quasi linear truth table reductions to p selective sets we show that if sat is quasi linear truth table reducible to a p selective set then np p as a consequence it follows that for a class k pp c p if every set in k is quasi linear truth table reducible to a p selective set then k p

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

query complexity of membership comparable sets this paper investigates how many queries to k membership comparable sets are needed in order to decide all k 1 membership comparable sets for k 2 this query complexity is at least linear and at most cubic as a corollary we obtain that more languages are o logn membership comparable than truth table reducible to p selective sets

hard promise problems and nonuniform complexity for every recursive set a let pp a denote the following promise problem input x and y promise x a y a property x a we show that if l is a solution of pp a then a p l poly from this result it follows that if a is t p hard for np then all solutions of pp a are hard for np under a reduction that generalizes both t p and t sn specifically if a is np hard then all solutions of pp a are generalized high2 bbs86b the main theorem that leads to this result states that if b is a self reducible set b t p a and a p l poly then 2 p b 2 p l several interesting connections between uniform and nonuniform complexity follow directly from this theorem

a self orthogonalizing efficient block adaptive filter this paper deals with the development of a unique self orthogonalizing block adaptive filter sobaf algorithm that yields efficient finite impulse response fir adaptive filter structures computationally the sobaf is shown to be superior to the least mean squares lms algorithm the consistent convergence performance which it provides lies between that of the lms and the recursive least squares rls algorithm but unlike the lms is virtually independent of input statistics the block nature of the sobaf exploits the use of efficient circular convolution algorithms such as the fft the rectangular transform rt the fermat number transform fnt and the fast polynomial transform fpt in performance the sobaf achieves the mean squared error mse convergence of a self orthogonalizing structure that is the adaptive filter converges under any input conditions at the same rate as an lms algorithm would under white input conditions furthermore the selection of the step size for the sobaf is straightforward as the range and the optimum value of the step size are independent of the input statistics

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

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

on the complexity of kings a king in a directed graph is a vertex from which each vertex in the graph can be reached through paths of length at most two there is a broad literature on tournaments completely oriented digraphs and it has been known for more than half a century that all tournaments have at least one king recently kings have proven useful in theoretical computer science in particular in the study of the complexity of reachability problems and semifeasible sets in this article we study the complexity of recognizing kings for each succinctly specified family of tournaments the king problem is already known to belong to p 2 p we prove that the complexity of kingship problems is a rich enough vocabulary to pinpoint every nontrivial many one degree in p 2 p that is we show that every set in p 2 p other than 0 and s is equivalent to a king problem under m p reductions indeed we show that the equivalence can even be realized by relatively simple padding and holds even if the notion of kings is redefined to refer to k kings for any fixed k 2 vertices from which all vertices can be reached through paths of length at most k in contrast we prove that for each succinctly specified family of tournaments the source problem the problem of deciding whether a given vertex v has the property that there exists a k such that v is a k king also falls within p 2 p yet cannot be p 2 p complete or even np hard unless p np using these and related techniques we obtain a broad range of additional results about the complexity of king problems diameter problems and radius problems it follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs which need not be tournaments is p 2 p complete we show that the radius problem for arbitrary succinctly represented graphs is s 3 p complete but that in contrast the diameter problem for arbitrary succinctly represented graphs or even tournaments is p 2 p complete

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 fast adaptive filter algorithm using eigenvalue reciprocals as stepsizes it is shown that there exist finite optimum update positions in the gradient direction of the least mean square lms algorithm the optimum stepsizes to reach these positions are in a discrete set on this basis a new adaptive filter adf algorithm is proposed the discrete cosine transform a fairly good approximation of the karhunen loeve transform for a large number of signal classes is used to estimate the optimum stepsizes a block averaging operation is also used for smoothing the gradient estimate computer simulations show that the proposed adf algorithm provides fast convergence rates when the input signal autocorrelation matrix has either large or small eigenvalue spread ratio of the largest to the smallest eigenvalues the number of multiplications required by the new adf is about 11 log sub 2 n 12 which is comparable to the 10 log sub 2 n 8 required by the fast lms algorithm

block realization of multirate adaptive digital filters in this paper we study multirate transversal adaptive digital filters adf s based on the block least mean square blms algorithms these include an adf with decimation adfd and an adf with interpolation adfi we first formulate these filters based on the block mean squared error bmse criterion and derive blms multi rate weight adjustment algorithms and then study efficient realization of those filters it is shown that the blms adfd can be realized efficiently in the direct form or in the filter bank structure both using the fast fourier transform fft and an appropriate sectioning procedure according to our analysis of computational complexity the fft realization of the blms adfd in the filter bank structure is more efficient than in the direct form and the two structures using the fft become more efficient in comparison to the lms adfd as the number of weights increases unlike the direct form realization the filter bank realization of the blms adfd using the fft becomes more efficient as the decimation ratio increases similar results have been obtained for the blms adfi finally we investigate by computer simulation the effects of different weight adjustment algorithms and several system parameters on the performances of adfd s the results of simulation indicate that as expected the convergence speed of the blms adfd can be significantly improved by self orthogonalization of weight adjustment in the frequency domain furthermore the convergence factor of the self orthogonalizing blms adfd can be chosen such that the steady state performance of the new adfd is the same as that of the existing lms adfd

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

on symmetric differences of np hard sets with weakly p selective sets the symmetric differences of np hard sets with weakly p selective sets are investigated we show that if there exist a weakly p selective set a and an np pm hard set h such that h a pbtt sparse and a h pm sparse then p np so no np pm hard set has sparse symmetric difference with any weakly p selective set unless p np the proof of our main result is an interesting application of the tree prunning techniques fortune 1979 mahaney 1982 in addition we show that there exists a p selective set which has exponentially dense symmetric difference with every set in pbtt sparse

generalized sliding fft and its application to implementation of block lms adaptive filters this paper has two contributions first the concept of the generalized sliding fast fourier transform gsfft as an efficient implementation of the hopping fft is introduced application of the gsfft is broad and not limited to what has been considered in this paper the frequency domain block lms fblms adaptive filters are then revised and their implementations for block lengths less than the length of the adaptive filter are studied the gsfft and the available pruned ffts are used to give an efficient implementation of these filters in the particular case of the block length equal to one where the fblms algorithm reduces to the frequency domain lms flms algorithm it is shown that the latter can be implemented with the order of m complexity where m is the length of the adaptive filter

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

performance of transform domain lms adaptive digital filters in this paper we analyze the performance particularly the convergence behavior of the transform domain least mean square lms adaptive digital filter adf using the discrete fourier transform and discrete orthogonal transforms such as discrete cosine and sine transforms we first obtain the optimum wiener solution and the minimum mean squared error mse in the transform domain it is shown that the two minimum mse s in the time and transform domains are identical independently of the transforms used we then study the convergence conditions and the steady state excess mse s of the transform domain lms trlms algorithms both for the cases of having a constant and a time varying convergence factors when a constant convergence factor is used the convergence behaviors of the lms and trlms adf s appear to be almost identical provided that each has an appropriate value of the convergence factor depending on the transform used also based on the concept of a self orthogonalizing algorithm in the transform domain it is shown that the convergence speed of the trlms adf can be improved significantly for the same excess mse as that of the lms adf in addition we compare the computational complexities of the lms and trlms adf s finally we investigate by computer simulation the effects of system parameter values and different transforms on the convergence behavior of the trlms adf

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

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

stable rank 2 reflexive sheaves on with small and applications we investigate the moduli spaces of stable rank two reflexive sheaves on p3 with small chern classes as an application to curves of low degree in p3 we prove the curve has maximal rank and that the corresponding hilbert scheme is irreducible and unirational introduction in the past few years the subject of vector bundles on projective spaces and in particular the case of rank 2 on p3 has received much attention several basic theorems have been proved e g the existence of the moduli space of stable vector bundles m though so far very few of these moduli spaces have been studied in detail the interest of vector bundles partly stems from their connection with curves however the class of curves obtained in this way is rather restricted recently hartshorne srs has focused attention on reflexive sheaves of rank 2 on p3 on the one hand most results proved for vector bundles also turn out to be true for reflexive sheaves on the other hand reflexive sheaves have two significant advantages first they correspond to quite general curves second and most importantly one has the reduction step introduced by hartshorne which is an effective tool in studying vector bundles by relating them to simpler reflexive sheaves in this paper we investigate the moduli spaces of stable rank 2 reflexive sheaves on p3 with c2 3 for c2 2 we prove the moduli spaces are irreducible nonsingular and rational we also classify the related unstable planes for c2 3 in most cases we show that the moduli space is irreducible and unirational as one of the applications to curves of low degree in p3 cf srs 4 1 we prove that the curve has maximal rank and that the corresponding hilbert scheme is irreducible and unirational hence as a corollary we conclude that the variety of moduli 1g of curves of genus g is unirational for g 5 6 7 8 in 1 we give facts on multiple lines and deduce a criterion for unstable planes 2 contains our classification of semistable sheaves with c2 2 3 is about sheaves with c2 3 4 gives the applications to curves acknowledgment i would like to thank my thesis advisor robin hartshorne he has been very generous in his advice and time in addition he attracted to received by the editors september 1 1982 and in revised form march 18 1983 1980 mathematics subject classification primary 14f05 secondary 14d22 i1984 american mathematical society 0002 9947 84 1 00 25 per page

p selectivity intersections and indices the p selective sets selman 1979 are those sets for which there is a polynomial time algorithm that given any two strings determines which is more likely to belong to the set if either of the strings is in the set the algorithm chooses one that is in the set we prove that for each k the k ary boolean connectives under which the p selective sets are closed are exactly those that are either completely degenerate or almost completely degenerate we determine the complexity of the index set of the r e p selective sets 30 complete

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

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

promise problems complete for complexity classes abstract a general framework is given to obtain hardness results for promise problems that derive from self reducible decision problems the principal theorem is that if a set a is d p equivalent to a disjunctive self reducible set in np then the natural promise problem associated with a is as hard to solve as it is to recognize a np hardness of the satisfiability promise problem follows and graph isomorphism hardness of a promise problem that derives from the graph isomorphism problem is proved

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

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

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

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

reducibility classes of p selective sets a set is p selective if there is a polynomial time semi decision algorithm for the set an algorithm that given any two strings decides which is more likely to be in the set this paper establishes a strict hierarchy among the various reductions and equivalences to p selective sets

self orthogonalizing adaptive equalization in the discrete frequency domain a self orthogonalizing discrete adaptive equalizer for synchronous data transmission is presented based on the overlapsave filtering technique self orthogonalization in the discrete frequency domain is adaptively performed by premultiplying by a diagonal matrix the mse gradient estimates before projecting by means of rosen s gradient projection method the diagonal of the matrix is the inverse of the power spectrum of the received sequence taken at equally spaced frequencies and estimates are obtained by using bartlett s procedure of periodograms averaging for spectrum estimation projection is accomplished by means of an off line derived projection matrix confidence of gradient estimates is improved by means of a block correlated estimation technique using available dft s of blocks of data this equalizer is compared to time domain self orthogonalizing algorithms as regards speed of convergence and ease of implementation during the short startup phase convergence is competitive with that of godard s algorithm which is the fastest algorithm known and in the decision directed mode fast convolution performed by blocks results in a considerable reduction of the number of multiplications with respect to the time domain algorithms a computationally simpler unconstrained startup algorithm is also examined obtained by removing the projection while retaining gradient block estimates in the dfd

unconstrained frequency domain adaptive filter an unconstrained frequency domain least mean squares uflms algorithm is presented the algorithm is based on the overlap save technique used in frequency domain filtering the proposed algorithm converges to the wiener solution without the time domain constraint on the impulse response as proposed by ferrara for a large number of taps geq32 the uflms algorithm offers a significant reduction in computation another advantage is its fast convergence for highly correlated input signals the performance of the algorithm is illustrated by a simulation

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

algebraic properties for selector functions the nondeterministic advice complexity of the p selective sets is known to be exactly linear regarding the deterministic advice complexity of the p selective sets i e the amount of karp lipton advice needed for polynomial time machines to recognize them in general the best current upper bound is quadratic k ko j comput system sci 26 1983 pp 209 221 and the best current lower bound is linear l hemaspaandra and l torenvliet theoret comput sci 154 1996 pp 367 377 we prove that every associatively p selective set is commutatively associatively p selective using this we establish an algebraic sufficient condition for the p selective sets to have a linear upper bound which thus would match the existing lower bound on their deterministic advice complexity if all p selective sets are associatively p selective then the deterministic advice complexity of the p selective sets is linear the weakest previously known sufficient condition was p np we also establish related results for algebraic properties of and advice complexity of the nondeterministically selective sets

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

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

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