affine automata and related techniques for generation of complex images abstract in this paper we introduce probabilistic affine automata paa which are probabilistic finite generators having transitions labeled with affine transformations it is shown that paa are capable of generating highly complex images barnsley s 1988 ifs method to generate fractals is a special case of paa when the automaton happens to have only a single state a number of theoretical results on paa are shown the relationship between paa mutually recursive function systems mrfs and affine regular sets is investigated

tree based picture generation the concept of tree based picture generation is introduced it is shown that there are equivalent tree based definitions of four picture generating devices known from the literature namely collage

tree based generation of languages of fractals the notion of p interpreted top down tree generators is introduced combining the nondeterministic nature of grammars as known from formal language theory with the infinite refinement of pictures studied in fractal geometry

global dynamics of one class of nonlinear nonautonomous systems with time varying delays abstract a class of nonautonomous systems of nonlinear delay differential equations was studied via construction of matrix inequalities and comparison techniques the results for the nonautonomous systems with time varying delays are novel e g the global stability of differential equations with nonlinear casual volterra operators is considered for the first time in the literature criteria obtained for permanence and global attractivity are explicit and hence are convenient for applying verifying in practice we illustrate applications of the results obtained to the nonautonomous and asymptotically autonomous nicholson type models

cancellation in context free languages enrichment by reduction the following problem is shown to be decidable given a context free grammar g and a string w x does there exist a string u l g such that w is obtained from u by deleting all substrings ui that are elements of the symmetric dyck set d 1 the intersection of any two context free languages can be obtained from only one context free language by cancellation either with the smaller semi dyck set d 1 d 1 or with d 1 itself also the following is shown here for the first time if the set eq xnx n n n d 1 is used for this cancellation then each recursively enumerable set can be obtained from linear context free languages previous work has shown that cancellation of substrings from the semi dyck language d 1 or from any of the former languages d 1 or eq allows one to obtain the following from the context free languages any terminal petri net language 14 16 the intersections of any context free with a terminal petri net language and the nested iterated substitution thereof 15 we generalize this characterization by showing that linear context free languages suffice for generating terminal petri net languages in proving this we obtain a new closure property of the family of petri net languages which is not shared by the context free sets

asymptotic stability for a linear difference equation with variable delay we shall obtain sufficient conditions for the uniform stability and the global asymptotic stability of the linear difference equation with variable delay xn 1 xn pnxn kn 0 n n where pn is a sequence of nonnegative real numbers kn is a sequence of nonnegative integers and there exists a nonnegative integer k such that kn k for n n

architectural bias in recurrent neural networks fractal analysis we have recently shown that when initialized with small weights recurrent neural networks rnns with standard sigmoid type activation functions are inherently biased toward markov models even prior to any training rnn dynamics can be readily used to extract finite memory machines hammer tino 2002 tino cernansk benuskova 2002a 2002b following christiansen and chater 1999 we refer to this phenomenon as the architectural bias of rnns in this article we extend our work on the architectural bias in rnns by performing a rigorous fractal analysis of recurrent activation patterns we assume the network is driven by sequences obtained by traversing an underlying finite state transition diagram a scenario that has been frequently considered in the past for example when studying rnn based learning and implementation of regular grammars and finite state transducers we obtain lower and upper bounds on various types of fractal dimensions such as box counting and hausdorff dimensions it turns out that not only can the recurrent activations inside rnns with small initial weights be explored to build markovian predictive models but also the activations form fractal clusters the dimension of which can be bounded by the scaled entropy of the underlying driving source the scaling factors are fixed and are given by the rnn parameters

generating self affine fractals by collage grammars abstract self affinity and self similarity are fundamental concepts in fractal geometry in this paper they are related to collage grammars syntactic devices based on hyperedge replacement that generate sets of collages essentially a collage is a picture consisting of geometric parts like line segments circles polygons polyhedra etc the overlay of all collages in a collage language yields a fractal pattern we show that collage grammars of a special type so called increasing generalized sierpinski grammars yield self affine fractals if one replaces the overlay by an intersection of all generated collages the same result holds for decreasing generalized sierpinski grammars here the converse also holds every self affine fractal can be generated by a decreasing generalized sierpinski grammar which provides a characterization of this class of fractals

two undecidability results for chain code picture languages it is undecidable whether or not two 1 retreat bounded regular languages describe exactly the same set of pictures or they describe a picture in common

word problems in elliott monoids abstract algorithmic issues concerning elliott local semigroups are seldom considered in the literature although these combinatorial structures completely classify af algebras in general the addition operation of an elliott local semigroup is partial but for every af algebra b whose murray von neumann order of projections is a lattice this operation is uniquely extendible to the addition of an involutive monoid e b let m 1 be the farey af algebra introduced by the present author in 1988 and rediscovered by f boca in 2008 the freeness properties of the involutive monoid e m 1 yield a natural word problem for every af algebra b with singly generated e b because b is automatically a quotient of m 1 given two formulas and in the language of involutive monoids the problem asks to decide whether and code the same equivalence of projections of b this mimics the classical definition of the word problem of a group presented by generators and relations we show that the word problem of m 1 is solvable in polynomial time and so is the word problem of the behncke leptin algebras a n k and of the effros shen algebras f for 0 1 q a real algebraic number or 1 e we construct a quotient of m 1 having a godel incomplete word problem and show that no primitive quotient of m 1 is godel incomplete

leftmove bound picture languages let fu d r lg be the chain code picture alphabet such that u d r l denotes the graphics command to move the drawing pen up down right left in the 2d cartesian plane it is known that the picture membership problem can be solved in polynomial time for each context free language over fu d rg and is np complete for a so called retreat bounded regular or reversalbounded linear language over imposing both retreat and reversal bounds on languages over results in the leftmove bounded languages whose words describe pictures by making no more than a bounded number of left moves the picture membership problem can be solved in polynomial time for each leftmove bounded context free language over and is np complete for a leftmoveunbounded but retreat bounded linear language over fu d lrg there exists a context sensitive language over fu d rg or fu d lrg for which the picture membership problem is undecidable c 2000 elsevier science b v all rights reserved

efficient implementation of regular languages using reversed alternating finite automata alternating finite automata afa provide a natural and succinct way to denote regular languages we introduce a bit wise representation of reversed afa r afa transition functions and describe an efficient implementation method for r afa and their operations using this representation experiments have shown that this implementation is much more efficient than for instance the grail dfa implementation in both space and time

rational probability measures abstract the theory of finite automata and coding has been linked to probability theory since its foundation by the work of shannon and its further enrichment especially by schutzenberger in a parallel development symbolic dynamical systems whose behaviour is governed by a finite automaton have been studied under several aspects especially by hell robertson binkowska and kaminski 4 13 17 weiss has given to them the name of sofic systems and he has shown that they constitute a generalization of topological markov chains 20 since then adler and his associates have shown how theoretical development in this field could lead to a remarkable treatment of engineering problems in coding 1 in this paper we continue the exploration of this borderline area and present some new results probability measures on words are studied in their relation with coding and finite automata we particularly insist on the notion of a rational probability measure which although perhaps not quite new has not yet received all the attention it deserves in a first section we fix the notation and prove some preliminary results in particular the basic properties of rational probability measures are established the second section deals with probability measures for which the sequences obtained after some coding are mutually independent it contains two main results the first one gives conditions under which the restriction of a probability measure to a given free submonoid is a bernoulli measure it extends a result due to schutzenberger the second result is due to langlois and is published here for the first time in the two remaining sections we consider the problem of transferring a probability measure from one free monoid into another one several new results are proved which complement the ones previously obtained in collaboration with blanchard 5 6

valuations of languages with applications to fractal geometry abstract valuations morphisms from to 0 1 are a simple generalization of bernoulli morphisms distributions measures as introduced by berstel and perrin 1988 berstel and reutenauer 1988 blanchard and hansel 1984 eilenberg 1974 and hansel and perrin 1983 1989 this paper shows that valuations are useful not only within the theory of codes but also when dealing with ambiguity especially in context free grammars or for defining outer measures on the space of words which are of some importance to the theory of fractals these connections yield new formulae to determine the hausdorff dimension of fractal sets especially in euclidean spaces defined via formal languages the class of fractals describable with context free languages strictly includes that of mrfs fractals introduced by culik and dube 1990 and mauldin and williams 1988 some of the results of this paper also appear as part of the author s ph d thesis and other works fernau 1992a b 1993

a theorem on the entropy of submonoids note abstract let x be a finite alphabet for l x let l denote the radius of convergence of the structure generating function s l n 0 card l x n t n of the language l the entropy of l is defined as hl logcardx l we shall prove the following proposition theorem let l be an arbitrary subset of x then for every 0 there is a finite subset u of l such that hl hu

global stability for separable nonlinear delay differential systems abstract in this paper we consider separable nonlinear delay differential systems and we establish conditions for global asymptotic stability of the zero solution applying these we offer improved 3 2 type criteria for global asymptotic stability of nonautonomous lotka volterra systems with delays

complexity and decidability for restricted classes of picture languages abstract the paper investigates complexity and decidability questions for two restricted classes of picture languages called three way picture languages and stripe picture languages it is shown that 1 the picture membership problem can be solved in deterministic polynomial time for three way context free picture languages 2 the equivalence and containment problems for picture languages are undecidable for a regular language l1 and a linear language l2 where both l1 and l2 describe three way stripe picture languages and 3 the intersection emptiness problem for picture languages is undecidable for two three way stripe linear picture languages and is also undecidable for a regular language l1 and a linear language l2 where l1 describes a three way stripe picture language and l2 describes a stripe picture language

global stability for separable nonlinear delay differential equations consider the following separable nonlinear delay differential equation dy t dt i 0ma i t g i y t i t t t 0y t f t t t 0 where we assume that there is a strictly monotone increasing function f x on such that f 0 0 00 0 i m andiff x x thenlim x f x orlim x f x isfinite in this paper to the above separable nonlinear delay differential equation we establish conditions of global asymptotic stability for the zero solution in particular for a special wide class of f x which contains a case of f x e x 1 we give more explicit conditions applying these we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays

perfectly quilted rectangular snake tilings we introduce a particular form of snake tilings to define picture languages and relate the obtained family to the recognizable picture languages as defined by wang tiles the correspondence for substitution tilings is even closer and hence is applicable to the hilbert curve

iterated function systems and control languages valuations morphisms from e to 0 1 are a generalization of bernoulli morphisms introduced by eilenberg automata languages and machines academic press new york 1974 here we show how to generalize the notion of entropy of a language in order to obtain new formulas to determine the hausdorff dimension of fractal sets also in euclidean spaces especially defined via regular languages by doing this we can sharpen and generalize earlier results in two ways first we treat the case where the underlying basic iterated function system contains noncontractive mappings and second we obtain results valid for nonregular languages as well

a transformation system for generating description languages of chain code pictures abstract a picture is a set of unit lines from the cartesian plane considered as a square grid to each connected picture a description is given by a word over the alphabet r l u d where r means go one unit line right from the current point and draw this unit line and l u d are interpreted analogously with left up and down respectively for a picture p it was shown in 9 that the description language i e all the words which describe the picture p is a regular one to construct such a regular language one needs geometric information about the picture in this paper we present a system of transformations which map one word over the alphabet r l u d to another word which describes the same picture it is also shown that exactly all the picture descriptions can be generated started by some word

uniform stability of nonlinear difference systems abstract we offer sufficient conditions for the uniform stability and uniformly asymptotic stability of the zero solution of the nonlinear difference system formula where 0 1 f n s r sis the set of all functions n k 0 rfor somek n andxn sis defined byxn m x n m form n k 0

undecidability of existential properties in picture languages abstract we are interested in the description of a set of pictures by string languages by using several semantics segments 12 segments with blank moves 8 and pixels 10 we give a method to code the post correspondence problem in rational picture languages in order to show the undecidability of existence of words satisfying a given property in a rational language in particular we study the properties like is a self avoiding word is a minimal picture word or describes a connected picture

complexity and decidability for chain code picture languages abstract a picture description is a word over the alphabet u d r l where u means go one unit line up from the current point and d r and l are interpreted analogously with down right and left instead of up by this a picture description describes a walk in the plane its trace is the picture it describes a set of picture descriptions describes a chain code picture language this paper investigates complexity and decidability questions for these picture languages thus it is shown that the membership problem is np complete for regular picture languages i e picture languages described by regular languages of picture descriptions and that it is undecidable whether two regular picture description languages describe a picture in common after this we investigate so called stripe picture languages all pictures are within a stripe defined by two parallel lines providing better complexity and decidability results membership is decidable in linear time for regular stripe picture languages emptiness of intersection and equivalence is decidable for regular stripe picture languages

3 2 type criteria for global attractivity of lotka volterra competition system without instantaneous negative feedbacks this paper deals with a two species lotka volterra competition model with discrete delays but without instantaneous negative feedbacks motivated by wright s 3 2 global attractivity result for the delayed scalar logistic equation we establish some new 3 type criteria for global attractivity of the positive equilibrium of the system these criteria provide convenient and better than some existing estimates for the diagonal delays

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

global attractivity of non autonomous lotka volterra competition system without instantaneous negative feedback abstract we consider a non autonomous lotka volterra competition system with distributed delays but without instantaneous negative feedbacks i e pure delay systems we establish some 3 2 type and m matrix type criteria for global attractivity of the positive equilibrium of the system which generalise and improve the existing ones

the membership and equivalence problems for picture languages abstract the paper describes two principal results 1 there is a nondeterministic polynomial time algorithm to determine whether an arbitrary picture is a member of a context free picture language and 2 the equivalence problem for regular picture languages is not decidable not even partially decidable this solves open questions described in earlier publications furthermore these results are best possible that is the membership problem for context free picture languages was earlier shown to be np hard and the complement of the equivalence problem for regular picture languages is easily seen to be partially decidable

dimensions and measures in infinite iterated function systems the hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated conformal measures for such systems reflecting geometric properties of the limit set are introduced proved to exist and to be unique the existence of a unique invariant probability equivalent to the conformal measure is derived our methods employ the concepts of the perron frobenius operator symbolic dynamics on a shift space with an infinite alphabet and the properties of the above mentioned ergodic invariant measure a formula for the hausdorff dimension of the limit set in terms of the pressure function is derived fractal phenomena not exhibited by finite systems are shown to appear in the infinite case in particular a variety of conditions are provided for hausdorff and packing measures to be positive or finite and a number of examples are described showing the appearance of various possible combinations for these quantities one example given special attention is the limit set associated to the complex continued fraction expansion in particular lower and upper estimates for its hausdorff dimension are given a large natural class of systems whose limit sets are dimensionless in the restricted sense is described

a global stability criterion in nonautonomous delay differential equations abstract consider the following nonautonomous nonlinear delay differential equation d y t d t a t y t i 0 m a i t g i y i t t t 0 y t t t t 0 where we assume that there is a strictly monotone increasing function f x on such that f 0 0 0 g i x f x 1 x 0 0 i m and if f x x then lim x f x or lim x f x is finite in this paper to the above nonautonomous nonlinear delay differential equation we establish conditions of global asymptotic stability for the zero solution in particular for a special wide class of f x which contains two cases f x e x 1 and f x x we give more explicit conditions which are some extension of the 3 2 type criterion applying these to discrete models of nonautonomous delay differential equations we also obtain new sufficient conditions of the global asymptotic stability of the zero solution

retreat bounded picture languages a well known string encoding of 2d pictures is to use the picture alphabet u d r l where u d r and l means draw a unit line by moving the pen up down right and left from the current point a word over is k retreat bounded k 0 if it describes a picture in such a manner that the maximum distance of left moves ignoring up and down moves from a rightmost point of any partially drawn picture is bounded by k a set of such words forms a k retreat bounded language a k retreat bounded picture language is a set of pictures described by a k retreat bounded language there is a 1 retreat bounded regular picture language 1 retreat bounded vertical stripe linear picture language for which the membership problem is np complete the membership problem can be solved in o n4 time for each k retreat bounded nonvertical stripe context free picture language the inclusion and intersection emptiness problems are undecidable for a 1 retreat bounded regular picture language and a 2 retreat bounded regular picture language the equivalence and ambiguity problems are undecidable for 2 retreat bounded regular picture languages the picture recognition algorithm presented is the first polynomial time algorithm in the literature for a reasonably large subclass of context free picture languages and the np completeness and undecidability results improve the previously known such results for regular picture languages with no structural restriction imposed

kolmogorov complexity and hausdorff dimension abstract in this paper various relationships between the kolmogorov complexity of infinite strings and measures of information content are given the general approach taken here is to bound the complexity of a maximally complex string in a given set of strings by the hausdorff dimension or the entropy of that set it turns out that hausdorff dimension yields lower bounds to the kolmogorov complexity whereas under certain recursiveness constraints on the structure of the respective sets their entropy yields upper bounds more detailed investigations result in a generalization of p martin lof s theorems on the kolmogorov complexity of random strings to maximally complex strings in regularly structured sets of infinite strings

an extension of stone duality to fuzzy topologies and mv algebras abstract in this paper we introduce the concept of mv topology a special class of fuzzy topological spaces and prove a proper extension of stone duality to the categories of limit cut complete mv algebras and stone mv spaces namely zero dimensional compact hausdorff mv topological spaces then we describe the category of limit cut complete mv algebras and show that it is a completion namely a reflective subcategory with faithful reflection of the category of semisimple mv algebra last we compose our duality with other known ones thus obtaining new categorical equivalences and dualities involving categories of mv algebras

using string languages to describe picture languages a picture is a set of unit lines from the cartesian plane considered as a square grid a word over the alphabet l r u d is a picture description in the sense that it represents a traversal of a picture where the interpretation of the symbols l r u d is l go one unit line to the left of the current point r go one unit line to the right of the current point u go one unit line up from the current point and d go one unit line down from the current point a set of picture descriptions forms a picture description language this paper investigates the basic properties of pictures and picture description languages from the formal language theory point of view

on reversal bounded picture languages abstract for an integer k 0 a k reversal bounded picture language is a chain code picture language which is described by a language l over the alphabet u d r l such that for every word x in l the number of alternating occurrences of r s and l s in x is bounded by k it is shown that the membership problem can be solved in o n4k 4 time for k reversal bounded regular picture languages for every k 1 and is np complete for 1 reversal bounded stripe linear picture languages the membership problem is known to be np complete for regular and context free picture languages without restriction on the number of reversals and solvable in o n time o n12 time for 0 reversal bounded regular context free picture languages whether the membership problem for stripe context free picture languages could be solved in polynomial time has been an open problem other basic properties of reversal bounded picture languages are also presented