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

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

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

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

random walks in weighted networks with a perfect trap an application of laplacian spectra trapping processes constitute a primary problem of random walks which characterize various other dynamical processes taking place on networks most previous works focused on the case of binary networks while there is much less related research about weighted networks in this paper we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node by utilizing the spectral graph theory we provide an exact formula for mean first passage time mfpt from one node to another based on which we deduce an explicit expression for average trapping time att in terms of the eigenvalues and eigenvectors of the laplacian matrix associated with the weighted graph where att is the average of mfpts to the trap over all source nodes we then further derive a sharp lower bound for the att in terms of only the local information of the trap node which can be obtained in some graphs moreover we deduce the att when the trap is distributed uniformly in the whole network our results show that network weights play a significant role in the trapping process to apply our framework we use the obtained formulas to study random walks on two specific networks trapping in weighted uncorrelated networks with a deep trap the weights of which are characterized by a parameter and levy random walks in a connected binary network with a trap distributed uniformly which can be looked on as random walks on a weighted network for weighted uncorrelated networks we show that the att to any target node depends on the weight parameter that is the att to any node can change drastically by modifying the parameter a phenomenon that is in contrast to that for trapping in binary networks for levy random walks in any connected network by using their equivalence to random walks on a weighted complete network we obtain the optimal exponent characterizing levy random walks which have the minimal average of atts taken over all target nodes

well posedness for the spatially homogeneous landau fermi dirac equation for hard potentials we study the cauchy problem for the spatially homogeneous landau equation for fermi dirac particles in the case of hard and maxwellian potentials we establish existence and uniqueness of a weak solution for a large class of initial data

effects of inhomogeneous activity of players and noise on cooperation in spatial public goods games we study the public goods game in the noisy case by considering the players with inhomogeneous activity of teaching on a square lattice it is shown that the introduction of the inhomogeneous activity of teaching the players can remarkably promote cooperation by investigating the effects of noise on cooperative behavior in detail we find that the variation of cooperator density c with the noise parameter displays several different behaviors c monotonically increases decreases with c first increases decreases with and then it decreases increases monotonically after reaching its maximum minimum value which depends on the amount of the multiplication factor r on whether the system is homogeneous or inhomogeneous and on whether the adopted updating is synchronous or asynchronous these results imply that the noise plays an important and nontrivial role in the evolution of cooperation

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

costly punishment and cooperation in the evolutionary snowdrift game abstract the role of punishments in promoting cooperation is an important issue we incorporate costly punishments into the snowdrift game sg by introducing a third punishing p character and study the effects the punishers who carry basically a cooperative c character are willing to pay a cost so as to punish a non cooperative d opponent by depending on the initial fractions of the characters and the cost to benefit ratio r in the sg the three character system evolves into a steady state consisting either only of c and p characters or only of c and d characters in a well mixed population the former situation represents an enhancement in cooperation relative to the sg while the latter is similar to the sg the dynamics in approaching these different steady states are found to be different analytically the key features in the dynamics and the steady states observed in simulations are captured by a set of differential equations the sensitivity to the initial distribution of characters is studied by depicting the flow in a phase portrait and analyzing the nature of fixed points the analysis also shows the role of p character agents in preventing a system from invasion by d character agents starting from a population consisting only of c and p agents a d character agent intended to invade the system cannot survive when the initial fraction of p agents is greater than r our model defined intentionally as a simulation algorithm can be readily generalized to incorporate many interesting effects such as those in a networked population

the landau equation in a periodic box the landau equation which was proposed by landau in 1936 is a fundamental equation to describe collisions among charged particles interacting with their coulombic force in this article global in time classical solutions near maxwellians are constructed for the landau equation in a periodic box our result also covers a class of generalized landau equations which describes grazing collisions in a dilute gas

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

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

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

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

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

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

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

cooperation for volunteering and partially random partnerships competition among cooperative defective and loner strategies is studied by considering an evolutionary prisoner s dilemma game for different partnerships in this game each player can adopt one of its coplayer s strategy with a probability depending on the difference of payoffs coming from games with the corresponding coplayers our attention is focused on the effects of annealed and quenched randomness in the partnership for fixed number of coplayers it is shown that only the loners survive if the four coplayers are chosen randomly mean field limit on the contrary on the square lattice all the three strategies are maintained by the cyclic invasions resulting in a self organizing spatial pattern if the fixed partnership is described by a regular small world structure then a homogeneous oscillation occurs in the population dynamics when the measure of quenched randomness exceeds a threshold value similar behavior with higher sensitivity to the randomness is found if temporary partners are substituted for the standard ones with some probability at each step of iteration

memory based snowdrift game on networks we present a memory based snowdrift game mbsg taking place on networks we found that when a lattice is taken to be the underlying structure the transition of spatial patterns at some critical values of the payoff parameter is observable for both four and eight neighbor lattices the transition points as well as the styles of spatial patterns can be explained by local stability analysis in sharp contrast to previously reported results cooperation is promoted by the spatial structure in the mbsg interestingly we found that the frequency of cooperation of the mbsg on a scale free network peaks at a specific value of the payoff parameter this phenomenon indicates that properly encouraging selfish behaviors can optimally enhance the cooperation the memory effects of individuals are discussed in detail and some nonmonotonous phenomena are observed on both lattices and scale free networks our work may shed some new light on the study of evolutionary games over networks

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

i o efficient algorithms for the minimum cut problem on unweighted undirected graphs the problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model first a lower bound of e v sort v on the number of i os is shown for the problem where v is the number of vertices and e is the number of edges then the following are presented for m b 2 1 a minimum cut algorithm that uses o c log e rm msf v e frac v b rm sort v i os here msf v e is the number of i os needed to compute a minimum spanning tree of the graph and c is the value of the minimum cut the algorithm performs better on dense graphs than the algorithm of 7 which requires o e c 2 v log v c i os when executed on the external memory model for a fat graph for 0 the maximum tree packing of the graph is at least 1 c 2 our algorithm computes a minimum cut in o c loge msf v e sort e i os 2 a randomized algorithm that computes minimum cut with high probability in o c log e cdot rm msf v e rm sort e log 2 v frac v b rm sort v log v i os 3 a 2 e minimum cut algorithm that requires o e v msf v e i os and performs better on sparse graphs than our exact minimum cut algorithm

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 evolutionary public goods game on scale free networks with heterogeneous investment abstract the public goods game pgg is generally considered as a suitable paradigm to explain ubiquitous cooperative behavior in this study we investigated the evolutionary pgg on scale free networks and studied the effect of individual heterogeneity by setting the cooperator x an investment value correlated to its degree as i x n k x j k j where k x is the degree of x j runs over all players and is a tunable parameter it is shown that the cooperation level is remarkably promoted by negative values of whereas it is highly depressed by positive values of moreover the effect of environmental noise has also been investigated our result may sharpen the understanding of cooperation induced by the individual diversity

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

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

evolutionary prisoner s dilemma game with dynamic preferential selection we study a modified prisoner s dilemma game taking place on two dimensional disordered square lattices the players are pure strategists and can either cooperate or defect with their immediate neighbors in the generations each player updates its strategy by following one of the neighboring strategies with a probability dependent on the payoff difference the neighbor selection obeys a dynamic preferential rule i e the more frequently a neighbor s strategy was adopted by the focal player in the previous rounds the larger probability it will be chosen to refer to in the subsequent rounds it is found that cooperation is substantially promoted due to this simple selection mechanism corresponding analysis is provided by the investigation of the distribution of the players impact weights persistence and correlation function

enhanced cooperation and harmonious population in an evolutionary n person snowdrift game abstract the n person evolutionary snowdrift game nesg is generalized to study the effects of the additional benefit to all agents in a competing group of size n resulting from an earlier completion of a task when more agents are willing to share the work following replicator dynamics an equation that can be used to solve for the steady state frequency of cooperation x in a well mixed population as a function of the parameters representing the cost to benefit ratio c b additional reward w b and n is derived cooperation is enhanced in general for w 0 and a stable state with all cooperative agents allc state emerges for small groups n and small c b in contrast such a harmonious allc state does not exist in the original nesg for c 0 the condition for the existence of an allc state is estimated to be n 1 c w and good agreement is found when compared with numerical solutions the observed disappearance of an allc state and the drop in x with n are discussed in terms of the change in the stability of the allc state as n increases previous results in nesg are recovered as the w 0 case of our model thus the pleasure of enjoying the completed task earlier promotes cooperation and the effects are more pronounced when small group collective interactions are considered

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

promotion of cooperation by payoff driven migration abstract migration plays an important role in the evolution of cooperation previous studies concerning mobile population often assumed that all agents move with the identical velocity in this paper we propose a payoff driven migration in which the velocity of an agent depends on his her payoff a parameter is introduced to adjust the influence of payoff when 0 means that agents all move with the identical velocity while 0 means that the lower the payoff is the faster the moving speed is and vice versa for the prisoner s dilemma game we find that in comparison with the case that agents all move with the same speed cooperation could be promoted strongly when payoff dependent velocity is considered remarkably the cooperation level is not a monotonic function of and there exists an optimal value of which can lead to the maximum cooperation level for the snowdrift game the cooperation level increases with

phase diagrams for an evolutionary prisoner s dilemma game on two dimensional lattices the effects of payoffs and noise on the maintenance of cooperative behavior are studied in an evolutionary prisoner s dilemma game with players located on the sites of different two dimensional lattices this system exhibits a phase transition from a mixed state of cooperators and defectors to a homogeneous one where only the defectors remain alive using monte carlo simulations and the generalized mean field approximations we have determined the phase boundaries critical points separating the two phases on the plane of the temperature noise and temptation to choose defection in the zero temperature limit the cooperation can be sustained only for those connectivity structures where three site clique percolation occurs

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

evolutionary prisoner s dilemma game on a square lattice a simplified prisoner s game is studied on a square lattice when the players interacting with their neighbors can follow two strategies to cooperate c or to defect d unconditionally the players updated in random sequence have a chance to adopt one of the neighboring strategies with a probability depending on the payoff difference using monte carlo simulations and dynamical cluster techniques we study the density c of cooperators in the stationary state this system exhibits a continuous transition between the two absorbing states when varying the value of temptation to defect in the limits stackrel ensuremath rightarrow c 0 and 1 we have observed critical transitions belonging to the universality class of directed percolation

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

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

preferential selection promotes cooperation in a spatial public goods game abstract we introduce a preferential selection mechanism into a spatial public goods game where players are located on a square lattice each individual chooses one of its neighbors as a reference with a probability proportional to exp p y a where p y is the neighbor s payoff and a 0 is a tunable parameter it is shown that the introduction of such a preferential selection can remarkably promote the emergence of cooperation over a wide range of the multiplication factor we find that the mean payoffs of cooperators along the boundary are higher than that of defectors and cooperators form larger clusters as a increases the extinction thresholds of cooperators and defectors for different values of noise are also investigated

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

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

cooperation enhanced by the difference between interaction and learning neighborhoods for evolutionary spatial prisoner s dilemma games we study an evolutionary prisoner s dilemma game with two layered graphs where the lower layer is the physical infrastructure on which the interactions are taking place and the upper layer represents the connections for the strategy adoption learning mechanism this system is investigated by means of monte carlo simulations and an extended pair approximation method we consider the average density of cooperators in the stationary state for a fixed interaction graph while varying the number of edges in the learning graph according to the monte carlo simulations the cooperation is modified substantially in a way resembling a coherence resonance like behavior when the number of learning edges is increased this behavior is reproduced by the analytical results

entropy dissipation estimates for the relativistic landau equation and applications abstract in this paper we study the cauchy problem for the spatially homogeneous relativistic landau equation with coulomb interactions despite its physical importance this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator in this paper we first largely decompose the structure of the relativistic landau collision operator after that we prove the global entropy dissipation estimate then we prove the propagation of any polynomial moment for a weak solution lastly we prove the existence of a true weak solution for a large class of initial data

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

impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies cooperation and defection may be considered to be two extreme responses to a social dilemma yet the reality is much less clear cut between the two extremes lies an interval of ambivalent choices which may be captured theoretically by means of continuous strategies defining the extent of the contributions of each individual player to the common pool if strategies are chosen from the unit interval where 0 corresponds to pure defection and 1 corresponds to the maximal contribution the question is what is the characteristic level of individual investments to the common pool that emerges if the evolution is guided by different benefit functions here we consider the steepness and the threshold as two parameters defining an array of generalized benefit functions and we show that in a structured population there exist intermediate values of both at which the collective contributions are maximal however as the cost to benefit ratio of cooperation increases the characteristic threshold decreases while the corresponding steepness increases our observations remain valid if more complex sigmoid functions are used thus reenforcing the importance of carefully adjusted benefits for high levels of public cooperation

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

the maximum flow problem is log space complete for p the space complexity of the maximum flow problem is investigated it is shown that the problem is log space complete for deterministic polynomial time thus the maximum flow problem probably has no algorithm which needs only o logk n storage space for any constant k another consequence is that there is probably no fast parallel algorithm for the maximum flow problem

coevolutionary games on networks we study agents on a network playing an iterated prisoner s dilemma against their neighbors the resulting spatially extended coevolutionary game exhibits stationary states which are nash equilibria after perturbation of these equilibria avalanches of mutations reestablish a stationary state scale free avalanche distributions are observed that are in accordance with calculations from the nash equilibria and a confined branching process the transition from subcritical to critical avalanche dynamics can be traced to a change in the degeneracy of the cooperative macrostate and is observed for many variants of this game

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

phase diagrams for three strategy evolutionary prisoner s dilemma games on regular graphs evolutionary prisoner s dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one the players follow one of the three strategies tit for tat unconditional cooperation and defection the simplified payoff matrix is characterized by two parameters the temptation b to choose defection and the cost c of inspection reducing the income of tit for tat the strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise using monte carlo simulations and the extended versions of pair approximation we have evaluated the b c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence absorbing and oscillatory states including continuous and discontinuous phase transitions by reasonable costs the tit for tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner s dilemma game irrespective of the connectivity structure we also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance

on spatially homogeneous solutions of a modified boltzmann equation for fermi dirac particles the paper considers a modified spatially homogeneous boltzmann equation for fermi dirac particles bfd we prove that for the bfd equation there are only two classes of equilibria the first ones are fermi dirac distributions the second ones are characteristic functions of the euclidean balls and they can be simply classified in terms of temperatures t 2 5tf and t 2 5tf where tf denotes the fermi temperature in general we show that the l bound 0 f 1 e derived from the equation for solutions implies the temperature inequality t 2 5tf and if t 2 5tf then f trend towards fermi dirac distributions if t 2 5tf then f are the second equilibria in order to study the long time behavior we also prove the conservation of energy and the entropy identity and establish the moment production estimates for hard potentials

diversity optimized cooperation on complex networks we propose a strategy for achieving maximum cooperation in evolutionary games on complex networks each individual is assigned a weight that is proportional to the power of its degree where the exponent is an adjustable parameter that controls the level of diversity among individuals in the network during the evolution every individual chooses one of its neighbors as a reference with a probability proportional to the weight of the neighbor and updates its strategy depending on their payoff difference it is found that there exists an optimal value of for which the level of cooperation reaches maximum this phenomenon indicates that although high degree individuals play a prominent role in maintaining the cooperation too strong influences from the hubs may counterintuitively inhibit the diffusion of cooperation other pertinent quantities such as the payoff the cooperator density as a function of the degree and the payoff distribution are also investigated computationally and theoretically our results suggest that in order to achieve strong cooperation on a complex network individuals should learn more frequently from neighbors with higher degrees but only to a certain extent