On the Size of Classes with Weak Membership Properties Marius Zimand ? Department of Computer Science University of Rochester Rochester , NY 14627 e-mail : zimand@cs.rochester.edu Abstract It is shown that the following classes have measure in E : the class of P - selective sets , the class of P-multiselective sets , the class of cheatable sets , the class of easily countable sets , the class of easily approximable sets , the class of near-testable sets , the class of nearly near-testable sets , the class of sets that are not P-bi-immune . These are corollaries of a more general result stating that the class of sets that are p-isomorphic to P-quasi-approximable sets has measure in E. By considering the recent approach of Allender and Strauss for measuring in subexponential classes , we obtain similar results with respect to P for classes having weak logarithmic time membership properties . ? Supported in part by grant NSF-CCR-8957604 , NSF-INT-9116781 / JSPS-ENG-207 and NSF - CCR-9322513 and by the Romanian Department of Education and Science grant 4975-92 . 1 Introduction For the common programmer , polynomial-time and exponential-time are the major classes in analyzing the complexity of algorithms . Although extremely rough , this taxonomy has the merit of being categorical . Indeed , by the deterministic time hierarchy , P is properly included in E , where P = Sk > = 0 DT IME -LSB- nk -RSB- and E = Sc > = 0 DT IME -LSB- 2cn -RSB- . It is natural to investigate the extent to which P and E are separated . For example , the statement that almost all sets in E are not in P is more informative than merely the fact that there is some set in E ? P. Also , if one replaces P by sets with a weaker polynomial-time computable membership property Q , our understanding of the dichotomy between P and E is enhanced . Our goal is to identify an as weak as possible such property Q for which a strong separation result holds , where \ strong '' is taken in the sense of the effective measure theory . Mayordomo -LSB- May92 -RSB- has obtained such a separation for Q being \ not P-bi-immune '' . It is easily noted that a set A is not P-bi-immune if there exists a polynomial-time algorithm that is allowed to answer \ do n't know '' and that for infinitely many inputs x decides whether x is in A or not . Clearly , this is just a weakening of the property that membership is polynomial-time computable for A . Many other classes with weak membership properties have been studied : P-selective sets , P-multiselective sets , cheatable sets , easily countable sets , near-testable sets , nearly-near testable sets and other -LRB- see -LSB- BKS94 -RSB- and -LSB- BHHT94 -RSB- -RRB- . The class of approximable sets has been introduced in -LSB- BKS94 -RSB- and also in -LSB- Ogi94 -RSB- . A set A is P-approximable if there exists some constant q such that for all q-tuple -LRB- x1 ; : : : ; xq -RRB- , one can exclude in polynomial time one possibility of how the characteristic function of A is defined on x1 ; : : : ; xq . We introduce an even weaker membership property . A set A is P-quasi-approximable if there exists a constant q and a polynomial-time algorithm M that is allowed to answer \ do n't know '' such that for infinitely many q-tuples -LRB- x1 ; : : : ; xq -RRB- , with xi +1 being the lexicographical successor of xi for i = 1 ; : : : ; q ? 1 , M outputs a q-long binary string that is different from A -LRB- x1 -RRB- : : : A -LRB- xq -RRB- . It is easy to see that P-quasiapproximability is no stronger than P-selectivity and the rest in the above list . We show that the closure under p-isomorphism of the class of P-quasi-approximable sets has measure zero in E . As an immediate corollary , we conclude that the following classes have measure in E : the class of P-selective sets , the class of P-multiselective sets , the class of cheatable sets , the class of easily countable sets , the class of easily approximable sets , the class of near-testable sets , the class of nearly near-testable sets , the class of sets that are not P-bi-immune . The above result can not be extended to the class of sets that are equivalent to some P-quasi-approximable set under one-one reducibility since this equivalence class is a superset of E . We observe that in the case of P-approximable sets , the main construction can be modified so that the constructed martingale has dependency set of arbitrarily low size above ! -LRB- 1 -RRB- . Roughly speaking , the dependency set of a martingale consists of the bits of the input that are actually used by the martingale in the computation . Allender and Strauss -LSB- AS94 -RSB- have shown that keeping the dependency set small is sufficient in order to construct measures that are significant in deterministic time classes below E . It follows from our construction that if C is any deterministic complexity class closed under taking the square , the class of C ffi log-approximable sets has measure zero in C. To our knowledge , this is the first non-trivial result showing measurability in the new setting developed by Allender and Strauss . 2 Definitions and notation The set of finite binary strings is denoted f0 ; 1g ? and f0 ; 1g1 is the set of infinite binary strings . The first set is considered to be ordered in the lexicographical order : > = < < 1 < 00 : : : , where > = is the empty word . Let si ; i > = 1 , be the i-th string in f0 ; 1g ? according to this ordering and pos -LRB- x -RRB- 2 N ? f0g be the rank of string x in this ordering -LRB- i.e. , pos -LRB- x -RRB- = i , x = si -RRB- . For x ; y 2 f0 ; 1g ? , jxj denotes the length of x , x -LRB- i -RRB- 2 f0 ; 1g is the i-th bit of x , x -LRB- i : j -RRB- is the string x -LRB- i -RRB- x -LRB- i + 1 -RRB- : : : x -LRB- j -RRB- and xy denotes the string obtained by concatenating x and y. For q 2 N ? f0g , a q-interval is a q-tuple of strings of the form -LRB- sl ; sl +1 ; : : : ; sl + q ? 1 -RRB- for some l > = 1 . For A ? f0 ; 1g ? and x 2 f0 ; 1g ? , A -LRB- x -RRB- = 1 if x 2 A and A -LRB- x -RRB- = if x 62 A . We introduce next the two concepts on which we focus in this work . In what follows , C is a set of time-constructible functions closed under squaring -LRB- i.e. , f -LRB- n -RRB- 2 C -RRB- f -LRB- n -RRB- 2 2 C -RRB- . Abusing notation , we refer to the complexity class Sf2C DT IME -LRB- f -RRB- also by C. Definition 2.1 A set A ? f0 ; 1g ? is C-approximable if there exists a constant q > = 1 and a C-computable function f : -LRB- f0 ; 1g ? -RRB- q ! f0 ; 1gq such that for all q-tuples -LRB- x1 ; : : : ; xq -RRB- 2 -LRB- f0 ; 1g ? -RRB- q ; f -LRB- x1 ; : : : ; xq -RRB- < > A -LRB- x1 -RRB- : : : A -LRB- xq -RRB- . Definition 2.2 A set A ? f0 ; 1g ? is C-quasi-approximable if there exists a constant q > = 1 and a C-computable function f : -LRB- f0 ; 1g ? -RRB- q ! f0 ; 1gq Sf?g with the properties : -LRB- i -RRB- for all q-tuples -LRB- x1 ; : : : ; xq -RRB- 2 -LRB- f0 ; 1g ? -RRB- q ; f -LRB- x1 ; : : : ; xq -RRB- 2 f0 ; 1gq implies f -LRB- x1 ; : : : ; xq -RRB- < > A -LRB- x1 -RRB- : : : A -LRB- xq -RRB- . -LRB- ii -RRB- there exist infinitely many q-intervals -LRB- sl ; sl +1 ; : : : ; sl + q ? 1 -RRB- with f -LRB- sl ; sl +1 ; : : : ; sl + q ? 1 -RRB- 2 f0 ; 1gq . It is straightforward to check that the class of P-approximable sets contains the following classes of sets that have been considered in the literature to model in various ways the idea of feasible weak membership : p-selective sets , p-multiselective sets , cheatable sets , easily countable sets , easily approximable sets . For completeness , we recall the definitions of these classes . Let FP denote the class of functions that are computable in polynomial time . A set A ? f0 ; 1g ? is p-selective -LSB- Sel82 -RSB- if there exists f 2 FP such that for all pair -LRB- x1 ; x2 -RRB- , f -LRB- x1 ; x2 -RRB- 2 fx1 ; x2g and -LRB- x1 2 A -RRB- _ -LRB- x2 2 A -RRB- -RRB- f -LRB- x1 ; x2 -RRB- 2 A . A set A ? f0 ; 1g ? is p-multiselective -LSB- HJRW -RSB- if there exists f 2 FP and a natural constant q > = 1 such that for all q-tuple -LRB- x1 ; : : : ; xq -RRB- , f -LRB- x1 ; : : : ; xq -RRB- 2 fx1 ; : : : ; xqg and -LRB- x1 2 A -RRB- _ : : : _ -LRB- xq 2 A -RRB- -RRB- f -LRB- x1 ; : : : ; xq -RRB- 2 A . A set A ? f0 ; 1g ? is cheatable -LSB- Bei87 -RSB- if there exists f 2 FP and a natural constant q > = 1 such that for all q-tuple -LRB- x1 ; : : : ; xq -RRB- , f -LRB- x1 ; : : : ; xq -RRB- outputs a set D ? f0 ; 1gq of size q which contains A -LRB- x1 -RRB- : : : A -LRB- xq -RRB- . A set A ? f0 ; 1g ? is easily countable -LSB- HN93 -RSB- if there exists f 2 FP and a natural constant q > = 1 such that for all q-tuple -LRB- x1 ; : : : ; xq -RRB- , f -LRB- x1 ; : : : ; xq -RRB- 2 f0 ; : : : ; qg and f -LRB- x1 ; : : : ; xq -RRB- is not equal to the cardinality of A \ fx1 ; : : : ; xqg . A set A ? f0 ; 1g ? is easily approximable -LSB- KS91 -RSB- and -LSB- BKS94 -RSB- if there exists f 2 FP and a natural constant q > = 1 such that for all q-tuple -LRB- x1 ; : : : ; xq -RRB- , f -LRB- x1 ; : : : ; xq -RRB- outputs a q-bit vector -LRB- y1 ; : : : ; yq -RRB- such at least half the numbers yi are equal to A -LRB- xi -RRB- . It is easy to observe that if one replaces in the above definitions \ each q-tuple '' by \ infinitely many q-intervals '' , the corresponding new obtained classes are contained in the class of P-quasi-approximable sets . It is also clear that the class of sets that are not P-bi-immune is included in the class of P-quasi-approximable sets . Also each near-testable set is P-quasi-approximable and the same holds for each nearly neartestable set . We recall the appropriate definitions . A set A is near testable -LSB- GHJY91 -RSB- if there is f 2 FP such that for each l 2 N , f -LRB- sl -RRB- computes the truth value of the predicate -LRB- sl 2 A -RRB- ? -LRB- sl +1 2 A -RRB- , where ? represents the \ exclusive or '' . A set A is nearly near testable -LSB- HH91 -RSB- if there is f 2 FP such that for each l 2 N , f -LRB- sl -RRB- outputs the truth value of one of the following two predicates : -LRB- a -RRB- sl 2 A or -LRB- b -RRB- -LRB- sl 2 A -RRB- ? -LRB- sl +1 2 A -RRB- . Departing somewhat from the idea of weak membership , we note that the class of P-creative sets with a length increasing productive function and the class of P - paddable sets are also included in the class of P-quasi-approximable sets . Indeed , any P-creative set with a length increasing productive function and each P-paddable set is P-bi-immune -LSB- Wan92 -RSB- . We next introduce the basic concepts of resource-bounded measure . The now consecrated method of defining resource-bounded measure is due to Lutz -LSB- Lut92 -RSB- building on earlier studies of Schnorr -LSB- Sch73 -RSB- and is based on resource-bounded martingales -LRB- which , roughly speaking , are betting strategies -RRB- . We sketch the method at an intuitive level . Suppose A is a set of infinite binary strings -LRB- equivalently , A is a set of real numbers in -LSB- 0 ; 1 -RSB- -RRB- and we want to build a sequence of intervals -LRB- Ii -RRB- i2N covering A such that ? i2NjIij < = 2 ? k -LRB- i.e. , 2 ? k is an arbitrarily small value -RRB- . The procedure runs in stages . We start with 2 ? k dollars invested in the whole interval -LSB- 0 ; 1 -RSB- and dollars on all others subintervals of -LSB- 0 ; 1 -RSB- . Thus initially , invest -LRB- -LSB- 0 ; 1 -RSB- -RRB- = 2 ? k and invest -LRB- I -RRB- = 0 , on all other subintervals I . At each stage , the investment on each interval I doubles its value and this new amount is reinvested on Il and Ir , the left half and the right half intervals of I , according to a recursive or resource-bounded betting strategy . The procedure continues a number of stages decided by us . Summarizing , at each stage the intervals on which we invest have half the length of their father and the available amount of money doubles . By induction on the stage number , it follows that ? I interval in -LSB- 0 ; 1 -RSB- jIj ? invest -LRB- I -RRB- = 2 ? k at each stage . Let us say that I is a winning interval if the final amount invested in I is greater or equal than 1 dollar . Then , if A is covered by winning intervals , we effectively found a covering of A of total length < = 2 ? k . Let ? be a complexity class for functions . According to the previous discussion , a set A has ? - measure if there is an investing strategy in ? that , for each natural k , starting with an initial investment of 2 ? k dollars , builds a covering of A by its winning intervals . In order to define a measure in a complexity class of the form Sf2C DT IME -LSB- f -RSB- for a nice class C of functions including exponentials , the martingale must be computable in time bounded by a function in ? -LRB- C -RRB- = ff ffi log : f 2 Cg . This is needed to guarantee that Sf2C DT IME -LSB- f -RSB- has ? -LSB- C -RSB- - measure one and thus the ? -LRB- C -RRB- - measure is meaningful for measuring the size of classes included in Sf2C DT IME -LSB- f -RSB- . Allender and Strauss -LSB- AS94 -RSB- have observed that one more constraint must be imposed on martingales for defining useful measures in subexponential time classes . Namely , there must exist a machine M computing the martingale in time bounded by function in ? -LRB- C -RRB- -LRB- which is sublinear time ! -RRB- and , additionally , the size of the dependency set of M on all inputs must also be bounded by a function in ? -LRB- C -RRB- . The dependency set of M on input w , denoted GM ; w , is defined as follows : GM ; w is the minimal subset of f1 ; 2 ; : : : ; jwjg such that for each i 2 GM ; w , M can compute M -LRB- w -LRB- 1 : i -RRB- -RRB- reading only the bits of w -LRB- 1 : i -RRB- located at addresses in GM ; w -LRB- this is slightly more liberal than in -LSB- AS94 -RSB- but all their theorems hold in this setting too -RRB- . If these conditions are fulfilled , we say that the martingale is ? -LRB- C -RRB- computable . Note that if C contains exponential functions 2cn with c > = 1 the condition on the dependency set is implicitely satisfied . We proceed with the formal definitions . A set A ? f0 ; 1g ? is identified with its infinite characteristic sequence A -LRB- s1 -RRB- A -LRB- s2 -RRB- : : : . In this way , classes of sets are subsets of f0 ; 1g1 . For w 2 f0 ; 1g ? , the cylinder Cw is defined by Cw = fx 2 f0 ; 1g1 : w is a prefix of xg . Definition 2.3 -LRB- i -RRB- A martingale is a function d : f0 ; 1g ? ! -LSB- 0 ; 1 -RRB- satisfying d -LRB- w -RRB- > = d -LRB- w0 -RRB- + d -LRB- w1 -RRB- 2 ; for all strings w 2 f0 ; 1g ? . -LRB- ii -RRB- The global value of a martingale d is d -LRB->=-RRB- . The set covered by a martingale d is S -LSB- d -RSB- = -LSB- w2f0 ; 1g ? ;d -LRB- w -RRB- > = 1 Cw : A density function d covers a set X ? f0 ; 1g1 if X ? S -LSB- d -RSB- . -LRB- iii -RRB- An 1-dimensional density system is a function d : N ? f0 ; 1g ? ! -LSB- 0 ; 1 -RRB- such that for all i 2 N , di is a martingale , where di is defined by di -LRB- x -RRB- = d -LRB- i ; x -RRB- . Definition 2.4 -LRB- i -RRB- A set A ? f0 ; 1g1 has ? -LRB- C -RRB- - measure zero , denoted ?? -LRB- C -RRB- -LRB- A -RRB- = , if there exists an 1-dimensional density system d such that for all k 2 N , dk covers A with global value dk -LRB->=-RRB- < = 2 ? k and dk is ? -LRB- C -RRB- - computable . -LRB- ii -RRB- A set A ? f0 ; 1g1 has ? -LRB- C -RRB- - measure one , denoted ?? -LRB- C -RRB- -LRB- X -RRB- = 1 , if the complement of A has ? -LRB- C -RRB- - measure zero . -LRB- iii -RRB- A set A ? f0 ; 1g1 has measure zero -LRB- one -RRB- in Sf2C DT IME -LSB- f -RSB- if A \ Sf2C DT IME -LSB- f -RSB- has ? -LRB- C -RRB- - measure zero -LRB- one -RRB- . 3 Results Theorem 3.1 The closure under p-isomorphism of the class of P-quasi-approximable sets has measure zero in E. Proof : Let -LRB- hi ; h ? 1 i -RRB- i2N be an enumeration of polynomial-time isomorphisms , hi ; h ? 1 i : N ! N , hi ffi h ? 1 i = hi ffi h ? 1 i = 1N , and for all naturals q > = 1 , -LRB- f qi -RRB- i2N an enumeration of polynomial-time computable functions f qi : -LRB- f0 ; 1g ? -RRB- q ! f0 ; 1gq . The closure under p-isomorphism of the class of P-quasi-approximable sets is equal to the union of the classes A < i ; q ; j > , i ; q ; j 2 N ; q > = 1 , where A < i ; q ; j > consists of the sets A that are p-isomorphic via hi to some set that is P-quasi-approximable via q and f qj . It is thus sufficient to design a 1-density system fd < i ; q ; j > ; kg for each class A < i ; q ; j > . We fix i ; q ; j and write more simply h ; f ; dk ; A instead of hi ; f qj ; d < i ; q ; j > ; k ; A < i ; q ; j > . The function dk will take advantage of the fact that from time to time -LRB- but for infinitely many l -RRB- , f -LRB- sl ; sl +1 ; : : : ; sl + q ? 1 -RRB- returns a q-long bit binary string that is different from A -LRB- h ? 1 -LRB- sl -RRB- -RRB- : : : A -LRB- h ? 1 -LRB- sl + q ? 1 -RRB- -RRB- . Consequently , dk bets on all strings x such that x -LRB- h ? 1 -LRB- sl ? 1 + j -RRB- -RRB- = f -LRB- sl ; sl +1 ; : : : ; sl + q ? 1 -RRB- -LRB- j -RRB- , for j = 1 ; : : : ; q and distributes the amount that becomes available to the other strings . The redistribution task must start well in advance of reaching the point where dk bets on strings x as above . Therefore it is convenient that as soon as a value l as above is found during the computation of dk on some input x , all further necessary bets -LRB- i.e. , the redistribution task -RRB- are made on the spot . The values of this antedated bets will be denoted by ^ dk -LRB- ? -RRB- and will be stored for further use in a data structure called LIST -LRB- x -RRB- which will be transferred to the offsprings of x and then to the offsprings of the offsprings and so on until the whole redistribution is finished . The strings x on which the redistribution task is performed will be marked active as opposed to the other strings which are marked inactive . We proceed to formally describe the computation of dk on input x = x1 : : : ; xn , where xi 2 f0 ; 1g ; i = 1 ; : : : ; n . If x = > = , then dk -LRB- x -RRB- = 2 ? k and x is marked inactive . Suppose x < > > = and let x0 = x1 : : : xn ? 1 . We first compute dk -LRB- y -RRB- for all strict prefixes y of x . There are two cases : Case 1 . x0 is marked inactive . Let sm = h -LRB- sn -RRB- . We check the following TEST : \ There is a natural l 2 -LSB- m ? q + 1 ; m -RSB- such that f -LRB- sl ; sl +1 ; : : : ; sl + q ? 1 -RRB- 2 f0 ; 1gq and all the strict prefixes of x among those of lengths pos -LRB- h ? 1 -LRB- sl -RRB- -RRB- ; pos -LRB- h ? 1 -LRB- sl -RRB- -RRB- ? 1 , pos -LRB- h ? 1 -LRB- sl +1 -RRB- -RRB- ; pos -LRB- h ? 1 -LRB- sl +1 -RRB- -RRB- ? 1 ; : : : ; pos -LRB- h ? 1 -LRB- sl + q ? 1 -RRB- -RRB- ; pos -LRB- h ? 1 -LRB- sl + q ? 1 -RRB- -RRB- ? 1 are marked inactive . '' Case 1.1 . The answer to the TEST is NO . Then dk -LRB- x -RRB- = dk -LRB- x0 -RRB- , LIST -LRB- x -RRB- = ; and x is marked inactive . Case 1.2 . The answer to the TEST is YES -LRB- i.e. , a good value l has beeen found and the antedated redistribution can be performed -RRB- . Let l -LRB- x -RRB- be the smallest value satisfying the TEST . We order in lexicographical order the set fh ? 1 -LRB- sl -RRB- ; : : : ; h ? 1 -LRB- sl + q ? 1 -RRB- g obtaining z1 < z2 < : : : < zq . We insert in LIST -LRB- x -RRB- , in order , the following q triplets : -LRB- z1 ; ^ dk -LRB- z1 -RRB- ; b1 -RRB- ; : : : ; -LRB- zq ; ^ dk -LRB- zq -RRB- ; bq -RRB- , where ^ dk -LRB- zi -RRB- = 1 2q ? 1 Pq ? 1 h = i 2h and bi = f -LRB- sl -LRB- x -RRB- ; : : : ; sl -LRB- x -RRB- + q ? 1 -RRB- -LRB- i -RRB- for i = 1 ; : : : ; q. Note that in the last triplet , ^ dk -LRB- zq -RRB- = 0 . A triplet -LRB- z ; ^ dk -LRB- z -RRB- ; b -RRB- 2 f0 ; 1g ? ? R ? f0 ; 1g signifies that when the computation of dk will reach a successor of x of length pos -LRB- z -RRB- , call it u , it will bet ^ dk -LRB- z -RRB- dk -LRB- x0 -RRB- on it if the bit pos -LRB- z -RRB- in u coincides with b and will bet 2q = -LRB- 2q ? 1 -RRB- if that bit does not coincide with b . The redistribution starts with x , so that , according to the strategy stated above , we mark x active and define : dk -LRB- x -RRB- = 8 > < > : 1 2q ? 1 -LRB- Pq ? 1 h = 1 2h -RRB- ? dk -LRB- x0 -RRB- if xn = f -LRB- sl -LRB- x -RRB- ; : : : ; sl -LRB- x -RRB- + q ? 1 -RRB- -LRB- 1 -RRB- 2q 2q ? 1dk -LRB- x0 -RRB- if xn < > f -LRB- sl -LRB- x -RRB- ; : : : ; sl -LRB- x -RRB- + q ? 1 -RRB- -LRB- 1 -RRB- Case 2 . x0 is marked active . Let LIST -LRB- x0 -RRB- = -LRB- -LRB- si1 ; ^ dk -LRB- si1 -RRB- ; b1 -RRB- ; : : : ; -LRB- siq ; ^ dk -LRB- siq -RRB- ; bq -RRB- -RRB- . Case 2.1 . One of -LRB- sn ; ^ dk -LRB- sn -RRB- ; xn -RRB- or -LRB- sn ; ^ dk -LRB- sn -RRB- ; 1 ? xn -RRB- is in LIST -LRB- x0 -RRB- . Then , dk -LRB- x -RRB- = 8 >> < >> : ^ dk -LRB- sn -RRB- dk -LRB- si1 ? 1 -RRB- if -LRB- sn ; ^ dk -LRB- sn -RRB- ; xn -RRB- 2 LIST -LRB- x0 -RRB- 2q 2q ? 1dk -LRB- si1 ? 1 -RRB- if -LRB- sn ; ^ dk -LRB- sn -RRB- ; 1 ? xn -RRB- 2 LIST -LRB- x0 -RRB- Next , if -LRB- sn ; ^ dk -LRB- sn -RRB- ; xn -RRB- or -LRB- sn ; ^ dk -LRB- sn -RRB- ; 1 ? xn -RRB- are not in the last position of LIST -LRB- x0 -RRB- , then LIST -LRB- x -RRB- = LIST -LRB- x0 -RRB- and x is marked active -LRB- the redistribution continues for the offsprings of x -RRB- . In case -LRB- sn ; ^ dk -LRB- sn -RRB- ; xn -RRB- or -LRB- sn ; ^ dk -LRB- sn -RRB- ; 1 ? xn -RRB- are in the last position of LIST -LRB- x0 -RRB- , then LIST -LRB- x -RRB- = ; and x is marked inactive -LRB- the redistribution task is finished -RRB- . Case 2.2 . Both -LRB- sn ; ^ dk -LRB- sn -RRB- ; xn -RRB- or -LRB- sn ; ^ dk -LRB- sn -RRB- ; 1 ? xn -RRB- are not in LIST -LRB- x0 -RRB- . Then dk -LRB- x -RRB- = dk -LRB- x0 -RRB- ; LIST -LRB- x -RRB- = LIST -LRB- x0 -RRB- and x is marked active . The following Claims show that dk achieves the purported goals . Claim 3.2 dk -LRB- x -RRB- can be computed in polynomial time . Proof : The computation of dk -LRB- x -RRB- involves an autonomous part and the computation of dk -LRB- y -RRB- for all strict prefixes y of x . Since there are jxj such prefixes , we only have to show that the autonomous part takes polynomial time . If Case 1 is entered , we have to compute h -LRB- sjxj -RRB- , check the TEST and , if Case 1.2 occurs , insert q elements in LIST -LRB- x -RRB- and do some easy computations . These operations take polynomial time . The operations required by Case 2 take polynomial time , since there are a constant number -LRB- namely q -RRB- of elements in LIST -LRB- x0 -RRB- . 2 Claim 3.3 dk -LRB- ? -RRB- is a density function . Proof : Let x = x1x2 : : : xn ; x0 = x1x2 : : : xn ? 1 and x00 = x0 -LRB- 1 ? xn -RRB- , where xi 2 f0 ; 1g for all i 2 f1 ; : : : ; ng . We show that dk -LRB- x0 -RRB- = dk -LRB- x -RRB- + dk -LRB- x00 -RRB- 2 ; -LRB- 1 -RRB- for all x 2 f0 ; 1g ? . We focus on the computation of dk -LRB- x -RRB- . If x0 is marked inactive , then dk -LRB- x00 -RRB- will also find x0 to be inactive and the TEST evaluates the same in the computation of dk -LRB- x -RRB- and dk -LRB- x00 -RRB- . Now , relation 1 can be easily checked . Suppose next that x0 is marked active and let LIST -LRB- x0 -RRB- = -LRB- -LRB- si1 ; ^ dk -LRB- si1 -RRB- ; b1 -RRB- ; : : : ; -LRB- siq ; ^ dk -LRB- siq -RRB- ; bq -RRB- -RRB- : It is clear that the same of Case 2.1 and Case 2.2 applies in both dk -LRB- x -RRB- and dk -LRB- x00 -RRB- . If Case 2.2 applies in both dk -LRB- x -RRB- and dk -LRB- x00 -RRB- , relation 1 is checked immediately . Suppose that Case 2.1 applies to both dk -LRB- x -RRB- and dk -LRB- x00 -RRB- , with -LRB- sn ; ^ dk -LRB- sn -RRB- ; xn -RRB- in LIST -LRB- x0 -RRB- -LRB- the other situation , -LRB- sn ; ^ dk -LRB- sn -RRB- ; 1 ? xn -RRB- in LIST -LRB- x0 -RRB- , is symmetric -RRB- . It follows that there is some p such that n = ip and xn = bp . We claim that p < > 1 -LRB- i.e. , -LRB- sn ; ^ dk -LRB- n -RRB- ; bn -RRB- is not the first triplet in LIST -LRB- x0 -RRB- -RRB- . Indeed , let x1 be the smallest strict prefix of x such that all prefixes of x between x1 and x are marked active -LRB- note that x1 < = x0 < x -RRB- . Since the predecessor of x1 is marked inactive and x1 is marked active , it follows that dk -LRB- x1 -RRB- is computed according to Case 1.2 . Let l -LRB- x1 -RRB- be the value selected when running Case 1.2 on dk -LRB- x1 -RRB- . We show that i1 = jx1j < n and thus p < > 1 . Otherwise , by the computation of dk -LRB- x1 -RRB- , it would follow that si1 = h ? 1 -LRB- sl -LRB- x1 -RRB- + t -RRB- , for some t 2 f0 ; : : : ; q ? 1g and i1 < jx1j -LRB- because si1 is the smallest in the set fh ? 1 -LRB- sl -LRB- x1 -RRB- -RRB- ; : : : ; h ? 1 -LRB- sl -LRB- x1 -RRB- + q ? 1 -RRB- g and sjx1j is in this set -RRB- . Note also that the prefixes of x of length i1 and i1 ? 1 are both marked inactive in the computation of dk -LRB- x1 -RRB- -LRB- because the TEST is answered YES in the computation of dk -LRB- x1 -RRB- -RRB- . Since h -LRB- si1 -RRB- = sl -LRB- x1 -RRB- + t , we get that pos -LRB- h -LRB- si1 -RRB- -RRB- = l -LRB- x1 -RRB- + t 2 I = -LSB- l -LRB- x1 -RRB- ; l -LRB- x1 -RRB- + q ? 1 -RSB- . One can check that for all values m 2 I , l -LRB- x1 -RRB- 2 -LSB- m ? q + 1 ; m -RSB- . Thus , l -LRB- x1 -RRB- satisfies the TEST in the computation of dk -LRB- x -LRB- 1 : i1 -RRB- -RRB- , which implies that the computation of dk -LRB- x -LRB- 1 : i1 -RRB- -RRB- enters Case 1.2 . This means that x -LRB- 1 : i1 -RRB- is marked active which contradicts the statement above . Therefore , p < > 1 . It is clear that for prefixes x -LRB- 1 : r -RRB- of x , with ip ? 1 < r < ip -LRB- if there are any -RRB- , dk -LRB- x -LRB- 1 : r -RRB- -RRB- is computed according to Case 2.2 and , thus , dk -LRB- x -LRB- 1 : r -RRB- -RRB- = dk -LRB- x -LRB- 1 : ip ? 1 -RRB- -RRB- . Now , either x0 = xip ? 1 or x0 is a prefix of x as above . In both cases dk -LRB- x0 -RRB- = dk -LRB- x0 -LRB- 1 : ip ? 1 -RRB- -RRB- = 1 = -LRB- 2q ? 1 -RRB- -LRB- q ? 1 X h =p ? 1 2h -RRB- ? dk -LRB- x -LRB- 1 : -LRB- i1 ? 1 -RRB- -RRB- -RRB- : Since dk -LRB- x -RRB- = 1 = -LRB- 2q ? 1 -RRB- -LRB- q ? 1 X h =p 2h -RRB- ? dk -LRB- x -LRB- 1 : -LRB- i1 ? 1 -RRB- -RRB- -RRB- and dk -LRB- x00 -RRB- = 2q = -LRB- 2q ? 1 -RRB- ? dk -LRB- x -LRB- 1 : -LRB- i1 ? 1 -RRB- -RRB- -RRB- ; relation 1 is verified . 2 Claim 3.4 For all k 2 N , A 2 S -LSB- dk -RSB- . Proof : We inductively define the infinite sequence of integers -LRB- li -RRB- i2N as follows . Let l0 = 1 and li +1 = \ the smallest value l > li that is selected as l -LRB- x -RRB- in Case 1.2 in the computation of dk on some x 2 f0 ; 1g ? . '' By the properties of f , it is clear that li is defined for all i. For a value l in the above sequence , let ml = minfpos -LRB- h ? 1 -LRB- sl -RRB- -RRB- ; : : : ; pos -LRB- h ? 1 -LRB- sl + q ? 1 -RRB- -RRB- g and Ml = maxfpos -LRB- h ? 1 -LRB- sl -RRB- -RRB- ; : : : ; pos -LRB- h ? 1 -LRB- sl + q ? 1 -RRB- -RRB- g . Since f -LRB- sl ; : : : ; sl + q ? 1 -RRB- < > A -LRB- h ? 1 -LRB- sl -RRB- -RRB- : : : A -LRB- h ? 1 -LRB- sl + q ? 1 -RRB- -RRB- ; it follows that dk -LRB- A -LRB- 1 : Ml -RRB- -RRB- = -LRB- 1 + 1 = -LRB- 2q ? 1 -RRB- -RRB- d -LRB- A -LRB- 1 : ml ? 1 -RRB- -RRB- : For T such that -LRB- 1 + 1 = -LRB- 2q ? 1 -RRB- -RRB- T > = 2k , we conclude that dk -LRB- A -LRB- 1 : MlT -RRB- -RRB- = -LRB- 1 + 1 = -LRB- 2q ? 1 -RRB- -RRB- T dk -LRB->=-RRB- > = 1 , because dk -LRB- A -LRB- 1 : mli ? 1 ? 1 -RRB- -RRB- = dk -LRB- A -LRB- 1 : Mli -RRB- -RRB- . 2 The proof of Theorem 3.1 is not valid for subexponential time measures . The problem is that , in the computation of dk -LRB- x -RRB- , one has to review the computations of dk -LRB- y -RRB- for all prefixes y of x and , in this process , all bits of x are read . Consequently , the dependency set has linear size . However , in the case of C-approximability , this difficulty is easily avoided . For example , one can do the redistribution task only among the strings in the set Sn = fx 2 f0 ; 1g ? : 2n < = jxj < = 2n + qg for all natural n and do neutral bets -LRB- i.e. ; d -LRB- x1 -RRB- = d -LRB- x0 -RRB- = d -LRB- x -RRB- -RRB- for the rest of the strings x. Hence , during the computation of dk -LRB- x -RRB- , only the bits 2i + j ; i = 1 ; : : : ; blog jxjc ; j = ; : : : ; q ? 1 are read and , thus , the dependency set has logarithmic size . Theorem 3.5 Let C be a deterministic time complexity class closed under squaring . Then the class of C ffi log-approximable sets has measure zero in C. Proof : By the above remarks and following the redistribution strategy in the proof of Theorem 3.1 -LRB- in an easier setting since there is no \ scrambling '' introduced by the isomorphism h -RRB- . 2 Observe first that the class of C-approximable sets is closed under many-one polynomial-time reductions -LRB- Proof : Let A ? f0 ; 1g ? be a set that is many-one reducible via a C-computable function g to a set B that is C-approximable . Let q > = 1 and f : -LRB- f0 ; 1g ? -RRB- q ! f0 ; 1gq be the constant and the function verifying the conditions in Definition 2.1 for B. For a q-tuple -LRB- x1 ; : : : ; xq -RRB- 2 -LRB- f0 ; 1g ? -RRB- q , there are two possibilities . Case 1 . jfg -LRB- x1 -RRB- ; : : : ; g -LRB- xq -RRB- gj = q . We take f -LRB- x1 ; : : : ; xq -RRB- = f -LRB- g -LRB- x1 -RRB- ; : : : ; g -LRB- xq -RRB- -RRB- . Case 2 . There exists s ; t 2 f1 ; : : : ; qg with g -LRB- xs -RRB- = g -LRB- xt -RRB- . We take f -LRB- x1 ; : : : ; xq -RRB- = b1 : : : bs : : : bt : : : bq with bs = 0 , bt = 1 and -LRB- arbitrarily -RRB- bi = for i = f1 ; : : : ; qg ? fs ; tg . It is immediate to check that A is C-approximable via q and f . 2 -RRB- . Consequently it does not make sense to consider such closures in Theorem 3.5 . On the other hand , we note that Theorem 3.1 can not be strenghthened by replacing \ P-isomorphism '' by \ equivalence under one-one polynomial-time reducibility '' . Theorem 3.6 The closure under one-one polynomial-time reducibility of the class of P-quasi-approximable sets is a superset of E and , thus , has measure one in E. Proof : Let fS1 < S2 < : : : g be the lexicographical ordering of f0 ; 1g ? ? f02n : n 2 Ng . For each A 2 E we define B -LRB- A -RRB- by : -LRB- 1 -RRB- si 2 A , Si 2 B -LRB- A -RRB- for all i 2 N and -LRB- 2 -RRB- 2n 2 B -LRB- A -RRB- , n 2 A for all n 2 N . Then A and B -LRB- A -RRB- are equivalent under the one-one polynomial-time reducibility and B -LRB- A -RRB- is not P-bi-immune and thus it is P-quasi-approximable . 2 . Since the classes with weak membership properties mentioned in Section 2 are all included in the class of P-quasi-approximable sets , Theorem 3.1 has the following immediate corollary . Corollary 3.7 The following classes have measure zero in E : -LRB- 1 -RRB- the class of P-selective sets , -LRB- 2 -RRB- the class of P-multiselective sets , -LRB- 3 -RRB- the class of cheatable sets , -LRB- 4 -RRB- the class of easily countable sets , -LRB- 5 -RRB- the class of easily approximable sets , -LRB- 6 -RRB- the class of near-testable sets , -LRB- 7 -RRB- the class of nearly near-testable sets , -LRB- 8 -RRB- the class of sets that are not P-bi-immune -LRB- -LSB- May92 -RSB- -RRB- . 4 Open Questions There is an intermediate equivalence relation between p-isomorphism and one-one polynomial-time equivalence , namely the equivalence generated by one-one polynomialtime reductions that are p-honest . In the attempt to identify the weakest membership property that generates classes having measure zero in E , it would be interesting to find out whether the class of sets that are equivalent to P-quasi-approximable sets under this type of reduction is measurable or not in E. 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I I O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

cheatable sets
easily approximable sets
easily countable sets

near-testable sets
nearly near-testable sets
p-bi-immune sets
p-multiselective sets
p-selective sets
resource-bounded measure