linear finite state machines .spielman obtained the best result for a general model of computation , by marrying the ideas of von neumann with reed - solomon ( rs ) codes .the fundamental contribution of this paper is to show existence of reliable memories built entirely from unreliable components and which have finite redundancies .we consider the adversarial failure model in which only a fixed fraction of the components fail at any given time and extend our results to the independent failure model using chernoff bounds .our memory architecture has lower redundancy compared to the tk scheme .our fault tolerant memory architecture is also based on ldpc codes but differs from the tk scheme in the decoding algorithm employed .the tk scheme can be shown to be an implementation of the gallager b decoding algorithm for ldpc codes ( the proof will be given in section [ tkscheme ] ) .we use the parallel bit flipping decoding algorithm proposed in the context of expander codes by sipser and spielman .expander codes are a class of asymptotically good error correcting codes with linear time decoding algorithms which can correct a linear fraction of errors .expander graph based arguments have been successfully applied for message passing algorithms by burshtein and miller in as well as for linear programming decoding by feldman _et.al _ in .at the time of their discovery , explicit construction of graphs with expansion required for parallel bit flipping algorithm were not known .et al_. recently gave an explicit construction of expander graphs based on randomness constructors .hence , our method can be seen as a constructive proof in contrast to taylor s method which is an existence proof .the rest of the paper is organized as follows . in section [ system ]we provide the necessary definitions and a brief overview of ldpc codes .we explain the proposed memory architecture and characterize it in terms of complexity and redundancy . in section[ main ] we introduce the model of failure of the components and prove our main result showing the existence of memories which can tolerate failures in all the components . in section [ results ]we provide a few numerical examples . in section [ tkscheme ]we establish an equivalence between the tk scheme and gallager b algorithm and extend our results to the independent failure model . in section [ discussion ]we discuss open questions and conclude with some interesting remarks .in this section , we give a detailed description of the memory system .we start by introducing the terminology used to characterize memories and proceed to discuss the importance of ldpc codes .we explain the coding scheme and the error correction scheme employed in the proposed memory architecture .we then calculate the redundancy and complexity associated with the memories .a memory is a device in which information is stored at some time and retrieved at a later time .the memories under consideration store information in form of bits and are built from registers ( memory elements ) each of which can store a single bit .the information storage capability of a memory is the number of information bits it stores .consider a memory built out of reliable registers . to build a memory with information storage capability of bitsrequires registers .such a memory is termed as an irredundant memory .now , consider the problem of information storage with unreliable memory elements . due to the component failures , the information read out of the memorymay not be identical to the information stored originally .hence , to ensure reliable storage , the information needs to be stored in coded form ( see for an excellent discussion on the importance of coded form ) . initially , a codeword from some error correcting code is stored in the memory .the unreliable nature of the memory elements introduces errors in the registers and the contents of the memory differ from the initial state . to ensure reliability ,a correcting circuit is employed which performs error correction and updates the contents of the registers with an estimate of the original codeword .hence , a fault - tolerant memory system ( referred to as memory system or simply memory henceforth ) consists of memory elements ( referred to as storage circuit ) and a correcting circuit .the correcting circuit is also built of unreliable components .the coding of information along with the correcting circuit introduce redundancy into the memory system .such redundant memories are characterized by two closely related parameters , namely , complexity and redundancy .the _ complexity _ of a memory is the number of components within the memory ( a component is a device which either performs an elementary operation or stores a single bit where an elementary operation is any boolean function of two binary operands ) .the _ redundancy _ of a memory is the ratio of the complexity of the memory to the complexity of an irredundant memory which has the same information storage capability .it should be noted that there can be many memory architectures with different complexities but the same information storage capability .another important characteristic of a memory is reliability .we say that arbitrarily reliable information storage is possible in a memory if the probability of memory failure can be made arbitrarily small .to quantify the reliability of a memory system , it is important to first define what constitutes a memory failure .let a memory failure be defined as an event in which the word read out of memory is not equal to the original codeword .arbitrarily reliable information storage is not possible with such a definition of memory failure .this is due to the fact that the probability of failure is lower bounded by the probability of failure of components in the final step of extracting the information bits .hence , we define a failure in the following manner . associated with each codeword in a code is a decoding equivalence class , i.e. , the set of words which decode to that particular codeword when decoded with a decoder built of reliable components .if the contents of the memory do not belong to the decoding equivalence class of the original codeword , we say a memory failure has occurred . the _ storage capacity, , of a memory is a number such that for all memory redundancies greater than , arbitrarily reliable information storage is possible .the memories under consideration store information in form of bits and therefore we restrict our attention to binary codes in this paper .an binary block code maps a message block of information bits to a binary -tuple .the rate of the code is given by .an binary linear block code , , is a subspace of of dimension  .a parity check matrix of is a matrix whose columns generate the orthogonal complement of , i.e. , an element of is a codeword of iff  .the information storage capability of a memory depends on the type of the code employed in the correcting circuit .hence , a memory employing an block code has information storage capability of bits .taylor in argues that no decoding scheme other than iterative decoding of ldpc codes can achieve non - zero storage capacity .ldpc codes are a class of linear block codes which can be defined by sparse bipartite graphs .let be a bipartite graph with two sets of nodes : variable ( bit ) nodes and check ( constraint ) nodes .the check nodes ( variable nodes ) connected to a variable node ( check node ) are referred to as its neighbors .the degree of a node is the number of its neighbors .this graph defines a linear block code of length and dimension at least in the following way : the variable nodes are associated to the coordinates of codewords .a vector is a codeword if and only if for each check node , the sum of its neighbors is zero .such a graphical representation of an ldpc code is called the tanner graph of the code .the adjacency matrix of gives , a parity check matrix of .an regular ldpc code has a tanner graph with variable nodes each of degree and check nodes each of degree .this code has length and rate  .it should be noted that the tanner graph is not uniquely defined by the code and when we say the tanner graph of an ldpc code , we only mean one possible graphical representation .   the complexity and redundancy of a fault - tolerant memory depend on the coding scheme as well as the decoding algorithm employed in updating the contents of the memory .we now explain our memory architecture in detail .  at time , a codeword from an regular binary ldpc code is written into the storage circuit consisting of registers each of which can store a single bit .the bits of the codeword correspond to the variable nodes in the tanner graph , , of the code .the contents of the registers are updated at times , .the update rules can be explained by defining messages along the edges in . for a variable node ( check node ) ,let ( ) denote the edges incident on ( ) .each edge is associated with a variable node and a check node .let and represent the messages passed on an edge from variable node to check node and check node to variable node at time respectively .let denote the value of variable node at time .then the update at time is given by the following algorithm :  * algorithm a *  * for each edge and corresponding variable node  * for each edge and corresponding check node  * for each variable node +   the algorithm can be interpreted in the following manner .every variable node sends an estimate of its value to the neighboring check nodes .a check node calculates an estimate of a neighboring variable node by computing the modulo two sum of all the remaining neighboring variable nodes .each variable node receives estimates , one from each neighboring check and the majority of these estimates is the updated value of the node ._ remarks : _ we assume that the update is instantaneous and use and to denote the value of variable just before and after the update respectively .we note that the algorithm presented above is a slight modification of the parallel bit flipping algorithm proposed in .ldpc codes can achieve non - zero capacity due to the fact that the redundancy of the ldpc codes memory increases linearly with the information storage capability .the complexity of the logic gates needed to perform decoding depend only on and and not on the length of the code .so the redundancy remains bounded even as the code length tends to infinity .we now calculate the complexity and redundancy associated with our fault - tolerant memory architecture .the storage circuit consists of registers each of which can store a single bit and hence has complexity .the correcting circuit consists of logic gates ( built from components ) needed to implement the update algorithm .the message sent from a check node to variable node involves computing the modulo two sum of bits which requires a -input xor gate which can be implemented using two input xor gates ( a two input xor gate calculates modulo two sum of two bits ) .each check node needs to compute such estimates .therefore , the total number of two input xor gates is each variable node is updated based on the majority of the estimates received from its neighbors .this requires a -input majority logic gate for every variable node whose complexity we denote by .hence , the complexity of the memory system is the memory has information storage capability of bits and the complexity of an irredundant memory with the same information storage capability is .the redundancy of the fault - tolerant memory is therefore storage capacity of a memory depends on the type of failures in the components .a logic gate is said to have failed if its output is flipped .a register is said to have failed if the bit stored in it is flipped . in this paperwe consider the adversarial failure model also referred to as bit flipping channel model . in the adversarial model ,the failures occur in the worst case fashion but no more than a fixed fraction of the components fail at any given time .in other words , the number of failures is bounded for a given number of components . as the number of components increasesso does the number of failures .we denote the fraction of memory element failures in a time interval by , fraction of two input xor gate failures for every use by and fraction of -input majority logic gate failures for every use by . as mentioned before, the component failures are transient and independent from one use to another .a memory system is said to tolerate a constant fraction of errors in all components if at any time at most a constant fraction of components can fail and no memory failure occurs in the system at all times .recall that , from our definition , a memory failure occurs if the contents in the memory do not belong to the decoding class of the originally stored codeword . in this section , we prove that the memory architecture proposed in section [ system ] can tolerate a constant fraction of failures in all the components .our proof is based on the expansion property of the underlying tanner graph , , of the code . a tanner graph of a ldpc code is a expander if for every subset of at most an variable nodes , at least check nodes are incident to .the definition of expander is much more general but we restrict our attention to tanner graphs of ldpc codes .sipser and spielman in proposed a class of asymptotically good error correcting codes based on expander graphs known as expander codes .they proposed two simple bit flipping algorithms , namely , serial and parallel and showed that when the underlying graph has sufficient expansion , these algorithms can correct a fixed fraction of errors .ldpc codes are a special case of expander codes in which the expander graph is the tanner graph of the ldpc code .we describe the parallel bit flipping algorithm and interested readers are referred to for details about serial bit flipping .we say that a constraint is satisfied by a setting of variables if the sum of the variables in the constraint is even ; otherwise , the constraint is unsatisfied .the set of variable nodes ( bits ) which differ from their original value are known as corrupt variables .  * parallel bit flipping algorithm *  * in parallel , flip each variable that is in more unsatisfied than satisfied constraints . * repeat until no such variable remains .the following theorem from gives the sufficient conditions for the parallel bit flipping algorithm to correct a constant fraction of errors .[ thm1 ] let be a expander over variable nodes , for any .then , the simple parallel decoding algorithm will correct any fraction of error after decoding rounds .also , if denotes the set of corrupt variables in the input and , then the parallel decoding algorithm produces a word with at most corrupt variables after one decoding round .see   from lemma [ thm1 ] , it is clear that a word belongs to the decoding class of a codeword as long as the fraction of corrupt variables ( bits ) is less than .note that algorithm a is a slight modification of one iteration of the parallel bit flipping algorithm of . in the parallel bit flipping algorithm , every check node indicates to its neighboring variable node if it is satisfied or not . in algorithma every check node gives an estimate of the variable node .theoretically , both the algorithms are equivalent but we use the algorithm a as it has lesser redundancy .we now state and prove our main theorem .[ thm2 ] let be a expander for any .the proposed memory architecture can tolerate constant fraction of errors in all the components if   at , a codeword from an ldpc code with tanner graph is written into the memory .the contents are updated at times , , by running algorithm a. we bound the number of corrupt variables at time .let denote the fraction of corrupt variables at time .we establish bounds on for all .we first prove the following .let , denote an infinitesimal duration of time .if then , let denote the set of corrupt variables at time . since , a decoder built with reliable gates outputs a word with at most corrupt variables ( by lemma [ thm1 ] ) .we now bound the number of errors introduced due to the faulty nature of the decoder .each xor gate failure can corrupt at most one variable and each majority logic gate failure can corrupt at most one variable .so , eq . [ eq2 ] bounds the number of corrupt variables at the end of correcting cycle .however , in the time interval , at most variables can get corrupted due to failures in memory elements .therefore , the time at which there are maximum number of corrupt variables is just before the start of a correcting cycle , i.e. , hence , it suffices to bound for . dividing eq .[ eq1 ] by gives since it follows that hence , since , the fraction of corrupt variables is less than , the contents of storage circuit correspond the decoding class of original codeword and hence , by our definition , no memory failure occurs .it is instructive to see the behavior of the memory in the absence of the correcting circuit . in any time interval of seconds , at most fraction of the memories may fail .after sufficiently long time , the fraction of corrupt variables becomes more than and a memory failure occurs .the presence of a correcting circuit ensures that at any time the number of corrupt variables remains less than the correcting capability of the code .however , for a given expander there is a loss in the tolerable memory failure due to the faulty nature of the gates as well as the iterative nature of the decoder . consider the case of where decoder is reliable and failures occur only once .the tolerable fraction of errors for a given expander is close to . in the case of memories with unreliable memory elements but reliable logic gates ,the tolerable fraction of memory errors is close to .the reduction by a factor of occurs due to the fact that decoder is iterative in nature and needs multiple rounds to converge to the codeword .one round of error correction decreases the errors by a factor of and new errors might be introduced due to memory failures . in the extreme case of we have a decoder which takes just one step to correct all the corrupt variables , in which case the tolerable failure rate is arbitrarily close to .the faulty nature of the decoder further reduces the tolerable memory failure rate .given the values of , a code based on graph with sufficient expansion can be chosen to build a fault tolerant memory .it is well known that a random graph is a good expander with high probability ( see and references therein ) . in the next section ,we illustrate this fact with a few examples .in this section , we illustrate with specific numerical examples the redundancies and tolerable failure rates associated with different values of and .we first make the following observations .the redundancy of a memory system depends on the parameters and of the ldpc code used .different values of and can result in same redundancy . to compare across different values of and , the values of and have to be chosen consistently .how and scale with depends on the technology and implementation .assuming that all gates are built out of universal nand gates also does not answer the question fully as different implementations can lead to different values .hence for the sake of illustration we consider a specific implementation .it should be noted that the subsequent discussion is for illustration purpose only .accurate analysis for a given case can be carried out along the lines of the method we present in this section . for a given implementation, we fix the the values of and thereby fixing the redundancy as well as and .we then use the bounds on the achievable expansion of a regular bipartite graph to find bounds on the value of .this in turn provides bounds on the value of for fixed and .[ redundancy9 ]    [ redundancy34 ]    [ alpha9 ]    [ alpha34 ]     recall that the redundancy of a memory system is given by for a fixed , is minimum for a certain depending on the value of .for example , if , then it can be shown that minimizes the redundancy .this implies that a rate code has the least redundancy for a given .[ redundancy9 ] and fig .[ redundancy34 ] show the dependence of the redundancy on for a given value of .we make use of the following theorem from to find an upper bound for a given and .  [theorem 25 , ] : let be a bipartite graph between -regular vertices and -regular vertices . for all , there exists a set of -regular vertices with at most it should be noted that the upper bound is tight for higher values of .  using this theorem, we can find an upper bound on for a given and .it should also be noted that we look for graphs which expand by at least a factor of .  the following proposition from addresses the issue of existence of expanders .  [ proposition 6 , ] : let and be any fixed constants , and let be such that is an integer which is at least 2 .then for any such that there is a tanner graph with variable nodes , check nodes , and regular left degree which is an -expander , where it should be noted that the notation for expanders is different in .also , the proof does not guarantee that all the check nodes have same degree .this proposition guarantees the existence of graphs with sufficient expansion and can be used to derive a lower bound on for given and .this in turn proves existence of memories which can tolerate and fraction of failures in respective components as long as [ alpha9 ] and [ alpha34 ] illustrate the upper bounds and lower bounds on for and respectively .we remark that the bounds have been derived numerically and we do not attempt to give closed form expressions for the bounds as the results are for illustration purpose only .as mentioned in the introduction , taylor , was the first to investigate the capacity and fault - tolerant architectures of storage systems built entirely from unreliable components .his results were refined by kuznetsov .the aim of taylor and kuznetsov ( tk ) was to derive results analogous to the ones derived by shannon on the capacity of communication systems .the spirit and methodology of taylor and kuznetsov s work , is similar to gallager sresults on ldpc codes .the bounds on probability of error are given for an ensemble of regular random ldpc codes of infinitely large length used in the correcting circuit .they are obtained under the assumptions that the bits in memory elements remain independent during the process of correction , i.e. , under the assumption that the girth ( the length of the shortest cycle ) of the tanner graph corresponding to a code is infinitely large .taylor and kuznetsov considered a failure model in which a faulty component , generally a logic gate or a memory element , is subject to transient faults , i.e. , faults that manifest themselves at particular time steps but do not necessarily persist for later times .it is also assumed that gates fail independently of each other , and that the defects are not permanent , i.e. , a gate that malfunctioned at some point in time may give correct output subsequently and that failure occurs by flipping the correct result with some probability , i.e. , if the correct result is `` 1 '' , the gate gives `` 0 '' and vice versa .such failure mechanism is referred to as von neumann type of error or as independent failure model .a faulty gate or memory element in this case can be modeled as as a binary symmetric channel ( bsc ) with crossover probability .the information to be stored is first encoded by a regular binary ldpc code .the stored codeword consist of bits referred also as variables .each variable bit is involved in parity - check equations by , where is an parity check matrix and all operations are in binary field .the degree of each check node is .the vector is called syndrome and corresponds to the value of parity - check sum for .parity check is said to be satisfied if and unsatisfied if . a set of parity checks involving bit is .after encoding , every coded bit is replaced with bit - copies of itself and stored in registers .all bit - copies initially have the same value .new estimates of each of these copies are obtained by using one combination of checks .note that there are exactly combinations .the estimates are obtained as follows .  1 .evaluate parity checks for each bit - copy ( exclude one distinct parity check from the original set of checks for each bit - copy ) .2 .  flip the value of a particular bit - copy if half or more of the parity checks are unsatisfied .3 .  iterate ( 1 ) and ( 2 ) .the tanner graph description of ldpc codes was unknown at the time of taylor s paper .it is easy to see that each bit copy corresponds to an edge in the tanner graph .the variable node corresponding to the edge is the corresponding bit and the check node is the parity check that is excluded in the estimation of that bit copy .if the update scheme is modified so that the check nodes indicate an estimate of the bit copy , then the update rule is an exact implementation of the hard decision message passing algorithm ( known as gallager b algorithm ) for iterative decoding of ldpc codes ( see for a more detailed discussion ) .such an equivalence is of great significance as expander graph arguments have been applied to message passing algorithms and allows us to extend these results to the case of unreliable gates also .the complexity and redundancy of the original tk scheme are given by    in this section , we extend our results to the independent failure model . by chernoff bounds , it follows that a code which can correct a fraction of errors achieves exponentially small probability of error on the bsc with crossover probability  . in other words , if there are components which can fail independently with probability , then the probability that more than fraction of the components fail at any time is bounded by where is the kullback - leibler divergence between bernoulli random variables with parameters and respectively .now consider a memory architecture built from unreliable components subject to independent failures .let denote the probability of failure of memory element in time interval , denote the probability of failure per use of an xor gate and a -input majority logic gate respectively .also , let be such that , and .let denote the probability of memory failure at time .for and satisfying the conditions in theorem [ thm2 ] , we now have the following theorem  [ thm3 ] the proposed memory architecture has the following parameters for the independent failure model :  1 .information storage capability [ one ] 2 . [ two ] 3 . [ three ]  ( [ one ] ) and ( [ two ] ) follow from our discussion in section [ system ] .a memory failure may occur if the fraction of components which fail at a time is more than the tolerable fraction of errors . in time steps ,the correcting circuit is run for times .the memory registers can fail times .hence , we have   the bound on the probability of memory failure given in theorem [ thm3 ] is a very weak bound and we do not try to improve it .theorem [ thm3 ] establishes the fact that in the proposed memory architecture , probability of memory failures decreases exponentially with the code length while the redundancy remains bounded .theorem [ thm3 ] has been stated in the same form as the main theorem in kuznetsov s paper .hence , the proposed memory architecture has exponentially decreasing probability of memory failure in code length and redundancy which is roughly times less the tk scheme .taylor in remarks that memories have an associated non - zero storage capacity but an explicit calculation of the capacity is , in general , a difficult problem . for a given failure mechanism ,finding storage capacity involves calculating the minimum redundancy to achieve arbitrarily low probability of error .the redundancy is a function of the coding scheme as well as the decoding algorithm .the tk scheme as well as the proposed memory architecture have finite redundancies and only give bounds on the storage capacity . in this paper , we have shown that there exist reliable memories with redundancies less than that of the tk scheme .this implies that the proposed memory architecture improves the bound by a factor of at least in a few cases .the explicit calculation of the storage capacity still remains an unsolved problem .while the proposed architecture has less redundancy , the tk scheme may achieve better error exponents as it employs message passing algorithm which is in general more powerful than the parallel bit flipping algorithm .it is worth noting that taylor in describes the parallel bit flipping algorithm as a scheme for the update rule .he remarks that such an algorithm leads to complex interrelation between the errors as on successive iterations the values of the bits involved in the estimation of new value of each bit depend on previous value of the bit .we overcome this problem in this paper by using expander arguments . also , extending the results from the adversarial model to the independent failure model using chernoff bounds results in very weak bounds on the probability of memory failure . using expander arguments directly for the independent failure model for both the proposed architecture and the tk scheme might result in better error exponents as well as lead to tighter bounds on the capacity .another problem which needs to be investigated is the bounds on the probabilities of failures of components , i.e. , what are the upper bounds on the probability of failure of various components .sipser and spielman in provided explicit construction of codes which can correct a certain fraction of errors .the fraction was later improved by zemor in .barg and zemor in proved that expander codes achieve capacity on the bsc under iterative decoding .guruswami and indyk in proposed linear time encodable and decodable codes which achieve optimal error correction performance .study of fault - tolerant memory architectures based on these codes can provide the required bounds .however , these codes do not directly imply a specific implementation as is the case with parallel bit flipping algorithm .we noted earlier that capalbo _et al_. gave an explicit construction of expanders .however , the redundancies associated with such expanders are typically very high .this serves as another reason to consider expander codes and other linear time decodable codes based on expanders .the proposed architecture as well as the tk scheme employ coding scheme based on regular ldpc codes .the works of richardson , urbanke and shokrollahi and luby , mitzenmacher , shokrollahi , and spielman show that well designed irregular codes perform close to capacity .burshtein and miller s work on expander graph arguments for message passing is also based on irregular graphs .investigating memory architectures based on irregular codes may serve as another avenue to study the storage capacity problem .the authors would like to thank milos ivkovic for fruitful discussions .j. v. neumann , _ probabilistic logics and the synthesis of reliable organisms from unreliable components _ , ser .automata studies.1em plus 0.5em minus 0.4emprinceton : princeton university press , 1956 , pp . 4398 .m. capalbo , o. reingold , s. vadhan , and a. wigderson , `` randomness conductors and constant - degree lossless expanders , '' in _stoc 02 : proceedings of the thiry - fourth annual acm symposium on theory of computing_.1em plus 0.5em minus 0.4emnew york , ny , usa : acm press , 2002 , pp . 659668 .a. shokrollahi , `` an introduction to low - density parity - check codes , '' in _ theoretical aspects of computer science : advanced lectures_.1em plus 0.5em minus 0.4emnew york , ny , usa : springer - verlag new york , inc . , 2002 , pp .175197 .b. vasic and s. k. chilappagari , `` an information theoretical framework for analysis and design of nano - scale fault - tolerant memories based on low - density parity - check codes , '' _ ieee trans .circuits syst .i , reg . papers _ , accepted for publication .m. g. luby , m. mitzenmacher , m. a. shokrollahi , and d. a. spielman , `` improved low - density parity - check codes using irregular graphs , '' _ ieee trans .inform . theory _47 , no . 2 ,585598 , feb .

Abstract: in this paper , memories built from components subject to transient faults are considered . a fault - tolerant memory architecture based on low - density parity - check codes is proposed and the existence of reliable memories for the adversarial failure model is proved . the proof relies on the expansion property of the underlying tanner graph of the code . an equivalence between the taylor - kuznetsov ( tk ) scheme and gallager b algorithm is established and the results are extended to the independent failure model . it is also shown that the proposed memory architecture has lower redundancy compared to the tk scheme . the results are illustrated with specific numerical examples .