accepting runs in a two way finite automaton abstract an accepting run in a two way finite automaton m is a sequence of states that m enters during some accepting computation the set of all such runs is denoted by l run m we study the complexity of l run m when m is a 2nfa 2dfa we also look at the complexity of position sampling the sequence of states that m enters in specified positions of some accepted input in a 2nfa in particular we give some language properties of sampled runs of 2nfas augmented with restricted unbounded storage

a note on the ziv lempel model for compressing individual sequences corresp the ziv lempel compression algorithm is a string matching and parsing approach to data compression the symbolwise equivalent for parsing models has been defined by rissanen and langdon and gives the same ideal codelength at the same cost in coding parameters by describing the context and coding parameter for each symbol an insight is provided into how the ziv lempel method achieves compression this treatment does not employ a probabilistic source for the data string the ziv lempel method effectively counts symbol instances within parsed phrases the coding parameter for each symbolwise context is determined by cumulative count ratios the code string length increase for a symbol y following substring s under the symbolwise equivalent is the log of the ratio of node counts in subtrees s and s cdot y of the ziv lempel parsing tree to demonstrate the symbolwise equivalent of the ziv lempel algorithm we extend the work of rissanen and langdon to incomplete parse trees the result requires the proper handling of the comma when one phrase is the prefix of another phrase

self avoiding walks that cross a square the authors consider self avoiding walks that traverse an l l square lattice whittington and guttmann 1990 have proved the existence of a phase transition in the infinite l limit at a critical value of the step fugacity they make several finite size scaling predictions for the critical region using the relation between self avoiding walks and the n vector model of magnetism adsorbing as well as nonadsorbing boundaries are considered the predictions are in good agreement with numerical data for l or 9

information rate of some classes of non regular languages an automata theoretic approach abstract we show that the information rate of the language accepted by a reversal bounded deterministic counter machine is computable for the nondeterministic case we provide computable upper bounds for the class of languages accepted by multi tape deterministic finite automata the information rate is computable as well

rate of pointwise approximation for locally bounded functions by szasz operators abstract in this paper the asymptotic behavior of szasz operators for locally bounded functions f is studied at points x where f x and f x exist an asymptotic estimate of this type approximation is obtained by using some techniques and results of probability theory the estimate essentially is the best possible for continuous points of f

transfer matrix study of the adsorption of a flexible self avoiding polymer chain in two dimensions the authors model the polymer as a self avoiding walk on a semi infinite square lattice with an energy contribution in from each step along the lattice boundary the critical adsorption temperature ta and the ordinary and special surface exponents of the walk above and at the adsorption temperature respectively are estimated with transfer matrix finite size scaling methods for the adsorption temperature they obtain exp in kta 2 041 or 0 002 and for the crossover and susceptibility exponents of the special transition phi sp 0 501 or 0 003 and gamma 1sp 1 454 or 0 004

quantifying communication in synchronized languages a mutual information rate is proposed to quantitatively evaluate inter process synchronized communication for finite state processes with implicit communication that can be described by a counting language it is shown that the mutual information rate is effectively computable when the synchronization always happens between the same two symbols at the same time or with a fixed delay the mutual information rate is computable in contrast when the delay is not fixed the rate is not computable finally it is shown that some cases exist where the mutual information rate is not computable

self avoiding walks which cross a square the authors consider self avoiding walks on the square lattice which are confined to lie in or on the boundary of a square with vertices at 0 0 0 l l 0 and l l they ask for the number of such walks which begin at the origin and end at the vertex l l especially in the large l limit similarly they ask for the mean number of steps in such walks as a function of l at fixed l the authors also associate a fugacity with the number of steps of the walk and ask how the system behaves as a function of this fugacity they provide some rigorous results in particular proving that there is a phase transition at some particular value of the fugacity and supplement these with the analysis of series data for the problem

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha

generalized discrete timed automata decidable approximations for safety verification we consider generalized discrete timed automata with general linear relations over clocks and parameterized constants as clock constraints and with parameterized durations we look at three approximation techniques i e the r reset bounded approximation the b bounded approximation and the b r crossing bounded approximation and derive automata theoretic characterizations of the binary reachability under these approximations the characterizations allow us to show that the safety analysis problem is decidable for generalized discrete timed automata with unit durations and for deterministic generalized discrete timed automata with parameterized durations an example specification written in astral is used to run a number of experiments using one of the approximation techniques

composite models of polygons a composite polygon is composed of a lattice polygon in the square lattice which contains in its interior an internal structure which may also be a lattice polygon or a lattice tree or a lattice animal or a lattice disc or a collection of these the properties of composite polygons are considered in this manuscript in particular i shall consider the growth constants and generating functions of these models as well as the statistical mechanics of interacting models of composite polygons it is shown that there is an adsorption transition of the internal structure on the containing polygon and a transition which corresponds to the inflation of the containing polygon by the internal structure

similarity in languages and programs we use an information theoretic notion namely shannon information rate to generalize common syntactic similarity metrics like hamming distance and longest common subsequences between strings to ones between languages we show that the similarity metrics between two regular languages are computable we further study self similarity of a regular language under various similarity metrics as far as semantic similarity is concerned we study the amplitude of an automaton which intuitively characterizes how much a typical execution of the automaton fluctuates finally we investigate through experiments how to measure similarity between two real world programs using lempel ziv compression on the runs at the assembly level

pushdown timed automata a binary reachability characterization and safety verification abstract we consider pushdown timed automata ptas that are timed automata with dense clocks augmented with a pushdown stack a configuration of a pta includes a state dense clock values and a stack word by using the pattern technique we give a decidable characterization of the binary reachability i e the set of all pairs of configurations such that one can reach the other of a pta since a timed automaton can be treated as a pta without the pushdown stack we can show that the binary reachability of a timed automaton is definable in the additive theory of reals and integers the results can be used to verify a class of properties containing linear relations over both dense variables and unbounded discrete variables the properties previously could not be verified using the classic region technique nor expressed by timed temporal logics for timed automata and ctl for pushdown systems the results are also extended to other generalizations of timed automata

sampling automata and programs we study sampling schemes for executions of automata and programs which can be used based on shannon information theory to measure how much information flows from one variable to another we show that the information rates of periodically sampled executions of a nondeterministic finite automaton and of a reversal bounded nondeterministic multicounter machine are computable finally through experiments we use lempel ziv compression algorithm to approximate information leakage in programs

approximation properties of bezier summation integral type operators based on polya bernstein functions in this article we introduce the bezier variant of summation integral type operators having polya and bernstein basis functions we give a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation

on two way fa with monotonic counters and quadratic diophantine equations we show an interesting connection between two way deterministic finite automata with monotonic counters and quadratic diophantine equations the automaton m operates on inputs of the form a1i1 anin for some fixed n and distinct symbols a1 an where i1 in are nonnegative integers we consider the following reachability problem given a machine m a state q and a presburger relation e over counter values is there i1 in such that m when started in its initial state on the left end of the input a1i1 anin with all counters initially zero reaches some configuration where the state is q and the counter values satisfy e in particular we look at the case when the relation e is an equality relation i e a conjunction of relations of the form ci cj we show that this case and variations of it are equivalent to the solvability of some special classes of systems of quadratic diophantine equations we also study the nondeterministic version of two way finite automata augmented with monotonic counters with respect to the reachability problem finally we introduce a technique which uses decidability and undecidability results to show separation between language classes

critical behaviour of self avoiding walks that cross a square consider the set of all self avoiding walks in the square lattice which start at 0 0 end at l l and are entirely contained in the square 0 l 0 l associate a fugacity x with each step of the walk whittington and guttmann 1990 showed that the dominant walks have o l steps when x is small and o l2 steps when x is large and they conjectured that there is a single transition point at x mu 1 where mu is the inverse of the connective constant for unconstrained self avoiding walks we present a rigorous proof of this conjecture and its analogue in higher dimensions we also discuss what can be said rigorously about two scaling exponents associated with this phase transition and compare this with analogous results that have been obtained exactly and rigorously on the discrete sierpinski gasket by hattori hattori and kusuoka 1990

regret minimization under partial monitoring we consider repeated games in which the player instead of observing the action chosen by the opponent in each game round receives a feedback generated by the combined choice of the two players we study hannan consistent players for these games that is randomized playing strategies whose per round regret vanishes with probability one as the number n of game rounds goes to infinity we prove a general lower bound of n 1 3 for the convergence rate of the regret and exhibit a specific strategy that attains this rate for any game for which a hannan consistent player exists

bounds for bernstein basis functions and meyer konig and zeller basis functions abstract in this paper the inequality estimates of bernstein basis functions and meyer konig and zeller basis functions are studied exact bounds for these two basis functions are obtained moreover some application results of the new estimates in estimating the rate of convergence of durrmeyer operators and meyer koning and zeller operators for functions of bounded variation are also given

nonparametric output prediction for nonlinear fading memory systems the authors construct a class of elementary nonparametric output predictors of an unknown discrete time nonlinear fading memory system their algorithms predict asymptotically well for every bounded input sequence every disturbance sequence in certain classes and every linear or nonlinear system that is continuous and asymptotically time invariant causal and with fading memory the predictor is based on k sub n nearest neighbor estimators from nonparametric statistics it uses only previous input and noisy output data of the system without any knowledge of the structure of the unknown system the bounds on the input or the properties of noise under additional smoothness conditions the authors provide rates of convergence for the time average errors of their scheme finally they apply their results to the special case of stable linear time invariant lti systems

rate of convergence of bernstein polynomials for functions with derivatives of bounded variation abstract for any x 0 1 we first prove that if x t t x on 0 1 then the bernstein polynomials of x satisfy the asymptotic relation k 0 n k n x k n x k 1 x n k 2x 1 x 1 2 1 n o 1 n this asymptotic relation is then used to study the rate of convergence of bernstein polynomials for functions with derivatives of bounded variation an estimate of the rate of convergence is given this estimate is asymptotically the best possible at points where is continuous

counter machines and verification problems we study various generalizations of reversal bounded multicounter machines and show that they have decidable emptiness infiniteness disjointness containment and equivalence problems the extensions include allowing the machines to perform linear relation tests among the counters and parameterized constants e g is 3x 5y 2d1 9d2 12 where x y are counters and d1 d2 are parameterized constants we believe that these machines are the most powerful machines known to date for which these decision problems are decidable decidability results for such machines are useful in the analysis of reachability problems and the verification debugging of safety properties in infinite state transition systems for example we show that binary forward and backward reachability and safety are solvable for these machines

descriptional complexity of two way pushdown automata with restricted head reversals two way nondeterministic pushdown automata 2pda are classical nondeterministic pushdown automata pda enhanced with two way motion of the input head in this paper the subclass of 2pda accepting bounded languages and making at most a constant number of input head turns is studied with respect to descriptional complexity aspects in particular the effect of reducing the number of pushdown reversals to a constant number is of interest it turns out that this reduction leads to an exponential blow up in case of nondeterministic devices and to a doubly exponential blow up in case of deterministic devices if the restriction on boundedness of the languages considered and on the finiteness of the number of head and pushdown turns is dropped the resulting trade offs are no longer bounded by recursive functions and so called non recursive trade offs are shown

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

deinterleaving finite memory processes via penalized maximum likelihood we study the problem of deinterleaving a set of finite memory markov processes over disjoint finite alphabets which have been randomly interleaved by a finite memory switch the deinterleaver has access to a sample of the resulting interleaved process but no knowledge of the number or structure of the component markov processes or of the switch we study conditions for uniqueness of the interleaved representation of a process showing that certain switch configurations as well as memoryless component processes can cause ambiguities in the representation we show that a deinterleaving scheme based on minimizing a penalized maximum likelihood cost function is strongly consistent in the sense of reconstructing almost surely as the observed sequence length tends to infinity a set of component and switch markov processes compatible with the original interleaved process furthermore under certain conditions on the structure of the switch including the special case of a memoryless switch we show that the scheme recovers all possible interleaved representations of the original process experimental results are presented demonstrating that the proposed scheme performs well in practice even for relatively short input samples

on the solvability of a class of diophantine equations and applications for 1 i k let ri denote pi y fi gi where pi y is a polynomial in y with integer coefficients and fi gi are linear polynomials in x1 xn with integer coefficients let p z1 zk be a presburger relation over the nonnegative integers we show that the following problem is decidable given r1 rk and a presburger relation p question are there nonnegative integer values for y x1 xn such that for these values r1 rk satisfies p we also give some applications to decision problems concerning counter machines

on synchronized multi tape and multi head automata motivated by applications to verification problems in string manipulating programs we look at the problem of whether the heads in a multi tape automaton are synchronized given an n tape pushdown automaton m with a one way read only head per tape and a right end marker on each tape and an integer k 0 we say that m is k synchronized if at any time during any computation of m on any input n tuple x 1 x n whether or not it is accepted no pair of input heads that are not on are more than k cells apart this requirement is automatically satisfied if one of the heads has reached note that an n tuple x 1 x n is accepted if m reaches the configuration where all n heads are on and m is in an accepting state the automaton can be deterministic dpda or nondeterministic npda and in the special case may not have a pushdown stack dfa nfa we obtain decidability and undecidability results for these devices for both one way and two way versions we also consider the notion of k synchronized one way and two way multi head automata and investigate similar problems

approximation for bounded functions and absolutely continuous functions by beta operators in this paper the pointwise approximation properties of beta operators n are studied to the bounded functions and the absolutely continuous functions respectively first we use the asymptotic form of the central limit theorem in probability theory to derive an asymptotic estimate on the rate of convergence of beta operators n for the bounded functions next we give the optimal estimate on the first order absolute moment of the beta operators b n email protected g by direct computations then we use this estimate and bojanic cheng khan s method combining with some analysis techniques to derive an asymptotically optimal estimate on the rate of convergence of beta operators n for the absolutely continuous functions

universal reinforcement learning we consider an agent interacting with an unmodeled environment at each time the agent makes an observation takes an action and incurs a cost its actions can influence future observations and costs the goal is to minimize the long term average cost we propose a novel algorithm known as the active lz algorithm for optimal control based on ideas from the lempel ziv scheme for universal data compression and prediction we establish that under the active lz algorithm if there exists an integer k such that the future is conditionally independent of the past given a window of k consecutive actions and observations then the average cost converges to the optimum experimental results involving the game of rock paper scissors illustrate merits of the algorithm

asymptotic behaviour of the number of self avoiding walks on finitely ramified fractals we study self avoiding walks saws on the checkerboard cb family of fractals each member of which can be labelled by an odd integer b 3 or b infinity so that the fractal dimension df tends to the euclidean value 2 when b to infinity by applying the exact renormalization group method for b 3 5 and 7 and the monte carlo renormalization group method for a sequence of fractals with 5 or b or 81 we have calculated the critical exponent gamma associated with the total number of distinct saws it turns out that gamma being always larger than the corresponding euclidean value 43 32 increases monotonically with b in order to learn the asymptotic behaviour of gamma for large b we have applied the finite size scaling fss method based on previous exact results for wedges of the two dimensional euclidean lattices and thereby we have shown that applying the same method fss we have also demonstrated that the critical exponent nu associated with the mean squared end to end distance of saws tends to the euclidean value 3 4 from below when b to infinity the obtained results extend and in a way weld together the previous studies of saws various families of finitely ramified fractals

exact transfer matrix enumeration and critical behaviour of self avoiding walks across finite strips we use a transfer matrix with suitably defined vertex weights to algebraically enumerate n step self avoiding walks confined to cross an l m rectangle on the square lattice we construct the exact generating functions for self avoiding walks from the south west to south east corners for l 1 2 3 4 5 and infinite height m corresponding to a half strip we also consider the number of n sided polygons rooted to the south west corner of the half strip and give a formulation to treat self avoiding walks across the full strip in each case the exact generating functions are ratios of polynomials in the step fugacity we investigate the singularity structure of the generating functions along with the finite size scaling in m of the singularity in the analogue of the heat capacity we find the critical exponents gamma 1 and gamma 2 for the half and open strip along with nu 1 2 these results are indicative of the one dimensional or gaussian nature of self avoiding walks in infinitely long but finitely wide strips

on context tree prediction of individual sequences motivated by the evident success of context tree based methods in lossless data compression we explore in this correspondence methods of the same spirit in universal prediction of individual sequences by context tree prediction we refer to a family of prediction schemes where at each time instant t after having observed all outcomes of the data sequence x1 xt 1 but not yet xt the prediction is based on a context or a state that consists of the k most recent past outcomes xt k xt 1 where the choice of k may depend on the contents of a possibly longer though limited portion of the observed past xt kmax xt 1 this is different from the study reported in the paper by feder merhav and gutman 1992 where general finite state predictors as well as markov finite memory predictors of fixed order were studied in the regime of individual sequences another important difference between this study and the work of feder is the asymptotic regime while in their work the resources of the predictor i e the number of states or the memory size were kept fixed regardless of the length n of the data sequence here we investigate situations where the number of contexts or states is allowed to grow concurrently with n we are primarily interested in the following fundamental question what is the critical growth rate of the number of contexts below which the performance of the best context tree predictor is still universally achievable but above which it is not we show that this critical growth rate is linear in n in particular we propose a universal context tree algorithm that essentially achieves optimum performance as long as the growth rate is sublinear and show that on the other hand this is impossible in the linear case

rate of convergence of lupas kantorovich operators based on polya distribution in the present paper we consider the kantorovich modification of lupas operators based on polya distribution we estimate the rate of convergence for absolutely continuous functions having a derivative coinciding a e with a function of bounded variation

self avoiding walks crossing a square we study a restricted class of self avoiding walks saws which start at the origin 0 0 end at l l and are entirely contained in the square 0 l 0 l on the square lattice z 2 the number of distinct walks is known to grow as l 2 o l 2 we estimate 1 744 550 0 000 005 as well as obtaining strict upper and lower bounds 1 628 1 it grows as l 2 h ere is the growth constant of unconstrained saws in z 2 f orx 1 we provide numerical evidence but no proof that the average walk length grows as l 4 3 another problem we study is that of saws as described above that pass through the central vertex of the square we estimate the proportion of such walks as a fraction of the total and find it to be just below 80 of the total number of saws we also consider hamiltonian walks under the same restriction they are known to grow as l 2 o l2

universal filtering via hidden markov modeling the problem of discrete universal filtering in which the components of a discrete signal emitted by an unknown source and corrupted by a known discrete memoryless channel dmc are to be causally estimated is considered a family of filters are derived and are shown to be universally asymptotically optimal in the sense of achieving the optimum filtering performance when the clean signal is stationary ergodic and satisfies an additional mild positivity condition our schemes are comprised of approximating the noisy signal using a hidden markov process hmp via maximum likelihood ml estimation followed by the use of the forward recursions for hmp state estimation it is shown that as the data length increases and as the number of states in the hmp approximation increases our family of filters attains the performance of the optimal distribution dependent filter an extension to the case of channels with memory is also established

on two way nondeterministic finite automata with one reversal bounded counter we show that the emptiness problem for two way nondeterministic finite automata augmented with one reversal bounded counter i e the counter alternates between nondecreasing and nonincreasing modes for a fixed number of times operating on bounded languages i e subsets of w 1 w k for some nonnull words w1 wk is decidable resolving an open problem the proof is a rather involved reduction to the solution of a special class of diophantine systems of degree 2 via a class of programs called two phase programs

on finite memory universal data compression and classification of individual sequences consider the case where consecutive blocks of letters of a semi infinite individual sequence over a finite alphabet are being compressed into binary sequences by some one to one mapping no a priori information about is available at the encoder which must therefore adopt a universal data compression algorithm it is known that if the universal lempel ziv lz data compression algorithm is successively applied to blocks then the best error free compression for the particular individual sequence is achieved as tends to infinity the best possible compression that may be achieved by any universal data compression algorithm for finite blocks is discussed it is demonstrated that context tree coding essentially achieves it next consider a device called classifier or discriminator that observes an individual training sequence the classifier s task is to examine individual test sequences of length and decide whether the test sequence has the same features as those that are captured by the training sequence or is sufficiently different according to some appropriate criterion here again it is demonstrated that a particular universal context classifier with a storage space complexity that is linear in is essentially optimal this may contribute a theoretical ldquoindividual sequencerdquo justification for the probabilistic suffix tree pst approach in learning theory and in computational biology

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

direct results for a certain family of summation integral type operators abstract in the present paper we establish some local and global direct results for a certain family of summation integral operators b n we also estimate the rate of convergence of b n f for a function f having derivatives of bounded variation

fractal dimension and logarithmic loss unpredictability we show that the hausdorff dimension equals the logarithmic loss unpredictability for any set of infinite sequences over a finite alphabet using computable feasible and finite state predictors this equivalence also holds for the computable feasible and finite state dimensions combining this with recent results of fortnow and lutz proc 15th ann conf on comput learning theory 2002 380 we have a tight relationship between prediction with respect to logarithmic loss and prediction with respect to absolute loss

crossover from the dilute to the dense phase for self repelling polymer chains finite size effects and relation to zero component landau ginzburg wilson theory using a recently established perturbative approach we analyse a single polymer chain or a few chains floating in a good solvent contained in a finite box with periodic boundary conditions we calculate to one loop order the partition function and the equation of state relating segment concentration to segment chemical potential and we discuss in detail the chain length distribution for a field theoretic ensemble of chains characterized by fixed our results obey finite size scaling and cover the whole crossover from the dilute to the dense limit where is the critical chemical potential the different limits evolve smoothly from one another the theoretical results for the chain length distribution are compared with monte carlo simulations of self avoiding walks on a cubic lattice we find a good agreement between our results and the simulation data

how to use expert advice we analyze algorithms that predict a binary value by combining the predictions of several prediction strategies called experts our analysis is for worst case situations i e we make no assumptions about the way the sequence of bits to be predicted is generated we measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence where the expectation is taken with respect to the randomization in the predictins we show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert and we give efficient algorithms that achieve this our upper and lower bounds have matching leading constants in most cases we then show how this leads to certain kinds of pattern recognition learning algorithms with performance bounds that improve on the best results currently know in this context we also compare our analysis to the case in which log loss is used instead of the expected number of mistakes

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

new decidability results concerning two way counter machines the authors study some decision questions concerning two way counter machines and obtain the strongest decidable results to date concerning these machines in particular it is shown that the emptiness containment and equivalence ece for short problems are decidable for two way counter machines whose counter is reversal bounded i e the counter alternates between increasing and decreasing modes at most a fixed number of times this result is used to give a simpler proof of a recent result in which the ece problems for two way reversal bounded pushdown automata accepting bounded languages i e subsets of w 1 ldots w k for some nonnull words w 1 ldots w k are decidable other applications concern decision questions about simple programs finally it is shown that nondeterministic two way reversal bounded multicounter machines are effectively equivalent to finite automata on unary languages and hence their ece problems are decidable also

minimax noiseless universal coding for markov sources upper and lower bounds are presented on the minimax redundancy for markov source noiseless block to variable universal coding the number of states block size and markov order are arbitrary but finite unlike earlier results the upper and lower bounds are absolute that is not merely asymptotic the upper bound is established by combinatorial bounds the lower bound is established by a bound on the average redundancy for an arbitrary distribution on the transition probabilities the bound can be optimized across the choice of distribution

complexity of strings in the class of markov sources shannon s self information of a string is generalized to its complexity relative to the class of finite state machine fsm defined sources unlike an earlier generalization the new one is valid for both short and long strings the definition is justified in part by a theorem stating that asymptotically the mean complexity provides a tight lower bound for the mean length of all so called regular codes this also generalizes shannon s noiseless coding theorem for a large subclass of fsm sources a simple algorithm is described for computing the complexity

gambling using a finite state machine sequential gambling schemes in which the amount wagered on the future outcome is determined by a finite state fs machine are defined and analyzed it is assumed that the fs machine determines the fraction of the capital wagered at each time instance i on the outcome at the next time instance i 1 and that wagers are paid at even odds the maximal capital achieved by any fs machine is found and its dependence on an empirical entropy measure h sup fs x defined as the finite state complexity of x is shown a specific gambling scheme is then proposed based on the lempel ziv method for universal compression the capital gained by this method is found and it is observed that asymptotically its exponential growth rate dominates the experimental growth rate achieved by gambling using any fs machine furthermore this specific scheme suggests a class of gambling methods based on a class of variable to variable length lossless compression methods in which the capital is doubled for every bit compressed these results emphasize the relation between gambling and data compression

a similarity measure for graphs with low computational complexity we present and analyze an algorithm to measure the structural similarity of generalized trees a new graph class which includes rooted trees for this we represent structural properties of graphs as strings and define the similarity of two graphs as optimal alignments of the corresponding property stings we prove that the obtained graph similarity measures are so called backward similarity measures from this we find that the time complexity of our algorithm is polynomial and hence significantly better than the time complexity of classical graph similarity methods based on isomorphic relations

universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence using finite memory is considered the finite state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite state fs predictor it is proven that this fs predictability can be achieved by universal sequential prediction schemes an efficient prediction procedure based on the incremental parsing procedure of the lempel ziv data compression algorithm is shown to achieve asymptotically the fs predictability some relations between compressibility and predictability are discussed and the predictability is proposed as an additional measure of the complexity of a sequence

on the rate of convergence of the generalized szasz type operators for functions of bounded variation abstract in this article we introduce the szasz bezier operatorsn which is the modified form of the szasz mirakyan operator by means of construction of suitable functions and the methods of bojanic vuillemier j approx theory31 1981 67 79 and cheng j approx theory40 1984 226 241 using some results and techniques of probability theory we obtain the rate of convergence ofsn for functions of bounded variation our results extend and sharpen the results of cheng j approx theory40 1984 226 241 guo and khan j approx theory58 1989 90 101 for the szasz mirakyan operator