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A Small Span Theorem Within P 1 Introduction A Small Span Theorem within P Wolfgang Lindner and Rainer Schuler Abteilung Theoretische Informatik Universitat Ulm Ob erer Eselsb erg Ulm GERMANY Abstract The development of Small Span Theorems for various complexity classes and re ducibili ties plays a basic role in resource b ounded measuretheoretic investigations of ecient reductions A Small Span Theorem for a complexity class C and reducibil ity is the assertion that for all sets A in C at least one of the cones b elowor r ab ove A is a negligible small class with resp ect to C where the cones b eloworabove A refer to the sets fB B Ag and fB A B g resp ectively That is a Small r r Span Theorem rules out one of the four p ossibilities of the size of upp er and lower cones for a set in C Here we use the recent formulation of resourceb ounded measure of Allender and Strauss whichallows meaningful notions of measure on p olynomialtime complexity classes Weshowtwo Small Span Theorems for p olynomialtime complexity classes and sublineartime reducibili ties namely a Small Span Theorem for P and Dlogtime NP uniform NC computable reductions and for P and Dlogtimetransformations Furthermore weshow that for every xed k the hard set for P under Dlogtime k uniform AC reductions of depth k and size n is a small class In contrast weshow that every upp er cone under Puniform NC reductions is not small Intro duction Resourceb ounded measure provides a to ol to investigate abundance phenomena in complexity classes Besides insights in the measuretheoretic structure of complexity classes resourceb ounded measure also enriches the measuretheoretic investigations of ecient reductions with its origin in the work of Bennet and Gill A unifying theme in this area is the developmentof Smal l Span Theorems for various complexity classes and reducibilities A rst Small Span Theorem for EXP and p olynomial time manyone reductions was shown by Juedes and Lutz and has subsequently ex tended to other reducibilities eg Briey a Small Span Theorem for a complexity class C is the assertion that for all sets A in C at least one of the cones b eloworabove A flindnerschulerginformatikuniulmde is a negligible small class with resp ect to C where the cones b eloworabove A refer to the sets reducible to A and the sets to which A can b e reduced resp ectively That is a Small Span Theorem rules out one of the four p ossibilities of the size of upp er and lower cones for a set in C As an immediate consequence the hard sets for C is a negligible small class with resp ect to C Furthermore there are sets for all of the three p ossibilities not ruled out by a Small Span Theorem which has b een further studied in For a recentoverview we refer to The formulation of resourceb ounded measure given by Lutz applies only to complexity classes at least containing E Recently Allender and Strauss provided meaningful notions of measure on P Here we concentrate on the most restricted notion the con servative Pmeasure Though some of intuitively small sub classes of P are in fact not measurable notably the pprintable sets and hence all sparse sets in P it satises all basic prop erties required by a reasonable notion of measure in P In particular it is p ossible to dene pseudorandom sets and to show that the ma jority of sets in P is pseudorandom Furthermore all pro ofs in this context relativize that is the denitions immediately NP apply to classes likeP In order to have a nontrivial degree structure in P without unproven assumptions we consider reductions computed by Dlogtimeuniform constant depth circuits see eg We show a Small Span Theorem for Dlogtimeuniform NC reductions in P In contrast we show that every upp er cone under Puniform NC reductions is not small It follows that a Small Span Theorem for Puniform NC reductions do es not hold A consequence of the Small Span Theorem is that the hard sets for P under Dlogtime uniform NC reductions is a small class We also show that this can b e improved to a restricted version of Dlogtimeuniform AC reductions of depth k As in the pro ofs in the main technical step in the pro of of the Small Span Theorem is to showthatevery reduction from a pseudorandom set can not decrease the length of its value to much In the case of p olynomialtime reductions and exp onentialtime classes this involves inverting p olynomialtime functions which can b e done in exp onential time But even Dlogtimeuniform NC computable functions can not b e inverted in p olynomial time unless P NP Thus we merely explore the fact that for a NC computable function there is some constant c such that each output bit dep ends on at most c dierent input bits In contrast we use the exp onential lower b ound on the size of a constant depth circuit for the parity function to show the result concerning the hard sets for P under restricted AC reductions However in the presence of an NPoracle Dlogtimetransformations are invertible This NP allows us to show a Small Span Theorem for Dlogtimetransformations within P with an adaption of the pro ofs in Preliminaries A circuit family is a sequence fC g n N where each C is an acyclic circuit with n n n Bo olean inputs x x as well as the constants and allowed as inputs and some n numb er of output gates y y fC g has size snifeachcircuitC has at most sn m n n gates it has depth dn if the length of the longest path from input to output in C is at n most dn A family fC g is uniform if the function n C is easy to compute in some n n sense We will consider Dlogtimeuniformity and Puniformity A function f is said to b e AC computable if there is a circuit family fC g of p olynomial n size and constant depth consisting of un b ounded fanin AND and OR and NOT gates such that for eachinput x of length n the output of C on input x is f x n A function f is said to b e NC computable if there is a circuit family fC g of p olynomial n size and constant depth consisting of fanin two AND and OR and NOT gates Note that for anyNC circuit family there is some constant c such that each output bit dep ends on at most c dierent input bits Note that a NC AC computable function f satises the restriction that jxj jy j jf xj jf y j A function g is an inverse of a function f if for all strings y y range f f g y y A pro of of the following can b e found in eg Prop osition PNPif and only if every length increasing Dlogtimeuniform NC computable function has a polynomialtime computable inverse A set A is NC AC reducible to a set B if A is manyone reducible to B via a p olynomially length b ounded NC AC computable function A function f is a Dlogtimetransformation if f is p olynomially length b ounded and the set fx i b the ith bit of f xisb f ggis decidable in logarithmic time Aset A is rprintable if there is a function computable within the resources sp ecied n byrwhich on input prints out the whole set of strings in A up to length n Measure on P In order to dene a reasonable notion of measure within sub exp onential time classes Allen der and Strauss considersublinear computations Here the underlying computation mo del is a Turing machine with randomaccess to its input via a sp ecial index tap e When M enters a sp ecial query state M receives the ith bit of the input where i is the content of the index tap e Furthermore M is given b oth w and the length of w as the input Given such a machine M and a string w letI w denote the set of bits queried by M M to the input w We assume that M queries the bits of the input w in paral lel that is the bits queried by M do not dep end on the actual input w but only on the length jw j Dene the dependency set D w f ng b e the unique minimal set containing I w M M and satisfying i D w I w i D w M M M Note that the queries to the length of w are not content of the dep endency set c A function f is n computable if it is computable byamachine M suchthat M runs c c in time O log jw j and has dep endency sets D w with size b ounded by O log jw j A M c function f is Pcomputable if f is n computable for some c N A martingale is a function d R satisfying the average law dx dx dx for all x A martingale succeeds onasetA if lim sup dAjz A n n c c class X is a n nul lset if there is a n computable martingale d which succeeds on c every set in X A class X is a Pnul lset if X is a n nullset for some c N Allender and Strauss show that the Pnullsets dene a reasonable notion of nullsets That is the Pmeasure corresp onds to P in the sense that all singletons of P are P nullsets but the whole space P is not a Pnullsets Moreover the collection of P nullsets is closed under subsets nite unions and arbitrary unions over the subcollection c of n nullsets The latter p ermits the denition of pseudorandom sets as the typical sets within P in c c the sense of More precisely dene a set A to b e n random if no n computable c martingale succeeds on AEquivalently A is n random if and only if the singleton fAg c c is a n nullset Then for each xed c all sets in P but a Pnullset are n random c c but no n random set p ossesses any prop ertywhich is sp ecic for only a n nullset c This gives us the following characterization of Pnullsets in terms of n random sets Prop osition Let X any class of sets The fol lowing areequivalent X is a Pnul lse t c For some c X contains no n random set c Mayordomo showed that for every xed c the class of nonDtimen biimmune sets is small in exp onential time The same pro of can b e used to show the following c A is biimmune for the class of
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