threshold circuits of small majority depth we investigate the complexity of computations with constant depth threshold circuits such circuits are composed of gates that determine if the sum of their inputs is greater than a certain threshold when restricted to polynomial size these circuits compute exactly the functions in the class tc sp0 these circuits are usually studied by measuring their efficiency in terms of their total depth using this point of view the best division and iterated multiplication circuits have depth three and four respectively in this thesis we propose a different approach since threshold gates are much more powerful than and or gates we allow the explicit use of and or gates and consider the main measure of complexity to be the majority depth of the circuit i e the maximum number of threshold gates on any path in the circuit using this approach we obtain division and iterated multiplication circuits of total depth four and five but of majority depth two and three the technique used is called chinese remaindering we present this technique as a general tool for computing functions with integer values and use it to obtain depth four threshold circuits of majority depth two for other arithmetic problems such as the logarithm and power series approximation we also consider the iterated multiplication problem for integers modulo q and for finite fields the notion of majority depth naturally leads to a hierarchy of subclasses of tc sp0 we investigate this hierarchy and show that it is closely related to the usual depth hierarchy

constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

p uniform circuit complexity much complexity theoretic work on parallelism has focused on the class nc which is defined in terms of logspace uniform circuits yet p uniform circuit complexity is in some ways a more natural setting for studying feasible parallelism in this paper p uniform nc punc is characterized in terms of space bounded auxpdas and alternating turing machines with bounded access to the input the notions of general purpose and special purpose computation are considered and a general purpose parallel computer for punc is presented it is also shown that nc punc if all tally languages in p are in nc this implies that the nc punc question and the nc p question are both instances of the aspace s n aspace time s n s n o 1 question as a corollary it follows that nc punc implies pspace dtime 2 no 1

bounded depth polynomial size circuits for symmetric functions abstract let f f 1 f 2 be a family of symmetric boolean functions where f n has n boolean variables for each n 1 let f n be the minimum number of variables of f n that each have to be set to constant values so that the resulting function is a constant function we show that the growth rate of f n completely determines whether or not the family f is good that is can be realized by a family of constant depth polynomial size circuits with unbounded fan in furthermore if f n log n k for some k then the family f is good however if f n n for some 0 then the family is not good

finite monoids and the fine structure of nc 1 recently a new connection was discovered between the parallel complexity class nc 1 and the theory of finite automata in the work of barrington on bounded width branching programs there nonuniform nc 1 was characterized as those languages recognized by a certain nonuniform version of a dfa here we extend this characterization to show that the internal structures of nc 1 and the class of automata are closely related in particular using therien s classification of finite monoids we give new characterizations of the classes ac 0 depth k ac 0 and acc the last being the ac 0 closure of the mod q functions for all constant q we settle some of the open questions in 3 give a new proof that the dot depth hierarchy of algebraic automata theory is infinite 8 and offer a new framework for understanding the internal structure of nc 1

a logic for constant depth circuits consider a family of boolean circuits c c2 c constructed by some uniform effective procedure operating on input n such a procedure provides a concise representation of a family of parallel algorithms for computing boolean values a formula of first order logic may also be viewed as a concise representation of a family of parallel algorithms for evaluating boolean functions the parallelism is implicit in the quantification a formula gx q x is true if and only if each of the formulas q a is true and all these formulas can be checked simultaneously and universes of different sizes give rise to boolean functions with different numbers of inputs the boolean values of the formula s predicates on various combinations of elements of the universe this note presents an extended first order logic designed to be exactly equivalent in expressiveness to polynomialsize constant depth unbounded fan in circuits constructed by turing machines of bounded computational complexity 1984 academic press inc

the binary network flow problem is logspace complete for p abstract it is shown that the problem of whether the maximum flow in a given network exceeds a given natural number is logspace many one complete for p if the edge capacities are presented in binary even if the problem is restricted to acyclic graphs this improves a result by goldschlager et al 1982 that this problem is logspace turing complete for p

methods for proving completeness via logical reductions abstract in this paper we illustrate five different techniques for showing that problems are complete for some complexity class via projection translations these are extremely weak logical reductions these techniques would not be available to us if we were to take the traditional view of complexity theory where decision problems are equated with sets of strings instead of sets of finite structures

an arithmetic model of computation equivalent to threshold circuits we define a new structured and general model of computation circuits using arbitrary fan in arithmetic gates over the characteristic two finite fields f2n these circuits have only one input and one output we show how they correspond naturally to boolean computations with n inputs and n outputs we show that if circuit sizes are polynomially related then the arithmetic circuit depth and the threshold circuit depth to compute a given function differ by at most a constant factor we use threshold functions with arbitrary weights however we show that when compared to the usual threshold model the depth measure of this generalised model differs only by at most a constant factor at polynomial size the fan in of our arithmetic model is also unbounded in the most generous sense circuit size is measured as the number of and gates there is no bound on the number of wires we show that these results are provable for any reasonable correspondence between strings of n bits and elements of f2n and we find two such distinct characterizations thus we show that arbitrary fan in arithmetic computations over f2n constitute a precise abstraction of boolean threshold computations with the pleasant property that various algebraic laws have been recovered

optimal bounds for decision problems on the crcw pram optimal o log n log log n lower bounds on the time for crcw prams with polynomially bounded numbers of processors or memory cells to compute parity and a number of related problems are proven a strict time hierarchy of explicit boolean functions of n bits on such machines that holds up to o log n log log n time is also exhibited that is for every time bound t within this range a function is exhibited that can be easily computed using polynomial resources in time t but requires more than polynomial resources to be computed in time t 1 finally it is shown that almost all boolean functions of n bits require log n log log n o 1 time when the number of processors is at most polynomial in n the bounds do not place restrictions on the uniformity of the algorithms nor on the instruction sets of the machines

regular languages defined with generalized quantifiers we study an extension of first order logic obtained by adjoining quantifiers that count with respect to an integer modulus it is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups we obtain an analogous result for regular languages and establish some connections with complexity theory for fixed depth families of circuits

using interior point methods for fast parallel algorithms for bipartite matching and related problems in this paper interior point methods for linear programming developed in the context of sequential computation are used to obtain a parallel algorithm for the bipartite matching problem this algorithm finds a maximum cardinality matching in a bipartite graph with n nodes and m edges in o sqrt m log 3 n time on a crcw pram the results here extend to the weighted bipartite matching problem and to the zero one minimum cost flow problem yielding o sqrt m log 2 n log nc algorithms where c 1 is an upper bound on the absolute value of the integral weights or costs in the two problems respectively the results here improve previous bounds on these problems and introduce interior point methods to the context of parallel algorithm design

root finding with threshold circuits we show that for any constant d complex roots of degree d univariate rational or gaussian rational polynomials given by a list of coefficients in binary can be computed to a given accuracy by a uniform tc 0 algorithm a uniform family of constant depth polynomial size threshold circuits the basic idea is to compute the inverse function of the polynomial by a power series we also discuss an application to the theory v tc 0 of bounded arithmetic

computing functions with parallel queries to np the class 2p of languages polynomial time truth table reducible to sets in np has a wide range of different characterizations we consider several functional versions of 2p based on these characterizations we show that in this way the three function classes fllognp fplognp and fp np are obtained in contrast to the language case the function classes seem to all be different we give evidence in support of this fact by showing that fllognp coincides with any of the other classes then l p and that the equality of the classes fplognp and fp np would imply that the number of nondeterministic bits needed for the computation of any problem in np can be reduced by a polylogarithmic factor and that the problem can be computed deterministically with a subexponential time bound of order 2no 1 log log n

the complexity of the parity function in unbounded fan in unbounded depth circuits abstract almost everything is known on the complexity of the parity function in fan in 2 circuits over various bases also the minimal depth of polynomial size unbounded fan in dl circuits for the parity function has been studied here the complexity without any depth restriction is considered for the basis dl almost optimal bounds and for the basis of nor gates and the basis of all threshold functions optimal bounds on the number of gates are obtained for the basis dl the minimal number of wires is determined for threshold circuits an exponential gap between synchronous and asynchronous circuits is proved the results not only answer open questions in complexity theory but also have implications for the real life circuit design

general purpose computation with neural networks a survey of complexity theoretic results we survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics the criteria of classification include the architecture of the network feedforward versus recurrent time model discrete versus continuous state type binary versus analog weight constraints symmetric versus asymmetric network size finite nets versus infinite families and computation type deterministic versus probabilistic among others the underlying results concerning the computational power and complexity issues of perceptron radial basis function winner take all and spiking neural networks are briefly surveyed with pointers to the relevant literature in our survey we focus mainly on the digital computation whose inputs and outputs are binary in nature although their values are quite often encoded as analog neuron states we omit the important learning issues

simulating threshold circuits by majority circuits we prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial size depth 2 majority circuit in general we show that a polynomial size depth d threshold circuit can be simulated uniformly by a polynomial size majority circuit of depth d 1 goldmann hastad and razborov showed in comput complexity 2 1992 pp 277 300 that a nonuniform simulation exists our construction answers two open questions posed by them we give an explicit construction whereas they use a randomized existence argument and we show that such a simulation is possible even if the depth d grows with the number of variables n their simulation gives polynomial size circuits only when d is constant

some two vertex resistances of the three towers hanoi graph formed by a fractal graph the resistance distance between two vertices of a simple connected graph g is equal to the resistance between two equivalent points on an electrical network constructed to correspond to g with each edge being replaced by a unit resistor in this study we apply the principle of substitution to three towers hanoi graph to procure an equivalent network that enables us to only use the series and parallel principles and the delta wye transformation to acquire the some two vertex resistance distances of three towers hanoi graph

the complexity of computing symmetric functions using threshold circuits abstract this paper considers size depth tradeoffs for threshold circuits computing symmetric functions the size measure used is the number of connections or edges in the threshold circuits as opposed to the number of gates in the circuits the main result is that for all d 2 and n 8 2 d there is a threshold circuit to compute any n input symmetric function which has size o 1 log n 2 d 1 n 1 1 2 d 1 and depth bounded by 6 d 8 as a consequence there is a threshold circuit for any n input symmetric function which has size o n and depth bounded by o log log n a somewhat simpler construction that contains many features of the general solution shows that for all d 1 and n 2 2 d 1 there is a threshold circuit for the n input parity function which has size bounded by 27 2 2 n 1 1 2 d 1 and depth bounded by 2 d

the expressiveness of a family of finite set languages abstract in this paper we characterize exactly the complexity of a set based database language called srl which presents a unified framework for queries and updates by imposing simple syntactic restrictions on it we are able to express exactly the classes p and logspace we also discuss the role of ordering in database query languages and show that the hom operator of machiavelli language in ohori et al 1989 does not capture all the order independent properties

bounds for the computational power and learning complexity of analog neural nets it is shown that high order feedforward neural nets of constant depth with piecewise polynomial activation functions and arbitrary real weights can be simulated for boolean inputs and outputs by neural nets of a somewhat larger size and depth with heaviside gates and weights from 1 0 1 this provides the first known upper bound for the computational power of the former type of neural nets it is also shown that in the case of first order nets with piecewise linear activation functions one can replace arbitrary real weights by rational numbers with polynomially many bits without changing the boolean function that is computed by the neural net in order to prove these results we introduce two new methods for reducing nonlinear problems about weights in multilayer neural nets to linear problems for a transformed set of parameters these transformed parameters can be interpreted as weights in a somewhat larger neural net as another application of our new proof technique we show that neural nets with piecewise polynomial activation functions and a constant number of analog inputs are probably approximately correct pac learnable in valiant s model for pac learning comm assoc comput mach 27 1984 pp 1134 1142

the worst parallel hanoi graphs abstract the towers of hanoi is a problem that has been extensively studied and frequently generalized in particular it has been generalized to be played on arbitrary directed graphs and using parallel moves of two types we ask what is the largest number of parallel moves in either of the two models that is required to move n disks from the starting node to the destination node not all directed graphs allow solving this problem we will call those graphs that do hanoi graphs in previous work we settled the question of what are the worst sequential hanoi graphs that is those graphs that require the largest number of sequential moves we also demonstrated that the characterization of sequential hanoi graphs carries over the parallel hanoi graphs here we determine the worst hanoi graphs provided parallel moves are allowed it turns out that for one of the two models of parallel moves the worst graphs are quite different from the worst sequential graphs while in the other model of parallelism there is little difference with the sequential situation

depth size tradeoffs for neural computation the tradeoffs between the depth i e the time for parallel computation and the size i e the number of threshold gates in neural networks are studied the authors focus the study on the neural computations of symmetric boolean functions and some arithmetic functions it is shown that a significant reduction in the size is possible for symmetric functions and some arithmetic functions at the expense of a small constant increase in depth in the process several neural networks which have the minimum size among all the known constructions have been developed results on implementing symmetric functions can be used to improve results about arbitrary boolean functions in particular it is shown that any boolean function can be computed in a depth 3 neural network with o 2 sup n sup 2 threshold gates it is also proven that at least omega 2 sup n sup 3 threshold gates are required

approximating threshold circuits by rational functions motivated by the problem of understanding the limitations of threshold networks for representing boolean functions we consider size depth trade offs for threshold circuits that compute the parity function using a fundamental result in the theory of rational approximation we show how to approximate small threshold circuits by rational functions of low degree we apply this result to establish an almost optimal lower bound of n2 ln2n on the number of edges of any depth 2 threshold circuit with polynomially bounded weights that computes the parity function we also prove that any depth 3 threshold circuit with polynomially bounded weights requires n1 2 ln5 3n edges to compute parity on the other hand we give a construction of a depth d threshold circuit that computes parity with n1 1 d edges where 1 5 2 is the golden ratio we conjecture that there are no linear size bounded depth threshold circuits for computing parity

a lower bound for integer multiplication with read once branching programs we prove that read once branching programs computing integer multiplication require size 2 omega sqrt n this is the first nontrivial lower bound for multiplication on branching programs that are not oblivious by the appropriate problem reductions we obtain the same lower bound for other arithmetic functions

the complexity of searching implicit graphs abstract the standard complexity classes of complexity theory do not allow for direct classification of most of the problems solved by heuristic search algorithms the reason is that almost always these are defined in terms of implicit graphs of state or problem reduction spaces while the standard definitions of all complexity classes are specifically tailored to explicit inputs to allow for more precise comparisons with standard complexity classes we introduce here a model for the analysis of algorithms on graphs given by vertex expansion procedures it is based on previously studied concepts of succinct representation techniques and allows us to prove pspace completeness or exptime completeness of specific natural problems on implicit graphs such as those solved by a ao and other best first search strategies

extensions to barrington s m program model abstract barrington s polynomal length program over a monoid is a model of computation which has been studied intensively in connection with the structure of the complexity class nc 1 barrington 1986 barrington and therien 1987 1988 mckenzie and therien 1989 peladeau 1989 here two extensions of the model are considered first with the use of nonassociative structures hence groupoids instead of associative monoids polynomial length program characterizations of complexity classes tc 0 nl and logcfl as well as new characterizations of nc 1 are given new word problems complete for logcfl for nl and for nc 1 under dlogtime reductions are obtained as corollaries second using monoids but permitting the use of a different monoid to handle each input length new complexity classes are defined combinatorial arguments are then developed to resolve the relationships between various such classes defined in terms of polynomial length programs over growing abelian monoid sequences then the orders of growing abelian group and monoid sequences required to accept specific languages defined in terms of the presence of a given substring are investigated finally the two extensions are combined to obtain characterizations of l and nl in terms of polynomial length programs defined over polynomially growing groupoid sequences it is further argued that such programs are generally no more powerful than logcfl

products of longitudinal pseudodifferential operators on flag varieties associated to each set s of simple roots of sl n c is an equivariant fibration x xs of the complete flag variety x of cn to each such fibration we associate an algebra js of operators on l2 x or more generally on l2 sections of vector bundles over x this ideal contains in particular the longitudinal pseudodifferential operators of negative order tangent to the fibres together they form a lattice of operator ideals whose common intersection is the compact operators thus for instance the product of negative order pseudodifferential operators along the fibres of two such fibrations x xs and x xt is a compact operator if s t is the full set of simple roots the construction of the ideals uses noncommutative harmonic analysis and hinges upon a representation theoretic property of subgroups of su n which may be described as essential orthogonality of subrepresentations

correctness of linear logic proof structures is nl complete we provide new correctness criteria for all fragments multiplicative exponential additive of linear logic we use these criteria for proving that deciding the correctness of a linear logic proof structure is nl complete

a note on some languages in uniform acc 0 acc is the class of languages recognized by circuit families with polynomial size constant depth and unbounded fan in where gates may calculate the and or or mod c function for constant c robust uniformity definitions for acc and related classes were given by barrington immerman and straubing 3 here we show that uniform acc contains all semi linear or rational sets of integer vectors using binary notation sharpening a result of ibarra jiang ravikumar and chang 10

polynomial size ω branching programs and their computational power abstract in the following new types of branching programs so called branching programs b 2 are introduced the complexity classes related to polynomial size branching programs will be completely classified in addition to identifying a new class p bp l poly between l poly and p poly new characterizations of such fundamental space complexity classes like nl poly co nl poly and p poly are obtained using these characterizations we relate the complexity of the mentioned classes to those of some extremely restricted problems resembling the graph accessibility problem

on a question of leiss regarding the hanoi tower problem the tower of hanoi problem is generalized in such a way that the pegs are located at the vertices of a directed graph g and moves of disks may be made only along edges of g leiss obtained a complete characterization of graphs in which arbitrarily many disks can be moved from the source vertex s to the destination vertex d here we consider graphs which do not satisfy this characterization hence there is a bound on the number of disks which can be handled denote by gn the maximal such number as g varies over all such graphs with n vertices and s d vary over the vertices answering a question of leiss finite hanoi problems how many discs can be handled congr numer 44 1984 221 229 we prove that gn grows sub exponentially fast moreover there exists a constant c such that gn cn1 2 log2n for each n on the other hand for each e 0 there exists a constant ce 0 such that gn cen 1 2 e log2n for each n

the complexity of planarity testing we clarify the computational complexity of planarity testing by showing that planarity testing is hard for l and lies in sl this nearly settles the question since it is widely conjectured that l sl the upper bound of sl matches the lower bound of l in the context of non uniform circuit complexity since l poly is equal to sl poly similarly we show that a planar embedding when one exists can be found in flsl previously these problems were known to reside in the complexity class ac1 via the o log n time crcw pram algorithm of ramachandran and reif although planarity checking for degree three graphs had been shown to be in sl chicago j theoret comput sci 1995 j acm 31 2 1984 401

one way functions and circuit complexity abstract a finite function f is a mapping of 0 1 n into 0 1 m where is a symbol to be thought of as undefined a family of finite functions is said to be one way in a circuit complexity sense if it can be computed with polynomial size circuits but every family of inverses of these functions cannot in this paper we show that provided functions that are not one to one are allowed one way functions exist if and only if the satisfiability problem sat does not have polynomial size circuits a family of functions fi x can be checked if some family of polynomial size circuits with inputs x and y can determine if fi x y a family of functions fi x can be evaluated if some family of polynomial size circuits with input x can compute fi x can all families of total functions that can be checked also be evaluated we show that this is true if and only if the nonuniform versions of the complexity classes p and up co up are equal a family of functions fi is one way for constant depth circuits if fi can be computed with unbounded famin circuits of polynomial size and constant depth but every family of inverses fi 1 cannot we give two provably one way functions in fact permutaions for constant depth circuits the second example has the stronger property that no bit of its inverse can be computed in polynomial size and constant depth

new lower bounds for parallel computation lower bounds are proven on the parallel time complexity of several basic functions on the most powerful concurrent read concurrent write pram with unlimited shared memory and unlimited power of individual processors denoted by priority it is proved that with a number of processors polynomial in n log n time is needed for addition multiplication or bitwise or of n numbers when each number has n bits hence even the bit complexity i e the time complexity as a function of the total number of bits in the input is logarithmic in this case this improves a beautiful result of meyer auf der heide and wigderson 22 they proved a log n lower bound using ramsey type techniques using ramsey theory it is possible to get an upper bound on the number of bits in the inputs used however for the case of polynomially many processors this upper bound is more than a polynomial in n an log n lower bound is given for priority with no 1 processors on a function with inputs from 0 1 namely for the function x1 xn nl 1 xlai where a is fixed and xi e 0 1 finally by a new efficient simulation of priority by unbounded fan in circuits that with less than exponential number of processors it is proven a priority cannot compute parity in constant time and with no 1 processors log n time is needed the simulation technique is of independent interest since it can serve as a general tool to translate circuit lower bounds into pram lower bounds further the lower bounds in 1 and 2 remain valid for probabilistic or nondeterministic concurrent read concurrent write prams

non uniform automata over groups abstract a new model non uniform deterministic finite automata nudfa s over general finite monoids has recently been developed as a strong link between the theory of finite automata and low level parallel complexity achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non uniform nc1 nudfa characterizations of several important subclasses of nc1 and a new proof of the old result that the dot dephth hierarchy is infinite using m sipser s 1983 in proceedings 15th acm symposium on the theory of computing association for computing machinery new york pp 61 69 work on constant depth circuits here we extend this theory to nudfa s over solvable groups nudfa s over non solvable groups have the maximum possible computing power we characterize the power of nudfa s over nilpotent groups and prove some optimal lower bounds for nudfa s over certain groups which are solvable but not nilpotent most of these results appeared in preliminary form in d a barrington and d therien 1987 in automata languages and programming 14th international colloquium springer verlag berlin pp 163 173

limits on the power of concurrent write parallel machines abstract the computation of specific functions using the most general form of concurrent read concurrent write parallel ram is considered it is shown that such a machine can compute any function of boolean inputs in time log n log log n o 1 given a polynomial number of processors and memory cells and that this bound is tight for integer addition despite this evidence of the power of the model we show that a very simple function namely parity requires time log n to compute given a polynomial bound on the number of processors independent of the number of memory cells

a first order isomorphism theorem we show that for most complexity classes of interest all sets complete under first order projections fops are isomorphic under first order isomorphisms that is a very restricted version of the berman hartmanis conjecture holds since natural complete problems seem to stay complete via fops this indicates that up to first order isomorphism there is only one natural complete problem for each nice complexity class

depth reduction for circuits of unbounded fan in we prove that constant depth circuits of size nlogo 1 n over the basis and or parity are no more powerful than circuits of this size with depth four similar techniques are used to obtain several other depth reduction theorems in particular we show every set in ac0 can be recognized by a family of depth three threshold circuits of size nlogo 1 n the size bound nlogo 1 n is optimal when considering depth reduction over and or and parity most of our results hold for both the uniform and the nonuniform case

the worst hanoi graphs the towers of hanoi is a problem that has been extensively studied and frequently generalized we are interested in its generalization to arbitrary directed graphs and ask how many moves are required in a given graph to move n disks from the starting peg to the destination peg not all directed graphs allow solving this problem we will call those graphs that do hanoi graphs we settle the question of what are the hanoi graphs that require the largest number of moves

on interpolation and automatization for frege systems the interpolation method has been one of the main tools for proving lower bounds for propositional proof systems loosely speaking if one can prove that a particular proof system has the feasible interpolation property then a generic reduction can usually be applied to prove lower bounds for the proof system sometimes assuming a usually modest complexity theoretic assumption in this paper we show that this method cannot be used to obtain lower bounds for frege systems or even for tc0 frege systems more specifically we show that unless factoring of blum integers is feasible neither frege nor tc0 frege has the feasible interpolation property in order to carry out our argument we show how to carry out proofs of many elementary axioms theorems of arithmetic in polynomial sized tc0 frege as a corollary we obtain that tc0 frege as well as any proof system that polynomially simulates it is not automatizable under the assumption that factoring of blum integers is hard we also show under the same hardness assumption that the k provability problem for frege systems is hard

multiplication division and shift instructions in parallel random access machines abstract we prove that polynomial time on a parallel random access machine pram with unit cost multiplication and division or on a pram with unit cost shifts is equivalent to polynomial space on a turing machine pspace this extends the result that polynomial time on a basic pram is equivalent to pspace to hold when the pram is allowed multiplication or division or unrestricted shifts it also extends to the pram the results that polynomial time on a random access machine ram with multiplication is equivalent to pspace and that polynomial time on a ram with shifts that is a vector machine is equivalent to pspace we obtain simulations of uniform circuits by rams with enhanced instruction sets and use the enhanced rams to simulate prams with enhanced instruction sets

resolution of hartmanis conjecture for nl hard sparse sets we resolve a conjecture of hartmanis from 1978 about sparse hard sets for nondeterministic logspace nl we show that there exists a sparse hard set s for nl under logspace many one reductions if and only if nl l deterministic logspace

rational approximation techniques for analysis of neural networks artificial neural networks are comprised of an interconnected collection of certain nonlinear devices examples of commonly used devices include linear threshold elements sigmoidal elements and radial basis elements we employ results from harmonic analysis and the theory of rational approximation to obtain almost tight lower bounds on the size i e number of elements of neural networks the class of neural networks to which our techniques can be applied is quite general it includes any feedforward network in which each element can be piecewise approximated by a low degree rational function for example we prove that any depth d 1 network of sigmoidal units or linear threshold elements computing the parity function of n variables must have r dn sup 1 d spl epsiv size for any fixed spl epsiv 0 in addition we prove that this lower bound is almost tight by showing that the parity function can be computed with o dn sup 1 d sigmoidal units or linear threshold elements in a depth d 1 network these almost tight bounds are the first known complexity results on the size of neural networks with depth more than two our lower bound techniques yield a unified approach to the complexity analysis of various models of neural networks with feedforward structures moreover our results indicate that the in the context of computing highly oscillating symmetric boolean functions networks of continuous output units such as sigmoidal elements do not offer significant reduction in size compared with networks of linear threshold elements of binary outputs

computing with noisy information this paper studies the depth of noisy decision trees in which each node gives the wrong answer with some constant probability in the noisy boolean decision tree model tight bounds are given on the number of queries to input variables required to compute threshold functions the parity function and symmetric functions in the noisy comparison tree model tight bounds are given on the number of noisy comparisons for searching sorting selection and merging the paper also studies parallel selection and sorting with noisy comparisons giving tight bounds for several problems