two proofs of pinsker s conjecture concerning arbitrarily varying channels m s pinsker 1990 conjectured the following theorem for an arbitrary varying channel avc every rate below the random code capacity is achievable with deterministic list codes of constant list size if the average error criterion is used two proofs of this theorem are given

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

common randomness in information theory and cryptography i secret sharing as the first part of a study of problems involving common randomness at distance locations information theoretic models of secret sharing generating a common random key at two terminals without letting an eavesdropper obtain information about this key are considered the concept of key capacity is defined single letter formulas of key capacity are obtained for several models and bounds to key capacity are derived for other models

programs over aperiodic monoids abstract we consider a model of computation where the execution of a program on an input corresponds to calculating a product in a finite monoid this model has recently been formalized to give an algebraic point of view on certain types of boolean circuits we focus our discussion on computations over aperiodic monoids in particular we exhibit examples of such monoids over which it is not possible to count modulo p

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 semilinear parabolic system arising in the theory of superconductivity arises in the theory of superconductivity of liquids proposed by abrahams and tsuneto in 11 system 1 has also been briefly treated as an example by chueh conley and smaller in 61 throughout this paper we shall assume that d is a bounded domain in r n 1 2 or 3 with smooth boundary x a denotes the laplacian all the results of the paper which hold for n 1 also hold when a is replaced by any second order symmetric elliptic differential expression with positive spectrum because of the very symmetric nature of l it is reasonable to expect solutions of 1 to have nice properties we show in this paper that this is in fact the case in section 2 we give a complete description of steady state solutions of 1 satisfying dirichlet or neumann boundary conditions we show that often all steady states are of the form u u t au for some constant a where u is the solution of a single differential equation it is possible using standard techniques for scalar nonlinear boundary value problems to obtain lots of information about the solutions of this single equation and so about solutions of 1 in 6 1 it is shown that all solutions of 1 are attracted towards the set u v u v 1 in section 3 we show that 1 has a lyapunov function and use this fact and the standard theory of w limit sets to prove that any solution of 1 tends as t 03 towards the family of all steady state solutions of 1 often in the case of scalar equations it can be shown that

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

dynamics of a predator prey model we describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator prey model proposed by r m may stability and complexity in model ecosystems princeton university press princeton nj 1974 in particular we give a qualitative description of the bifurcation curve when two limit cycles collapse on a semistable limit cycle and disappear moreover we show that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator prey models

finite semigroup varieties defined by programs abstract we study the regular languages recognized by polynomial length programs over finite semigroups belonging to product varieties v li where v is a variety of finite monoids and li is the variety of finite locally trivial semigroups in the case where the semigroup variety has a particular closure property with respect to programs we are able to give precise characterizations of these regular languages as a corollary we obtain new proofs of the results of barrington compton straubing and therien characterizing the regular languages in certain circuit complexity classes

on the computational power of depth 2 circuits with threshold and modulo gates abstract we investigate the computational power of depth 2 circuits consisting of mod r gates at the bottom and a threshold gate with arbitrary weights at the top for short threshold mod r circuits and circuits with two levels of mod gates mod p mod q circuits in particular we will show the following results 1 i for all prime numbers p and integers q r it holds that if p divides r but not q then all threshold mod q circuits for mod r have exponentially many nodes 2 ii for all integers r all problems computable by depth 2 and or not circuits of polynomial size have threshold mod r circuits with polynomially many edges 3 iii there is a problem computable by depth 3 and or not circuits of linear size and constant bottom fan in which for all r needs threshold mod r circuits with exponentially many nodes 4 iv for p r different primes and q 2 k positive integers where r does not divide q every mod p k mod q circuit for mod r has exponentially many nodes results i and iii imply the first known exponential lower bounds on the number of nodes of threshold mod r circuits r 2 they are based on a new method for estimating the minimum length of threshold realizations over predefined function bases which in contrast to previous related techniques goldmann et al 1992 bruck and smolensky 1990 kailath et al 1991 goldmann 1993 grolmusz 1993 works even if the weight of the realization is allowed to be unbounded and if the bases are allowed to be nonorthogonal the special importance of result iii consists of the fact that the known spectral theoretically based lower bound methods for threshold xor circuits bruck and smolensky 1990 kailath et al 1991 can provably not be applied to ac 0 functions thus by ii result iii is sharp it gives a partial negative answer to the open question whether there exist simulations of ac 0 circuits by small depth threshold circuits which are more efficient than that given by yao s important result that acc functions have depth 3 threshold circuits of quasipolynomial weight yao 1990 finally we observe that our method works also for mod p mod q circuits if p is a power of a prime iv above see barrington et al 1990 krause and waack 1991 yan and parberry 1994 for related results a preliminary version of this paper appeared in krause and pudlak 1993

lyapunov functions for the global stability of competing predators in this paper we study the global stability of an exploitative competition system between two predator species and a single prey species a sufficient condition for the global stability is given which is applicable to most of three dimensional competing predators system then we restrict our attentions to the case in which the prey is of logistic growth and predators have holling s type ii functional responses

the complexity of solving equations over finite groups we study the computational complexity of solving systems of equations over a finite group an equation over a group g is an expression of the form w1 w2 wk 1g where each wi is either a variable an inverted variable or a group constant and 1g is the identity element of g a solution to such an equation is an assignment of the variables to values in g which realizes the equality a system of equations is a collection of such equations a solution is then an assigmnent which simultaneously realizes each equation we show that the problem of determining if a single equation has a solution is np complete for all nonsolvable groups g for nilpotent groups this same problem is shown to be in p the analogous problem for systems of such equations is shown to be np complete if g is non abelian and in p otherwise finally we observe some connections between these problems and the theory of nonuniform automata

the method of types information theory the method of types is one of the key technical tools in shannon theory and this tool is valuable also in other fields in this paper some key applications are presented in sufficient detail enabling an interested nonspecialist to gain a working knowledge of the method and a wide selection of further applications are surveyed these range from hypothesis testing and large deviations theory through error exponents for discrete memoryless channels and capacity of arbitrarily varying channels to multiuser problems while the method of types is suitable primarily for discrete memoryless models its extensions to certain models with memory are also discussed

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

the maximal error capacity of arbitrarily varying channels for constant list sizes the capacity of an arbitrarily varying channel avc for list codes of arbitrarily small list rate under the maximal error criterion has previously been determined the authors prove that for an avc a any rate r below the list code capacity c sub 1 a is achievable with a particular list size dependent on a r and y where y is the output alphabet

learning expressions and programs over monoids we study the problem of learning an unknown function represented as an expression or a program over a known finite monoid as in other areas of computational complexity where programs over algebras have been used the goal is to relate the computational complexity of the learning problem with the algebraic complexity of the finite monoid indeed our results indicate a close connection between both kinds of complexity we focus on monoids which are either groups or aperiodic and on the learning model of exact learning from queries for a group g we prove that expressions over g are efficiently learnable if g is nilpotent and impossible to learn efficiently under cryptographic assumptions if g is nonsolvable we present some results for restricted classes of solvable groups and point out a connection between their efficient learnability and the existence of lower bounds on their computational power in the program model for aperiodic monoids our results seem to indicate that the monoid class known as da captures exactly learnability of expressions by polynomially many evaluation queries when using programs instead of expressions we show that our results for groups remain true while the situation is quite different for aperiodic monoids

global dynamics of a mathematical model of competition in the chemostat general response functions and differential death rates a model of exploitative competition of n species in a chemostat for a single essential nonreproducing growth limiting resource is considered s b hsu siam j appl math 34 1978 pp 760 763 applies lasalle s extension theorem of lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by michaelis menten dynamics g j butler and g s k wolkowicz siam j appl math 45 1985 pp 138 151 on the other hand allow more general response functions including monotone and nonmonotone functions but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate and hence can be ignored by means of lyapunov stability theory the global dynamics of the model for a large class of response functions are studied including both monotone and nonmonotone functions though it is not as general as the class studied by butler and wolkowicz and the results in

global asymptotical stability of a n 1 dimensional leslie gower predator prey model abstract by constructing suitable lyapunov function we obtain the global asymptotical stability of n 1 dimensional leslie gower predator prey models it is shown that the prey refuge and the proportional harvesting have no influence on the global asymptotical stability of the system but they can change the position of the unique interior equilibrium finally mathematic analysis shows that if we choose and control the harvesting efforts e e i appropriately we can not only harvest the biological resources and achieve the economic interest but also protect the biological resources

a strongly coupled predator prey system with modified holling tanner functional response in this paper a strongly coupled system of partial differential equations in a bounded domain with the homogeneous neumann boundary condition which models a predator prey system with modified holling tanner functional response is considered first the authors study the stability of the positive constant solution sufficient conditions are derived for the global stability of the positive equilibrium by constructing a suitable lyapunov function by using the leray schauder theorem the authors prove a number of existence and non existence results about the non constant steady states of the system

spatiotemporal dynamics in a delayed diffusive predator model in this paper we investigate the spatiotemporal dynamics of a delayed reaction diffusion leslie gower model based on the stability analysis we demonstrate that delayed feedback may generate hopf and turing instability under some conditions resulting in spatial patterns one of the most interesting findings is that the model exhibits complex pattern replication pure turing instability gives birth to spots spots stripes mixture stripes stripes holes mixture and holes patterns pure hopf instability to spiral wave pattern and hopf turing instability to chaotic wave pattern our results well extend the findings of spatiotemporal dynamics in the delayed reaction diffusion predator prey model and indicate that time delay play an important roles in pattern formation

list decoding for arbitrarily varying multiple access channel revisited list configuration and symmetrizability in a recent work s nitinawarat obtained a lower bound and an upper bound for the minimum list size in list decoding for an arbitrarily varying multiple access channel avmac for which the interior of the capacity region of deterministic list codes is nonempty in the same paper he proved that for a binary avmac the minimum list size is finite if and only if the interior of the capacity region of random correlated codes is nonempty the goal of this paper is to close the gap between the two bounds for the minimum list size we find a necessary and sufficient condition for the list codes for an avmac to have a capacity region with a nonempty interior in terms of bipartite graphs therefore we determine the minimum list size moreover we prove that for any avmac the minimum list size is finite if and only if the interior of the capacity region of random correlated codes is nonempty

qualitative analysis of a predator prey model with holling type ii functional response incorporating a prey refuge abstract we study a predator prey model with holling type ii functional response incorporating a prey refuge under homogeneous neumann boundary condition we show the existence and non existence of non constant positive steady state solutions depending on the constant m 0 1 which provides a condition for protecting 1 m u of prey u from predation moreover we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions

learning behaviors of automata from multiplicity and equivalence queries we consider the problem of identifying the behavior of an unknown automaton with multiplicity in the field ratviii of rational numbers ratviii automaton from multiplicity and equivalence queries we provide an algorithm which is polynomial in the size of the ratviii automaton and in the maximum length of the given counterexamples as a consequence we have that ratviii automata are probably approximately correctly learnable pac learnable in polynomial time when multiplicity queries are allowed a corollary of this result is that regular languages are polynomially predictable using membership queries with respect to the representation of unambiguous nondeterministic automata this is important since there are unambiguous automata such that the equivalent deterministic automaton has an exponentially larger number of states

positive steady state solutions of a leslie gower predator prey model with holling type ii functional response and density dependent diffusion abstract in this paper we consider a leslie gower predator prey model with holling type ii functional response and density dependent diffusion under zero dirichlet boundary condition using degree theory bifurcation theory energy estimates and asymptotical behavior analysis the existence and multiplicity of positive steady state solutions were shown under certain conditions on the parameters

dynamics in a diffusive plant herbivore model with toxin determined functional response abstract a diffusive plant herbivore model with the toxin determined functional response and homogeneous neumann boundary condition is considered the stability and hopf bifurcation of the constant positive steady state are investigated by using the linearization method

dynamics of nonlinear parabolic systems with time delays abstract the dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system the same conclusions are obtained for a coupled system of parabolic ordinary equations with time delays applications are given to three model problems arising from ecology and nuclear engineering these model problems possess multiple steady state solutions and sufficient conditions are given to ensure the stability and instability of these solutions

classification of finite monoids the language approach abstract the concept of a variety of congruences is introduced and related to variety of languages and variety of monoids a systematic construction of increasingly complex varieties of congruences is presented this construction is powerful enough to generate all monoids containing solvable groups some hierarchies occurring through this process are shown to correspond to well known hierarchies of monoids thus indicating that our construction is natural from an algebraic point of view some problems that remain open are also discussed

qualitative analysis of a ratio dependent predator prey system with diffusion ratio dependent predator prey models are favoured by many animal ecologists recently as they better describe predator prey interactions where predation involves a searching process when densities of prey and predator are spatially homogeneous the so called michaelis menten ratio dependent predator prey system which is an ordinary differential system has been studied by many authors the present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous neumann boundary condition its main purpose is to study qualitative properties of solutions to this reaction diffusion partial differential system in particular we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary differential equation dynamics non constant positive steady states exist for the partial differential equation model this demonstrates that stationary patterns arise as a result of diffusion

stability analysis of diffusive predator prey model with modified leslie gower and holling type iii schemes abstract the stability of a diffusive predator prey model with modified leslie gower and holling type iii schemes is investigated a threshold property of the local stability is obtained for a boundary steady state and sufficient conditions of local stability and un stability for the positive steady state are also obtained furthermore the global asymptotic stability of these two steady states are discussed our results reveal the dynamics of this model system

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

non constant positive steady states of a diffusive predator prey system in homogeneous environment abstract in this paper we investigate the existence and non existence of non constant positive steady states of a diffusive predator prey interaction system under homogeneous neumann boundary condition in homogeneous environment we show that the predator prey model with leslie gower functional response has no non constant positive solution but the system with a general functional response may have at least one non constant positive steady state under some conditions

languages defined with modular counting quantifiers we prove that a regular language defined by a boolean combination of generalized 1 sentences built using modular counting quantifiers can be defined by a boolean combination of 1 sentences in which only regular numerical predicates appear the same statement with 1 replaced by first order is equivalent to the conjecture that the nonuniform circuit complexity class acc is strictly contained in nc1 the argument introduces some new techniques based on a combination of semigroup theory and ramsey theory which may shed some light on the general case

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

the membership problem in aperiodic transformation monoids the problem of testing membership in aperiodic or group free transformation monoids is the natural counterpart to the well studied membership problem in permutation groups the class a of all finite aperiodic monoids and the class g of all finite groups are two examples of varieties the fundamental complexity units in terms of which finite monoids are classified the collection of all varieties v forms an infinite lattice under the inclusion ordering with the subfamily of varieties that are contained in a forming an infinite sublattice for each v a the associated problem memb v of testing membership in transformation monoids that belong to v is considered remarkably the computational complexity of each such problem turns out to look familiar moreover only five possibilities occur as v ranges over the whole aperiodic sublattice with one family of np hard exceptions whose exact status is still unresolved any such memb v is either pspace complete np complete p complete or in ac 0 these results thus uncover yet another surprisingly tight link between the theory of monoids and computational complexity theory

global stability of a model for competing predators an extension of the ardito ricciardi lyapunov function global stability of a model for competing predators an extension of the ardito ricciardi lyapunov function

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

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

diffusion has no influence on the global asymptotical stability of the lotka volterra prey predator model incorporating a constant number of prey refuges by constructing a suitable lyapunov function we obtain the global asymptotical stability of the lotka volterra prey predator model incorporating a constant number of prey refuges and diffusion it is shown that diffusion has no influence on the global asymptotical stability of the system with homogeneous neumann boundary condition

learning regular sets from queries and counterexamples the problem of identifying an unknown regular set from examples of its members and nonmembers is addressed it is assumed that the regular set is presented by a minimamy adequate teacher which can answer membership queries about the set and can also test a conjecture and indicate whether it is equal to the unknown set and provide a counterexample if not a counterexample is a string in the symmetric difference of the correct set and the conjectured set a learning algorithm l is described that correctly learns any regular set from any minimally adequate teacher in time polynomial in the number of states of the minimum dfa for the set and the maximum length of any counterexample provided by the teacher it is shown that in a stochastic setting the ability of the teacher to test conjectures may be replaced by a random sampling oracle ex a polynomial time learning algorithm is shown for a particular problem of context free language identification

formulas regular languages and boolean circuits abstract this article surveys the links between regular languages and the class nc1 showing their importance in the classification of the fine structure of this parallel complexity class logical characterizations of these two classes are given in which the only difference lies in the use of arbitrary numerical predicates in the circuit case it is also shown that there is a unique maximal class of numerical predicates which allows us to define only regular languages several characterizations of this class are given

dynamics of a diffusive predator prey model with modified leslie gower and holling type iii schemes abstract the diffusive predator prey system with modified leslie gower and holling type iii schemes is considered here firstly stability analysis of the equilibrium for a reduced ode system is discussed secondly we obtain that the system is permanent thirdly sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by using the method of lyapunov function finally we establish the existence and nonexistence of nonconstant positive steady states of this reaction diffusion system which indicates the effect of large diffusivity

global asymptotic stability of a diffusive predator prey model with ratio dependent functional response abstract this paper is concerned with a diffusive leslie gower predator prey system with ratio dependent holling type iii functional response under homogeneous neumann boundary conditions the uniform persistence of the solutions semiflows the existence of global attractors local and global asymptotic stability of the positive constant steady state of the reaction diffusion model are discussed by using comparison principle the linearization method and the lyapunov functional method respectively the global asymptotic stability of the positive constant steady state shows that the prey and predator will be spatially homogeneously distributed as time converges to infinities

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

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

learning functions represented as multiplicity automata we study the learnability of multiplicity automata in angluin s exact learning model and we investigate its applications our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its hankel matrix with this theorem in hand we present a new simple algorithm for learning multiplicity automata with improved time and query complexity and we prove the learnability of various concept classes these include among others the class of disjoint dnf and more generally satisfy o 1 dnf the class of polynomials over finite fields the class of bounded degree polynomials over infinite fields the class of xor of terms certain classes of boxes in high dimensions in addition we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over gf 2 while multiplicity automata are shown to be useful to prove the learnability of some subclasses of dnf formulae and various other classes we study the limitations of this method we prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general dnf formulas or even k term dnf for k log n or satisfy s dnf formulas for s 1 these results are proven by exhibiting functions in the above classes that require multiplicity automata with super polynomial number of states

logspace and logtime leaf languages the computation tree of a nondeterministic machinemwith inputxgives rise to aleaf stringformed by concatenating the outcomes of all the computations in the tree in lexicographical order we may characterize problems by considering for a particular leaf language y the set of allxfor which the leaf string ofmis contained iny in this way in the context of polynomial time computation leaf languages were shown to capture many complexity classes in this paper we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation we show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages capturing also many classes withinp in contrast logtime leaf languages basically behave like logtime reducibilities both cases are more subtle to handle than the polynomial time case we also raise the issue of balanced versus nonbalanced computation trees underlying the leaf language we indicate that it is a nontrivial problem to obtain information about the leaf string of a nonbalanced computation tree and present conditions under which it does not matter whether the computation tree is balanced or not

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