polynomial time 1 turing reductions from ph to p in this paper we investigate relative complexity between p and other classes of functions our particular interest is to compare p with ph and with pfh by using polynomial time reducibility and to demonstrate that a weaker notion of polynomial time reducibility is sufficiently powerful for reducing ph functions to p functions our main result is stated as follows every function in ph is polynomial time 1 turing reducible to some function in p that is ph pf p 1 some consequences of this result are as follows every function in pfh is polynomial time 1 turing reducible to some function in p if pf p 1 ph then ph collapses to a finite level furthermore if either p pfh or pfh p then ph collapses to a finite level we also give an affirmative answer to an open question posed by valiant 1979 and we show a generalized result about p rankability by hemachandra 1987

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

infinitely often autoreducible sets a set a is autoreducible if one can compute for all x the value a x by querying a only at places y neq x furthermore a is infinitely often autoreducible if for infinitely many x the value a x can be computed by querying a only at places y neq x for all other x the computation outputs a special symbol to signal that the reduction is undefined it is shown that for polynomial time turing and truth table autoreducibility there are a b c in the class exp of all exponential time computable sets such that a is not infinitely often turing autoreducible b is turing autoreducible but not infinitely often truth table autoreducible and c is truth table autoreducible with g n 1 queries but not infinitely often turing autoreducible with g n queries here n is the length of the input g is nondecreasing and there exists a polynomial p such that p n bounds both the computation time and the value of g at input of length n furthermore connections between notions of infinitely often autoreducibility and notions of approximability are investigated the hausdorff dimension of the class of sets which are not infinitely often autoreducible is shown to be 1

approximable sets much structural work on np complete sets has exploited sat s d self reducibility in this paper we exploit the additional fact that sat is a d cylinder to show that np complete sets are p superterse unless p np in fact every set that is np hard under polynomial time no 1 tt reductions is p superterse unless p np in particular no p selective set is np hard under polynomial time no 1 tt reductions unless p np in addition no easily countable set is np hard under turing reductions unless p np self reducibility does not seem to suffice far our main result in a relativized world we construct a d self reducible set in np p that is polynomial time 2 tt reducible to a p selective set

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

classical and bayesian inference in neuroimaging theory this paper reviews hierarchical observation models used in functional neuroimaging in a bayesian light it emphasizes the common ground shared by classical and bayesian methods to show that conventional analyses of neuroimaging data can be usefully extended within an empirical bayesian framework in particular we formulate the procedures used in conventional data analysis in terms of hierarchical linear models and establish a connection between classical inference and parametric empirical bayes peb through covariance component estimation this estimation is based on an expectation maximization or em algorithm the key point is that hierarchical models not only provide for appropriate inference at the highest level but that one can revisit lower levels suitably equipped to make bayesian inferences bayesian inferences eschew many of the difficulties encountered with classical inference and characterize brain responses in a way that is more directly predicated on what one is interested in the motivation for bayesian approaches is reviewed and the theoretical background is presented in a way that relates to conventional methods in particular restricted maximum likelihood reml this paper is a technical and theoretical prelude to subsequent papers that deal with applications of the theory to a range of important issues in neuroimaging these issues include i estimating nonsphericity or variance components in fmri time series that can arise from serial correlations within subject or are induced by multisubject i e hierarchical studies ii spatiotemporal bayesian models for imaging data in which voxels specific effects are constrained by responses in other voxels iii bayesian estimation of nonlinear models of hemodynamic responses and iv principled ways of mixing structural and functional priors in eeg source reconstruction although diverse all these estimation problems are accommodated by the peb framework described in this paper

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

enumerative counting is hard abstract an n variable boolean formula may have anywhere from 0 to 2n satisfying assignments can a polynomial time machine given such a formula reduce this exponential number of possibilities to a small number of possibilities we call such a machine an enumerator and prove that if there is a good polynomial time enumerator for p i e one where for every boolean formula f the small set has at most o f 1 e numbers then p np p p and probabilistic polynomial time equals polynomial time furthermore we show that p polynomial time turing reduces to enumerating p

sparse sets lowness and highness we develop the notions of generalized lowness for sets in ph the union of the polynomial time hierarchy and of generalized highness for arbitrary sets also we develop the notions of extended lowness and extended highness for arbitrary sets these notions extend the decomposition of np into low sets and high sets developed by schoning 15 and studied by ko and schoning 9 we show that either every sparse set in ph is generalized high or no sparse set in ph is generalized high further either every sparse set is extended high or no sparse set is extended high in both situations the former case corresponds to the polynomial time hierarchy having only finitely many levels while the latter case corresponds to the polynomial time hierarchy extending infinitely many levels

an implicit enumeration algorithm to generate tests for combinational logic circuits the d algorithm dalg is shown to be ineffective for the class of combinational logic circuits that is used to implement error correction and translation ecat functions podem path oriented decision making is a new test generation algorithm for combinational logic circuits podem uses an implicit enumeration approach analogous to that used for solving 0 1 integer programming problems it is shown that podem is very efficient for ecat circuits and is significantly more efficient than dalg over the general spectrum of combinational logic circuits a distinctive feature of podem is its simplicity when compared to the d algorithm podem is a complete algorithm in that it will generate a test if one exists heuristics are used to achieve an efficient implicit search of the space of all possible primary input patterns until either a test is found or the space is exhausted

the boolean hierarchy and the polynomial hierarchy a closer connection we show that if the boolean hierarchy collapses to level k then the polynomial hierarchy collapses to bh 3 k where bh 3 k is the k th level of the boolean hierarchy over sigma p 2 this is an improvement over the known results which show that the polynomial hierarchy would collapse to p np np o log n this result is significant in two ways first the theorem says that a deeper collapse of the boolean hierarchy implies a deeper collapse of the polynomial hierarchy also this result points to some previously unexplored connections between the boolean and query hierarchies of delta p 2 and delta p 3 namely begin eqnarray bh k rm co bh k implies bh 3 k rm co bh 3 k jot p rm np k p rm np k 1 implies p rm np rm np k 1 p rm np rm np k 2 end eqnarray

recursion theoretic properties of frequency computation and bounded queries the notion of frequency computation captures the class of all sets a such that for some n the n fold characteristic function of a can be computed with fewer than n errors we consider the recursion theoretic properties of with special emphasis on recursively enumerable sets

reducibilities among equivalence relations induced by recursively enumerable structures in this paper we investigate the dependence of recursively enumerable structures on the equality relation which is fixed to a specific r e equivalence relation we compare r e equivalence relations on the natural numbers with respect to the amount of structures they permit to represent from a given class of structures such as algebras permutations and linear orders in particular we show that for various types of structures represented there are minimal and maximal elements

assessing the significance of pedobarographic signals using random field theory traditional pedobarographic statistical analyses are conducted over discrete regions recent studies have demonstrated that regionalization can corrupt pedobarographic field data through conflation when arbitrary dividing lines inappropriately delineate smooth field processes an alternative is to register images such that homologous structures optimally overlap and then conduct statistical tests at each pixel to generate statistical parametric maps spms the significance of spm processes may be assessed within the framework of random field theory rft rft is ideally suited to pedobarographic image analysis because its fundamental data unit is a lattice sampling of a smooth and continuous spatial field to correct for the vast number of multiple comparisons inherent in such data recent pedobarographic studies have employed a bonferroni correction to retain a constant family wise error rate this approach unfortunately neglects the spatial correlation of neighbouring pixels so provides an overly conservative albeit valid statistical threshold rft generally relaxes the threshold depending on field smoothness and on the geometry of the search area but it also provides a framework for assigning p values to suprathreshold clusters based on their spatial extent the current paper provides an overview of basic rft concepts and uses simulated and experimental data to validate both rft relevant field smoothness estimations and rft predictions regarding the topological characteristics of random pedobarographic fields finally previously published experimental data are re analysed using rft inference procedures to demonstrate how rft yields easily understandable statistical results that may be incorporated into routine clinical and laboratory analyses

modified bounds for covering codes the covering radius of binary codes is studied bounds on k n r the minimum cardinality of any binary code of length n and covering radius r are found modifications of the van wee lower bounds are proved for k n r the minimal number of codewords in any binary code of length n and covering radius r the first of the two van wee bounds is based on studying the hamming spheres of radius 1 centered at the points which have distance r to the code c the points covered by more than one codeword are divided into several classes and better estimates for some of these classes are obtained using a suitable averaging process the lower bound for k n r when r or 2 is improved the second van wee bound studies spheres of radius 2 centered at the points which have distance r 1 or r to the code c these points are divided essentially into two classes the points that are covered by only one codeword of c and the points that are covered by more than one codeword

quantifying the amount of verboseness we study the fine structure of the classification of sets of natural numbers a according to the number of queries which are needed to compute the n fold characteristic function of a a complete characterization is obtained relating the question to finite combinatorics in order to obtain an explicit description we consider several interesting combinatorial problems

atpg for heat dissipation minimization during test application a automatic test pattern generator atpg algorithm is proposed that reduces switching activity between successive test vectors during test application the main objective is to permit safe and inexpensive testing of low power circuits and bare die that might otherwise require expensive heat removal equipment for testing at high speeds three new cost functions namely transition controllability observability and test generation costs have been defined it has been shown for a fanout free circuit under test that the transition test generation cost for a fault is the minimum number of transitions required to test a given stuck at fault the proposed algorithm has been implemented and the generated tests are compared with those generated by a standard podem implementation for the larger iscas85 benchmark circuits the results clearly demonstrate that the tests generated using the proposed atpg can decrease the average number of weighted transitions between successive test vectors by a factor of 2 to 23

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

a structural property of regular frequency computations the notion of an m n computation was already introduced in 1960 by rose and further investigated by trakhtenbrot in 1963 it has been extended to finite automata by kinber in 1976 and he has shown an analogue of trakhtenbrot s result the class of languages m n recognizable by deterministic finite automata is equal to the class of regular languages if and only if 2m n furthermore for a unary alphabet the class of m n recognizable languages coincides with the class of regular languages for all m and n in this paper we will present the first structural property of m n recognizable languages which is valid for all 1 m n and for all alphabets kinber s result for unary alphabets becomes a corollary this property is also used to show that certain non unary languages are not m n regular and that the class of all m n recognizable languages is not closed under the reversal operation however this class forms a boolean algebra

two remarks on the power of counting the relationship between the polynomial hierarchy and valiant s class p is at present unknown we show that some low portions of the polynomial hierarchy namely deterministic polynomial algorithms using an np oracle at most a logarithmic number of times can be simulated by one p computation we also show that the class of problems solvable by polynomial time nondeterministic turing machines which accept whenever there is an odd number of accepting computations is idempotent that is closed under usage of oracles from the same class

on the reducibility of sets inside np to sets with low information content this paper studies for various natural problems in np whether they can be reduced to sets with low information content such as branches p selective sets and membership comparable sets the problems that are studied include the satisfiability problem the graph automorphism problem the undirected graph accessibility problem the determinant function and all logspace self reducible languages some of these are complete for complexity classes within np but for others an exact complexity theoretic characterization is not known reducibility of these problems is studied in a general framework introduced in this paper prover verifier protocols with low complexity provers it is shown that all these natural problems indeed have such protocols this fact is used to show for certain reduction types that these problems are not reducible to sets with low information content unless their complexity is much less than what it is currently believed to be the general framework is also used to obtain a new characterization of the complexity class l l is the class of all logspace self reducible sets in ll sel

on sets truth table reducible to sparse sets we study sets that are truth table reducible to sparse sets in polynomial time the principal results are as follows 1 for every integer k 0 there is a set l and a sparse set s such that l

polynomial time membership comparable sets this paper studies a notion called polynomial time membership comparable sets for a function g a set a is polynomial time g membership comparable if there is a polynomial time computable function f such that for any x 1 cdots x m with m geq g max x 1 cdots x m outputs b in 0 1 m such that a x 1 cdots a x m neq b the following is a list of major results proven in the paper 1 polynomial time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments 2 polynomial time membership comparable sets have polynomial size circuits 3 for any function f and for any constant c 0 if a set is leq p f n tt reducible to a p selective set then the set is polynomial time 1 c log f n membership comparable 4 for any cal c chosen from rm pspace up fewp np c p pp mod 2 p mod 3 cdots if cal c subseteq rm p mc c log n for some c 1 then cal c rm p as a corollary of the last two results it is shown that if there is some constant c 1 such that all of cal c are polynomial time n c truth table reducible to some p selective sets then cal c rm p which resolves a question that has been left open for a long time

further results on the covering radius of codes a number of upper and lower bounds are obtained for k n r the minimal number of codewords in any binary code of length n and covering radius r several new constructions are used to derive the upper bounds including an amalgamated direct sum construction for nonlinear codes this construction works best when applied to normal codes and we give some new and stronger conditions which imply that a linear code is normal an upper bound is given for the density of a covering code over any alphabet and it is shown that k n 2 r 1 leq k n r holds for sufficiently large n

improved sphere bounds on the covering radius of codes the sphere bound is a trivial lower bound on k n r the minimal cardinality of any binary code of length n and with covering radius r by simple arguments it is considerably improved to k n 1 or 2 sup n n for n even a table of lower and upper bounds on k n r for n

regular frequency computations an m n computation of a function f is given by a deterministic turing machine which on n pairwise different inputs produces n output values where at least m of the n values are in accordance with f in such a case we say that the turing machine computes f with frequency m n the most prominent result for frequency computations is due to trakhtenbrot the class of m n computable functions equals the class of computable functions if and only if 2m n via characteristic functions the definition of m n computability carries over to sets here trakhtenbrot s result reads as the class of m n computable sets equals the class of recursive sets if and only if 2m n the notion of frequency computation can be extended to other models of computation for resource bounded computations the behavior is completely different for e g whenever n m n m it is known that under any reasonable resource bound there are sets m n computable but not m n computable however scaling down to finite automata the analogue of trakhtenbrot s result holds again we show here that the class of languages m n recognizable by deterministic finite automata equals the class of regular languages if and only if 2m n this was originally stated by kinber but his proof has a flaw as pointed out by tantau conversely for 2m n the class of languages m n recognizable by deterministic finite automata is uncountable for a two letter alphabet when restricted to a one letter alphabet then every m n recognizable language is regular this was also shown by kinber we give a new and more direct proof for this result

polynomial terse sets abstract let a be a set and k n be such that we wish to know the answers to x1 a x2 a xk a for various k tuples x1 x2 xk if this problem requires k queries to a in order to be solved in polynomial time then a is called polynomial terse or pterse we show the existence of both arbitrarily complex pterse and non pterse sets and that p np iff every np complete set is pterse we also show connections with p immunity p selective p generic sets and the boolean hierarchy in our framework unique satisfiability and a variation of it called ksat is in some sense close to satisfiability

some connections between bounded query classes and non uniform complexity let a x be the characteristic function of a consider the function cka x1 xk a x1 a xk we show that if cka can be computed in polynomial time with fewer than k queries to some set x then a p poly a generalization of this result has applications to bounded query classes circuits and enumerability in particular we obtain the following 1 assuming 3p 3p there are functions computable using f n 1 queries to sat that are not computable using f n queries to sat for f n o log n 2 if cka restricted to length n inputs can be computed by an unbounded fanin oracle circuit of size s n and depth d n with k 1 queries to some set x then a can be computed with an unbounded fanin non oracle circuit of size no k s n and depth d n o 1 3 assuming that ph 4p 4p and e 1 sat is not 2ne enumerable

robust smoothness estimation in statistical parametric maps using standardized residuals from the general linear model the assessment of significant activations in functional imaging using voxel based methods often relies on results derived from the theory of gaussian random fields these results solve the multiple comparison problem and assume that the spatial correlation or smoothness of the data is known or can be estimated end results i e p values associated with local maxima clusters or sets of clusters critically depend on this assessment which should be as exact and as reliable as possible in some earlier implementations of statistical parametric mapping spm spm94 spm95 the smoothness was assessed on gaussianized t fields gt f that are not generally free of physiological signal this technique has two limitations first the estimation is not stable the variance of the estimator being far from negligible and second physiological signal in the gt f will bias the estimation in this paper we describe an estimation method that overcomes these drawbacks the new approach involves estimating the smoothness of standardized residual fields which approximates the smoothness of the component fields of the associated t field knowing the smoothness of these component fields is important because it allows one to compute corrected p values for statistical fields other than the t field or the gt f e g the f map and eschews bias due to deviation from the null hypothesis we validate the method on simulated data and demonstrate it using data from a functional mri study

the global power of additional queries to random oracles abstract it is shown that for every k 0 and every fixed algorithmically random language b there is a language that is polynomial time truth table reducible in k 1 queries to b but not truth table reducible in k queries in any amount of time to any algorithmically random language c in particular this yields the separation pk tt rand p k 1 tt rand where rand is the set of all algorithmically random languages

frequency computation and bounded queries there have been several papers over the last ten years that consider the number of queries needed to compute a function as a measure of its complexity the following function has been studied extensively in that light faa x1 xa a x1 a xa we are interested in the complexity in terms of the number of queries of approximating faa let b a and let f be any function such that faa x1 xa and f x1 xa agree on at least b bits for a general set a we have matching upper and lower bounds on f that depend on coding theory these are applied to get exact bounds for the case where a is semirecursive a is superterse and assuming p np a sat we obtain exact bounds when a is the halting problem using different methods

test set generation with a large number of unspecified bits using static and dynamic techniques this work presents two new methods for the generation of test sets with a small number of specified bits such type of test sets have been proven beneficial to a large number of test related applications such as deterministic bist low power testing and test set enrichment the first technique is static since it considers an initial test set which attempts to relax via test replacement with tests of similar coverage but with fewer specified bits the second technique is dynamic it generates a test set from a zero base using a hierarchical fault compatibility algorithm both methods are applicable to any enumerative fault method linear to the circuit size the experiments performed using the stuck at fault model demonstrate the superiority of the proposed methods over comparable existing techniques in reducing the total number of specified bits per generated test set the applicability of the generated relaxed test sets is demonstrated for one out of the many possible applications that of deterministic test set embedding a general framework that integrates the proposed relaxation methods in two popular lfsr based test set embedding schemes full and partial reseeding along with a systematic exploration of related parameters is proposed the obtained results show significant reductions in seed storage requirements

optimal series parallel trade offs for reducing a function to its own graph consider the problem of computing a function given only an oracle for its graph for this problem we present optimal trade offs between serial and parallel queries in particular we give a fuction for which parallel access to its own graph is exponentially more expensive than sequential access

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

terse superterse and verbose sets abstract let a be a subset of the natural numbers and let f a n x 1 x n a x 1 a x n where a is the characteristic function of a an oracle turing machine with oracle a could certainly compute f a n with n queries to a there are some sets a e g the halting set for which f a n can be computed with substantially fewer than n queries one key reason for this is that the questions asked to the oracle can depend on previous answers i e the questions are adaptive we examine when it is possible to save queries a set a is terse if the computation of f a n from a requires n queries a set a is superterse if the computation of f a n from any set requires n queries a set a is verbose if f a 2 n 1 can be computed with n queries to a the range of possible query savings is limited by the following theorem f a n cannot be computed with only log n queries to a set x unless a is recursive in addition we produce the following 1 a verbose set in each truth table degree and a superterse set in each nonzero truth table degree and 2 an r e verbose set in each r e truth table degree and an r e terse set in each nonzero r e turing degree

bi immunity results for cheatable sets abstract an oracle a is k cheatable if there is a polynomial time algorithm to determine the answers to 2k parallel queries to a from the answers to only k queries to some other oracle b it is known that 1 cheatable sets cannot be bi immune for p in contrast we construct 2 cheatable sets that are bi immune for arbitrary time complexity classes in addition for each k we construct a set that is k 1 cheatable but not k cheatable we show that this separation does not hold with bi immunity we show that if a recursive set a is bi immune for p then there exists a nontrivial 1 cheatable set that is polynomial time m reducible to a consequently if np contains a set that is bi immune for p then np contains a set that is not polynomial time turing equivalent to a self reducible set

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

new insights into the plantar pressure correlates of walking speed using pedobarographic statistical parametric mapping pspm this study investigates the relation between walking speed and the distribution of peak plantar pressure and compares a traditional ten region subsampling 10rs technique with a new technique pedobarographic statistical parametric mapping pspm adapted from cerebral fmri methodology pspm is a digital image processing technique that registers foot pressure images such that homologous structures optimally overlap thereby enabling statistical tests to be conducted at the pixel level following previous experimental protocols we collected pedobarographic records from 10 subjects walking at three different speeds slow normal and fast walking speed was recorded and correlated with the peak pressures extracted from the 10 regions and subsequently with the peak pixel data extracted after pspm preprocessing both methods revealed significant positive correlation between peak plantar pressure and walking speed over the rearfoot and distal forefoot after bonferroni correction for multiple comparisons the 10rs analysis found positive correlation in the midfoot and medial proximal forefoot but the pixel data exhibited significant negative correlation throughout these regions p 5 10 5 comparing the statistical maps from the two approaches shows that subsampling may conflate pressure differences evident in pixel level data obscuring or even reversing statistical trends the negative correlation observed in the midfoot implies reduced longitudinal arch collapse with higher walking speeds we infer that this results from pre or early stance phase muscle activity and speculate that preferred walking speed reflects in part a balance between the energy required to tighten the longitudinal arch and the apparent propulsive benefits of the stiffened arch

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

mixed effects and fmri studies abstract this note concerns mixed effect mfx analyses in multisession functional magnetic resonance imaging fmri studies it clarifies the relationship between mixed effect analyses and the two stage summary statistics procedure holmes a p friston k j 1998 generalisability random effects and population inference neuroimage 7 s754 that has been adopted widely for analyses of fmri data at the group level we describe a simple procedure based on restricted maximum likelihood reml estimates of covariance components that enables full mixed effects analyses in the context of statistical parametric mapping using this procedure we compare the results of a full mixed effects analysis with those obtained from the simpler two stage procedure and comment on the situations when the two approaches may give different results

on the structure of bounded queries to arbitrary np sets kadin siam j comput 17 1988 pp 1263 1282 showed that if the polynomial hierarchy ph has infinitely many levels then for all k text p operatorname sat k subset text p operatorname sat k 1 this paper extends kadin s technique and shows that a proper query hierarchy is not an exclusive property of np complete sets in fact for any a in text np widehat text low 3 if ph has infinitely many levels then text p a k subset text p a k 1 moreover for the case of parallel queries text p a k 1 is not even contained in text p operatorname sat k these same techniques can be used to explore some other questions about query hierarchies for example if there exists any a such that text p a 2 text p operatorname sat 1 then ph collapses to delta 3 p

bounded queries to sat and the boolean hierarchy we study the complexity of decision problems that can be solved by a polynomial time turing machine that makes a bounded number of queries to an np oracle depending on whether we allow some queries to depend on the results of other queries we obtain two probably different hierarchies we present several results relating the bounded np query hierarchies to each other and to the boolean hierarchy we also consider the similarly defined hierarchies of functions that can be computed by a polynomial time turing machine that makes a bounded number of queries to an np oracle we present relations among these two hierarchies and the boolean hierarchy in particular we show for all k that there are functions computable with 2k parallel queries to an np set that are not computable in polynomial time with k serial queries to any oracle unless p np as a corollary k 1 parallel queries to an np set allow us to compute more functions than are computable with only k parallel queries to an np set unless p np the same is true of serial queries similar results hold for all tt self reducible sets using a mind change technique we show that 2k 1 parallel queries to an np set allow us to accept in polynomial time exactly the same sets as can be accepted in polynomial time with k serial queries to an np set in fact the same is true for any class in place of np that is closed under polynomial time positive bounded truth table reductions this contrasts with the expected result for function computations with an np oracle beigel 1988 in addition we show that the boolean hierarchy and the bounded query hierarchies of languages either stand or collapse together finally we show that if the boolean hierarchy collapses to any level but the zeroth deterministic polynomial time then for all k there are functions computable in polynomial time with k parallel queries to an np set that are not computable in polynomial time with k 1 serial queries to any set np complete sets are p superterse

time bounded frequency computations 1 we obtain two new results concerning the inclusion problem of polynomial time frequency classes with equal numbers of errors 1 m m d p m 1 m d 1 pform 2d 2 m m d p m 1 m d 1 pform c d wherec d is large enough this disproves a conjecture of kinber 2 we give a transparent proof of a generalization of kinber s result that there exist arbitrarily complex problems admitting a polynomial time frequency computation several corollaries provide more insight into the structure of the hierarchy of polynomial time frequency classes 3 the relationships between polynomial time frequency classes and selectivity classes are studied

binary search and recursive graph problems abstract a graph g v e is recursive if every node of g has a finite number of neighbors and both v and e are recursive i e decidable we examine the complexity of identifying the number of connected components of an infinite recursive graph and other related problems both when an upper bound to that value is given a priori or not the problems that we deal with are unsolvable but are recursive in some level of the arithmetic hierarchy our measure of the complexity of these problems is precise in two ways the turing degree of the oracle and the number of queries to that oracle although they are in several different levels of the arithmetic hierarchy all problems addressed have the same upper and lower bounds for the number of queries as the binary search problem both in the bounded and in the unbounded case