algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

numerosities of point sets over the real line we consider the possibility of a notion of size for point sets i e subsets of the euclidean spaces e d r of all d tuples of real numbers that satisfies the fifth common notion of euclid s elements the whole is larger than the part clearly such a notion of numerosity can agree with cardinality only for finite sets we show that numerosities can be assigned to every point set in such a way that the natural cantorian definitions of the arithmetical operations provide a very good algebraic structure contrasting with cardinal arithmetic numerosities can be taken as nonnegative elements of a discretely ordered ring where sums and products correspond to disjoint unions and cartesian products respectively actually our numerosities form suitable semirings of hyperintegers of nonstandard analysis under mild set theoretic hypotheses e g cov b c n we can also have the natural ordering property that given any two countable point sets one is equinumerous to a subset of the other extending this property to uncountable sets seems to be a difficult problem

a note on the circuit complexity of pp in this short note we show that for any integer k there are languages in the complexity class pp that do not have boolean circuits of size nk

a classification of complexity core lattices abstract lynch 1975 has shown that every recursive set a not in p contains an infinite polynomial complexity core a set of elements c a such that any algorithm deciding a needs more than polynomial time almost everywhere on c actually any a not in p contains infinitely many different cores the collection of which forms a lattice under inclusion we study the structure of this lattice proving that surprisingly there are only three possibilities assuming the lattice is not trivial which happens if a is in p its shape depends only on whether a is almost p immune or not it is known that the natural intractable sets usually do not have this property

the probability distribution of word maps on finite groups abstract word maps provide a wealth of information about finite groups we examine the connection between the probability distribution induced by a word map and the underlying structure of a finite group we show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size moreover we show that probability distributions themselves are sufficient to identify nilpotent groups and these same distributions can be used to determine abelian groups up to isomorphism in addition we answer a question of amit and vishne

some consequences of cryptographical conjectures for s 1 2 and ef we show that there is a pair of disjoint np sets whose disjointness is provable in s 2 1 and which cannot be separated by a set in p poly if the cryptosystem rsa is secure further we show that factoring and the discrete logarithm are implicitly definable in any extension of s 2 1 admitting an np definition of primes about which it can prove that no number satisfying the definition is composite

kolmogorov complexity and degrees of tally sets we show that either epm tally epbtt tally or epm tally ep1 tt tally ep2 tt tally ep3 tt tally where erp tally denotes the class of sets which are equivalent to a tally set under rp reductions furthermore the question of whether or not emp tally ebttp tally is equivalent to the question of whether or not ne predicates can be solved in deterministic exponential time the proofs use the techniques of generalized kolmogorov complexity as corollaries to some of the main results we obtain new results about the kolmogorov complexity of sets in p

circuit lower bounds for merlin arthur classes we show that for each k 0 mathsf ma 1 mathsf ma with 1 bit of advice does not have circuits of size n k this implies the first superlinear circuit lower bounds for the promise versions of the classes mathsf ma mathsf am and mathsf zpp parallel mathsf np we extend our main result in several ways for each k we give an explicit language in mathsf ma cap mathsf coma 1 which does not have circuits of size n k we also adapt our lower bound to the average case setting i e we show that mathsf ma 1 cannot be solved on more than 1 2 1 n k fraction of inputs of length n by circuits of size n k furthermore we prove that mathsf ma does not have arithmetic circuits of size n k for any k as a corollary to our main result we obtain that derandomization of mathsf ma o 1 implies the existence of pseudorandom generators computable using o 1 bits of advice

coding into ramsey sets in 6 w t gowers formulated and proved a ramsey type result which lies at the heart of his famous dichotomy for banach spaces he defines the notion of weakly ramsey set of block sequences of an infinite dimensional banach space and shows that every analytic set of block sequences is weakly ramsey we show here that gowers result follows quite directly from the fact that all g sets are weakly ramsey if the banach space does not contain c0 and from the fact that all f sets are weakly ramsey in the case of an arbitrary banach space we also show that every result obtained by the application of gowers theorem to an analytic set can also be obtained by applying the theorem to a f set or to a g set if the space does not contain c0 this fact explains why the only known applications of this technique are based on very low ranked borel sets open closed f or g

weak completeness in e and e 2 abstract the notions of weak m p completeness for the complexity classes e dtime 2 linear and e 2 dtime 2 polynomial are compared an element c of one of these classes is weakly m p complete for the class if the set p m c consisting of all languages a m p c does not have measure 0 in the class the following two results are proven 1 i every problem that is weakly m p complete for e is weakly m p complete for e 2 2 ii there is a problem in e that is weakly m p complete for e 2 but not for e

hall s relations in finite groups let h be a finite group and let be an automorphism of h whose order divides n hall proved that the number of elements x of h that satisfy the relation xx x 2 x n 1 1 is a multiple of gcd n h for a prime factor p of gcd n h if such a number is not a multiple of gcd pn h then a sylow p subgroup of h is exceptional

a generalization of resource bounded measure with application to the bpp vs exp problem we introduce resource bounded betting games and propose a generalization of lutz s resource bounded measure in which the choice of the next string to bet on is fully adaptive lutz s martingales are equivalent to betting games constrained to bet on strings in lexicographic order we show that if strong pseudorandom number generators exist then betting games are equivalent to martingales for measure on e and exp however we construct betting games that succeed on certain classes whose lutz measures are important open problems the class of polynomial time turing complete languages in exp and its superclass of polynomial time turing autoreducible languages if an exp martingale succeeds on either of these classes or if betting games have the finite union property possessed by lutz s measure one obtains the nonrelativizable consequence mbox bpp neq mbox exp we also show that if mbox exp neq mbox ma then the polynomial time truth table autoreducible languages have lutz measure zero whereas if mbox exp mbox bpp they have measure one

measure stochasticity and the density of hard languages the main theorem of this paper is that for every real number alpha 1 e g alpha 0 99 only a measure 0 subset of the languages decidable in exponential time are leq p n alpha tt reducible to languages that are not exponentially dense thus every leq p n alpha tt hard language for e is exponentially dense this strengthens watanabe s 1987 result that every leq p o log n tt hard language for e is exponentially dense the combinatorial technique used here the sequentially most frequent query selection also gives a new simpler proof of watanabe s result the main theorem also has implications for the structure of np under strong hypotheses ogiwara and watanabe 1991 have shown that the hypothesis p ne np implies that every leq p btt hard language for np is nonsparse i e not polynomially sparse their technique does not appear to allow significant relaxation of either the query bound or the sparseness criterion it is shown here that a stronger hypothesis namely that np does not have measure 0 in exponential time implies the stronger conclusion that for every real alpha 1 every leq p n alpha tt hard language for np is exponentially dense evidence is presented that this stronger hypothesis is reasonable the proof of the main theorem uses a new very general weak stochasticity theorem ensuring that almost every language in e is statistically unpredictable by feasible deterministic algorithms even with linear nonuniform advice

new collapse consequences of np having small circuits we show that if a self reducible set has polynomial size circuits then it is low for the probabilistic class zpp np as a consequence we get a deeper collapse of the polynomial time hierarchy ph to zpp np under the assumption that np has polynomial size circuits this improves on the well known result in karp and lipton proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 stating a collapse of ph to its second level sigmap 2 under the same assumption furthermore we derive new collapse consequences under the assumption that complexity classes like up fewp and c p have polynomial size circuits finally we investigate the circuit size complexity of several language classes in particular we show that for every fixed polynomial s there is a set in zpp np which does not have o s n size circuits

computation times of np sets of different densities abstract in this paper we study the computational complexity of sets of different densities in np we show that the deterministic computation time for sets in np can depend on their density if and only if there is a collapse or partial collapse of the corresponding higher nondeterministic and deterministic time bounded complexity classes we show also that for np sets of different densities there exist complete sets of the corresponding density under polynomial time turing reductions finally we show that these results can be interpreted as results about the complexity of theorem proving and proof presentation in axiomatized mathematical systems this interpretation relates fundamental questions about the complexity of our intellectual tools to basic structural problems about p np conp and pspace discussed in this paper

martin s maximum saturated ideals and non regular ultrafilters part ii we prove assuming the existence of a huge cardinal the consistency of fully non regular ultrafilters on the successor of any regular cardinal we also construct ultrafilters with ultraproducts of small cardinality part ii is logically independent of part i

hard core theorems for complexity classes nancy lynch proved that if a decision problem a is not solvable in polynomial time then there exists an infinite recursive subset x of its domain on which the decision is almost everywhere complex in this paper general theorems of this kind that can be applied to several well known automata based complexity classes including a common class of randomized algorithms are proved

the size of spp derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class spp 1 p spp 0 2 ph spp in other words spp is small by being a negligible subset of exponential time or large by containing the entire polynomial time hierarchy this addresses an open problem about the complexity of the graph isomorphism problem it is not weakly complete for exponential time unless ph is contained in spp it is also shown that the polynomial time hierarchy is contained in sppnp if np does not have p measure 0

a note on dimensions of polynomial size circuits in this paper we use resource bounded dimension theory to investigate polynomial size circuits we show that for every i 0 p poly has ith order scaled p3 strong dimension 0 we also show that p polyi o has p3 dimension 1 2 and p3 strong dimension 1 our results improve previous measure results of lutz almost everywhere high nonuniform complexity j comput syst sci 44 2 1992 220 258 and dimension results of hitchcock and vinodchandran dimension entropy rates and compression in proc 19th ieee conf computational complexity 2004 pp 174 183 j comput syst sci to appear additionally we establish a supergale dilation theorem which extends the martingale dilation technique introduced implicitly by ambos spies terwijn and zheng resource bounded randomness and weakly complete problems theoret comput sci 172 1 2 1997 195 207 and made explicit by juedes and lutz weak completeness in e and e2 theoret comput sci 143 1 1995 149 158

cosmological lower bound on the circuit complexity of a small problem in logic an exponential lower bound on the circuit complexity of deciding the weak monadic second order theory of one successor ws1s is proved circuits are built from binary operations or 2 input gates which compute arbitrary boolean functions in particular to decide the truth of logical formulas of length at most 610 in this second order language requires a circuit containing at least 10125 gates so even if each gate were the size of a proton the circuit would not fit in the known universe this result and its proof due to both authors originally appeared in 1974 in the ph d thesis of the first author in this article the proof is given the result is put in historical perspective and the result is extended to probabilistic circuits

if np has polynomial size circuits then ma am abstract it is shown that the assumption of np having polynomial size circuits implies apart from a collapse of the polynomial time hierarchy as shown by karp and lipton that the classes am and ma of babai s arthur merlin hierarchy coincide this means that also a certain inner collapse of the remaining classes of the polynomial time hierarchy occurs

supercompact cardinals sets of reals and weakly homogeneous trees it is shown that if there exists a supercompact cardinal then every set of reals which is an element of l r is the projection of a weakly homogeneous tree as a consequence of this theorem and recent work of martin and steel martin d a steel j r 1988 proc natl acad sci usa 85 6582 6586 it follows that if there is a supercompact cardinal every set of reals in l r is determined

a conjecture of frobenius and the sporadic simple groups ii a conjecture of frobenius which has been reduced to the classification of finite simple groups is verified for the sporadic simple groups let g be a finite group and n be a positive integer dividing igi let ln g x e g xn 1 then by a theorem of frobenius 6 one knows that iln g i cnn for some integer cn frobenius conjectured that ln g forms a subgroup of g provided iln g i n see 2 zemlin 25 has reduced the conjecture to the classification of finite simple groups which is now complete see 8 the author has verified the conjecture for the fischer griess monster f1 and the fischer baby monster f2 in 24 the purpose of this note is to prove the following theorem the conjecture of frobenius is true for all the sporadic simple groups the proof of our theorem has been carried out in the following way with the use of a computer let g be one of the sporadic simple groups by 24 we may assume that g f1 and g f2 let f g t be the number of elements of order t in g and ord g order of x i x e g tables of f g t are given in the appendix see the supplements section at the end of this issue for f g t the reader is referred to the following papers ml1 m22 m23 burgoyne and fong 1 m12 m24 frobenius 5 ii janko 14 hj j2 hall and wales 8 hjm j3 janko 15 j4 janko 16 his frame 4 suz wright 23 mcl 3 finkelstein 3 rud rudvalis 19 hhm held 10 lys lyons 17 received june 21 1982 revised march 19 1984 and july 11 1984 1980 mathematics subject classification primary 20d05

numerosities of labelled sets a new way of counting abstract the notions of labelled set and numerosity are introduced to generalize the counting process of finite sets the resulting numbers called numerosities are then used to develop nonstandard analysis the existence of a numerosity function is equivalent to the existence of a selective ultrafilter hence it is independent of the axioms of zfc

the complexity and distribution of hard problems measure theoretic aspects of the leq rm p rm m reducibility structure of the exponential time complexity classes e dtime 2 rm linear and e 2 rm dtime 2 rm polynomial are investigated particular attention is given to the complexity measured by the size of complexity cores and distribution abundance in the sense of measure of languages that are leq rm p rm m hard for e and other complexity classes tight upper and lower bounds on the size of complexity cores of hard languages are derived the upper bound says that the leq rm p rm m hard languages for e are unusually simple in the sense that they have smaller complexity cores than most languages in e it follows that the leq rm p rm m complete languages for e form a measure 0 subset of e and similarly in e 2 this latter fact is seen to be a special case of a more general theorem namely that it every pmr degree e g the degree of all pmr complete languages for np has measure 0 in e and in ep

counting classes are at least as hard as the polynomial time hierarchy in this paper it is shown that many natural counting classes such as pp c p and text mod k text p are at least as computationally hard as ph the polynomial time hierarchy in the following sense for each bf k of the counting classes above every set in bf k ph is polynomial time randomized many one reducible to a set in bf k with two sided exponentially small error probability as a consequence of the result it is seen that all the counting classes above are computationally harder than ph unless ph collapses to a finite level some other consequences are also shown

the density and complexity of polynomial cores for intractable sets let a be a recursive problem not in p lynch has shown that a then has an infinite recursive polynomial complexity core this is a collection c of instances of a such that every algorithm deciding a needs more than polynomial time almost everywhere on c we investigate the complexity of recognizing the instances in such a core and show that every recursive problem a not in p has an infinite core recognizable in subexponential time we further study how dense the core sets for a can be under various assumptions about the structure of a our main results in this direction are that if p np then np complete problems have polynomially nonsparse cores recognizable in subexponential time and that exptime complete problems have cores of exponential density recognizable in exponential time

a complexity theory based on boolean algebra a projection of a boolean function is a function obtained by substituting for each of its variables a variable the negation of a variable or a constant reducibilities among computational problems under this relation of projection are considered it is shown that much of what is of everyday relevance in turing machine based complexity theory can be replicated easily and naturally in this elementary framework finer distinctions about the computational relationships among natural problems can be made than in previous formulations and some negative results are proved

the existence and density of generalized complexity cores if c is a class of sets and a is not in c then an infinite set h is a proper hard core for a with respect to c if h a and for every c e c such that c a c c h is finite it is shown that if c is a countable class of sets of strings that is closed under finite union and finite variation then every infinite set not in c has a proper hard core with respect to c in addition the density of such generalized complexity cores is studied

circuit principles and weak pigeonhole variants this paper considers the relational versions of the surjective partial surjective and multifunction weak pigeonhole principles for pv 1 b 1 b and b 1 b formulas as well as relativizations of these formulas to higher levels of the bounded arithmetic hierarchy we show that the partial surjective weak pigeonhole principle for 1 b formulas implies that for each k there is a string of length 2 2 n k which is hard to block recognize by circuits of size n k these principles in turn imply the partial surjective principle for 1 b formulas we show that the surjective weak pigeonhole principle for b 1 b formulas in s 2 1 implies our hard string principle which in turn implies the surjective weak pigeonhole principle for 1 b formulas we introduce a class of predicates corresponding to poly log length iterates of polynomial time computable predicates and show that over s 2 1 the multifunction weak pigeonhole principle for such predicates is equivalent to an iterative circuit block recognition principle a consequence of this is that if s 2 1 proves this principle then rsa is vulnerable to polynomial time attacks

completeness and weak completeness under polynomial size circuits this paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for espace under nonuniform many one reductions that are computed by polynomial size circuits p poly many one reductions every weakly p poly many one complete problem is shown to have a dense exponential nonuniform complexity core an exponential lower bound on the space bounded kolmogorov complexities of weakly p poly turing complete problems is established more importantly the p poly many one complete problems are shown to be unusually simple elements of espace in the sense that they obey nontrivial upper bounds on nonuniform complexity size of nonuniform complexity cores and space bounded kolmogorov complexity that are violated by almost every element of espace more generally a small span theorem for p poly many one reducibility in espace is proven and used to show that every p poly many one degree including the complete degree has measure 0 in espace in contrast almost every element of espace is weakly p many one complete all upper and lower bounds are shown to be tight

circuit size lower bounds and non reducibility to sparse sets as remarked in cook towards a complexity theory of synchronous parallel computation univ of toronto 1980 a nonlinear lower bound on the circuit size of a language in p or even in np is not known the best known published lower bound seems to be due to paul proceedings 7th acm symposium on theory of computing 1975 in this paper it is shown first that for each nonnegative integer k there is a language l k in 2 2 of the meyer and stockmeyer proceedings 13th ieee symposium on switching and automata theory 1972 hierarchy which does not have o n k size circuits using the same techniques one is able to prove several similar results for example it is shown that for each nonnegative integer k there is a language l k in np that does not have o n k size uniform circuits this follows as a corollary of a stronger result shown in the paper this result like the others to follow is not provable by direct diagonalization it thus points to the most interesting feature of the techniques used hereby using the polynomial time hierarchy they are able to prove results about np that cannot seem to proved by direct diagonalization finally it is noted that existence of small circuits is in suitable contexts equivalent to being reducible to sparse sets using this one is able to prove for example that for any time constructible superpolynomial function f n ntime f n contains a language which is not many to one p time reducible to any sparse set

on fixed polynomial size circuit lower bounds for uniform polynomials in the sense of valiant we consider the problem of fixed polynomial lower bounds on the size of arithmetic circuits computing uniform families of polynomials assuming the generalised riemann hypothesis grh we show that for all k there exist polynomials with coefficients in ma having no arithmetic circuits of size o n k over c allowing any complex constant we also build a family of polynomials that can be evaluated in am having no arithmetic circuits of size o n k then we investigate the link between fixed polynomial size circuit bounds in the boolean and arithmetic settings in characteristic zero it is proved that np size n k or ma size n k or np ma imply lower bounds on the circuit size of uniform polynomials in n variables from the class vnp over c assuming grh in positive characteristic p uniform polynomials in vnp have circuits of fixed polynomial size if and only if both vp vnp over f p and mod p p has circuits of fixed polynomial size

non abelian sylow subgroups of finite groups of even order we have been able to prove that every nonabelian sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution the proof depends upon the consequences of the classification of finite simple groups the purpose of this note is to announce 4 main theorem every nonabelian sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution let g be a finite group and g the prime graph of g g is the graph such that the vertex set is the set of prime divisors of g and two distinct vertices p and r are joined by an edge if and only if there exists an element of order pr in g let n g be the number of connected components of g and dg p r the distance between two vertices p and r of g it has been proved that n g 6 in 12 10 11 9 theorem 1 let g be a finite group of even order and p a prime divisor of g if dg 2 p 2 then a sylow p subgroup of g is abelian theorem 1 is a restatement of the main theorem in terms of the prime graph g corollary 1 let g be a finite group of even order and p a prime divisor of g if is a connected component of g p not containing 2 then a sylow r subgroup of g is abelian for any r there is a certain relation between a subgraph g p of g and brauer characters of p modular representations of g see 3 theorem 2 let g be a finite nonabelian simple group and p an odd prime divisor of g then dg 2 p 1 or 2 provided dg 2 p the significance of the prime graphs of finite groups can be found in 1 3 5 6 7 8 14 15 we apply the classification of finite simple groups see 1 2 4 7 10 13 it has been proved that a minimal counterexample to theorem 1 received by the editors october 20 1997 1991 mathematics subject classification primary 20d05 20d06 20d20

on inefficient special cases of np complete problems abstract every intractable set a has a polynomial complexity core a set h such that for any p subset s of a or of s h is finite a complexity core h of a is proper if h a it is shown here that if p np then every currently known i e either invertibly paddable or k creative np complete set a and its complement have proper polynomial complexity cores that are nonsparse and are accepted by deterministic machines in time 2 cn for some constant c turning to the intractable class d ext c 0 d time 2 cn it is shown that every set that is p m complete for d ext has an infinite proper polynomial complexity core that is nonsparse and recursive

near coherence of filters ii applications to operator ideals the stone cech remainder of a half line order ideals of sequences and slenderness of groups the set theoretic principle of near coherence of filters ncf is lnown to be neither provable nor refutable from the usual axioms of set theory we show that ncf is equivalent to the following statements among others 1 the ideal of compact operators on hilbert space is not the sum of two smaller ideals 2 the stone tech remainder of a half line has only one composant this was first proved by j mioduszewski 3 the partial ordering of slenderness classes of abelian groups minus its top element is directed upward and in fact has a top element thus 11 these statements are also consistent and independent

quasi selective ultrafilters and asymptotic numerosities abstract we isolate a new class of ultrafilters on n called quasi selective because they are intermediate between selective ultrafilters and p points under the continuum hypothesis these three classes are distinct the existence of quasi selective ultrafilters is equivalent to the existence of asymptotic numerosities for all sets of tuples a n k such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets most notably they maintain the structure of ordered semiring and in a precise sense they allow for a natural extension of asymptotic density to all sets of tuples of natural numbers

robust simulations and significant separations abstract we define a new notion of robust simulations between complexity classes which is intermediate between the traditional notions of infinitely often and almost everywhere as well as a corresponding notion of significant separations a language l has a robust simulation in a complexity class c if there is a language in c which agrees with l on arbitrarily large polynomial stretches of input lengths we show that various implications in complexity theory such as the collapse of ph if np p and the karp lipton theorem have analogues for robust simulations we then use these results to prove that most known separations in complexity theory can be strengthened to significant separations though in each case an almost everywhere separation is unknown proving our results requires several new ideas including a completely different proof of the hierarchy theorem for non deterministic polynomial time than the ones previously known

on a notion of breadth in the sense of frobenius abstract given a finite group g and an integer e 1 dividing the order of g the size of the set l e g x g x e 1 was studied originally by frobenius in order to find restrictions on the structure of g the aim of the present paper is to classify groups by b g max l e g e e div exp g where div exp g is the set of all divisors of the exponent exp g of g we will show general statements regarding the center and the central quotient of g by looking at b g this improves some recent contributions of w meng and j shi in 7

on embedding certain partial orders into the p points under rudin keisler and tukey reducibility the study of the global structure of ultrafilters on the natural numbers with respect to the quasi orders of rudin keisler and rudin blass reducibility was initiated in the 1970s by blass keisler kunen and rudin in a 1973 paper blass studied the special class of p points under the quasiordering of rudin keisler reducibility he asked what partially ordered sets can be embedded into the p points when the p points are equipped with this ordering this question is of most interest under some hypothesis that guarantees the existence of many p points such as martin s axiom for centered posets in his 1973 paper he showed under this assumption that both 1 and the reals can be embedded analogous results were obtained later for the coarser notion of tukey reducibility we prove in this paper that martin s axiom for centered posets implies that the boolean algebra p fin equipped with its natural partial order can be embedded into the p points both under rudin keisler and tukey reducibility consequently the continuum hypothesis implies that every partial order of size at most continuum embeds into the p points under both notions of reducibility

the perfect order subset conjecture for simple groups abstract this paper verifies the perfect order subset conjecture for simple groups for all but one family of finite simple groups specifically for each nonabelian finite simple group g there is some n such that the cardinality of the nonempty subset of all elements of order n in g does not divide the order of g unless g is an orthogonal group of plus type in dimension 4n for some n 2

p printable sets p printable sets arise naturally in the studies of generalized kolmogorov complexity and data compression as well as in other areas we present new characterizations of the p printable sets and present necessary and sufficient conditions for the existence of sparse sets in p that are not p printable as a corollary to one of our results we show that the class of sets of small generalized kolmogorov complexity is exactly the class of sets which are p isomorphic to a tally language

category and measure in complexity classes this paper presents resource bounded category and resource bounded measure two new tools for computational complexity theory and some applications of these tools to the structure theory of exponent

nonuniform acc circuit lower bounds the class acc consists of circuit families with constant depth over unbounded fan in and or not and modm gates where m 1 is an arbitrary constant we prove the following nexp the class of languages accepted in nondeterministic exponential time does not have nonuniform acc circuits of polynomial size the size lower bound can be slightly strengthened to quasipolynomials and other less natural functions enp the class of languages recognized in 2o n time with an np oracle doesn t have nonuniform acc circuits of 2no 1 size the lower bound gives an exponential size depth tradeoff for every d m there is a 0 such that enp doesn t have depth d acc circuits of size 2n with modm gates previously it was not known whether expnp had depth 3 polynomial size circuits made out of only mod6 gates the high level strategy is to design faster algorithms for the circuit satisfiability problem over acc circuits then prove that such algorithms entail these lower bounds the algorithms combine known properties of acc with fast rectangular matrix multiplication and dynamic programming while the second step requires a strengthening of the author s prior work

equivalence of families of functions on the natural numbers we present some consequences of the inequality u g among cardinal invariants of the continuum which has previously been shown to be consistent relative to zfc we are interested in its effects on two orderings of families of functions on the natural numbers in particular we show that under u g there are exactly five equivalence classes for both orderings excludint the families bounded by a fixed constant function this implies under the same hypothesis the existence of exactly four classes of rarefaction of measure zero sets

souslin partitions of products of finite sets abstract to every infinite sequence of positive integers m m i i we associate two fields of sets a field cl m of subsets of and a field pcl m of subsets of their relevance to ramsey theory is based on the fact that for every cl m measurable partition c 2 there is a sequence h i i with hi mi such that c is constant on i hi similarly for every pcl m measurable partition c 2 there is h and a sequence h i i of sets with hi and hi mi such that c is constant on i hi h in di prisco et al j combin theory ser a 93 2001 333 combinatorica to appear it is shown that cl m and pcl m are fields that contain all closed sets and therefore all borel subsets of their corresponding domains we show here that they are in fact closed under souslin operation and that under suitable assumptions they contain all reasonably definable subsets of their corresponding domains these results are then used to show that the classical partition relation is not equivalent to its polarized version solving thus a long standing problem in this area see j symbolic logic 58 1998 860 notas de logica matematica vol 39 universidad nacional del sur bahia blanca argentina 1994 pp 89 94

on the notion of infinite pseudorandom sequences abstract three definitions of infinite pseudorandom sequences with respect to polynomial time and space complexity are introduced and compared with each other it is shown that the first two definitions based on martin lof s notion of sequential tests and levin and schnorr s notion of monotonic operator complexity are equivalent with respect to polynomial space complexity while both are strictly stronger than the third definition which is derived from von mises s notion of collectives

perfect set properties inl r u abstract we study perfect set properties in the modell r u whenl r is elementarily equivalent to a solovay model anduis a selective ultrafilter on the integers generic overl r

cook versus karp levin separating completeness notions if np is not small under the hypothesis that np does not have p measure 0 roughly that np contains more than a negligible subset of exponential time it is shown that there is a language that is t p complete cook complete but not m p complete karp levin complete for np this conclusion widely believed to be true is not known to follow from p np or other traditional complexity theoretic hypotheses