for years , there has been strong interest in the distinction between the idealized world of computation and complexity and that of its real - world applications , particularly in problems or algorithms where we find a separation between the worst - case complexity ( or , more broadly , difficulty ) and the worst cases actually encountered in practice .the simplex algorithm for linear programming is the classic example ; there is a family of examples on which the algorithm takes exponential time , yet in practice , every problem actually encountered is solved within polynomial time bounds .even more extreme examples are known , including several problems in group theory ( including some variants of the word problem ) that are non - computable in general , but for which a low - complexity algorithm solves all examples encountered in practice . in complexity theory , current methods for exploring such structure include the average - case complexity introduced by gurevich and levin , though this is sensitive to one s choice of probability measure , as well as the smoothed analysis of spielman and tang ; however , none of these methods have been adapted to computability theory , and it may well be that none are well - suited to such problems .  taking a more direct approach ,several researchers have begun work on the question of whether an algorithm s problematic behavior might be restricted to a negligible set .this is clearly related to the analysts notion of `` almost everywhere '' , whereby one works modulo sets of measure 0 so as to disregard problematic variations with no practical effect . in a sense , this study is motivated by envy of their methods in recent years , we have discovered problems that seem to be `` computable almost everywhere '' , and are working to find the right definition for the phrase . in this paper , we take the direct approach , studying a new definition of negligibility as applied to the non - negative integers ; we will spend most of our time fitting this idea into its proper computability - theoretic context , and then lay the foundations for further investigation into our motivating problem .the essential difficulty in defining `` computable almost everywhere '' is that there is no uniform probability measure on the integers , and thus no natural notion of a null set . instead ,if we want a uniform measurement of the size of a subset of , we are forced to abandon countable additivity and fall back to pseudo - measures .one of the most practical is asymptotic density .let , where is the set of natural numbers .for every , we denote by .we define the -th _ partial density _ of as the _ lower density _ of is and the _ upper density _ of is   if the limit of the partial densities exists ( i.eifnextchar .. , ) , then we say that has _( asymptotic ) density _  of course , for all .in an unfortunate collision of terms , at least for computability - theoretic work , a set is said to be _ generic _ if it has density 1 ( equivalently , ) .the name is motivated by the fact that given a generic set , the probability that a random integer selected from will lie in approaches 1 as increases ; thus , in some sense , such a set contains all generic integers .the complement of this notion is more useful for our purposes :  a set is said to be _negligible _ if it has density 0 ( equivalently , if ) .  in 2003 ,kapovich , myasnikov , schupp , and shpilrain introduced generic - case complexity , considering problems modulo sets of density 0 .they showed that this captured the phenomenon observed in several group - theoretic problems that are known to have non - computable instances while being simple to solve for every case encountered in practice ; for instance , they demonstrated that for any in an extremely large class of groups , the word problem for has linear - time generic - case complexity .myasnikov , in collaboration with hamkins , went on to apply these ideas to turing s halting problem , and proved that ( for reasons having to do with the prevalence of trivially halting or trivially non - halting programs in many models of computation ) the halting problem is `` generic - case decidable '' in said models .this was later refined by rybalov , who proved that the halting problem is _ not _ `` strongly generic - case decidable '' ( that is , decidable modulo sets with partial density converging to 0 exponentially fast ) ; this proof , by contrast , is valid for all turing - machine models of computation .jockusch and schupp have since defined and begun the study of the computability theory corresponding to generic - case complexity , and more generally the relations between asymptotic density and computability .their work has been further developed in collaboration with downey and mcnicholl , and refined in specific cases by igusa and bienvenu , hlzl , and day .  however , we return the focus to the notion of negligibility , since one would expect such a definition to have interesting ties to classical computability theory . for one ,a negligible set might be said to be `` small '' , `` sparse '' , or even `` thin '' .such `` thinness '' properties have historically been of great interest in computability ; they were the focus of post s program , the first attempt to construct an incomplete c.eifnextchar .. set , and have since proven to be of interest for unrelated reasons .  negligibility ( in the sense of asymptotic density ) is closed downwards under the subset relation ; any subset of a negligible set is itself negligible .it seems natural that it should be in the same family as the classical immunity properties , which provide the unifying computability - theoretic model for `` thinness '' .however , negligibility does not lend itself to the same analysis that we apply to immunity . choosing an alternate coding for the parameters of a membership problem is equivalent to applying a computable permutation to the underlying set , which can dramatically alter its asymptotic density .the most extreme example comes when we consider the class of infinite , co - infinite computable sets ; the resulting consequences for c.eifnextchar .. and co-c.eifnextchar .. sets are essential to the remainder of this paper .we will need one standard definition of computability theory to incorporate a result of downey , jockusch , and schupp : we say that a real is left- ( left- ) if its left cut is ( ) .  [ prop : computabledensity ] suppose is an infinite , co - infinite computable set . for any left- real and any left- real with , there is a computable permutation such that has lower density and upper density .we note first that there is an infinite , co - infinite computable set with lower density and upper density .in fact , this is nearly a theorem of downey , jockusch , and schupp , which states that for any and meeting our preconditions , there is a computable set with lower density and upper density .unless or , this already ensures that is both infinite and co - infinite ; if considering one of these cases , let be the set of perfect squares or its complement , respectively .since the infinite , co - infinite computable sets form an orbit under computable permutations , there is a computable permutation such that ; therefore , has lower density and upper density .[ cor : cedensity ] if is infinite and c.eifnextchar .. , there is a computable permutation such that has density 1 .  since is infinite and c.eifnextchar .. , has an infinite ( and co - infinite ) computable subset . by proposition[ prop : computabledensity ] , there is a computable permutation such that has density 1 .since , , so must also have density 1 .[ cor : cocedensity ] if is co - infinite and co-c.eifnextchar .. , there is a computable permutation such that has density 0 .since any infinite c.eifnextchar .. set has density 1 under some computable permutation , any problem that is decidable on some infinite c.eifnextchar .. subset of is in fact generic - case decidable if we choose the `` correct '' coding of the input .the corresponding coding is usually highly artificial , having little to do with the problem at hand .  in short ,due to the sensitivity of asymptotic density to computable permutation , generic - case computability is sensitive to the coding we choose for a given problem . as some of the great strengths of turing computability come from its invariance under choice of coding, we might hope to strengthen generic - case computability in such a way as to recover this invariance . to do so, we need to develop a stronger concept of negligibility , considering not only the upper and lower densities of a set , but those of all its images under computable permutations of .in section [ sec : intrinsicdensity ] , we follow this approach and obtain a new pseudo - measure , _ intrinsic density _ , which is invariant under computable permutations of .we discuss various classes of sets that have intrinsic density , including the 1-random sets , which provide the foundation for our investigations in the rest of this paper .  for the remainder of the paper , we turn our focus to the new properties of intrinsic density . in section[ sec : idimmunity ] , we begin with intrinsic density 0 , the natural notion of being _ intrinsically negligible _ , discussing it in the context of classical computability theory .in fact , intrinsic density 0 is an immunity property , fitting naturally into the hierarchy between immunity and cohesiveness , and we determine its place in the hierarchy for both unrestricted and sets . in order to complete our description , we improve on a result of downey , jockusch , and schupp , constructing a strongly hyperhyperimmune set with upper density at least below .  in section [ sec : idrandomness ] , we reflect on the relation between intrinsic density and randomness , and the connection it provides between classical computability and randomness .in fact , intrinsic density provides a continuum from immunity to stochasticity , as any intrinsic density from the range is a version of stochasticity ( modulo a fixed bias ) , while intrinsic density 0 is a form of immunity ( as discussed in the previous section ) .in fact , this correspondence can be reversed to extract various strengthenings of asymptotic density from the assorted notions of stochasticity some of which may prove fruitful topics of interest for future research .lastly , in section [ sec : intrinsiccomputability ] , we return to our motivating problem : the task of strengthening generic - case computability . after discussing some additional reasons for considering computably invariant notions of generic - case computability , we propose four such definitions , varying in degree of uniformity .all are strictly weaker than ordinary turing computability , but even the weakest of our notions does not consider the halting problem ( or , in fact , any nontrivial index set ) to be computable .  in the remainder of the introduction ,we collect notation and definitions that will be used for the rest of this paper .we will denote the -th partial computable function by .we will routinely identify a set with its characteristic function , , and also with the infinite binary sequence defining its characteristic function , .by , we mean either or the string consisting of the initial bits of the infinite sequence ; which notation we are using at a given moment will be made clear by context .two sets and have _ finite difference _, denoted , if for all sufficiently large .given two finite binary strings and ,we say is a _ prefix _ of , denoted , if there is a binary string such that the concatenation of followed by is ( i.eifnextchar .. , ) ; this definition extends to infinite binary sequences in the natural way .  the prefix - free kolmogorov complexity of a binary string is denoted as ; we refer to downey and hirschfeldt or nies for the details of its definition and properties , but note that it does relativize : we can consider the prefix - free kolmogorov complexity of with respect to , denoted .both of these books also provide many equivalent characterizations of a 1-random set ; for this paper , we will take the characterization in terms of the kolmogorov complexity of initial segments as our definition .a set is _1-random _ if there is some constant such that for all .this definition inherits a natural relativization from prefix - free complexity : is 1-random relative to if there is some such that for all .a set is _1-generic _ if , for every c.eifnextchar .. set of finite binary strings , there is some initial segment such that either or for every .let . the _ absolute lower density _ of is and the _ absolute upper density _ of is where is taken to vary over the set of computable permutations .if the absolute upper and lower densities are equal , then we say that has _ intrinsic ( asymptotic ) density _ , where in this case , not only does have a density , but its density is fixed under all computable permutations. we can develop analogous definitions for lower and upper densities ; if for all computable permutations , we say that has _ intrinsic lower density _ , and similarly for upper density .if a set has intrinsic density 0 , we say it is _ intrinsically negligible_.  by proposition [ prop : computabledensity ] , all infinite , co - infinite computable sets have absolute lower density 0 and absolute upper density 1 .thus , they are `` as far as possible '' from having an intrinsic density , at least in the sense that , under computable permutations , their densities range as widely as possible.  however , some might argue that 1-generic sets are further from having an intrinsic density than computable sets .it is simple to show that all 1-generic sets have lower density 0 and upper density 1 .since the class of 1-generic sets is closed under computable permutation , we can conclude that all 1-generic sets in fact have _ intrinsic _ lower density 0 and _ intrinsic _ upper density 1 .this puts them `` as far as possible '' from having an intrinsic density , in the sense that no computable permutation can bring their upper and lower densities together .  for the rest of our work in this paper, we will focus primarily on sets that have an intrinsic density , rather than classes of sets that do not . with a few examples , we begin to establish the connections between intrinsic density and other computability - theoretic properties , and ( particularly in discussing sets with intrinsic density strictly between 0 and 1 ) lay the groundwork for our later results .we start with the r - cohesive and r - maximal sets .recall that an infinite set is _ r - cohesive _ if there is no computable set such that and are both infinite , while a c.eifnextchar .. set is _ r - maximal _ if its complement is r - cohesive .[ thm : rcohesivedensity ] every r - cohesive set has intrinsic density 0 .we note that if a set is r - cohesive , then its image under any computable permutation of is also r - cohesive ; it thus suffices to prove that every r - cohesive set has density 0 .if we have a finite computable partition of ( i.eifnextchar .. , computable and pairwise disjoint , with union ) , must have finite intersection with all but one of these , say . by the finite subadditivity of upper density , . if we take , we have that , so . since was an arbitrary natural number , the upper density of must be 0 , so .every r - maximal set has intrinsic density 1 .however , sets of intermediate intrinsic density ( strictly between 0 and 1 ) provide a more versatile basis for further investigation ; as such , the 1-random sets will be essential to certain constructions later in this paper .  [prop : randomdensity ] every 1-random set has intrinsic density .any 1-random set obeys the law of large numbers , in the sense that it has density . ( prop .3.2.13 ) since the class of 1-random sets is closed under computable permutations of , every 1-random set has intrinsic density  .we can use 1-randoms to construct sets of other intermediate intrinsic densities as well , by means of the following lemma and its corollaries .[ lem : randomthinning ] if has density , and is 1-random relative to , then has density .  interpreting as a binary sequence ,consider the -computable subsequence selected by the rule `` if , select . ''since is 1-random relative to , we see that must be an unbiased sequence ; in other words , .  however , by the definition of and asymptotic density , so therefore , has density .since being 1-random is invariant under computable permutation , we obtain one more pair of corollaries :  if has intrinsic density , and is 1-random relative to , then has intrinsic density .[ cor : randomintersectiondensity ] if are mutually relatively 1-random sets ( i.eifnextchar .. , each set is 1-random relative to the join of the others ) , then has intrinsic density .having established a few tools to use in controlling the intrinsic density of sets ( in this paper , largely useful for the construction of counterexamples ) , we can now proceed to consider intrinsic density in a broader context .as discussed in the introduction , asymptotic density was defined as a substitute for a probability measure on a countable space .its use in generic - case computability ( and other topics ) is in defining a density-0 set to be negligible , in the sense that its elements are eventually scarce .this provides one of the more practical notions of a `` small '' or `` thin '' subset of the integers , in some senses more natural than asserting that a set has no infinite c.eifnextchar .. subset ( i.eifnextchar .. , is _ immune _ ) .unfortunately , having density 0 is not computably invariant . from the perspective of computability theory , set properties that vary under computable permutation have limited applications . by addressing this one issue , having intrinsic density 0proves to be more powerful ; for example , any infinite set having intrinsic density 0 ( or , in fact , any intrinsic lower density other than 1 ) must be immune .any infinite non - immune set has density 1 under some computable permutation .this immediately follows from corollary [ cor : cedensity ] .if is infinite and not immune , it contains an infinite c.eifnextchar .. subset . by corollary [ cor : cedensity ] , has density 1 for some computable permutation . since , , so must also have density 1 .[ cor : id0immune ] any infinite set with intrinsic lower density 0 , and hence , any infinite set with intrinsic density 0 , is immune .it is clear that the upper density of a set bounds the upper density of any of its subsets , so intrinsic density 0 is closed downwards under the subset relation . since having intrinsic density 0 is a computably invariant property , closed under subsets , and implies immunity , intrinsic density 0 ( here abbreviated id0 ) is a natural new immunity property , describing a strong notion of `` thinness '' .we therefore seek to determine its relation to the classical immunity properties :  a c.eifnextchar .. list of pairwise disjoint finite sets ( indexed as finite sets , so that the sets and the function are computable ) is said to be a _ strong array_. similarly , a uniformly c.eifnextchar .. list of pairwise disjoint c.eifnextchar .. sets is a _weak array_. there appears to be no standard name for the intermediate concept , which we here term a _computable array _ : a c.eifnextchar .. list of pairwise disjoint computable sets ( indexed appropriately ) , with union also computable .an infinite set is _ hyperimmune _ ( sometimes abbreviated h - immune ) if for every strong array , there is some such that ; in this case , we say that fails to meet .similarly , is said to be _ strongly hyperimmune _( sh - immune ) if no computable array meets , and _ strongly hyperhyperimmune _ ( shh - immune ) if no weak array meets . in a slight generalization, we say is _ finitely strong hyperimmune _( fsh - immune ) if no computable array with all finite meets , and _ hyperhyperimmune _ if no weak array with all finite meets .it was quickly noted that a set is hyperimmune iff no computable function dominates its principal function ; that is , for all computable functions , we have infinitely often . strengthening this , we say that an infinite set is _ dense immune _ if its principal function dominates all computable functions : for all computable functions and all sufficiently large , we have .an infinite set is _ cohesive _ if , for all c.eifnextchar .. sets , either or is finite .we can weaken this in a few standard ways : is _ r - cohesive _ if the same property holds for computable sets , or _ quasicohesive _ ( q - cohesive ) if is a finite union of cohesive sets .these properties are organized in a natural hierarchy of implication , shown as figure [ fig : immunityimplications ] .chapter xi.1 of soare provides an excellent reference for this hierarchy ( though focused on co-c.eifnextchar .. sets ) .we note that in the general case , the lack of implication between cohesiveness and dense immunity is witnessed by the existence of a non - high cohesive set , as constructed by jockusch and stephan .also , shh - immunity and hh - immunity are distinguishable in the general case , but by a remark of cooper , are equivalent for sets ( and thus for co-c.eifnextchar .. sets ) .  .35  .35  by theorem [ thm : rcohesivedensity ], r - cohesiveness implies intrinsic density 0 .this has a simple corollary , included here for completeness :  every quasi - cohesive set has intrinsic density 0 .  as a finite union of cohesive ( and thus r - cohesive ) sets ,any quasi - cohesive set is a finite union of sets of intrinsic density 0 .since density is finitely subadditive , must also have intrinsic density 0 .it is slightly more interesting to note that dense immunity also implies intrinsic density 0 . to show this, we note that dense immunity is computably invariant , and that a certain domination property is equivalent to having density 0 .  for any infinite set , where is the principal function of .consider the sequence .we note that is an infinite subsequence in fact , for all  so the sequence may increase only at in the image of ( and thus at points also in the subsequence ) , so these limits must be equal .[ prop : density0domination ] an infinite set has density 0 iff its principal function dominates all linear functions .( in standard asymptotic [ `` big - o '' ] notation , has density 0 iff . ) has density 0iff , and by the preceding lemma , however , for all , iff for all , dominates ; therefore , iff dominates for all and .[ prop : denseimmuneinvariant ] if the set is dense immune , is also dense immune for any computable permutation .let be a computable permutation , and consider a computable function .we define since is a computable function , the principal function of must dominate ; that is , for all but finitely many . in other words , for all sufficiently large , there are fewer than elements of less than .however , by construction of , every element of less than must come from an element of below .since for almost all , there are fewer than elements of below , we see that dominates . since both and were arbitrary , every computable permutation of is dense immune .if is dense immune , has intrinsic density 0 .  by proposition [ prop : denseimmuneinvariant ] ,dense immunity is a computably invariant property .it therefore suffices to show that dense immunity implies density 0 .however , this is an immediate consequence of proposition [ prop : density0domination ] , as all linear functions are bounded by computable functions , and so are dominated by the principal function of any dense immune set .none of the remaining standard immunity properties imply intrinsic density 0 .in fact , as demonstrated by downey , jockusch , and schupp , for every , there is a strongly hyperhyperimmune set with upper density at least ( though none have upper density 1 ) , constructed by mathias forcing .we here extend their result , using a direct -computable construction , to show that we may assume these sets to be .of course , per the aforementioned remark of cooper , any hh - immune set is in fact shh - immune ; this apparently does not simplify the argument required , so we will make no use of this fact .[ thm : hhimmune ] for all , there is a ( s)hh - immune set with upper density at least .we assume that is rational ; if not , we can replace it by any rational less than .we will construct as a -computable set , consulting as an oracle during our otherwise - computable construction .we work to satisfy the requirements : where is a listing of the uniformly c.eifnextchar .. sequences of sets .( in other words , every uniformly c.eifnextchar .. sequence of sets is of the form for some . )the negative requirements together assert that every weak array fails to meet ; this is the definition of shh - immunity .our negative requirements should , in principle , be simple to satisfy .we simply omit a set from each weak array of small lower density , thus leaving us with a set with high upper density .there are slight complications in ensuring that taking all of these requirements still can not force s upper density to fall below , but these are easily addressed .after all , at any given point , only disjoint sets can have partial density exceeding ; therefore , for any weak array , there must be some with lower density less than .the trouble comes in attempting to identify the in question . is incapable of resolving whether a c.eifnextchar .. set has lower density below some bound ; in fact , this problem is -hard , as it would allow us to distinguish finite c.eifnextchar .. sets from total c.eifnextchar .. sets .therefore , we can not search directly for such a in our weak array , and must use more indirect methods .  towards this end, we will make heavy use of the following sentence , for varying values of , , and :  will be defined in the course of our construction , but is c.eifnextchar ...the s are taken from our listing of uniformly c.eifnextchar .. sequences of sets .as all sets involved are c.eifnextchar .. , and we only ask whether a c.eifnextchar .. set has partial density exceeding some lower bound , is a sentence , and thus decidable by .  putting in context , we understand it to say that there are elements of our weak array , not including any with index less than , which all have high partial density ( exceeding ) at a single point , where is chosen from some c.eifnextchar .. set of possibilities . clearly , this sentence is vacuously true for , and ( presuming our s to be disjoint ) necessarily false for .therefore , for any fixed and , the maximum for which this sentence holds is computable in by a simple bounded search ; let us refer to it as .if we have such elements of a weak array , we refer to them together as a maximal tuple for that array under the conditions , , and .if we have a maximal tuple for our weak array , and is not among its members , then we know that for some with .otherwise , would be satisfied with . this will be our primary tool for controlling the lower density of the sets we omit as we build to avoid meeting the weak array .* _ organization _ * :  as we combine multiple negative requirements , we allow finite injury of each negative requirement by higher - priority requirements , though never revoking any previous decisions as to whether . we activate the requirements in order of decreasing priority , activating at most one at each stage . at stage , each active requirement independently decides whether to allow into ; we put into if none of these requirements object .for convenience , we denote by .each requirement will maintain a c.eifnextchar .. set of elements such that , if , then will be satisfied . for internal reference , we will also keep track of , which determines which element of the weak array we are actually restricting out of .lastly , we will maintain a c.eifnextchar .. set of locations where the partial density of is known to be strictly less than , while guaranteeing that at all stages . s basic goal is to prevent the weak array from meeting , while ensuring that the lower density of its restricted set does not exceed .to do so , will repeatedly consult regarding . in context, we can now see that we choose to ensure that the density of the set we omit for falls below at the same time as the densities of the previously - chosen sets fall below their critical values ; this will make certain that the density of our set rises above its goal of .  * _ module for _ * :  _ on activation at stage : _ we first consult , asking whether the sets are in fact pairwise disjoint ( i.eifnextchar .. , is a weak array ) . if not , then is trivially satisfied . in this case, will never restrict anything out of ; it simply maintains and at all stages , while voting to allow all elements into .if is a weak array , we define to be the least such that .we then set , and let , as we do not yet know of any locations where ._ at stage : _ we assume that has already been determined , and consider only whether to allow into . before making this decision, we must first determine whether we can still believe that we can restrict out of while keeping close to 1 .in fact , we want to verify that will again appear to have partial density less than at some point where the partial density of ( for all ) is also small .  if ( a -computable question ) , we have already verified this at some previous stage .we simply define , set , and let .we then allow into iff .  if , though , we must attempt to verify that the partial density of will fall below at some point in the future . we know that can not answer this question directly , asit can not determine whether a c.eifnextchar .. set will ever have partial density less than some critical value .we instead use to attack from a different angle .we will need to reference several times in the remainder of the procedure ; for simplicity s sake , we will abbreviate it by .we first determine , the greatest value of for which holds with .since at most disjoint sets can have partial density exceeding at the same location , this is a bounded search on a parameter of a statement ; thus , suffices to compute .we then ask whether holds with and .if so , then we have a maximal tuple ( within the array for ) in which every set has high partial density at the same point .since we can not add to this collection , we must have .we define to be the set of all for which there is such a collection ( along with all for which ) , set and , and allow into iff .if this case occurs immediately following injury or initialization of at stage , we say that was a `` recovery stage '' for ; otherwise , we deactivate all lower - priority requirements ( ) , as has changed .otherwise , the does not hold with and strictly greater than . in this case, we have no way to verify that the density of again drops below , and so consider to be injured .we vote to allow into , and deactivate all lower - priority requirements ( ) .we then effectively reset our procedure for ; we define to be the least such that , set and let .  * _ verification of the basic module : _ *  suppose that the module for is at some point activated and never again deactivated ( i.eifnextchar .. , does not change at any later stage ) .we assume that is in fact a weak array ; if it is not , is trivially satisfied , has partial density identically 0 for all , and does not change at any later stage .  by the construction of , we know that exists , andconsists of all elements restricted out of by .we will show that there is some stage at which is infinite , thus ensuring that will not change at any later stage and preventing future injury to .this will also guarantee that and .given such an , since does not intersect , we have satisfied .furthermore , the partial density of approaches that of , as the sets agree on all ; therefore , if the partial density of drops below infinitely often , the partial density of must be less than at all but finitely many of the same points .we note that the sequence is nonincreasing , as we monotonically reduce the set of witnesses for at successive stages .in fact , the sequence must decrease each time changes ( except at recovery stages ) ; this can only happen when we have run out of witnessing collections of size .as for recovery stages , they can only occur immediately after initialization of , or immediately after an injury to ; since injuries cause to decrease , any recovery stage is still associated with a corresponding decrease in . since for , is vacuously true , is always a non - negative integer and so can not decrease infinitely often . therefore , there must be some stage such that for all , which is only possible if is infinite .lastly , the module must force .whenever we choose a new , we always choose a value such that , and redefine accordingly to remain disjoint from .when we keep the same , we allow elements into iff they are not in . therefore ,as long as is active , we are assured that for all ; since , we will always have .  * _ construction of : _ *  at stage 0 , begin by activating .  at stage , check whether . if so , determine the highest - priority inactive requirement .if was deactivated in stage , do not activate any requirements ; otherwise , activate .( this delay in s reactivation ensures that is not activated during a recovery stage for . )  next , consult all active requirements in priority order .if any restrict out of , we declare that ; if all allow to enter , we put into .  * _ verification : _ *  nothing can deactivate , so is permanently activated . by the correctness of the basic module ,if the module for is permanently activated , will be satisfied .furthermore , there is some stage after which does not change , so that will never again be deactivated .therefore , as long as there are infinitely many stages at which we activate some inactive requirement , every module will be permanently activated at some point , and thus every will be satisfied .suppose , towards a contradiction , that some requirement is never permanently activated .let be the highest - priority such requirement , so that only modules through are permanently activated .we consider the construction at stage , after the last such module has been permanently activated and has stopped changing ( and is infinite ) .  at this stage, can never again be deactivated , so since is not permanently activated , must never again be activated .this can only be because the construction will never reach another stage where it activates the highest - priority inactive requirement ; therefore , it must be that contains all elements except those in , and for all .for all sufficiently large , we have that implies ; thus , for all sufficiently large .  however , .recall that for all but finitely many , we have , so since is infinite , this is a contradiction ; therefore , all modules must be permanently activated eventually .finally , since every module is eventually activated , we must activate a new module infinitely often .this can only happen if infinitely often , so every requirement is also satisfied ; must have upper density at least .we have yet to consider implications in the other direction ; what immunity properties are implied by intrinsic density 0 ?the first such result is simple ; as established above in corollary [ cor : id0immune ] , intrinsic density 0 at least implies immunity for infinite sets .  on the other hand , we already know that hyperimmunity ( even shh - immunity ) does not imply intrinsic density 0 .we can further prove that a set of intrinsic density 0 need not be hyperimmune ; we can construct counterexamples , and in fact will build a counterexample below every 1-random set .  for every 1-random set , there is an infinite set with intrinsic density 0 that is not hyperimmune .suppose that for all .by van lambalgen s theorem , given a 1-random set , there exists a uniformly -computable sequence of sets that are mutually relatively 1-random .in fact , defining , we have that for all , where is uniformly computable from and ; this can be shown by a simple inspection of a proof of van lambalgen s theorem .let . since is 1-random ,it has intrinsic density . given and using the incompressibility of , we can compute such that , ensuring that . since is computable , we can use as the first partition in a weak array that will witness that the set we construct is not hyperimmune .we then define since , and and are mutually relatively 1-random , must have intrinsic density by corollary [ cor : randomintersectiondensity ] . using and along with the incompressibility of ( relative to ) ,we can compute such that .  repeating this process, we see that is computable in , since . for all , we have that , so is infinite .furthermore , since the s are uniformly computable from , an integer , this partition of is in fact computable , demonstrating that is not hyperimmune .finally , for all .since , and the s are mutually relatively 1-random , has intrinsic density ; therefore , must have intrinsic density 0 .  as a convenient side effect , this theorem immediately gives us some information on the turing degrees of infinite sets with intrinsic density 0 : such sets exist below every 1-random turing degree , but can not be computable . among other things, this implies that there are infinite id0 sets in non - computable , low , and even hyperimmune - free degrees ( where our construction of a non - hyperimmune becomes rather superfluous , though it at least ensures that is infinite ) .it still eliminates any hopes we might have of further positive implications between the immunity properties and intrinsic density , as we will discuss in our summary below . by falling back to intrinsic lower density, we can recover one more positive implication , as shown by jockusch in private correspondence .every hyperimmune set has intrinsic lower density 0 .  since hyperimmunity is computably invariant , it suffices to show that every hyperimmune set has lower density 0 .suppose is hyperimmune .consider the strong array since is hyperimmune , for infinitely many .for all such , we have that ; therefore , .since this occurs infinitely often , we conclude that .  the above results , along with earlier work  , will suffice to disprove all other potential implications between intrinsic density 0 , intrinsic lower density 0 , and the standard immunity properties .we first repeat , per jockusch and schupp , that any 1-generic set has lower density 0 and upper density 1 ; since 1-genericity is computably invariant , 1-generics in fact have intrinsic lower density 0 and intrinsic upper density 1 .therefore , intrinsic lower density 0 does not imply intrinsic density 0 , even for sets .in addition , all 1-random sets are immune ; otherwise , there would be a 1-random with an infinite computable subset , which admits a trivial computable martingale that succeeds on .since 1-randoms have intrinsic density , immunity does not imply intrinsic lower density 0 , even for sets .lastly , theorem [ thm : hhimmune ] above demonstrates that for every , there is a shh - immune set ( in fact , a hh - immune set ) with upper density at least .in particular , shh - immunity does not imply intrinsic density 0 , even for sets .combining these counterexamples with our results above , we exhaust all possible implications between intrinsic density 0 , intrinsic lower density 0 , and the standard immunity properties .the graph of the resulting implications for infinite sets is shown in figure [ fig : densityimmunityimplications ] ; all implications depicted are strict , and counterexamples are discussed above for all arrows not present in the diagram .unfortunately , in the co-c.eifnextchar .. case ( well - studied due to post s program ) , the majority of our proofs of failures of implication collapse . since hh - immunity does imply dense immunity for co-c.eifnextchar .. sets , it seems unlikely that our proof method from theorem [ thm : hhimmune ] will help separate the higher immunity properties from intrinsic density 0 .in fact , most of our other failures of implication are exhibited by 1-generics or derived from 1-randoms , examples that are inherently not co-c.eifnextchar ... this leaves the co-c.eifnextchar .. diagram incomplete , with a few interesting open questions .is there an infinite c.eifnextchar .. set with intrinsic density 1 that is not hypersimple ?( f)sh - simple ?  is there a hypersimple set with lower density less than 1 ?equal to 0 ? for all , is there a hypersimple set with lower density less than ?let us move from the extremes of density ( density 0 or 1 ) to the intermediate densities , as exemplified by density  .the notion of `` density  '' is easily recognized as the law of large numbers , as applied to a sequence of flips of a fair coin .we might hope that having density  would be in some way related to a randomness - theoretic property , and stochasticity is the obvious candidate .this follows von mises in establishing the existence of limiting frequencies as the key property of a random sequence and , more specifically , the preservation of limiting frequencies under place - selection rules that determine the next bit sampled based only on the values previously sampled .if is such a class of selection rules , we say that a sequence is -stochastic if no selection rule in can select a biased ( non - density- ) subsequence from .we say that a selection rule is monotonic if the places it selects are always in increasing order , and oblivious if the places it selects are independent of , the sequence subject to the selection rule .there are several standard notions of stochasticity that will be useful to keep in mind .church - stochastic sequences are stochastic under computable monotonic selection rules , whereas von mises - wald - church - stochastic sequences are stochastic under partial computable monotonic selection rules . by this definition , sets with density might be termed `` trivially stochastic '' ; that is to say , they are unbiased under the single selection rule that selects all positions in order .however , this is rarely considered , as stochasticity is generally taken to require selection of proper subsequences .passing to intrinsic density , we find something more practical : stochasticity under the class of all computable permutations , represented as oblivious selection rules .in fact , this is the class of non - monotonic oblivious selection rules that must eventually select every position .the corresponding notion of randomness , that no computable martingale succeeds on the sequence of bits selected by such a rule , is permutation randomness as defined by miller and nies ; intrinsic density  is thus the natural notion of permutation stochasticity .  as mentioned above, stochasticity is generally taken to require selection of proper subsequences to preserve density  ; this would seem to be an obstacle to considering intrinsic density  as a valid notion of stochasticity . fortunately , permutation stochasticity in fact ensures that many proper subsequences are also unbiased , including all computably - sampled subsequences .we can be fully precise about this with one more definition in hand , and a combinatorial lemma to relate it to our previous work .  given a total computable injection and an infinite binary sequence , we say that the _ subsequence of sampled by _ is abusing notation as noted in the introduction , we can apply this directly to any set . in set notation , this gives   however , even though this new method of sampling generalizes our previous method of considering sets under computable permutations of , it has no additional power as far as density is concerned .[ lem : injectiondensity ] given any total computable injection , there is a computable permutation such that , for any set , has upper and lower density equal to those of .  given a total computable injection , we define a computable permutation by assigning values in increasing order of . if is a non - square integer , and has not already been assigned to for some , define .otherwise , define to be the least value not assigned to any with .the sizes of and differ by at most .thus , differs from by less than .therefore , and   from this minor lemma , we note that in fact , any set with intrinsic density has constant density not only under all computable permutations of , but also under all computable `` samplings '' of . to be more precise :  a set has intrinsic density iff for every total computable injection .the reverse direction is obvious by definition , since computable permutations of are also total computable injections .the forward direction is , at this point , also quite straightforward .fix a total computable injection .by lemma [ lem : injectiondensity ] , there is a computable permutation such that has the same upper and lower densities as for any set , and in particular for . since has intrinsic density , we know that , and so that .this corollary reveals that intrinsic density  coincides with another form of stochasticity : stochasticity under all computable injections , or equivalently the class of all oblivious non - monotonic selection rules .the corresponding notion of randomness is injection randomness , also as defined by miller and nies .thus , we see that :  permutation stochasticity and injection stochasticity coincide , and are both equivalent to intrinsic density  .  considering this interpretation of intermediate intrinsic densities ( strictly between 0 and 1 ) as a form of stochasticity, we find that intrinsic density provides an interesting link between the immunity properties and randomness - theoretic ideas . as discussed above, intrinsic density 0 is an immunity - type property , and so intrinsic density 1 is a form of co - immunity ( or , as it is called for c.eifnextchar .. sets , simplicity ) .thus , intrinsic density illustrates the relations between immunity , randomness , and simplicity , and provides a continuum of intermediate concepts , all of which follow in the spirit of stochasticity as established by von mises .this calls our attention to the fact that all of these properties are , in essence , descriptions of unpredictability : a set is immune if it is sufficiently difficult for a computable enumeration to stay within the set , co - immune if it is difficult to avoid the set , and stochastic if it is difficult to achieve any sort of persistent pattern of biased intersection with the set or its complement .of course , all of this relies fundamentally on our use of _ intrinsic _ density . considering asymptotic density alone , we find no useful connection to computability or randomness . a set with density 0 need not be immune in any useful sense , as is made clear by considering the computable set of perfect squares . taking the complement, we obtain a set with density 1 that is trivial to avoid .moreover , density  is a poor notion of randomness , as recognized by and before von mises , carrying no real implications for unpredictability ; for instance , the set of even numbers is `` stochastic '' in this sense , and yet is trivially predictable .having begun by investigating the implications of adding computable invariance to asymptotic density , we end by returning to the motivating problem with which we began : strengthening jockusch and schupp s generic - case computability to obtain similar invariance .they defined generic - case computability as follows :  a partial function is a _ partial description _ of if whenever converges .we say that is _ computable in the generic case _ , or _generic - case computable _ , if has a computable partial description with density-1 domain .we call such a description a _ generic - case description_.   in practical terms , the weakness of generic - case computability was shown by hamkins and miasnikov , who demonstrated that , in several reasonable codings , the halting problem is in fact decidable on a set of asymptotic density 1 , due to the density of trivially non - halting programs .this suggests that we should strengthen generic - case computability , to avoid rendering the halting problem `` decidable '' for trivial reasons .rybalov has shown that if we insist on convergence on a set with density exponentially approaching 1 ( also known as strong generic - case computability ) , then the halting problem is instead undecidable .of course , his analysis makes use of asymptotic density on the set of turing programs , considering the programs with at most non - final states ; strong generic - case computability is not directly applicable to arbitrary subsets of , so we must look for an alternative approach .furthermore , corollary [ cor : cedensity ] has a somewhat unfortunate consequence for generic - case computability .for any problem , if there is an algorithm that converges on an infinite set of inputs , that algorithm becomes a generic - case solution for the problem under some alternate coding of the input .after all , the domain of the algorithm is necessarily c.eifnextchar .. ; there is therefore some coding of the underlying problem ( corresponding to a permutation of ) under which the algorithm converges on a set of density 1 . in other words ,most natural problems have generic - case computable solutions ( as defined by jockusch and schupp ) under some computable permutation .this gives us another reason to use a stricter notion of generic - case computability .returning to the original definition of generic - case complexity for group - theoretic problems , from kapovich , myasnikov , schupp , and shpilrain , we note that the authors defined a problem in a finitely generated group to have generic - case complexity if and only if this complexity is independent of the choice of generating set .they specifically state that , though the worst - case complexity for most group - theoretic problems does not depend on one s choice of generating set , there is no reason to assume that this should also hold for generic - case complexity . as this choice directly corresponds to a coding of the input to the generic - case algorithm, a natural translation would require that our set be generic - case computable under every computable permutation of .equivalently , by the myhill isomorphism theorem , should not be considered generic - case decidable unless all of the 1-equivalent sets are as well . fortunately , this coincides with the standard idea that most computability - theoretic definitions are ( or `` should be '' ) invariant under computable permutation .we will call this new notion intrinsic generic - case computability , as it must be preserved under computable permutations of .below , we propose four definitions , varying in degree of uniformity .our weakest candidate notion of intrinsic generic - case computability is the direct translation of the definition by kapovich , myasnikov , schupp , and shpilrain :  a set is _ ( weakly ) intrinsically generic - case computable _iff is generic - case computable for every computable permutation .note that we place no requirements on the relationships between the generic - case descriptions for each such image ; the algorithms may be essentially unrelated .  insisting on a bare minimum of uniformity, we obtain our next candidate definition :  a set is _ ( uniformly ) intrinsically generic - case computable _ iff there is a uniformly computable family of functions such that , if is a computable permutation , is a generic - case description of ; that is , has density-1 domain and wherever converges , it converges to .on the other hand , allowing our description to require an index may weaken our notion of uniformity ; after all , this means that our description can not be given only a black - box oracle specifying the computable permutation , but actually requires knowledge of _ how _ the permutation can be computed and in particular may depend on the specific program provided to compute the permutation .requiring the description to work with only an oracle might seem a trivial variation , but significant differences have been observed in analogous situations ; specifically , in computable model theory , the index - based definition of uniform computable categoricity has been shown to be strictly weaker ( and less natural ) than the definition providing only an oracle . ( in general , any oracle - based definition must be at least as strong as the corresponding index - based definition , since it is well - established that there is a turing - machine procedure allowing us to convert an index into an effective oracle . )we therefore include this option in our list of candidate notions . in this case, we would say that :  a set is _( oracle ) intrinsically generic - case computable _ iff there is a turing functional such that , for any computable permutation ( represented as a set of pairs ) , is a generic - case description of .finally , we might insist on complete uniformity , and require that a single algorithm provide a description of on a set that has density 1 under all computable permutations ; in other words , that the algorithm converge on a set of _ intrinsic _ density 1 .  a set is _ ( strongly ) intrinsically generic - case computable _ iff it has a description that converges on a set of intrinsic density 1 .( equivalently , is a generic - case description of for all computable permutations . )since r - maximal sets are c.eifnextchar .. and have intrinsic density 1 , any r - maximal set is in fact strongly intrinsically generic - case computable .this provides a convenient demonstration that even this strongest definition is weaker than ordinary computability .more work will be required to distinguish these definitions of intrinsic generic - case computability , and some of them may prove to be equivalent .at this point , though , there are no reasons to presume any equivalences .the author personally expects that the uniform and strong definitions of intrinsic generic - case computability will be the most useful of these four .  on the other hand, even our weakest definition has a certain demonstrable strength .a set is said to be an _ index set _ if for all where and are indices for equivalent turing machines .theorem states that the only computable index sets are and .we can easily extend this to intrinsic generic - case computability , showing that no non - trivial index set can be weakly intrinsically generic - case computable .therefore , the halting problem is not intrinsically generic - case computable under any of these definitions .[ thm : indexsetsnotigc ] suppose is an index set ( i.eifnextchar .. , for all such that ) . is generic - case computable for all computable permutations iff is computable , and thus iff or .the reverse implication is obvious ; we will only consider the forward implication .  by the padding lemma for turing machines , for any , we can enumerate a set such that for all .consider the computable sets , and note that ; these , along with the singleton , comprise a partition of .one can easily construct a computable permutation of such that for all , and is filled with the `` waste '' of the process .suppose is generic - case computable , with generic - case description .since has positive density for all , there must be some for which converges . to determine whether , wait until converges for some such ; we then have , since .this shows that is computable , so by rice s theorem , or .the halting problem is not ( weakly ) intrinsically generic - case computable .the halting problem is 1-equivalent to a non - computable index set ( e.gifnextchar .. , ) . by the myhill isomorphism theorem , this means that its image under some computable permutation is a non - computable index set .composing this with the permutation from the proof of theorem [ thm : indexsetsnotigc ] , we obtain a computable permutation under which the image of the halting problem is not generic - case computable .thus , the halting problem is not even weakly intrinsically generic - case computable .significant thanks are owed to denis hirschfeldt and robert soare for innumerable helpful discussions on both the broad strokes and details of the work found herein ; in particular , the proof of theorem [ thm : hhimmune ] is based on an approach suggested by hirschfeldt .the author is also grateful to carl jockusch , paul schupp , and rod downey , for the work that inspired this research and for several conversations on the specifics of this investigation , and to an anonymous referee , whose suggestions have added substantial clarity to this paper .rodney g. downey , carl g. jockusch , jr . , timothy h. mcnicholl , and paul e. schupp , asymptotic density and the ershov hierarchy , _ mathematical logic quarterly _ , to appear .available at ` arxiv:1309.0137 ` .ilya kapovich , alexei myasnikov , paul e. schupp , and vladimir shpilrain , generic - case complexity , decision problems in group theory , and random walks , _ journal of algebra _ * 264*(2 ) ( 2003 ) , pp . 665694 .victor l. klee , jr . andgeorge j. minty , how good is the simplex algorithm ?, in : _ inequalities , iii ( proc .third sympos .california , los angeles , calif . , sept . 19 , 1969 ) _( oved shisha , ed . ) , academic press , new york , 1972 , pp . 159175 .

Abstract: in 2012 , inspired by developments in group theory and complexity , jockusch and schupp introduced generic computability , capturing the idea that an algorithm might work correctly except for a vanishing fraction of cases . however , we observe that their definition of a negligible set is not computably invariant ( and thus not well - defined on the 1-degrees ) , resulting in some failures of intuition and a break with standard expectations in computability theory .  to strengthen their approach , we introduce a new notion of intrinsic asymptotic density , with rich relations to both randomness and classical computability theory . we then apply these ideas to propose alternative foundations for further development in ( intrinsic ) generic computability .  toward these goals , we classify intrinsic density 0 as a new immunity property , specifying its position in the standard hierarchy from immune to cohesive for both general and sets , and identify intrinsic density as the stochasticity corresponding to permutation randomness . we also prove that rice s theorem extends to all intrinsic variations of generic computability , demonstrating in particular that no such notion considers to be `` computable '' .


the minority game ( mg ) , introduced by challet and zhang  as a generalisation of arthur s ` el farol ' problem  , has gained much popularity as a simple yet realistic model of the workings of financial markets .a great variety of work has since been done on the mg  , leading eventually to an analytic solution  ; these insights have proven particularly useful in justifying the mg as a realistic model of real market situations  .  the interactive minority game ( img ) proposes to further this use of the mg in modelling real markets by using the mg to try to gain insight into the behaviour of humans in a simplified market situation .individual humans compete against the traditional computer - controlled agents , whose characteristics can be fine - tuned through our understanding of the traditional mg : the human player can be presented with a very inefficient and easy - to - predict market , or with one that is much more complex and difficult .  in creating an interface for the game , our emphasis was on simplicity and ease of use: the game should be playable by anyone , from a seasoned mg researcher to someone who comes across the website at random .a number of aspects of the interface may therefore seem a little strange to those who have experience of the mg , and so in this paper we explain the mechanical workings - behind - the - scenes that go into the img : firstly , the interface , and secondly , the setup and parameters behind the different choices of game .fundamentally , the interactive minority game is a minority game like any other .a group of agents must choose between two actions , ( ` buy ' ) or ( ` sell ' ) , with the intention to make the opposite choice to the majority of agents ; agents decisions are based on the knowledge of which was the minority or majority action of the last turns .however , in order to make the game a little easier to play , we have had to change the presentation ( though not the underlying game ! ) a little for the sake of the human player .the human player not only has the advantage of a greater memory ( denoted by ) than the computer players , but is also given a choice between two different methods of presentation for the information about the game history ; in addition , rather than asking the human player for an action ( ` buy ' or ` sell ' ) , we ask them for a prediction on the direction the market is going to take , which corresponds to an action in the underlying game . in this section we explain ( and , we hope , justify ) in detail all these aspects of the interface .imagine being confronted by the string of 1 s and 0 s that is given to the regular agents !it is not a particularly pleasant thought , and the human player will clearly demand more .we have provided two different solutions to the problem . before we continue , let us recall a few aspects of the mg . at each timestep , each agent takes an an action , which allows us to define the action :    this allows us to define the first , simpler view of the game history : the _ binary viewpoint _ , a set of spins corresponding to in the the last turns of the game .this is of course identical in information content to the aforementioned string of 1 s and 0 s , but looks nicer !the more interesting viewpoint follows from an idea suggested by johnson _ et . of a price level , , in the mg , defined by    where is the market liquidity .we have used this in our _ chart viewpoint _ , literally a price - chart recording the past values of ( see fig . [ fig:2 ] ) .  the price - chart view of the game .the price changes proportionally to the global action of agents ( eq . [ eq:2 ] ) ; the player must predict whether the market will rise or fall .note that the last action is highlighted for the player . ]  clearly , the binary viewpoint is more suited to games where the player s gain is also binary , i.e. where , and the price - chart view is more suited to games with a linear payoff , .this is the practise we have followed in our ready - made games .however , the option to create a custom game allows the user to ignore our advice if s / he feels like it !the reader may have noticed that , rather than being asked to make a choice ( ` buy ' or ` sell ' , or , ) , the human player is asked to ` _ _ predict which way you think the market is going to go _ _ ' .that is , the user is asked to _ predict the majority action _ , and their own action ( the opposite ) is inferred from this we ask the human player for a prediction and an amount bet on this prediction , ( their _ weight _ or _ market impact _ ) , and their action is then taken to be .we apologise if this is perhaps a little confusing for those familiar with the mg ! our concern , as we noted in the introduction , was to make the game simpler for those who have never encountered the mg before . ` what way do you think the market will go ? ' seemed to us a simpler question than , say , ` would you like to buy or sell ( do the opposite of what you think everyone else will do ) ? ' in addition , those unused to the mg might reasonably ask , like andersen and sornette in a recent paper  , questions such as , ` why does buying hurt me if the market keeps rising ? ' we would rather avoid such problems !  in summary : although it is a little hidden from view , the player still has to try and perform the minority action .;-)   the game interface consists of a series of php files that interact with an underlying c program ; the latter calculates the actions of all the computer - controlled agents , and calculates all the relevant and interesting quantities ( such as predictability ) .we hope that the reader will forgive us if we are coy about revealing any more ( in particular , about revealing source code ) ; we are willing to reveal the workings - behind - the - scenes but there are a few rabbits - up - our - sleeve that we would like to keep from players for now , so as to make the game more interesting !the interactive minority game offers a number of possibilities for the human player to pick from . as well as a variety of ` standard ' , ready - made games, there is the option to create a custom game by choosing from various parameters . for those well - versed in mg lore, this could no doubt do with some extra explanation .parameters particular to the human player are denoted by a subscript asterisk ( ) .   to begin with ,let us summarise the basic parameters of the mg : we have a homogeneous population of agents with memory , each with strategies .this gives us as the size of the space of histories given to agents , and we can then define the control parameter . it is this parameter which determines the type of market : a low value creates an extremely inefficient market , easily predictable by the smart human , while a high value of creates a market whose behaviour is close to randomness .the most efficient market ( and therefore most difficult to play ) , i.e. the market with the minimal value of standard deviation , lies somewhere in the middle at the critical point  . for the purposes of our games ( custom or standard ) ,we have chosen to make and variables , with other parameters being calculated from these two .as we have already noted , the human player enjoys the benefit of a larger memory , , than the computer - controlled agents ; we have chosen to set for the binary view and for the price - chart view .we have also added a parameter particular to the img for the human player : the player s _ weight _ or _market impact_. recall from [ sec : predict ] that at each time the player is asked for a prediction on the direction the market will take , and the amount they bet on this , .their action is then taken to be ,   put simply , this amounts to the human player taking an action rather than ; i.e. , the player chooses to have a market impact equivalent to that of normal agents .the player s gain per turn is then also proportional to .depending on the market phase , this can be an advantage or a disadvantage : it brings the potential of a greater reward if the player can correctly predict the market , but with the potential of a far greater loss if s / he can not .we have also used the concept of _ minimum weight _ , , where we require that .we can force the human player to act on a larger scale than the typical agent equivalent , perhaps , to the actions of a large investment bank rather than an individual trader .the other new parameter introduced , for custom games , is the length of the _ introductory period _ , .this is the length of time that the computer - controlled agents play by themselves before the human player joins the game .a long value allows the agents to adapt to the statistically stationary state of the game , while if is low , the players start off `` stupid '' , not knowing which of their strategies is better . in the more difficult games ( efficient markets ) , a such a low value is essential as a high value of makes it completely impossible for a human to compete !the img allows the player both to pick from a variety of ` standard ' , ready - made games and to create their own custom game .the total choices available to the player are as follows : their _ minimum weight _, ; the _ market phase _ , determined by ; the _ agent memory _ , ; the display mode ( see [ sec:2 - 1 ] ) ; the payoff ( binary or linear ) ; and , in the custom game , the length of the introductory period , .the * custom game * allows the choice of a full range of parameters , including some that are not used in our standard games .minimum weight can be small ( ) , medium ( ) or large ( ) absolute values ; the market phase varies from extremely crowded ( ) to crowded ( or ) , the critical point ( ) and anticrowded ( or ) . can be 1 , 2 , 3 , 4 , 6 or 8 ; the introductory period can be the full period ( ) , or it can be short ( 200 ) or very short ( 5 ) .the * standard games * were set up to create a variety of different market experiences for the human player ._ beginners _ is exactly that : a nice and easy game , with , and .this is designed simply as a friendly introduction to the img and should not be taken too seriously !  the _ newbie traders _ game is a little trickier , with with and as before , but now with ._ day traders _ is again trickier , with , and , while _ chartists _ ups the ante still further with ._ wall street gurus _ throws the player into a difficult market at the critical point , with , and ._ mission impossible _ is literally that the player must struggle with a large and an anticrowded market ( ) full of very intelligent agents ( ) .all standard games use the price - chart view with a linear payoff .while clearly the fundamental aim of the img ( apart from being a fun distraction for the econophysics community ! ) is to investigate the behaviour of humans in market situations , we should perhaps specify a bit more precisely what we are looking for . hereare a few of the questions we have in mind :  * what sort of performance do we see from humans in different types of market ( crowded , anticrowded , critical ) ? * how do humans make their decisions ?do they stick to one strategy or vary their behaviour often ? * how much information do human players give to the market ? *do they take into account their market impact ?  in conclusion, we hope that the img , while a very simple and easy game , might provide a fascinating window into what is actually going on inside the heads of real , human , economic agents surely one of the ultimate econophysical goals !999999999 d. challet and y .- c .zhang , _ physica _ * 246a * , 407 ( 1997 ) .

Abstract: the _ interactive _ minority game ( img ) is an online version of the traditional minority game in which human players can enter into competition with the traditional computer - controlled agents . through the rich ( and , importantly , analytically understood ) behaviour of the mg , we can explore humans behaviour in different kinds of market crowded , efficient , critical with a high degree of control . to make the game easily understandable even to those who are encountering it for the first time , we have presented the game with a rather simplified interface ; in this working paper we explain the underlying technical aspects for those who have experience with the traditional mg .


standardized assessment tests that allow researchers to compare the performance of students taught according to various curricula are highly desirable .such comparisons provide information about the effectiveness of different curricula and , as a result , can improve methods of teaching , learning trajectories and , ultimately , student learning . appropriately designed diagnostics not only reveals common student difficulties but can also help to determine to what extent students understand the content .  as of the present day, there are several research - based conceptual tests that serve as instruments to assess and identify students difficulties in lower - division courses ( e.g. , the force concept inventory , the conceptual survey of e&m , the basic electricity and magnetism survey ) .data from these tests help to determine , among other things , where students lack a conceptual understanding of the material and help to correlate this with various methods of teaching .it also allows teachers and researchers to find out if these difficulties are present more universally .  at the upper - division level , assessing students difficulties is a much more challenging task .several research groups are currently working on such tests ( e.g. , the colorado upper - division electrostatics , colorado upper - division electrodynamics test , the quantum mechanics assessment tool , the survey of quantum mechanics concepts ) .these upper - division assessments are relatively new and thus they have only been employed at a few institutions .  in the paradigms in physics program at osu , we instituted a radical reform of all the upper - division physics courses that led to extensive reordering of the content . in traditional curricula , courses focus on a particular subfield of physics ( e.g. , classical mechanics , electricity and magnetism , quantum mechanics ) . at osu , courses called paradigms revolve around concepts underlying those fields ( e.g. , energy , symmetry , forces , wave motion ) .therefore , the content is arranged differently in e&m we spend more time on direct integration and curvilinear coordinates , less time on separation of variables , we cover potentials before electric fields and magnetostatics in vacuum before electrostatics in matter .the gravitational analogue of electrostatics is covered at the same time as electrostatics rather than in a classical mechanics course .moreover , we use a large variety of active engagement strategies , such as individual small whiteboard questions , small group problem - solving , computer visualization and kinesthetic activities .the paradigms courses , taken in the junior year , are then followed by capstone courses , which have a more traditional , lecture - based structure . our students thus represent an important test case to examine the versatility of new assessment tools . in this paper, we will focus on the colorado upper - division electrostatics diagnostic .we have found significant value in this new assessment tool . using the cue in our own classeshas already pointed out several possibilities for the improvement of the curriculum at osu .however , this new measure is still in the developmental stage and requires some fine tuning when used outside the university of colorado .our goal is to help generalize the cue to be accessible and relevant at a range of institutions .c c c || c & senior courses + fall & winter & spring & fall +  ' '' ''  & & & +the cue was originally designed as a free - response conceptual survey of electrostatics ( and some magnetostatics ) for the first semester of an upper - division level e&m sequence .it is designed in a pre / post format .the 20-minute pre - test contains 7 out of the 17 post cue questions that junior - level students might reasonably be expected to solve based on their introductory course experience .the post - test is intended to be given at the end of the first upper - division semester in a single 50-minute lecture . instead of actually solving problems ,students are asked to explain how they would solve them .they are rated for both choosing the appropriate method and the correctness of their reasoning in deciding on a given method .the instructions for the first half of the post - test are as follows :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for each of the following , give a brief outline of the easiest method that you would use to solve the problem .methods used in this class include but are not limited to : direct integration , ampere s law , superposition , gauss law , method of images , separation of variables , and multipole expansion ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    , blue dotted pattern ) and for cu ( , purple hatched pattern).,scaledwidth=70.0% ]  we have been using the cue to assess students understanding of junior - level e&m over a period of 5 years ( from 2009 to 2013 ) . at the beginning of the fall term of each year , junior - level students enrolled in symmetries and idealizations paradigm ( ph320 ) took the full cue pre - test ( see table [ tab : course_desc ] for the osu course schedule and figure [ fig : emscheme ] for a timeline of the cue at osu and cu ) .the same group of students was given the mid - test ( a subset of 12 post - test questions we chose to conform to our course goals ) at the end of the static vector fields paradigm ( ph422/522 ) .in the following year , those students who successfully finished their junior year were again given two tests within the electromagnetism capstone course ( ph 431 ) .there was a second mid - test at the beginning of the term ( with the same set of 12 questions as in the first mid - test ) and the full cue post - test at the end of the term .the necessity of introducing the mid - test arose due to the different course structure at osu . since not everything that the cue tests is covered by the end of the fall quarter, the results from a full cue post - test would not have been appropriate .we also note that , although osu students have had more contact hours in e&m ( 72 hours ) at the time they take the post - test than cu students ( 45 hours ) , most of the additional hours are on the more advanced content from e&m(ii ) at cu .the cue post - test was administered three times between the fall term of 2010 and the fall term of 2013 ( with the exclusion of the fall term of 2012 ) .the timing of the test was consistent throughout that period each time it was given in the last week of the term .a total number of students took the cue post - test , out of whom two students were excluded from the research ( one student was a member of the per group at osu and participated in meetings where the cue was discussed , the other student took only some of osu s e&m courses and therefore did not take a sequence of at least two tests ) .students at osu scored on average ( compared to at cu reported in ref . ) , with the spread of their performance ranging from about to . to provide a measure of student improvement over time we used the ( non-)normalized gain proposed in ref .the normalized gain is defined as the ratio of the actual average gain to the maximum possible average gain , where by we denote the average score of a given student from the subset of post - test questions which match the pre - test problems . for the students who took both the pre- and post - tests, we found an average normalized gain of ( non - normalized . ] ) , which is similar to gains of ( normalized ) and ( non - normalized ) reported in ref .thus , although students at osu on the average scored about lower than students at cu on both the pre- and post - tests at osu ( for ) and at cu ( for ) , see ref ., they showed similar learning _gains _ to students from other institutions taught in per - based courses and higher gain than were observed in standard lecture - based courses ( see figure 7 in ref . ) .figure [ fig : post_total ] shows comparison of the average performance on each question between students from osu ( blue dotted plot ) and cu ( purple hatched plot ) .one of the most striking features of this plot is the similarity of the overall pattern of students scores both on the high- and low - scored questions . with the exception of two questions ( q1 and q15 ), the averages agree to within on the first 12 questions and to within thereafter .it is also worth noting that , despite the low number of students taking the cue post - test in individual years , this pattern is still preserved when we plot the average scores of each question by year .this affirms the reliability of the cue across the two very different curricula .moreover , the surprisingly low average scores on some questions from both institutions suggest that the cue is a very challenging test in general , regardless of the curriculum .although the overall pattern in figure [ fig : post_total ] from both institutions is very similar , there are some significant differences that need to be addressed . in particular , osu students scores differ by over on question q1 regarding finding the potential ( or field ) inside an insulating sphere and by almost on question q15 regarding selecting boundary conditions to solve for the potential on a charged spherical surface . for reference, the full problems are reproduced in figure [ fig : q1+q15 ] .both of these questions are intended to test whether students can set up the solution to a problem involving partial differential equations ( i.e , recognizing separation of variables as an appropriate problem - solving technique and/or defining appropriate boundary conditions ) . to address these discrepancieswe now look more closely at the learning goals for courses at osu .    in a traditional curriculum ,as defined by the standard e&m text by david griffiths , students are first exposed to the application of separation of variables in physics in their e&m course , before they take quantum mechanics . at osu , however , students are exposed to the separation of variables mainly in the context of the schrdinger equation first in the waves in one dimension paradigm and in the central forces paradigm in the winter term of the junior year and then in the mathematical methods capstone in the fall term of the senior year .separation of variables is being discussed in multiple courses before students take the capstone in e&m and thus not much time is devoted to this topic in the e&m capstone . to be precise ,there is only one day ( typically the second day of the first week ) spent on laplace s equation , followed by 2 or 3 homework problems . as a consequence , students have much more experience with separation of variables in the context of quantum mechanics , long before they see it as part of e&m , and even then the structure of the capstone does not provide them with many opportunities to practice it in the e&m context .low scores on the two other questions involving separation of variables and boundary conditions ( bcs ) : q11 ( finding bcs in a specific scenario ) and q13 ( recognizing the form of solutions that match given bcs ) supports our suspicion that students are not getting enough exposure to these topics in the context of e&m .due to the significantly restructured curriculum at osu , our findings provide valuable data for comparison with reported results from cu s more moderately reformed curriculum and from institutions with a more traditional ( lecture ) format .it is intriguing that the across curricula difficulty pattern for most questions is preserved , even though the sample of students is quite different .this result confirms the overall robustness of the cue .in addition , strong differences in scores on a few specific questions shows that this assessment test is capable of helping to distinguish between different programs of study and uncovering important gaps in a curriculum .it is crucial to understand the causes for the big discrepancies between osu and cu scores on q1 and q15 . as we indicated above , one of the reasonsmight be the current organization of courses at osu .while restructuring the junior- and senior - level courses at osu , it was assumed that once exposed to certain techniques of solving problems in one context students will be able to transfer their knowledge of its applicability from one subfield of physics to another .as the cue has revealed , however , this is not happening and the separation of variables procedure does not become a natural e&m problem - solving technique for students once they depart from the quantum world .osu has made a recent change in the schedule of the paradigms and capstones moving the mathematical methods capstone , as well as , the central forces paradigm to the spring term of the junior year .this rearrangement gives us an opportunity to test whether the inclusion of more examples where separation of variables and boundary conditions are explicitly used to solve e&m problems can impart the generality of the techniques to the students and subsequently be reflected in higher cue scores on the relevant questions .we will discuss this change in a later publication .supported in part by nsf due 1023120 and 1323800 .we would like to thank steve pollock and bethany wilcox for conversations about the design and grading of the cue and stephanie chasteen for helping us with the cu test data .

Abstract: the colorado upper - division electrostatics ( cue ) diagnostic is an exam developed as part of the curriculum reform at the university of colorado , boulder ( cu ) . it was designed to assess conceptual learning within upper - division electricity and magnetism ( e&m ) . using the cue , we have been documenting students understanding of e&m at oregon state university ( osu ) over a period of 5 years . our analysis indicates that the cue identifies concepts that are generally difficult for students , regardless of the curriculum . the overall pattern of osu students scores reproduces the pattern reported by chasteen et al . at cu . there are , however , some important differences that we will address . in particular , our students struggle with the cue problems involving separation of variables and boundary conditions . we will discuss the possible causes for this , as well as steps that may rectify the situation .   address = department of physics , oregon state university , 301 weniger hall , corvallis , or 97331  address = department of physics , oregon state university , 301 weniger hall , corvallis , or 97331