horizontal and vertical self organization of braided rivers toward a critical state self organization in an experimental braided river is studied it is shown that the experimental braided river self organizes into a critical state where it shows dynamic scaling that is small and large parts of the river evolve statistically identically after proper renormalization of space and time the dynamic scaling emerges during the process of approaching the critical state which involves self adjustment of both profile vertical self organization and braiding pattern horizontal self organization the obtained result corroborates the hypothesis suggested by the authors earlier sapozhnikov and foufoula georgiou 1997 that braided rivers are self organized critical systems the results are also important for understanding and statistically predicting the behavior of natural braided rivers because owing to external conditions e g sudden streamflow changes some of them may be driven out of the critical state and therefore may show deviation from dynamic scaling

computing the solution of the korteweg de vries equation with arbitrary precision on turing machines in this paper we answer an open question raised by pour el and richards is the solution operator of the korteweg de vries kdv equation computable the initial value problem of the kdv equation posed on the real line r ut uux uxxx o t x r u x 0 x defines a nonlinear map kr from the space hs r to the space c r hs r for real numbers s 0 we prove that for any integer s 3 the map kr hs r c r hs r is turing computable

representations versus numberings on the relationship of two computability notions abstract this paper gives an answer to weihrauch s computability springer berlin 1987 question whether and if not always when an effective map between the computable elements of two represented sets can be extended to a partial computable map between the represented sets examples are known showing that this is not possible in general a condition is introduced and for countably based topological t 0 spaces it is shown that exactly the partial effective maps meeting the requirement are extendable for total effective maps the extra condition is satisfied in the standard cases of effectively given separable metric spaces and continuous directed complete partial orders in which the extendability is already known in the first case a similar result holds also for partial effective maps but not in the second

fine hierarchies and m reducibilities in theoretical computer science this is a survey of results about versions of fine hierarchies and many one reducibilities that appear in different parts of theoretical computer science these notions and related techniques play a crucial role in understanding complexity of finite and infinite computations we try not only to present the corresponding notions and facts from the particular fields but also to identify the unifying notions techniques and ideas

effectiveness of the global modulus of continuity on metric spaces abstract in the e definition of continuity of a function between metric spaces the value of depends on e and function as well this kind of dependence can be described by a function so called global modulus of continuity which maps the triple e to corresponding by a recent result of repovs and semenov proc int conf topol trieste 1993 g gentili ed rent ist mat univ trieste vol 25 1993 pp 441 446 there is a continuous global modulus of continuity for the function space c x y of all continuous functions from a locally compact metric space x to an arbitrary metric space y based on weihrauch s framework on computable metric spaces k weihrauch theoret comput sci 113 1993 191 210 we show that there is a computable global modulus of continuity for c x y if x is an effectively locally compact metric space and y is a computable metric space the proof is a direct construction not depending on the proof of repovs and semenov

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

computability on the probability measures on the borel sets of the unit interval while computability theory on many countable sets is well established and for computability on the real numbers several mutually non equivalent definitions are applied for most other uncountable sets in particular for measures no generally accepted computability concepts at all ha ve been available until now in this contribution we introduce computability on the set m of probability measures on the borel subsets of the unit interval 0 1 its main purpose is to demonstrate that this concept of computability is not merely an ad hoc definition but has very natural properties although the definitions and many results can of course be transferred to more general spaces of measures we restrict our attention to m in order to keep the technical details simple and concentrate on the central ideas in particular we show that simple obvious reqirements exclude a number of similar definitions that the definition leads to the expected computability results that there are other natural definitions inducing the same computability theory and that the theory is embedded smoothly into classical measure theory as background we consider tte type 2 theory of effectivity kw84 kw85 which provides a frame for very realistic computability definitions in this approach computability is defined on finite and infinite sequences of symbols explicitly by turing machines and on other sets by means of notations and representations canonical representations are derived from information structures wei97 we introduce a standard representation delta m subseteq sum omega to m via some natural information structure defined by a subbase the atomic properties of some topology on m and a standard notation of while several modifications of m suggesting themselves at first glance violate simple and obvious requirements m has several very natural properties and hence should induce an important computability theory many interesting functions on measures turn out to be computable in particular linear combination integration of continuous functions and any transformation defined by a computable iterated function system with probabilities some other natural representations of m are introduced among them a cauchy representation associated with the hutchinson metric and proved to be equivalent to m as a corollary the final topology of m is the well known weak topology on m

computability theory of generalized functions the theory of generalized functions is the foundation of the modern theory of partial differential equations pde as computers are playing an ever larger role in solving pdes it is important to know those operations involving generalized functions in analysis and pde that can be computed on digital computers in this article we introduce natural concepts of computability on test functions and generalized functions as well as computability on schwartz test functions and tempered distributions type 2 turing machines are used as the machine model weihrauch 2000 it is shown here that differentiation and integration on distributions are computable operators and various types of fourier transforms and convolutions are also computable operators as an application it is shown that the solution operator of the distributional inhomogeneous three dimensional wave equation is computable

computability on subsets of euclidean space i closed and compact subsets in this paper we introduce and compare computability concepts on the set of closed subsets of euclidean space we use the language and framework of type 2 theory of effectivity tte which supplies a concise language for distinguishing a variety of effectivity properties and which admits highly effective versions of classical theorems in particular type 2 theory of effectivity allows to separate topological from computational aspects of effectivity we consider three different computability concepts on the set of closed subsets each of which is characterized by several representations which are proved to be equivalent the three induced types of computable closed sets have already been considered by many authors however under different and partly inconsistent names our characterizations show that they can be regarded as straightforward generalizations of the r e co r e and recursive subsets of natural numbers therefore we suggest to call them the recursively enumerable the co recursively enumerable and the recursive closed subsets of euclidean space open subsets obtain the dual names we extend the investigation by introducing several natural representations of the compact subsets of euclidean space and proving equivalences the paper extends and generalizes earlier definitions adds new ones and compares them in a single framework the resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets

recursion and topology on 2 ω for possibly infinite computations in the context of possibly infinite computations yielding finite or infinite binary outputs the space 2 2 u 2 appears to be one of the most fundamental spaces in computer science though underconsidered next to 2 this space can be viewed section 3 5 2 as the simplest compact space native to computer science in this paper we study some of its properties involving topology and computability though 2 can be considered as a computable metric space in the sense of computable analysis a direct and self contained study based on its peculiar properties related to words is much illuminating it is well known that computability for maps 2 2 reduces to continuity with recursive modulus of continuity with 2 things get less simple maps 2 2 or 2 2 induced by input output behaviours of turing machines on finite or infinite words which we call semicomputable maps are not necessarily continuous but merely lower semicontinuous with respect to the prefix partial ordering on 2 continuity asks for a stronger notion of computability we prove for semi continuous and semi computable maps f f o with f o 2 2 a detailed representation theorem theorem 81 via functions f 2 2 following two approaches bottom up from f to f and top down from f to f

type 2 computational complexity of functions on cantor s space abstract continuity and computability on cantor s space c has turned out to be a very natural basis for a type 2 theory of effectivity tte in particular the investigation of type 2 computational complexity e g complexity in analysis requires the study of oracle machines which w l g operate on cantor s space however no general type 2 complexity theory has been developed so far in this paper we lay a foundation for this theory by investigating computational complexity of functions c 0 1 and c c and the related concepts of dependence and input lookahead in the former case results of gordon and shamir are extended and generalized for continuous functions c c the relation between the three concepts is studied in detail compact sets are proved to be natural domains of resource bounded functions on c finally we demonstrate that an optimization of the input lookahead used by a machine may result in a rapid increase of computation time

admissible representations of effective cpo s abstract the elements of a cpo complete partial order d are abstract objects in general a concrete machine cannot operate with abstract objects but only with names of objects in this paper the set f of total functions from n to n is suggested as the set of names admissible representations f d for effective cpo s are defined by two axioms which generalize the axioms for godel numberings of the partial recursive functions topological and recursion theoretic considerations show that the definition is very natural it is also proved that p the set of subsets of n is not as suitable as a set of names finally it is proved that for admissible representations the computable functions over cpo s are represented by the computable extensional functions over the names

on the computability of walsh functions the haar and the walsh functions are proved to be computable with respect to the fine metric df which is induced from the infinite product 0 1 1 2 with the weighted product metric dc of the discrete metric on 0 1 although they are discontinuous functions on 0 1 with respect to the euclidean metric they are continuous functions on dc and on 0 1 df on dc computable real valued cylinder functions which include the walsh functions become computable and every computable function can be approximated effectively by a computable sequence of cylinder functions the metric space 0 1 df is separable but not complete nor effectively complete we say that a function on 0 1 is uniformly fine computable if it is sequentially computable and effectively uniformly continuous with respect to the metric df it is proved that a uniformly fine computable function is essentially a computable function on it is also proved that walsh fourier coefficients of a uniformly fine computable function f form a computable sequence of reals and there exists a subsequence of the walsh fourier series which fine converges effectively uniformly to f

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

equality is a jump abstract we define a notion of degree of unsolvability for subsets of r n where r is a real closed archimedean field and prove that in contrast to type 2 computability the presence of exact equality in the bss model forces exactly one jump of the unsolvability degree of decidable sets

computability noncomputability and hyperbolic systems abstract in this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point these manifolds are the essential feature which characterizes a hyperbolic system having many applications in physical sciences and other fields we show that i locally these manifolds can be computed but ii globally they cannot since their degree of computational unsolvability lies on the second level of the borel hierarchy we also show that smale s horseshoe the first example of a hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit is computable

a binary independent component analysis approach to tree topology inference using multicast probes to infer network topologies and internal link node characteristics is an attractive approach due to its bandwidth efficiency and suitability for large scale measurements in this paper we propose a new approach to tree topologies inference by exploiting dependence among end point receivers we first show that under the assumption of independent failure of intermediate nodes or links inferring tree topology is a special instance of the more general problem of binary independent component analysis bica and thus is amiable to existing analytical results and algorithms for bica then we propose the seqbica algorithm that is tailored for tree topology inference evaluation study shows that the proposed algorithm outperforms existing approaches in convergence speed and accuracy even when the number of measurements is small

eigenvectors of network complement reveal community structure more accurately abstract we propose a general spectral method to find communities of a network based on network complement and anti community concepts analytical and numerical results show that the eigenspace of matrices corresponding to a network complement reveals the community structure of a network more accurately than the eigenspace of matrices corresponding to the network itself it is shown that the laplacian eigenspace is the best candidate for spectral community detection especially in networks with a heterogeneous community structure the method is applied to some computer generated and real world networks with known community structures

effective symbolic dynamics random points statistical behavior complexity and entropy we consider the dynamical behavior of martin lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure we use computable partitions to define a sort of effective symbolic model for the dynamics through this construction we prove that such points have typical statistical behavior the behavior which is typical in the birkhoff ergodic theorem and are recurrent we introduce and compare some notions of complexity for orbits in dynamical systems and prove i that the complexity of the orbits of random points equals the kolmogorov sinai entropy of the system ii that the supremum of the complexity of orbits equals the topological entropy

computability of probability measures and martin lof randomness over metric spaces in this paper we investigate algorithmic randomness on more general spaces than the cantor space namely computable metric spaces to do this we first develop a unified framework allowing computations with probability measures we show that any computable metric space with a computable probability measure is isomorphic to the cantor space in a computable and measure theoretic sense we show that any computable metric space admits a universal uniform randomness test without further assumption

wadge hierarchy of omega context free languages abstract the main result of this paper is that the length of the wadge hierarchy of omega context free languages is greater than the cantor ordinal e 0 and the same result holds for the conciliating wadge hierarchy defined by duparc j symbolic logic to appear of infinitary context free languages studied by beauquier ph d thesis universite paris 7 1984 in the course of our proof we get results on the wadge hierarchy of iterated counter languages which we define as an extension of classical finitary iterated counter languages to languages

a note on ω regular languages abstract this paper is a continuation of the work by landweber 5 to clarify the relationship between classes of languages accepted by finite automata we study their properties in the topological framework introduced by boasson and nivat 1 we characterize the classes in terms of borel hierarchy and show some interrelation of them

on omega context free languages which are borel sets of infinite rank this paper is a continuation of the study of topological properties of omega context free languages cfl we proved in topological properties of omega context free languages theoretical computer science 262 1 2 2001 669 697 that the class of cfl exhausts the finite ranks of the borel hierarchy and in borel hierarchy and omega context free languages theoretical computer science to appear that there exist some cfl which are analytic but non borel sets we prove here that there exist some omega context free languages which are borel sets of infinite but not finite rank giving additional answer to questions of lescow and thomas logical specifications of infinite computations in a decade of concurrency j w de bakker et al eds springer lncs 803 1994 583 621

towards a descriptive set theory for domain like structures this is a survey of results in descriptive set theory for domains and similar spaces with the emphasis on the algebraic domains we try to demonstrate that the subject is interesting in its own right and is closely related to some areas of theoretical computer science since the subject is still in its beginning we discuss in detail several open questions and possible future development we also mention some relevant facts of effective descriptive set theory

characterization of ω regular languages by first order formulas abstract first order formulas are used to specify various ways of acceptance of languages by deterministic finite automata and we study the relationship between the arithmetic hierarchy of the formulas in prenex normal form and the topological hierarchy of the accepted languages among other things it is proved that the languages accepted by finite automata under the accepting conditions specified by 1 type formulas are precisely open regular languages those accepted under the conditions specified by 2 type formulas coincide with the regular languages which are denumerable unions of closed sets and that as long as the accepting conditions are specified by first order formulas the accepted languages remain to be regular

ambiguity in omega context free languages we extend the well known notions of ambiguity and of degrees of ambiguity of finitary context free languages to the case of omega context free languages cfl accepted by buchi or muller pushdown automata we show that these notions may be defined independently of the buchi or muller acceptance condition which is considered we investigate first properties of the subclasses of omega context free languages we get in that way giving many examples and studying topological properties of cfl of a given degree of ambiguity

on syntactic congruences for ω languages for languages several notions of syntactic congruence were defined the present paper investigates relationships between the so called simple because it is a simple translation from the usual definition in the case of finitary languages syntactic congruence and its infinitary refinements investigated by arnold ar85 we show that in both cases not every language having a finite syntactic monoid is regular and we give a characterization of those languages having finite syntactic monoids as the main result we derive a condition which guarantees that the simple syntactic congruence and arnold s syntactic congruence coincide and show that all languages in the borel class f g satisfy this condition finally we define an alternative canonical object for languages namely a family of right congruence relations using this object we give a necessary and sufficient condition for a regular language to be accepted by its minimalstate automaton

embedding metric spaces into cpo s abstract we show that metric spaces with polish topology mp can be represented very well by bounded complete algebraic cpo s with continuous weight and distance cwd an operator cwd mp is defined such that 1 and 2 hold 1 1 d can be embedded topologically into d such that a partial function on d is metrically continuous if it can be extended to a total continuous function on d 2 2 every m mp is isometric to some d therefore the continuous cpo functions are exactly adequate for describing the continuous functions on the corresponding metric spaces the approach could be refined in order to study computability on metric spaces with polish topology examples of application are the euclidean space the space of total functions on the natural numbers baire space and spaces of infinite trees

from axiomatics to intrinsic characterization some open problems in computable analysis abstract five open problems are presented the framework for these problems is the notion of a computability structure on a banach space which is discussed below the study of banach spaces in particular the study of hilbert space is an important topic in mathematics and its applications problems a and b are concerned with transfer principles for obtaining effective versions of classical theorems of mathematics problems c and d deal with separability and effective separability problem e is concerned with computability for classes of mathematical structures other than banach spaces

x automata on ω words abstract for any storage type x the languages accepted by x automata are investigated six accepting conditions including those introduced by landweber are compared for x automata the inclusions between the corresponding six families of languages are essentially the same as for finite state automata apart from unrestricted automata also real time and deterministic automata are considered the main tools for this investigation are 1 a characterization of the languages accepted by x automata in terms of inverse x transductions of finite state languages and 2 the existence of topological upper bounds on some of the families of accepted languages independent of the storage type x

borel hierarchy and omega context free languages we give in this paper additional answers to questions of lescow and thomas a decade of concurrency lecture notes in computer science vol 803 springer berlin 1994 pp 583 621 proving topological properties of omega context free languages cfl which extend those of o finkel theoret comput sci 262 1 2 2001 669 697 there exist some cfl which are non borel sets and one cannot decide whether an cfl is a borel set we give also an answer to a question of niwinski problem on powers posed in the proceedings of the workshop logics and recognizable sets 1990 and of simonnet automates et th orie descriptive ph d thesis universit paris 7 1992 about powers of finitary languages giving an example of a finitary context free language l such that l is not a borel set then we prove some recursive analogues to preceding properties in particular one cannot decide whether an cfl is an arithmetical set finally we extend some results to context free sets of infinite trees

two characterizations of rational adherences resume nous presentons deux caracterisations des adherences en termes de langages finis d homomorphismes strictement alphabetiques et d homomorphismes uniformes inverses nous en deduisons une caracterisation similaire pour les langages rationnels

the turing closure of an archimedean field a bss machine is uniform if it does not use exact tests such machines are equivalent modulo parameters to type 2 turing machines we define a notion of closure related to turing machines for archimedean fields and show that such fields admit nontrivial uniformly decidable sets iff they are not turing closed then the partially ordered set of turing closed fields is proved isomorphic to the ideal completion of unsolvability degrees

recursion theory on the reals and continuous time computation abstract we define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers corresponding to a conceptual analog computer that operates in continuous time this class turns out to be surprisingly large and includes many functions which are uncomputable in the traditional sense we stratify this class of functions into a hierarchy according to the number of uses of the zero finding operator at the lowest level are continuous functions that are differentially algebraic and computable by shannon s general purpose analog computer at higher levels are increasingly discontinuous and complex functions we relate this hierarchy to the arithmetical and analytical hierarchies of classical recursion theory

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

theory of representations abstract an approach for a simple general and unified theory of effectivity on sets with cardinality not greater than that of the continuum is presented a standard theory of effectivity on f f n n has been developed in a previous paper by representations f m this theory is extended to other sets m topological and recursion theoretical properties of representations are studied where the final topology of a representation plays an essential role it is shown that for any separable t0 space an up to equivalence unique admissible representation can be defined which reflects the topological properties correctly

recursive quasi metric spaces in computable analysis recursive metric spaces play an important role since these are roughly speaking spaces with computable metric and limit operation unfortunately the concept of a metric space is not powerful enough to capture all interesting phenomena which occur in computable analysis some computable objects are naturally considered as elements of asymmetric spaces which are not metrizable nevertheless most of these spaces are t0 spaces with countable bases and thus at least quasi metrizable we introduce a definition of recursive quasi metric spaces in analogy to recursive metric spaces we show that this concept leads to similar results as in the metric case and we prove that the most important spaces of computable analysis can be naturally considered as recursive quasi metric spaces especially we discuss some hyper and function spaces

a combinatorial approach to the theory of ω automata a combinatorial lemma is proved and used here to derive new results on automata and to give simpler proofs of known ones in particular we reprove mcnaughton s fundamental theorem characterizing the regular sequence sets without having to construct a sophisticated automaton the theorem is obtained by coding the behaviour of automata in a second order language and a simple application of the lemma in close analogy now referring to a first order language a theory of counter free automata is developed it is shown that these automata are appropriate for characterizing the star free sequence sets finally the lemma is applied in mathematical logic here new normal form theorems and also decidability results are proved for the first order and the monadic second order theory of certain structures over the ordering of natural numbers

computability on continuous lower semi continuous and upper semi continuous real functions in this paper we extend computability theory to the spaces of continuous upper and lower semi continuous real functions we apply the framework of tte type 2 theory of effectivity where not only computable objects but also computable functions on the spaces can be considered first some basic facts about tte are summarized for each of the function spaces we introduce several natural representations based on different intiuitive concepts of effectivity and prove their equivalence computability of several operations on the function spaces is investigated among others limits mappings to open sets images of compact sets and preimages of open sets maximum and minimum values the positive results usually show computability in all arguments negative results usually express non continuity several of the problems have computable but not extensional solutions since computable functions map computable elements to computable elements many previously known results on computability are obtained as simple corollaries

computable banach spaces via domain theory this paper extends the order theoretic approach to computable analysis via continuous domains to complete metric spaces and banach spaces we employ the domain of formal balls to define a computability theory for complete metric spaces for banach spaces the domain specialises to the domain of closed balls ordered by reversed inclusion we characterise computable linear operators as those which map computable sequences to computable sequences and are effectively bounded we show that the domain theoretic computability theory is equivalent to the well established approach by pour el and richards

projection lemmas for ω languages abstract the paper presents projection lemmas relating the infinite behaviour of deterministic and unboundedly branching nondeterministic infinite automata

community detection based on the clumpiness matrix in complex networks abstract the clumpiness matrix of a network is used to develop a method to identify its community structure a projection space is constructed from the eigenvectors of the clumpiness matrix and a border line is defined using some kind of angular distance in this space the community structure of the network is identified using this borderline and or hierarchical clustering methods the performance of our algorithm is tested on some computer generated and real world networks the accuracy of the results is checked using normalized mutual information the effect of community size heterogeneity on the accuracy of the method is also discussed

infinite behaviour of petri nets abstract the infinite behaviour of a petri net is the set of infinite labelled or unlabelled firing sequences for the well known notion of i acceptance for i 1 1 2 2 3 two hierarchies are established and their interrelation is studied

effective properties of sets and functions in metric spaces with computability structure we consider an abstract metric space with a computability structure and an effective separating set in this article we also introduce an effectively compact space the computability of real valued functions on such a space is defined it is shown that some of typical propositions in a metric space namely baire category theorem tietze s extension theorem and decomposition of unity can be effectivized it is also proved that computable functions are dense in continuous functions

infinite word languages and continuous mappings abstract languages with infinite words have often been studied as subsets of a topological space the topology used for this purpose was always the natural topology of sequences this paper suggests that another topology may be more useful it is obtained by taking lim w as the derived set of w the resulting topological space is shown to be completely regular strongly zero dimensional and not normal the languages appearing in the theory of automata are shown to be closely related to functionally closed sets continuous inverse images of the set 0 as a consequence a number of known results can be expressed and proved in terms of continuous mappings

some remarks on non algebraic adherences we present some properties of non algebraic adherences of languages which can be consistently called context sensitive adherences because it is proved that in the chomsky hierarchy of adherences context sensitive and recursive enumerable adherences coincide concerning the center mapping the traversing from algebraic to non algebraic takes simple recursive to not necessarily recursive enumerable except for the last equality the chomsky hierarchy of adherences is proper radh cfadh csadh readh and moreover it can be refined in the non algebraic case by considering the mcnaughton nivat adherences regarding the programming context adherences occur when we consider the set of divergent computations shapes of a program or a program scheme in this respect the interpretation is powerful enough to translate radh to csadh

effective metric spaces and representations of the reals based on standard notions of classical recursion theory a natural model of approximate computability for partial functions between effective metric spaces is presented it generalizes the ko friedman approach to computability of real functions by means of oracle turing machines follows the main ideas of weihrauch s type 2 theory of effectivity but it avoids the explicit use of representations the topological arithmetical hierarchy is introduced and shown to be strict if the underlying space contains an effectively discrete sequence the domains of computable functions are exactly the 2 sets of this hierarchy if the space admits a finitary stratification finally this framework is used to investigate and characterize the standard representations of the real numbers they are just those functions from the name space onto the reals which have both computable extensions and inversions that are computable as relations

fixed point characterization of context free languages several concepts of context freeness of sets of finite infinite words are characterized by means of greatest solutions of systems of equations of the form xi gi i 1 n where gi is a not necessarily finite union of monomials consideration of the systems with the components gi context free regular or finite leads to characterizations of the following classes of languages the kleene closure of the family of context free languages algebraic languages infinitely generated by context free grammars in the sense of nivat 1977 and cantor like topological closures of context free languages respectively

shift invariant topologies for the cantor space x the space of one sided infinite words plays a crucial rle in several parts of theoretical computer science usually it is convenient to regard this space as a metric space the cantor space it turned out that for several purposes topologies other than the one of the cantor space are useful e g for studying fragments of first order logic over infinite words or for a topological characterisation of random infinite words it is shown that these topologies refine the topology of the cantor space moreover from common features of these topologies we extract properties which characterise a large class of topologies it turns out that for this general class of topologies the corresponding closure and interior operators respect the shift operations and also to some extent the definability of sets of infinite words by finite automata

anisotropic scaling in braided rivers an integrated theoretical framework and results from application to an experimental river dynamic scaling in braided rivers is reexamined under an extended theoretical framework developed herein which explicitly incorporates the self affinity scaling anisotropy in the spatial structure of braided rivers it is shown that in structures exhibiting anisotropic spatial scaling dynamic scaling if present is necessarily anisotropic through analysis of the behavior of an experimental braided river the presence of anisotropic dynamic scaling in braided rivers was revealed this implies that there exists a pair of dynamic exponents zx and zy enabling one to rescale space differently in the direction x of the slope and in the perpendicular direction y and time such that the evolution of a smaller part of a braided river looks statistically identical to that of a larger one the presence of such a space time scale invariance provides an integrated framework for describing simultaneously the spatial and temporal structure of braided rivers and may be explored toward statistical prediction of large and rare changes from the statistics of smaller and frequent ones

study of self similar and self affine objects using logarithmic correlation integral we suggest a logarithmic correlation integral z x y as a good tool for investigating self affine and self similar objects first it enables us to extract fractal exponents vx and vy from one pattern of an object having any topology second we show that the integral z x y which completely characterizes a monofractal object provides more information on the density correlation properties of the object than just the exponents vx and vy we quantify this additional information by introducing two parameters delta characterizing the object s anisotropy of a nonscaling nature and k characterizing the curvature of the logarithmic correlation integral of the object we demonstrate that the four parameters vx vy delta and kappa provide an effective parametrization of the logarithmic correlation integral of a self affine monofractal object we give some examples of self affine objects having the same fractal exponents vx and vy but different parameters delta and kappa indicating the differences in the correlation properties of the objects we demonstrate that even a self similar object showing isotropic scaling vx vy may have the non scaling anisotropy parameter delta different from zero which indicates that the object has an asymmetric integral z x y and therefore different correlation properties in different directions it is shown that the equality kappa 0 outlines a class of objects for which the exponents v and v are not defined uniquely for instance such objects can be treated as both self similar and self affine if k is close to zero estimation of the exponents vx and vy may become problematic relationships connecting the exponents vx vy and fractal dimensions of the projection and cross section of an object are established

exact real number computations relative to hereditarily total functionals we show that the continuous existential quantifier is not definable in escardo s real pcf from all functionals equivalent to a given total one in a uniform way we further prove that relative to any total functional of type i i i which gives the maximum value for any total input we may given a computable total functional of type r r r find a real pcf definable total equivalent to

topological properties of omega context free languages abstract this paper is a study of topological properties of omega context free languages cfl we first extend some decidability results for the deterministic ones dcfl proving that one can decide whether an dcfl is in a given borel class or in the wadge class of a given regular language we prove that cfl exhaust the hierarchy of borel sets of finite rank and that one cannot decide the borel class of an cfl giving an answer to a question of lescow and thomas a decade of concurrency springer lecture notes in computer science vol 803 springer berlin 1994 pp 583 621 we give also a partial answer to a question of simmonet automates et theorie descriptive ph d thesis universite paris 7 march 1992 about omega powers of finitary languages we show that buchi landweber s theorem cannot be extended to even closed cfl in a gale stewart game with a closed cfl winning set one cannot decide which player has a winning strategy from the proof of topological properties we derive some arithmetical properties of cfl

recursive characterization of computable real valued functions and relations abstract corresponding to the definition of recursive functions we introduce a class of recursive relations in metric spaces such that each relation is generated from a class of basic relations by a finite number of applications of some specified operators we prove that our class of recursive relations essentially coincides with our class of densely computable relations defined via turing machines in the special case of the real numbers our subclass of recursive functions coincides with the classical class of computable real valued functions defined via turing machines by grzegorczyk lacombe and others

computability on computable metric spaces in previous papers e g weihrauch 1987 type 2 theory of effectivity tte has been shown to be a very general and powerful computer oriented theory for studying effectivity in mathematics in this contribution computability on certain computable separable metric spaces is studied in detail by applying the framework of tte computationally admissible representations of metric spaces are introduced after showing that four different effective representations are computationally equivalent for extending computability to the set of continuous functions several effective namings which are related to definitions of continuity are introduced and compared the definitions are used to prove effective g extension theorems for continuous functions and an effective g characterization of the domains of strongly continous functions which generalize the known properties of real functions kreitz 1984

computability on subsets of metric spaces the notions recursively enumerable and recursive are the basic notions of effectivity in classical recursion theory in computable analysis these notions are generalized to closed subsets of euclidean space using their metric distance functions we study a further generalization of these concepts to subsets of computable metric spaces it appears that different characterizations which coincide in case of euclidean space lead to different notions in the general case however under certain additional conditions such as completeness and effective local compactness the situation is similar to the euclidean case we present all results in the framework of type 2 theory of effectivity which allows to express effectivity properties in a very uniform way instead of comparing properties of single subsets we compare corresponding representations of the hyperspace of closed subsets such representations do not only induce a concept of computability for single subsets but they even yield a concept of computability for operations on hyperspaces such as union intersection etc we complete our investigation by studying the special situation of compact subsets

real number computation through gray code embedding we propose an embedding g of the unit open interval to the set 0 1 1 of infinite sequences of 0 1 with at most one undefined element this embedding is based on gray code and it is a topological embedding with a natural topology on 0 1 1 we also define a machine called an indeterministic multihead type 2 machine which input output sequences in 0 1 1 and show that the computability notion induced on real functions through the embedding g is equivalent to the one induced by the signed digit representation and type 2 machines we also show that basic algorithms can be expressed naturally with respect to this embedding

an effective riemann mapping theorem the main results of the paper are two effective versions of the riemann mapping theorem the first uniform version is based on the constructive proof of the riemann mapping theorem by cheng and by bishop and bridges and formulated in the computability framework developed by kreitz and weihrauch it states which topological information precisely one needs about a nonempty proper open connected and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from the unit disk onto this set and vice versa which topological information about the set can be obtained from a description of a holomorphic bijection the second version which is derived from the first by considering the sets and the functions with computable descriptions characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection from the unit disk this gives a partial answer to a problem posed by pour el and richards we also show that this class of sets is strictly larger than a class of sets considered by zhou which solves an open problem posed by him in preparation recursively enumerable open subsets and closed subsets of euclidean spaces are considered and several effective results in complex analysis are proved

type 2 recursion theory abstract an attempt is made to lay a basis for a general unified concise and simple theory of computable and continuous functions from f to f or n where f f n n the theoy is formally very similar to ordinary recursion theory it splits into a purely topological version and more special theory of computability the basic definitions are given and fundamental properties are proved as an example it is shown how the theory of recursively enumerable subsets of n can be transferred to a theory of open and a theory of computably open subsets of f

topology on words we investigate properties of topologies on sets of finite and infinite words over a finite alphabet the guiding example is the topology generated by the prefix relation on the set of finite words considered as a partial order this partial order extends naturally to the set of infinite words hence it generates a topology on the union of the sets of finite and infinite words we consider several partial orders which have similar properties and identify general principles according to which the transition from finite to infinite words is natural we provide a uniform topological framework for the set of finite and infinite words to handle limits in a general fashion

three concepts of decidability for general subsets of uncountable spaces there is no uniquely standard concept of an effectively decidable set of real numbers or real n tuples here we consider three notions decidability up to measure zero m w parker undecidability in rn riddled basins the kam tori and the stability of the solar system phil sci 70 2 2003 359 382 which we abbreviate d m z recursive approximability or r a k i ko complexity theory of real functions birkhauser boston 1991 and decidability ignoring boundaries d i b w c myrvold the decision problem for entanglement in r s cohen et al eds potentiality entanglement and passion at a distance quantum mechanical studies fo abner shimony vol 2 kluwer academic publishers great britain 1997 pp 177 190 unlike some others in the literature these notions apply not only to certain nice sets but to general sets in rn and other appropriate spaces we consider some motivations for these concepts and the logical relations between them it has been argued that d m z is especially appropriate for physical applications and on rn with the standard measure it is strictly stronger than r a m w parker undecidability in rn riddled basins the kam toil and the stability of the solar system phil sci 70 2 2003 359 382 here we show that this is the only implication that holds among our three decidabilities in that setting under arbitrary measures even this implication fails yet for intervals of non zero length and more generally convex sets of non zero measure the three concepts are equivalent

finite acceptance of infinite words abstract in this paper we consider the following two types of finite acceptance of infinite words by finite automata an infinite word is accepted if and only if there is a run on input for which 1 1 an accepting state is visited at least once or 2 2 an accepting state is visited at least once but only finitely often the resulting classes of regular languages are characterized by language theoretic means and they are positioned into the known hierarchies of regular languages