one way permutations computational asymmetry and distortion abstract computational asymmetry i e the discrepancy between the complexity of transformations and the complexity of their inverses is at the core of one way transformations we introduce a computational asymmetry function that measures the amount of one wayness of permutations we also introduce the word length asymmetry function for groups which is an algebraic analogue of computational asymmetry we relate combinational circuits to words in a thompson monoid over a fixed generating set in such a way that circuit size is equal to word length moreover combinational circuits have a representation in terms of elements of a thompson group in such a way that circuit size is polynomially equivalent to word length we show that circuits built with gates that are not constrained to have fixed length inputs and outputs are at most quadratically more compact than circuits built from traditional gates with fixed length inputs and outputs finally we show that the computational asymmetry function is closely related to certain distortion functions the computational asymmetry function is polynomially equivalent to the distortion of the path length in schreier graphs of certain thompson groups compared to the path length in cayley graphs of certain thompson monoids we also show that the results of razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions

existence and stability of nonlinear functional differential equations existence and stability results for a class of nonlinear functional differential equations which contains the generalised pantograph equation as a special case are obtained

when and how can a cellular automaton be rewritten as a lattice gas both cellular automata ca and lattice gas automata lg provide finite algorithmic presentations for certain classes of infinite dynamical systems studied by symbolic dynamics it is customary to use the terms cellular automaton and lattice gas for a dynamic system itself as well as for its presentation the two kinds of presentation share many traits but also display profound differences on issues ranging from decidability to modeling convenience and physical implementability following a conjecture by toffoli and margolus it had been proved by kari that any invertible ca at least up to two dimensions can be rewritten as an isomorphic lg but until now it was not known whether this is possible in general for noninvertible ca which comprise almost all ca and represent the bulk of examples in theory and applications even circumstantial evidence whether in favor or against was lacking here for noninvertible ca a we prove that an lg presentation is out of the question for the vanishingly small class of surjective ones we then turn our attention to all the rest noninvertible and nonsurjective which comprise all the typical ones including conway s game of life for these b we prove by explicit construction that all the one dimensional ones are representable as lg and c we present and motivate the conjecture that this result extends to any number of dimensions the tradeoff between dissipation rate and structural complexity implied by the above results have compelling implications for the thermodynamics of computation at a microscopic scale

time space trade offs for reversible computation a reversible turing machine is one whose transition function is 1 1 so that no instantaneous description id has more than one predecessor using a pebbling argument this paper shows that for any varepsilon 0 ordinary multitape turing machines using time t and space s can be simulated by reversible ones using time o t 1 varepsilon and space o s log t or in linear time and space o st varepsilon the former result implies in particular that reversible machines can simulate ordinary ones in quadratic space these results refer to reversible machines that save their input thereby insuring a global 1 1 relation between initial and final ids even when the function being computed is many to one reversible machines that instead erase their input can of course compute only 1 1 partial recursive functions and indeed provide a godel numbering of such functions the time space cost of computing a 1 1 function on such a machine is equal within a small polynomial to the cost of co

efficient and exact quantum compression we present a divide and conquer based algorithm for optimal quantum compression decompression using o n log 4n loglogn elementary quantum operations our result provides the first quasi linear time algorithm for asymptotically optimal in size and fidelity quantum compression and decompression we also outline the quantum gate array model to bring about this compression in a quantum computer our method uses various classical algorithmic tools to significantly improve the bound from the previous best known bound of o n 3 for this operation

reversible arithmetic coding for quantum data compression we study the problem of compressing a block of symbols a block quantum state emitted by a memoryless quantum bernoulli source we present a simple to implement quantum algorithm for projecting with high probability the block quantum state onto the typical subspace spanned by the lending eigenstates of its density matrix we propose a fixed rate quantum shannon fano code to compress the projected block quantum state using a per symbol code rate that is slightly higher than the von neumann 1955 entropy limit finally we propose quantum arithmetic codes to efficiently implement quantum shannon fano 1948 codes our arithmetic encoder and decoder have a cubic circuit and a cubic computational complexity in the block size both the encoder and decoder are quantum mechanical inverses of each other and constitute an elegant example of reversible quantum computation

genotypes of irreducible representations of finite p groups for any characteristic zero coefficient field an irreducible representation of a finite p group can be assigned a roquette p group called the genotype this has already been done by bouc and kronstein in the special cases q and c a genetic invariant of an irrep is invariant under group isomorphism change of coefficient field galois conjugation and under suitable inductions from subquotients it turns out that the genetic invariants are precisely the invariants of the genotype we shall examine relationships between some genetic invariants and the genotype as an application we shall count galois conjugacy classes of certain kinds of irreps

monotonic solutions of a convolution functional integral equation in this paper we show that a very general convolution functional integral equation has at least one monotonic solution in the space of lebesgue integrable functions on an unbounded interval a suitable combination of the technique associated with measures of noncompactness in both the weak and the strong sense and the darbo fixed point is the main tool in our analysis

rush hour is psapce complete or why you should generously tip parking lot attendants rush hour is a children s game that consists of a grid board several cars that are restricted to move either vertically or horizontally but not both a special target car and a single exit on the perimeter of the grid the goal of the game is to find a sequence of legal moves that allows the target car to exit the grid we consider a slightly generalized version of the game that uses an nn grid and assume that we can place the single exit and target car at any location we choose on initialization of the game in this work we show that deciding if the target car can legally exit the grid is pspace complete our constructive proof uses a lazy form of dual rail reversible logic such that movement of output cars can only occur if logical combinations of input cars can also move emulating this logic only requires three types of devices two switches and one crossover thus our proof technique can be easily generalized to other games and planning problems in which the same three primitive devices can be constructed

information distance while kolmogorov 1965 complexity is the accepted absolute measure of information content in an individual finite object a similarly absolute notion is needed for the information distance between two individual objects for example two pictures we give several natural definitions of a universal information metric based on length of shortest programs for either ordinary computations or reversible dissipationless computations it turns out that these definitions are equivalent up to an additive logarithmic term we show that the information distance is a universal cognitive similarity distance we investigate the maximal correlation of the shortest programs involved the maximal uncorrelation of programs a generalization of the slepian wolf theorem of classical information theory and the density properties of the discrete metric spaces induced by the information distances a related distance measures the amount of nonreversibility of a computation using the physical theory of reversible computation we give an appropriate universal antisymmetric and transitive measure of the thermodynamic work required to transform one object in another object by the most efficient process information distance between individual objects is needed in pattern recognition where one wants to express effective notions of pattern similarity or cognitive similarity between individual objects and in thermodynamics of computation where one wants to analyze the energy dissipation of a computation from a particular input to a particular output

reversible simulation of irreversible computation abstract computer computations are generally irreversible while the laws of physics are reversible this mismatch is penalized by among other things generating excess thermic entropy in the computation computing performance has improved to the extent that efficiency degrades unless all algorithms are executed reversibly for example by a universal reversible simulation of irreversible computations all known reversible simulations are either space hungry or time hungry the leanest method was proposed by bennett and can be analyzed using a simple reversible pebble game the reachable reversible simulation instantaneous descriptions pebble configurations of such pebble games are characterized completely as a corollary we obtain the reversible simulation by bennett and moreover show that it is a space optimal pebble game we also introduce irreversible steps and give a theorem on the tradeoff between the number of allowed irreversible steps and the memory gain in the pebble game in this resource bounded setting the limited erasing needs to be performed at precise instants during the simulation the reversible simulation can be modified so that it is applicable also when the simulated computation time is unknown

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

a class of recursive permutations which is primitive recursive complete abstract we focus on total functions in the theory of reversible computational models we define a class of recursive permutations dubbed reversible primitive permutations rpp which are computable invertible total endo functions on integers so a subset of total reversible computations rpp is generated from five basic functions identity sign change successor predecessor swap two notions of composition sequential and parallel one functional iteration and one functional selection rpp is closed by inversion and it is expressive enough to encode cantor pairing and the whole class of primitive recursive functions

oscillations of first order linear retarded differential equations the oscillatory behavior of differential equations with deviating arguments has been studied by many authors in particular several authors have studied the oscillations caused by the deviating arguments the study of such oscillations is a relatively new field and is very interesting in applications for some contributions in this area see the papers by arino gyori and jawhari 11 bowcock and yu 2 gyiiri 4 5 j hunt and yorke 6 kulenovic ladas and meimaridou 7 ladas s ladas sficas and stavroulakis 9 lo ladas and stavroulakis 111 sugie 141 tramov 15 and yan 16 we also refer to the recent book by ladde lakshmikantham and zhang 123 the purpose of this paper is to investigate the oscillatory behavior of the solutions of first order linear retarded differential equations with variable coefftcients and nonconstants delays consider the first order retarded differential equation

on a certain inequality arising in the theory of differential equations abstract in the present paper we establish some new inequalities related to a certain integral inequality arising in the theory of differential equations the inequalities that we propose here can be used as handy tools in the theory of certain new classes of differential and integral equations

on some new inequalities related to certain inequalities in the theory of differential equations in the present paper we establish some new integral inequalities related to a certain inequality arising in the theory of differential equations the discrete analogues of the main results are also given the analysis used in the proofs is elementary and the results established here provide new estimates for these types of inequalities

topological transversality application to an integrodifferential equation 2 k t s x f f t g s w x t s x g r r 3 q where f g h g c r r and w belongs to the class r of continuous q q nondecreasing functions on r with w r 0 for r 0 and satisfying q h dsrw s s 1 under assumptions 2 and 3 we show that 1 has a solution on rq and we give bounds on that solution our discussion is based on topological transversality arguments the last section is devoted to an application in the mathematical theory of water percolation phenomena

note monotone iterative technique for a nonlinear integral equation abstract a result for the existence of a positive solution to a nonlinear integral equation is proved using the monotone iterative technique and an application in the mathematical theory of water percolation phenomena is given

time and space bounds for reversible simulation we prove a general upper bound on the trade off between time and space that suffices for the reversible simulation of irreversible computation previously only simulations using exponential time or quadratic space were known the trade off shows for the first time that we can simultaneously achieve subexponential time and subquadratic space the boundary values are the exponential time with hardly any extra space required by the lange mckenzie tapp method and the log 3 th power time with square space required by the bennett method we also give the first general lower bound on the extra storage space required by general reversible simulation this lower bound is optimal in that it is achieved by some reversible simulations

existence of solutions to some retarded integrodifferential equations via the topological transversality theorem abstract we prove the existence of solutions to some retarded integrodifferential equations under some suitable conditions on the involved functions some applications of our results are also provided

bounds on monotone switching networks for directed connectivity we separate monotone analogues of l and nl by proving that any monotone switching network solving directed connectivity on n vertices must have size at least n lg n

quantum formulas a lower bound and simulation we show that nechiporuk s method i wegener the complexity of boolean functions teubner wiley new york 1987 for proving lower bounds for boolean formulas can be extended to the quantum case this leads to an omega n 2 log 2 n lower bound for quantum formulas computing an explicit function the only known previous explicit lower bound for quantum formulas a yao proceedings of 34th ieee symposium on foundations of computer science ieee computer society press los alamitos ca 1993 pp 352 361 states that the majority function does not have a linear size quantum formula we also show that quantum formulas can be simulated by boolean circuits of almost the same size

rational representations and permutation representations of finite groups we investigate the question which mathbb q valued characters and characters of mathbb q representations of finite groups are mathbb z linear combinations of permutation characters this question is known to reduce to that for quasi elementary groups and we give a solution in that case as one of the applications we exhibit a family of simple groups with rational representations whose smallest multiple that is a permutation representation can be arbitrarily large

asymptotic properties of solutions to some nonlinear integral equations of convolution type for a class of nonlinear integral equations of convolution type we give necessary and sufficient conditions for the boundedness of nonnegative solutions moreover conditions for the solution to converge asymptotically to a determined limit are obtained

integral inequalities for retarded volterra equations integral inequalities are very useful in the qualitative analysis of differential and integral equations starting with o lipovan a retarded gronwall like inequality and its applications j math anal appl 252 2000 389 401 several recent investigations see o lipovan a retarded integral inequality and its applications j math anal appl 285 2003 436 443 b g pachpatte explicit bounds on certain integral inequalities j math anal appl 267 2002 48 61 b g pachpatte on some retarded integral inequalities and applications j inequal pure appl math 3 2002 article 18 b g pachpatte on a certain retarded integral inequality and its applications j inequal pure appl math 5 2004 article 19 b g pachpatte on some new nonlinear retarded integral inequalities j inequal pure appl math 5 2004 article 80 were devoted to retarded integral inequalities in this paper we consider the case of retarded volterra integral equations we establish bounds on the solutions and by means of examples we show the usefulness of our results in investigating the asymptotic behaviour of the solutions

one complexity theorist s view of quantum computing the complexity of quantum computation remains poorly understood while physicists attempt to find ways to create quantum computers we still do not have much evidence one way or the other as to how useful these machines will be the tools of computational complexity theory should come to bear on these important questions quantum computing often scares away many potential researchers in computer science because of the apparent background need in quantum mechanics and the alien looking notation used in papers on the topic this paper will give an overview of quantum computation from the point of view of a complexity theorist we will see that one can think of bqp as yet another complexity class and study its power without focusing on the physical aspects behind it

continuability of solutions of perturbed differential equations on considere les systemes d equations differentielles suivants e x f t x p x f t x g t x per x f t x h t ou x f g h sont des n vecteurs f t x g t x sont continues sur 0 r n et h t est continue sur 0 on demontre des theoremes sur la perturbation de la continuite des solutions de e a p et per

a retarded gronwall like inequality and its applications we prove a gronwall like inequality and present some of its applications to the qualitative study of retarded differential equations the problems of global continuation of the solutions and the existence of nonoscillatory solutions are considered by means of examples we show the usefulness of our results

the word problem of the brin thompson group is conp complete abstract we prove that the word problem of the brin thompson group nv over a finite generating set is conp complete for every n 2 it is known that n v n 1 is an infinite family of infinite finitely presented simple groups we also prove that the word problem of the thompson group v over a certain infinite set of generators related to boolean circuits is conp complete

explicit lower bounds on strong quantum simulation we consider the problem of classical strong amplitude wise simulation of n qubit quantum circuits and identify a subclass of simulators we call monotone this subclass encompasses almost all prominent simulation techniques we prove an unconditional i e without relying on any complexity theoretic assumptions and explicit n 2 2 n 3 1 lower bound on the running time of simulators within this subclass assuming the strong exponential time hypothesis seth we further remark that a universal simulator computing any amplitude to precision 2 n 2 must take at least 2 n o n time we then compare strong simulators to existing sat solvers and identify the time complexity below which a strong simulator would improve on state of the art general sat solving finally we investigate clifford t quantum circuits with t t gates using the sparsification lemma we identify a time complexity lower bound of 2 2 2451 times 10 8 t below which a strong simulator would improve on state of the art 3 sat solving this also yields a conditional exponential lower bound on the growth of the stabilizer rank of magic states

pspace complete problems for subgroups of free groups and inverse finite automata we investigate the complexity of algorithmic problems on finitely generated subgroups of free groups margolis and meakin showed how a finite monoid synt h can be canonically and effectively associated with such a subgroup h we show that h is pure that is closed under radical if and only if synt h is aperiodic we also show that testing for this property of h is pspace complete in the process we show that certain problems about finite automata which are pspace complete in general remain pspace complete when restricted to injective and inverse automata with single accept state whereas they are known to be in nc for permutation automata with single accept state

the complexity of reversible cellular automata we study the orbits of reversible one dimensional cellular automata it is shown that the turing degree structure of the orbits of these automata is the same as for general cellular automata in particular there are reversible cellular automata whose orbits have arbitrary recursively enumerable degree

a retarded integral inequality and its applications we prove a retarded nonlinear integral inequality and present some applications of it to the global existence of solutions to differential equations with time delay