extended formulations nonnegative factorizations and randomized communication protocols an extended formulation of a polyhedron p p is a linear description of a polyhedron q q together with a linear map pi such that pi q p q p these objects are of fundamental importance in polyhedral combinatorics and optimization theory and the subject of a number of studies yannakakis factorization theorem yannakakis in j comput syst sci 43 3 441 466 1991 provides a surprising connection between extended formulations and communication complexity showing that the smallest size of an extended formulation of p p equals the nonnegative rank of its slack matrix s s moreover yannakakis also shows that the nonnegative rank of s s is at most 2 c 2c where c c is the complexity of any deterministic protocol computing s s in this paper we show that the latter result can be strengthened when we allow protocols to be randomized in particular we prove that the base 2 2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation using yannakakis factorization theorem this implies that the base 2 2 logarithm of the smallest size of an extended formulation of a polytope p p equals the minimum complexity of a randomized communication protocol computing the slack matrix of p p in expectation we show that allowing randomization in the protocol can be crucial for obtaining small extended formulations specifically we prove that for the spanning tree and perfect matching polytopes small variance in the protocol forces large size in the extended formulation

alternating multihead finite automata abstract we define alternating multihead finite automata a generalization of nondeterministic multihead finite automata based on the alternating turing machine model introduced by chandra kozen and stockmeyer 1981 we study the relationships between the classes of languages accepted by alternating multihead finite automata and the classes accepted by deterministic and nondeterministic multihead finite automata and pushdown automata we also examine basic questions about alternating multihead finite automata for example are k 1 heads better than k we conclude by placing upper bounds on the deterministic time and space complexity of the classes of languages accepted by alternating multihead finite automata as corollaries to our results about alternating multihead finite automata we obtain several facts about multihead pushdown automata indicating that the study of alternating multihead finite automata may lead to useful results about nonalternating automata

nonregular languages in the kicked rotor a symbolic dynamics of the markov chain model for trajectories of the kicked rotor standard map in sticky regions near resonant orbits is developed two sets of trajectories are shown to correspond to a context free and a context sensitive language respectively

alternating pushdown and stack automata the classes of languages accepted by alternating pushdown automata alternating stack automata and alternating nonerasing stack automata both with and without an auxiliary space bounded worktape are characterized in terms of complexity classes defined by time bounded deterministic turing machines it is also shown that alternating 2 way finite state machines accept only regular languages

properties of probabilistic pushdown automata properties of probabilistic as well as probabilistic plus nondeterministic pushdown automata and auxiliary pushdown automata are studied these models are analogous to their counterparts with nondeterministic and alternating states complete characterizations in terms of well known complexity classes are given for the classes of languages recognized by polynomial time bounded logarithmic space bounded auxiliary pushdown automata with probabilistic states and with probabilistic plus nondeterministic states also complexity lower bounds are given for the classes of languages recognized by these automata with unlimited running time it follows that by fixing an appropriate mode of computation the difference between classes of languages such as p and pspace nl and sac 1 pl and diff sac 1 is characterized as the difference between the number of stack symbols that is whether the stack alphabet contains one versus two distinct symbols

testing computability by width two obdds property testing is concerned with deciding whether an object e g a graph or a function has a certain property or is far for a prespecified distance measure from every object with that property in this work we consider the property of being computable by a read once width 2ordered binary decision diagram obdd also known as a branching program in two settings in the first setting the order of the variables is fixed and given to the algorithm while in the second setting it is not fixed that is while in the first setting we should accept a function f if it is computable by a width 2 obdd with a given order of the variables in the second setting we should accept a function f if there exists an order of the variables according to which a width 2 obdd can compute f width 2 obdds generalize two classes of functions that have been studied in the context of property testing linear functions over gf 2 and monomials in both these cases membership can be tested by performing a number of queries that is independent of the number of variables n and is linear in 1 e where e is the distance parameter in contrast we show that testing computability by width 2 obdds when the order of variables is fixed and known requires a number of queries that grows logarithmically with n for a constant e and we provide an algorithm that performs email protected logn e queries for the case where the order is not fixed we show that there is no testing algorithm that performs a number of queries that is sublinear in n

prolongational limit sets of control systems abstract this article introduces the notions of prolongations and prolongational limit sets of control systems it is shown how the prolongational limit sets play in studying recursive and dispersive concepts of control systems

on the distributional complexity of disjointness we prove that the distributional communication complexity of the predicate disjointness with respect to a very simple measure on inputs is n

exponential lower bounds for polytopes in combinatorial optimization we solve a 20 year old problem posed by yannakakis and prove that no polynomial size linear program lp exists whose associated polytope projects to the traveling salesman polytope even if the lp is not required to be symmetric moreover we prove that this holds also for the cut polytope and the stable set polytope these results were discovered through a new connection that we make between one way quantum communication protocols and semidefinite programming reformulations of lps

communication lower bounds using directional derivatives we study the set disjointness problem in the most powerful model of bounded error communication the k party randomized number on the forehead model we show that set disjointness requires n 2kk bits of communication where n is the size of the universe our lower bound generalizes to quantum communication where it is essentially optimal proving this bound was a longstanding open problem even in restricted settings such as one way classical protocols with k 4 parties wigderson 1997 the proof contributes a novel technique for lower bounds on multiparty communication based on directional derivatives of protocols over the reals

approximation of boolean functions by combinatorial rectangles this paper deals with the number of monochromatic combinatorial rectangles required to approximate a boolean function on a constant fraction of all inputs where each rectangle may use its own partition of the input variables the main result of the paper is that the number of rectangles required for the approximation of boolean functions in this model is very sensitive to the allowed error there is an explicitly defined sequence of boolean functions fn on n variables such that fn has rectangle approximations with a constant number of rectangles and one sided error 13 o 1 or two sided error 14 o 1 but on the other hand fn requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant as applications of this result the following separation results for read once branching programs are obtained the functions from the main result require only linear size for nondeterministic read once branching programs and randomized read once branching programs with two sided error 13 o 1 while randomized read once branching programs with constant two sided error smaller than 13 and unambiguous nondeterministic read once branching programs require exponential size

attraction and lyapunov stability for control systems on vector bundles abstract let e b be a finite dimensional vector bundle whose base space is compact in this paper we study attraction and lyapunov stability for control systems on e we prove that under certain conditions the concepts of conley attractor uniform attractor attractor exponential attractor asymptotically stable set and stable set are equivalent for the zero section of e b

communication complexity method for measuring nondeterminism in finite automata while deterministic finite automata seem to be well understood surprisingly many important problems concerning nondeterministic finite automata nfa s remain open one such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata in this paper the concept of communication complexity is applied in order to achieve progress in this problem area the main results are as follows 1 deterministic communication complexity provides lower bounds on the size of nfa s with bounded unambiguity applying this fact the proofs of several results about nfa s with limited ambiguity can be simplified and presented in a uniform way 2 there is a family of languages konk2 with an exponential size gap between nfa s with polynomial leaf number ambiguity and nfa s with ambiguity k this partially provides an answer to the open problem posed by b ravikumar and o ibarra 1989 siam j comput 18 1263 1282 and h leung 1998 siam j comput 27 1073 1082

complete lyapunov functions of control systems abstract in this paper the notion of complete lyapunov function of control systems is introduced the purpose is to determine a continuous real valued function that describes the global structure of the system the existence of complete lyapunov functions is proved for certain classes of affine control systems on compact manifolds

critical properties of the s4 model on diamond type hierarchial lattices we study the critical properties of the s 4 model on diamond type hierarchical lattices in the presence of an external magnetic field it is assumed that for this type of inhomogenous fractal lattice the gaussian distribution constant the four spins interaction parameter and the external magnetic field in the s4 model depend on the coordination number of the site on the fractal lattices by combining the real space renormalization group scheme with the cumulative expansion method we obtain the critical points and further calculate critical exponents according to the scaling theory the results show that on diamond type hierarchical lattices with branches m 4 of 2 bonds the critical point of the s4 model is just the gaussian fixed point and therefore critical exponents are in full agreement with those of the gaussian model and that on those with m 4 the wilson fisher fixed point as well as the gaussian fixed point is obtained the wilson fisher fixed point has a decisive influence on the critical behavior of the system and critical exponents are related to the fractal dimensionality we found that the s4 model on the fractal lattices and that on the translation symmetric lattices show similar behaviors in the dependence of the critical properties on the dimensionality c 2005 elsevier b v all rights reserved

on morse decompositions of control systems in this article both notions of morse decomposition and dynamic morse decomposition of control systems are studied the main objective is to present the basic conditions which assure that a morse decomposition is a dynamic morse decomposition and vice versa

approximation limits of linear programs beyond hierarchies we develop a framework for proving approximation limits of polynomial size linear programs lps from lower bounds on the nonnegative ranks of suitably defined matrices this framework yields unconditional impossibility results that are applicable to any lp as opposed to only programs generated by hierarchies using our framework we prove that o n1 2 approximations for clique require lps of size 2n this lower bound applies to lps using a certain encoding of clique as a linear optimization problem moreover we establish a similar result for approximations of semidefinite programs by lps our main technical ingredient is a quantitative improvement of razborov s 38 rectangle corruption lemma for the high error regime which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix

real space renormalization group study of the phase transition in a gaussian model of fractals in this paper the phase transition of the gaussian model on m sheet fractals msg mathrm l and mdh mathrm l is investigated by the real space renormalization group method i e decimation following a spin rescaling the latter is introduced to keep the parameter b constant fixed points of the renormalization group transformation are found and discussed our results show the existence of different properties of phase transition between the gaussian model and the ising model on fractals in addition we find that the critical point k b 4 in a regular sierpinski gasket is identified with result of k b d d is the coordination number in euclidean space this indicates that the critical point of the gaussian model may be uniquely determined by the coordination number whether on homogeneous fractals or translationally invariant lattices

a parallel repetition theorem we show that a parallel repetition of any two prover one round proof system mip 2 1 decreases the probability of error at an exponential rate no constructive bound was previously known the constant in the exponent in our analysis depends only on the original probability of error and on the total number of possible answers of the two provers the dependency on the total number of possible answers is logarithmic which was recently proved to be almost the best possible u feige and o verbitsky proc 11th annual ieee conference on computational complexity ieee computer society press los alamitos ca 1996 pp 70 76

critical behavior of the gaussian model on a diamond type hierarchical lattice with periodic and aperiodic interactions abstract the critical behavior of the gaussian models on a diamond type hierarchical lattice with periodic and aperiodic interactions according to a class of substitution sequences are studied by the exact renormalization group approach and spin rescaling for the periodic case with a two letter general fibonacci sequence given by the generating substitutions a ab b ab and the aperiodic case with the generating substitutions a ab b ba the critical behaviors are found unchanged with respect to the uniform case and belong to the same universality of the uniform case for the other aperiodic case with the generating substitutions a ab b aa the geometric fluctuation is relevant by luck s criterion and the critical phenomena of the uniform and periodic cases are found to disappear

efficient analysis of probabilistic programs with an unbounded counter we show that a subclass of infinite state probabilistic programs that can be modeled by probabilistic one counter automata poc admits an efficient quantitative analysis we start by establishing a powerful link between poc and martingale theory which leads to fundamental observations about quantitative properties of runs in poc in particular we provide a divergence gap theorem which bounds a positive non termination probability in poc away from zero using these observations we show that the expected termination time can be approximated up to an arbitrarily small relative error in polynomial time and the same holds for the probability of all runs that satisfy a given regular property encoded by a deterministic rabin automaton

quantum entanglement and communication complexity we consider a variation of the communication complexity scenario where the parties are supplied with an extra resource particles in an entangled quantum state we note that quantum nonlocality can be naturally expressed in the language of communication complexity these are communication complexity problems where the output is embodied in the correlations between the outputs of the individual parties without entanglement the parties must communicate to produce the required correlations whereas with entanglement no communication is necessary to produce the correlations in this sense nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication we show how to transform examples of nonlocality into more traditional communication complexity problems where the output is explicitly determined by each individual party the resulting problems require communication with or without entanglement but the required communication is less when entanglement is available all these results are a noteworthy contrast to the well known fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties

on the power of randomized multicounter machines one way two counter machines represent a universal model of computation here we consider the polynomial time classes of multicounter machines with a constant number of reversals and separate the computational power of nondeterminism randomization and determinism for instance we show that polynomial time one way multicounter machines with error probability tending to zero with growing input length can recognize languages that cannot be accepted by polynomial time nondeterministic two way multicounter machines with a bounded number of reversals a similar result holds for the comparison of determinism and one sided error randomization and of determinism and las vegas randomization

graph distances in the data stream model we explore problems related to computing graph distances in the data stream model the goal is to design algorithms that can process the edges of a graph in an arbitrary order given only a limited amount of working memory we are motivated by both the practical challenge of processing massive graphs such as the web graph and the desire for a better theoretical understanding of the data stream model in particular we are interested in the trade offs between model parameters such as per data item processing time total space and the number of passes that may be taken over the stream these trade offs are more apparent when considering graph problems than they were in previous streaming work that solved problems of a statistical nature our results include the following 1 spanner construction there exists a single pass tilde o tn 1 1 t space tilde o t 2n 1 t time per edge algorithm that constructs a 2t 1 spanner for t omega log n log log n the algorithm satisfies the semistreaming space restriction of o n operatorname polylog n and has per edge processing time o operatorname polylog n this resolves an open question from j feigenbaum et al theoret comput sci 348 2005 pp 207 216 2 breadth first search bfs trees for any even constant k we show that any algorithm that computes the first k layers of a bfs tree from a prescribed node with probability at least 2 3 requires either greater than k 2 passes or tilde omega n 1 1 k space since constructing bfs trees is an important subroutine in many traditional graph algorithms this demonstrates the need for new algorithmic techniques when processing graphs in the data stream model 3 graph distance lower bounds any t approximation of the distance between two nodes requires omega n 1 1 t space we also prove lower bounds for determining the length of the shortest cycle and other graph properties 4 techniques for decreasing per edge processing we discuss two general techniques for speeding up the per edge computation time of streaming algorithms while increasing the space by only a small factor

characterization of uniform attractors for control systems this paper is devoted to the characterisation of uniform attractors for control systems by means of lyapunov functions we consider a uniform attractor that is compact and positively invariant by the system we present the relationship between the concept of uniform attractor and the conley concept of attractor

global attractors for semigroup actions abstract in this paper the notions of global attractor and global uniform attractor are extended to the setting of general semigroup actions we study the necessary and sufficient conditions for the existence of the global attractor for semigroups acting on noncompact metric spaces we present conditions for the equivalence between the global attractor and global uniform attractor

the sign rank of ac 0 the sign rank of a matrix a a ij with pm1 entries is the least rank of a real matrix b b ij with a ij b ij 0 for all i j we obtain the first exponential lower bound on the sign rank of a function in mathsf ac 0 namely let f x y bigwedge i 1 dots m bigvee j 1 dots m 2 x ij wedge y ij we show that the matrix f x y x y has sign rank exp omega m this in particular implies that sigma 2 cc not subseteq mathsf upp cc which solves a longstanding open problem in communication complexity posed by babai frankl and simon proceedings of the 27th symposium on foundations of computer science focs 1986 pp 337 347 our result additionally implies a lower bound in learning theory specifically let phi 1 dots phi r 0 1 n to mathbb r be functions such that every dnf formula f 0 1 n to 1 1 of polynomial size has the representation f equiv mathrm sgn a 1 phi 1 dots a r phi r for some reals a 1 dots a r we prove that then r geqslant exp omega n 1 3 which essentially matches an upper bound of exp tilde o n 1 3 due to klivans and servedio j comput system sci 68 2004 pp 303 318 finally our work yields the first exponential lower bound on the size of threshold of majority circuits computing a function in mathsf ac 0 this substantially generalizes and strengthens the results of krause and pudlak theoret comput sci 174 1997 pp 137 156

on the power of las vegas ii two way finite automata abstract the investigation of the computational power of randomized computations is one of the central tasks of complexity and algorithm theory while for one way finite automata the power of different computational modes was successfully determined one does not have any nontrivial result relating the power of determinism las vegas and nondeterminism for two way finite automata the main results of this paper are as follows i if for a regular language l there exist small two way nondeterministic finite automata for both l and l then there exists a small two way las vegas finite automaton for l ii there is a quadratic gap between nondeterminism and las vegas for two way finite automata iii for every k n there is a regular language s k such that s k can be accepted by a two way las vegas finite automaton with o k states but every two way deterministic finite automaton recognizing s k has at least k 2 log 2 k states