a very hard log space counting class abstract we consider the logarithmic space counting and optimization classes l span l and opt l which are defined analogously to their polynomial time counterparts we obtain complete functions for these three classes in terms of graphs and finite automata we show that l and opt l are both included in nc 2 but that surprisingly span l seems to be a much harder class than l and opt l we demonstrate that span l functions can be computed in polynomial time if and only if p p and all the classes of the polynomial time hierarchy are included in p this result follows from the fact that span l and p are very similar span l p and any function in p can be represented as the difference of a function in fl and a function in span l nevertheless the inclusion p span l would imply nl p np we furthermore investigate restrictions of the classes opt l and span l

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

a new pebble game that characterizes parallel complexity classes a new two person pebble game that models parallel computations is defined this game extends the two person pebble game defined by dymond and tompa j comput system sci 30 1985 pp 149 161 and is used to characterize two natural parallel complexity classes namely logcfl and text ac 1 the characterizations show a fundamental way in which the computations in these two classes differ this game model also unifies the proofs of some well known results of complexity theory

a circuit based proof of toda s theorem abstract we present a simple proof of toda s result toda 1989 in proceedings 30th annual ieee symposium on foundations of computer science pp 514 519 which states that p is hard for the polynomial hierarchy under randomized reductions our approach is circuit based in the sense that we start with uniform circuit definitions of the polynomial hierarchy and apply the valiant vazirani lemma on these circuits valiant and vazirani 1986 thoeret comput sci 47 85 93

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

response of nonlinear systems in probability domain using stochastic averaging determination of probability density function pdf of the response for strongly nonlinear single degree of freedom system subjected to both multiplicative and additive random excitations using stochastic averaging technique is faced with few difficulties firstly the size of excitations should be small such that the response of the system converges weakly to a markov process secondly the excitations should preferably be broad banded so that the analytical results are reasonably accurate and finally the nonlinear functions should be integrable if closed formed expressions for the result are desired the above issues are examined in the paper with a view to show a limiting values of the size parameter of excitations for which stochastic averaging technique can be applied to obtain reasonable estimates of pdf and mean square value of response b the effect of the nature of the frequency contents of excitation on the response and c the use of the stochastic averaging method employing generalized harmonic functions for nonintegrable functions representing the dynamic system it is shown that the stochastic averaging procedure provides results which compare very well with those obtained from simulation analysis not only for wide band excitations but also for narrow band excitations for a wide range of the size parameter of excitation further the procedure can be easily implemented for highly irregular functions by employing a numerical scheme with fft

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

some exact complexity results for straight line computations over semirings the problem of computing polynomials in certain semmngs is considered precise bounds are obtained on the number of multiplications required by straight hne algorithms which compute such functions as iterated matrix multiplication iterated convolution and permanent usmg these bounds tt is shown that the use of branching can exponentially speed up computations using the min operations and that subtraction can exponentially speed up arithmetic computations these results can be interpreted as denying the existence of fast universal algorithms for computing certain polynomials k v wol os and prmases artthmeuc complexity convexity theory farkas lemma minimax algebra straight hne algorithm

the complexity of iterated multiplication abstract for a monoid g the iterated multiplication problem is the computation of the product of n elements from g by refining known completeness arguments we show that as g varies over a natural series of important groups and monoids the iterated multiplication problems are complete for most natural low level complexity classes the completeness is with respect to first order projections low level reductions that do not obscure the algebraic nature of these problems

circuit definitions of nondeterministic complexity classes we consider restictions on boolean circuits and use them to obtain new uniform circuit characterizations of nondeterministic space and time classes we also obtain characterizations of counting classes based on nondeterministic time bounded computations on the arithmetic circuit model it is shown how the notion of semiunboundedness unifies the definitions of many natural complexity classes

randomness optimal unique element isolation withapplications to perfect matching and related problems in this paper we precisely characterize the randomness complexity of the unique element isolation problem a crucial step in the rnc algorithm for perfect matching due to mulmuley vazirani and vazirani combinatorica 7 1987 pp 105 113 and in several other applications given a set s and an unknown family cal f subseteq 2 s with cal f leq z we present a scheme to assign polynomially bounded weights to the elements of s using only o log z log s random bits such that the minimum weight set in cal f is unique with high probability this generalizes the solution of mulmuley vazirani and vazirani who use o s log s bits independent of z we also provide a matching lower bound for the randomness complexity of this problem the new weight assignment scheme yields a randomness efficient rnc 2 algorithm for perfect matching which uses o log z log n random bits where z is any given upper bound on the number of perfect matchings in the input graph this generalizes the result of grigoriev and karpinski proc ieee symposium on foundations of computer science 1987 pp 166 172 who present an nc 3 algorithm when z is polynomial and also improves the running time in this case the worst case randomness complexity of our algorithm is o n log m n random bits improving on the previous bound of o m log n our scheme also gives randomness efficient solutions for several problems where unique element isolation is used such as rnc algorithms for variants of matching and basic problems on linear matroids we obtain a randomness efficient random reduction from sat to usat the language of uniquely satisfiable formulas which can be derandomized in the case of languages in fewp to yield new proofs of the results fewp subseteq oplus p and fewp subseteq c p

asymptotic l p stability for transition fronts in cahn hilliard systems abstract we consider the asymptotic behavior of perturbations of transition front solutions arising in cahn hilliard systems on r such equations arise naturally in the study of phase separation processes and systems describe cases in which three or more phases are possible when a cahn hilliard system is linearized about a transition front solution the linearized operator has an eigenvalue at 0 due to shift invariance which is not separated from essential spectrum in cases such as this nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate our main result asserts that if initial perturbations are small in l 1 l then spectral stability a necessary condition for stability defined in terms of an appropriate evans function implies asymptotic nonlinear stability in l p for all 1 p

on threshold circuits and polynomial computation a threshold circuit consists of an acyclic digraph of unbounded fanin where each node computes a threshold function or its negation this paper investigates the computational power of threshold circuits a surprising relationship is uncovered between threshold circuits and another class of unbounded fanin circuits which are denoted finite field z p n circuits where each node computes either multiple sums or products of integers modulo a prime p n in particular it is proved that all functions computed by threshold circuits of size s n x2265 n and depth d n can also be computed by z p n circuits of size o s n log s n np n log p n and depth o d n furthermore it is shown that all functions computed by z p n circuits of size s n and depth d n can be computed by threshold circuits of size o 1 x2208 2 s n log p n 1 x2208 and depth o 1 x2208 5 d n these are the main results of this paper there are many useful and quite surprising consequences of this result for example an integer reciprocal can be computed in size n o 1 m and depth o 1 more generally any analytic function with a convergent rational polynomial power series such as sine cosine exponentiation square root and logarithm can be computed within accuracy 2 nc for any constant c by threshold circuits of polynomial size and constant depth in addition integer and polynomial division fff polynomial interpolation chinese remaindering all the elementary symmetric functions banded matrix inverse and triangular toeplitz matrix inverse can be exactly computed by threshold circuits of polynomial size and constant depth all these results and simulations hold for polytime uniform circuits this paper also gives a corresponding simulation oflogspace uniform z p n circuits by logspace uniform threshold circuits requiring an additional multiplying factor of o log log log p n depth finally purely algebraic methods forlowerbounds for zp n circuits are developed using degree arguments a depth hierarchy theorem for z p n circuits is proved for any s n x2265 n d n o s n c for some constant c p n where 6 s n d n d n p n c 2 n there exist explicitly constructible functions computable by z p n circuits of size s n and depth d n but provably not computable by z p n circuits of size s n c and depth o d n for any constant c x2265 1

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

non commutative arithmetic circuits depth reduction and size lower bounds abstract we investigate the phenomenon of depth reduction in commutative and non commutative arithmetic circuits we prove that in the commutative setting uniform semi unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree and unrestricted depth earlier proofs did not work in the uniform setting this also provides a unified proof of the circuit characterizations of the class logcfl and its counting variant logcfl we show that ac1 has no more power than arithmetic circuits of polynomial size and degree no log log n improving the trivial bound of no log n connections are drawn between tc1 and arithmetic circuits of polynomial size and degree then we consider non commutative computation we show that over the algebra max concat arithmetic circuits of polynomial size and polynomial degree can be reduced to o log2 n depth and even to o log n depth if unbounded fanin gates are allowed this establishes that optlogcfl is in ac1 this is the first depth reduction result for arithmetic circuits over a non commutative semiring and it complements the lower bounds of kosaraju and nisan showing that depth reduction cannot be done in the general non commutative setting we define new notions called short left paths and short right paths and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible this class also can be characterized using the auxpda model finally we characterize the languages generated by efficient circuits over the semiring 2 union concat in terms of simple one way machines and we investigate and extend earlier lower bounds on non commutative circuits

spectral analysis for transition front solutions in multidimensional cahn hilliard systems abstract we consider the spectrum associated with the linear operator obtained when a cahn hilliard system on r n is linearized about a planar transition front solution in the case of single cahn hilliard equations on r n it s known that under general physical conditions the leading eigenvalue moves into the negative real half plane at a rate 3 where is the fourier transform variable corresponding with components transverse to the wave moreover it has recently been verified that for single equations this spectral behavior implies nonlinear stability in the current analysis we establish that the same cubic rate law holds for a broad range of multidimensional cahn hilliard systems the analysis of nonlinear stability will be carried out separately

heteroclinic connections for multiple well potentials the anisotropic case we investigate the existence of solutions to systems of n differential equations representing connections between minima of potentials with several equal depths in r using variational techniques and in particular a method introduced in af we first prove such existence for n 2 and two minima dealing next with symmetric potentials corresponding to free bulk energies in crystals we establish existence for n 2 in various cases of more than two minima finally we obtain a sufficient condition establishing existence of connections to not necessarily symmetric potentials for arbitrary n and three minima

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

dynamics of offshore structures part ii stochastic averaging analysis this is the second of two companion papers concerned with an investigation into the non linear dynamics of drag dominated offshore structures excited by waves the equations of motion are approximated by a linear system with a time varying damping term the excitation however is a non linear function of the wave particle kinematics to study the effect of the time varying damping the resonant response is here examined by means of a narrow band idealization a stochastic averaging of the governing equations is applied and the fokker planck equation for the response amplitude probability densities is thereby obtained from the stationary solution of this equation second order statistics are derived together with an expression for the hydrodynamic damping which is equivalent to the time varying term numerical results are given to illustrate the magnitude of this term for a typical framed structure and to provide comparisons with results obtained by the conventional technique of linearization

a uniform circuit lower bound for the permanent the authors show that uniform families of acc circuits of subexponential size cannot compute the permanent function this also implies similar lower bounds for certain sets in pp this is one of the very few examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition it is still unknown if there is any set in ntime 2 n o 1 that does not have nonuniform acc circuits

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

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

skew circuits of small width abstract a celebrated result of barrington 1985 proved that polynomial size width 5 branching programs bp are equivalent in power to a restricted form of branching programs polynomial sized width 5 permutation branching programs pbp which in turn capture all of nc 1 on the other hand it is known that width 3 pbps require exponential size to compute the and function no such lower bound is known for width 4 pbps however it is widely conjectured that width 4 pbps will not capture all of nc 1 in this work we study the power of bounded width branching programs by comparing them with bounded width skew circuits it is well known that branching programs of bounded width have the same power as skew circuit of bounded width the naive approach converts a bp of width w to a skew circuit of width w 2 we improve this bound and show that bp of width w 5 can be converted to a skew circuit of width 7 this also implies that skew circuits of bounded width are equal in power to skew circuits of width 7 for the other way we prove that for any w 2 a skew circuit of width w can be converted into an equivalent branching program of width w we prove that width 2 skew circuits are not universal while width 3 skew circuits are universal and that any polynomial sized cnf or dnf is computable by width 3 skew circuits of polynomial size it is known that parity does not have small cnfs or dnfs it is easy to see that parity has width 4 skew circuits we prove that a width 3 skew circuit computing parity requires exponential size this gives an exponential separation between the power of width 3 skew circuits and width 4 skew circuits

asymptotic stability analysis for transition front solutions in cahn hilliard systems abstract we consider the asymptotic behavior of perturbations of transition front solutions arising in cahn hilliard systems on r such equations arise naturally in the study of phase separation and systems describe cases in which three or more phases are possible when a cahn hilliard system is linearized about a transition front solution the linearized operator has an eigenvalue at 0 due to shift invariance which is not separated from essential spectrum in cases such as this nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate our main result asserts that spectral stability a necessary condition for stability defined in terms of an appropriate evans function implies nonlinear stability

fast parallel computation of polynomials using few processors it is shown that any multivariate polynomial of degree d that can be computed sequentially in c steps can be computed in parallel in o log d log c log d steps using only cd o 1 processors

heteroclinic travelling waves of gradient diffusion systems we establish existence of travelling waves to the gradient system ut uzz w u connecting two minima of w when u r 0 r n that is we establish existence of a pair u c c 2 r n 0 satisfying uxx w u c ux u a where a are local minima of the potential w c 2 r n with w a w a 0 and n 1 our method is variational and based on the mini mization of the functional ec u r r n 1 2 ux 2 w u o e cx dx in the appro priate space setup following alikakos fusco a f we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior which we later remove we provide variational characterizations of the travel ling wave and the speed in particular we show that ec u 0

a finite element method for the statistics of non linear random vibration the transitional probability density function for the random response of a certain class of non linear system satisfies the fokker planck kolmogorov equation this paper concerns the numerical solution of the stationary form of this equation yielding the stationary probability density function of response the weighted residual statement for the problem is integrated by parts to yield the weak form of the equations which are then solved by the finite element method the method is applied to a duffing oscillator and good agreement is found with the exact result and the method is compared favourably with a galerkin solution method given by bhandari and sherrer 1 also the method is applied to the ship rolling problem and good agreement is found with an approximate analytical result due to roberts 2

amplifying lower bounds by means of self reducibility we observe that many important computational problems in nc1 share a simple self reducibility property we then show that for any problem a having this self reducibility property a has polynomial size tc0 circuits if and only if it has tc0 circuits of size n1 for every 0 counting the number of wires in a circuit as the size of the circuit as an example of what this observation yields consider the boolean formula evaluation problem bfe which is complete for nc1 and has the self reducibility property it follows from a lower bound of impagliazzo paturi and saks that bfe requires depth d tc0 circuits of size n1 d if one were able to improve this lower bound to show that there is some constant 0 independent of the depth d such that every tc0 circuit family recognizing bfe has size at least n1 then it would follow that tc0 nc1 we show that proving lower bounds of the form n1 is not ruled out by the natural proof framework of razborov and rudich and hence there is currently no known barrier for separating classes such as acc0 tc0 and nc1 via existing natural approaches to proving circuit lower bounds we also show that problems with small uniform constant depth circuits have algorithms that simultaneously have small space and time bounds we then make use of known time space tradeoff lower bounds to show that sat requires uniform depth d tc0 and ac0 6 circuits of size n1 c for some constant c depending on d