upward separations and weaker hypotheses in resource bounded measure we consider resource bounded measure in double exponential time complexity classes in contrast to complexity class separation translating downwards we show that measure separation translates upwards for example m p np email protected email protected e ne email protected email protected e x p nexp 0 we also show that if ne does not have e measure 0 then the np machine hypothesis holds we give oracles relative to which the converses of these statements do not hold therefore the hypothesis on the e measure of ne is relativizably weaker than the often investigated p measure hypothesis on np but it has many of the same consequences

a note on measuring in p we revisit the problem of generalising lutz s resource bounded measure rbm to small complexity classes and propose a definition of a random based rbm on p k n dtime o nk which we argue as being a good generalisation to p of lutz s rbm we cannot unconditionally prove the existence of such a measure but we give sufficient and necessary conditions for its existence we also revisit an rbm for p defined by strauss inform comput 136 1 1997 1 and correct an erroneous claim concerning the relations between and random sets a correction to this mistake is then proposed which is a less powerful but accurate relation between and random sets in order to obtain these results we introduce a mathematical structure called a measuring system which is a general setting that can be used to compare different rbms on any fixed complexity class through a partial ordering relation

convergence of path measures arising from a mean field or polaron type interaction we discuss the limiting path measures of markov processes with either a mean field or a polaron type interaction of the paths in the polaron type situation the strength is decaying at large distances on the time axis and so the interaction is of short range in time in contrast in the mean field model the interaction is weak but of long range in time donsker and varadhan proved that for the partition functions there is a transition from the polaron type to the mean field interaction when passing to a limit by letting the strength tend to zero while increasing the range the discussion of the path measures is more subtle we treat the mean field case as an example of a differentiable interaction and discuss the transition from the polaron type to the mean field interaction for two instructive examples

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

a comparison of polynomial time completeness notions ladner lynch and selman 1975 showed the differences among the power of several types of polynomial time reductions for dext c 0dtime 2cn but questions concerning complete degrees w r t these reductions have remained open here we show in dext almost all the possible differences between the completeness notions w r t any pair of the following reductions pt ptt pp n tt pc pd pbtt pk tt pm differences between several types of truth table completenesses we introduce the notion f easy subset for each pr reduction we first prove the property that all pr complete sets for dext have f easy subsets where the class f depends on each reduction type r then the construction of a ps complete set which is not pr complete becomes a simple diagonalization argument these constructions also work for all deterministic superpolynomial complexity classes which have a pm complete set in order to construct a set which is pt complete for dext but not ptt complete for dext we use strings which are not polynomially producible from small sized inputs these strings are said to have large generalized kolmogorov complexity cf hartmanis 1983

separation of np completeness notions we use hypotheses of structural complexity theory to separate various np completeness notions in particular we introduce an hypothesis from which we describe a set in np that is mbox leq rm p rm t complete but not mbox leq rm p tt complete we provide fairly thorough analyses of the hypotheses that we introduce

a heuristic for finding embedded network structure in mathematical programmes abstract an intuitive description is given of how to recognise when a mathematical programme can be converted to a network flow problem in practice it is more likely that a mathematical programme with a high degree of network structure will have additional complicating constraints thus a heuristic is presented for identifying a maximal subset of the constraints which can be converted to a network flow problem computational results show that the heuristic is highly effective and that much larger networks are identified by seeking a general network structure rather than limiting the search to a subset of rows giving at most one 1 and one 1 in each column as has been done previously

martingale families and dimension in p we introduce a new measure notion on small complexity classes called f measure based on martingale families that gets rid of some drawbacks of previous measure notions it can be used to define dimension because martingale families can make money on all strings and it yields random sequences with an equal frequency of 0 s and 1 s on larger complexity classes e and above f measure is equivalent to lutz resource bounded measure as applications to f measure we answer a question raised in e allender m strauss measure on small complexity classes with application for bpp in proc of the 35th ann ieee symp on found of comp sci 1994 pp 807 818 by improving their result to for almost every language a decidable in subexponential time p a bpp a we show that almost all languages in pspace do not have small non uniform complexity we compare f measure to previous notions and prove that martingale families are strictly stronger than c measure e allender m strauss measure on small complexity classes with application for bpp in proc of the 35th ann ieee symp on found of comp sci 1994 pp 807 818 we also discuss the limitations of martingale families concerning finite unions we observe that all classes closed under polynomial many one reductions have measure zero in exp iff they have measure zero in subexp we use martingale families to introduce a natural generalization of lutz resource bounded dimension j h lutz dimension in complexity classes in proceedings of the 15th annual ieee conference on computational complexity 2000 pp 158 169 on p which meets the intuition behind lutz s notion we show that p dimension lies between finite state dimension and dimension on e we prove an analogue of a theorem of eggleston in p i e the class of languages whose characteristic sequence contains 1 s with frequency a has dimension the shannon entropy of a in p

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

complexity measures for public key cryptosystems a general theory of public key cryptography is developed that is based on the mathematical framework of complexity theory two related approaches are taken to the development of this theory and th

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

almost every set in exponential time is p bi immune a set a is p bi immune if neither a nor its complement has an infinite subset in p we investigate here the abundance of p bi immune languages in linear exponential time e using resource bounded measure we prove that the class of p bi immune sets has measure 1 in e this implies that almost every language in e is p bi immune a bit further we show that every p random pseudorandom language is e bi immune regarding the existence of p bi immune sets in np we show that if np does not have measure 0 in e then np contains a p bi immune set another consequence is that the class of 1 tt p complete languages for e has measure 0 in e in contrast using the approach of resource bounded category it is shown that in e the class of p bi immune languages lacks the property of baire the baire category analogue of lebesgue measurability

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

non mitotic sets we study the question of the existence of non mitotic sets in np we show under various hypotheses that 1 tt mitoticity and m mitoticity differ on np t autoreducibility and t mitoticity differ on np this contrasts the situation in the recursion theoretic setting where ladner showed that autoreducibility and mitoticity coincide 2 tt autoreducibility does not imply weak 2 tt mitoticity from this it follows that autoreducibility and mitoticity are not equivalent for all reducibilities between 2 tt and t although the notions coincide for m and 1 tt reducibility

the zero one law holds for bpp abstract we show that bpp has p measure zero if and only if bpp differs from exp the same holds when we replace bpp by any complexity class c that contains bpp and is closed under tt reductions the zero one law for each of these classes c follows within exp c has either measure zero or else measure one

reductions on np and p selective sets abstract p selective sets are used to distinguish polynomial time bounded reducibilities on np in particular we consider different kinds of positive reductions these preserve membership in np and are not a priori closed under complements we show that the class of all sets which are both p selective and have positive reductions to their complements is p this is used to show that if dext next then there exists a set in np p that is not positive reducible to its complement various similar results are obtained we also show that p is the class of all sets which are both p selective and positive truth table self reducible from this result it follows that various naturally defined apparently intractible problems are not p selective unless p np

large deviations for many brownian bridges with symmetrised initial terminal condition consider a large system of n brownian motions in mathbb r d with some non degenerate initial measure on some fixed time interval 0 with symmetrised initial terminal condition that is for any i the terminal location of the i th motion is affixed to the initial point of the i th motion where is a uniformly distributed random permutation of 1 n such systems play an important role in quantum physics in the description of boson systems at positive temperature 1 in this paper we describe the large n behaviour of the empirical path measure the mean of the dirac measures in the n paths and of the mean of the normalised occupation measures of the n motions in terms of large deviations principles the rate functions are given as variational formulas involving certain entropies and fenchel legendre transforms consequences are drawn for asymptotic independence statements and laws of large numbers in the special case related to quantum physics our rate function for the occupation measures turns out to be equal to the well known donsker varadhan rate function for the occupation measures of one motion in the limit of diverging time this enables us to prove a simple formula for the large n asymptotic of the symmetrised trace of rm e beta mathcal h n where mathcal h n is an n particle hamilton operator in a trap

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

genericity and randomness over feasible probability measures this paper investigates the notion of resource bounded genericity developed by ambos spies fleischhack and huwig ambos spies neis and terwijn have recently shown that every language that is t n random over the uniform probability measure is t n generic it is shown here that in fact every language that is t n random over any strongly positive t n computable probability measure is t n generic roughly speaking this implies that when genericity is used to prove a resource bounded measure result the result is not specific to the underlying probability measure

complete distributional problems hard languages and resource bounded measure we say that a distribution is reasonable if there exists a constant s 0 such that x x n 1 ns we prove the following result which suggests that all distnp complete problems have reasonable distributions if np contains a dtime 2n bi immune set then every distnp complete set has a reasonable distribution it follows from work of mayordomo 19 that the consequent holds if the p measure of np is not zero cai and selman 6 defined a modification and extension of levin s notion of average polynomial time to arbitrary time bounds and proved that if l is p bi immune then l is distributionally hard meaning that for every polynomial time computable distribution the distributional problem l is not polynomial on the average we prove the following results which suggest that distributional hardness is closely related to more traditional notions of hardness 1 if np contains a distributionally hard set then np contains a p immune set 2 there exists a language l that is distributionally hard but not p bi immune if and only if p contains a set that is immune to all p printable sets the following corollaries follow readily 1 if the p measure of np is not zero then there exists a language l that is distributionally hard but not p bi immune 2 if the p2 measure of np is not zero then there exists a language l in np that is distributionally hard but not p bi immune

identification of special structure constraints in linear programs in large lps it may be that a major fraction of the constraints has a particularly simple form frequently this simple form may be exploited by either decomposition or implicit representation methods in these cases the effective size and computational difficulty are more closely related to the number of remaining nonspecial constraints than to the number of special constraints a unified computational procedure is presented for mechanically identifying near maximal sets of special structure constraints for most of the kinds of special structures which have been identified thus far as being exploitable

query monotonic turing reductions we study reductions that limit the extreme adaptivity of turing reductions in particular we study reductions that make a rapid structured progression through the set to which they are reducing each query is strictly longer shorter than the previous one we call these reductions query increasing query decreasing turing reductions we also study query nonincreasing query nondecreasing turing reductions these are turing reductions in which the sequence of query lengths is nonincreasing nondecreasing we ask whether these restrictions in fact limit the power of reductions we prove that query increasing and query decreasing turing reductions are incomparable with that is are neither strictly stronger than nor strictly weaker than truth table reductions and are strictly weaker than turing reductions in addition we prove that query nonincreasing and query nondecreasing turing reductions are strictly stronger than truth table reductions and strictly weaker than turing reductions despite the fact that we prove query increasing and query decreasing turing reductions to in the general case be strictly weaker than turing reductions we identify a broad class of sets a for which any set that turing reduces to a will also reduce to a via both query increasing and query decreasing turing reductions in particular this holds for all tight paddable sets where a set is said to be tight paddable exactly if it is paddable via a function whose output length is bounded tightly both from above and from below in the length of the input we prove that many natural np complete problems such as satisfiability clique and vertex cover are tight paddable

diagonalizations over polynomial time computable sets abstract a formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets the class of polynomial time computable functions respectively like e g p immunity it is shown that there are sets called p generic which have all properties enforceable by such diagonalizations we study the behaviour and the complexity of p generic sets in particular we show that the existence of p generic sets in np is oracle dependent even if we assume p np

comparing reductions to np complete sets under the assumption that np does not have p measure 0 we investigate reductions to np complete sets and prove the following 1 adaptive reductions are more powerful than nonadaptive reductions there is a problem that is turing complete for np but not truth table complete 2 strong nondeterministic reductions are more powerful than deterministic reductions there is a problem that is snp complete for np but not turing complete 3 every problem that is many one complete for np is complete under length increasing reductions that are computed by polynomial size circuits the first item solves one of lutz and mayordomo s twelve problems in resource bounded measure 1999 we also show that every many one complete problem for ne is complete under one to one length increasing reductions that are computed by polynomial size circuits

baire categories on small complexity classes and meager comeager laws we introduce two resource bounded baire category notions on small complexity classes such as p quasipoly subexp and pspace and on probabilistic classes such as bpp which differ on how the corresponding finite extension strategies are computed we give an alternative characterization of small sets via resource bounded banach mazur games as an application of the first notion we show that for almost every language a i e all except a meager class computable in subexponential time p a bpp a we also show that almost all languages in pspace do not have small nonuniform complexity we then switch to the second baire category notion called locally computable and show that the class sparse is meager in p we show that in contrast to the resource bounded measure case meager comeager laws can be obtained for many standard complexity classes relative to locally computable baire category on bpp and pspace another topic where locally computable baire categories differ from resource bounded measure is regarding weak completeness we show that there is no weak completeness notion in p based on locally computable baire categories i e every p weakly complete set is complete for p we also prove that the class of complete sets for p under turing logspace reductions is meager in p if p is not equal to dspace logn and that the same holds unconditionally for quasipoly finally we observe that locally computable baire categories are incomparable with all existing resource bounded measure notions on small complexity classes which might explain why those two settings seem to differ so fundamentally

on polynomial time turing and many one completeness in pspace meyer and paterson 1979 investigated pm reducibility of pspace complete sets to sparse sets we point out that their observation indicates the difference between the powers of pt and pm reducibilities and establish a machinery that uses this difference for separating pt completeness from pm completeness in pspace by using this machinery we show the difference between pt and pm completeness notions in pspace from each of the following assumptions i a randomized completeness notion differs from a deterministic one in pspace and ii pspace has a dense set almost every element of which is hard to produce by any polynomial time computation

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

completeness approximation and density polynomial time approximations to exponential time computable problems exp are considered from the point of view of structure infinitely often speedable and almost everywhere complex problems are studied using the notions of polynomial time productivity and immunity in particular the existence of a polynomial time immune set which is not polynomial time approximable at all but which is polynomial time tt complete in exp is proven the relationship between completeness and approximability is also studied it is shown that being polynomial time m complete in exp does not provide any control of the probability of erroneous approximations

phase uniqueness for the mallows measure on permutations for a positive number q the mallows measure on the symmetric group is the probability measure on sn such that pn q is proportional to q to the power inv where inv equals the number of inversions inv equals the number of pairs i j one may consider this as a mean field model from statistical mechanics the weak large deviation principle may replace the gibbs variational principle for characterizing equilibrium measures in this sense we prove the absence of phase transition i e phase uniqueness

on polynomial time bounded truth table reducibility of np sets to sparse sets it is proved that if text p ne text np then there exists a set in text np that is not polynomial time bounded truth table reducible in short leqq text btt text p reducible to any sparse set in other words it is proved that no sparse leqq text btt text p hard set exists for text np unless text p text np by using the technique proving this result the intractability of several number theoretic decision problems i e decision problems defined naturally from number theoretic problems is investigated it is shown that for these number theoretic decision problems if it is not in text p then it is not leqq text btt text p reducible to any sparse set

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

reductions between disjoint np pairs disjoint np pairs are pairs a b of nonempty disjoint sets in np we prove that all of the following assertions are equivalent there is a many one complete disjoint np pair there is a strongly many one complete disjoint np pair there is a turing complete disjoint np pair such that all reductions are smart reductions there is a complete disjoint np pair for one to one invertible reductions the class of all disjoint np pairs is uniformly enumerable let a b c and d be nonempty sets belonging to np a smart reduction between the disjoint np pairs a b and c d is a turing reduction with the additional property that if the input belongs to a b then all queries belong to c d we prove under the reasonable assumption that up co up has a p bi immune set that there exist disjoint np pairs a b and c d such that a b is truth table reducible to c d but there is no smart reduction between them this paper contains several additional separations of reductions between disjoint np pairs we exhibit an oracle relative to which distnp has a truth table complete disjoint np pair but has no many one complete disjoint np pair

compressibility and resource bounded measure we give a new definition of resource bounded measure based on compressibility of infinite binary strings we prove that the new definition is equivalent to the one commonly used this new characterization offers us a different way to look at resource bounded measure shedding more light on the meaning of measure zero results and providing one more tool to prove such results the main contribution of the paper is the new definition and the proofs leading to the equivalence result we then show how this new characterization can be used to prove that the class of linear autoreducible sets has p measure 0 we also prove that the class of sets that are truth table reducible to a p selective set has p measure 0 and that the class of sets that turing reduce to a subpolynomial dense set has p measure 0 this strengthens various results

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

the global power of additional queries to p random oracles we consider separations of reducibilities by random sets first we show a result on polynomial time bounded reducibilities that query their oracle nonadaptively for every p random set r there is a set that is reducible to r with k 1 queries but is not reducible to any other p random set with at most k queries this result solves an open problem stated in a recent survey paper by lutz and mayordomo eatcs bulletin 68 1999 pp 64 80 second we show that the separation result above can be transferred from the setting of polynomial time bounds to a setting of rec random sets and recursive reducibilities this extends the main result of book lutz and martin inform and comput 120 1995 pp 49 54 who by using different methods showed a similar separation with respect to martin lof random sets moreover in both settings we obtain similar separation results for truth table versus bounded truth table reducibility

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

max3sat is exponentially hard to approximate if np has positive dimension under the hypothesis that np has positive p dimension we prove that any approximation algorithm a for max3sat must satisfy at least one of the following 1 for some 0 a uses at least 2n time 2 for all e 0 a has performance ratio less than 7 8 e on an exponentially dense set of satisfiable instances as a corollary this solves one of lutz and mayordomo s twelve problems on resource bounded measure bull european assoc theoret comput sci 68 1999 64 80

large deviations for trapped interacting brownian particles and paths are defined in terms of transformed path measures on finite time intervals under a trap hamiltonian and two respective pair interaction hamiltonians the first pair interaction exhibits a particle repellency while the second one imposes a path repellency we analyze both models in the limit of diverging time with fixed number n of brownian motions in particular we prove large deviations principles for the normalized occupation measures the minimizers of the rate functions are related to a certain associated operator the hamilton operator for a system of n interacting trapped particles more precisely in the particle repellency model the minimizer is its ground state and in the pathrepellency model the minimizers are its ground product states in the case of path repellency we also discuss the case of a dirac type interaction which is rigorously defined in terms of brownian intersection local times we prove a large deviation result for a discrete variant of the model this study is a contribution to the search for a mathematical formulation of the quantum system of n trapped interacting bosons as a model for bose einstein condensation motivated by the success of the famous 1995 experiments recently lieb et al described the large n behavior of the ground state in terms of the well known gross pitaevskii formula involving the scattering length of the pair potential we prove that the large n behavior of the ground product states is also described by the gross pitaevskii formula however with the scattering length of the pair potential replaced by its integral

resource bounded randomness and weakly complete problems we introduce and study resource bounded random sets based on lutz s concept of resource bounded measure 5 6 we concentrate on nc randomness c 2 which corresponds to the polynomial time bounded p measure of lutz and which is adequate for studying the internal and quantative structure of e dtime 2lin first we show that the class of nc random sets has p measure 1 this provides a new simplified approach to p measure 1 results next we compare randomness with genericity in the sense of 1 2 and we show that nc 1 random sets are nc generic whereas the converse fails from the former we conclude thatnc random sets are not p btt complete for e our technical main results describe the distribution of the nc random sets under p m reducibility we show that every nc random set in e has nk random predecessors in e for any k 1 whereas the amount of randomness of the successors is bounded we apply this result to answer a question raised by lutz 8 we show that the class of weakly complete sets has measure 1 in e and that there are weakly complete problems which are not p btt complete for e

a zero one law for rp and derandomization of am if np is not small we show that if rp does not have p measure zero then zpp exp as corollaries we obtain a zero one law for rp in exp and that both probabilistic classes zpp and rp have the same measure in exp we also prove that if np does not have p measure zero then np am

the practical conversion of linear programmes to network flow models a practical algorithm for converting linear and integer programs to minimum cost network flow models is described reasons for the importance of such conversions are given the method depends on the representation of a network by one of its polygon matrices in this representation which is described polygons circuits of a network are given in terms of a spanning tree

bi immunity separates strong np completeness notions we prove that if for some e 0 np contains a set that is dtime 2ne bi immune then np contains a set that is 2 turing complete for np hence 3 truth table complete but not 1 truth table complete for np thus this hypothesis implies a strong separation of completeness notions for np lutz and mayordomo theor comput sci 164 1996 141 163 and ambos spies and bentzien j comput syst sci 61 3 2000 335 361 previously obtained the same consequence using strong hypotheses involving resource bounded measure and or category theory our hypothesis is weaker and involves no assumptions about stochastic properties of np

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