Ludwig Staiger

Martin Luther University Halle-Wittenberg | MLU · Institute of Computer Science

Infinite words are often considered as limits of finite words. As topological methods have been proved to be useful in the theory of ω -languages it seems to be providing to include finite and infinite words into one (topological) space. In most cases this results in a poor topological structure induced on the subspace of finite words. In the prese...

A quasiperiod of a finite or infinite string is a word whose occurrences cover every part of the string. An infinite string is referred to as quasiperiodic if it has a quasiperiod. We present a characterisation of the set of infinite strings having a certain word q as quasiperiod via a finite language Pq consisting of prefixes of the quasiperiod q....

In this paper we study various notions of bi-immunity over alphabets with b≥2 elements and recursive transformations between sequences on different alphabets which preserve them. Furthermore, we extend the study from sequence bounded by a constant to sequences over the alphabet of all natural numbers, which may or may not be bounded by a recursive...

Solid codes provide outstanding fault-tolerance when used for information transmission through a noisy channel involving not only symbol substitutions, but also synchronisation errors and black-outs. In this paper we provide an automaton theoretic characterisation of solid codes which takes this fault-tolerance into account. The fault-tolerance aff...

It is shown that, for every length \(l\ge 3\), a quasiperiod of the form \(a^{n}ba^{n}\) (or \(a^{n}bba^{n}\) if l is even) generates the largest language Q of words having this word as quasiperiod. As a means of comparison we use the growth of the function which counts the number of words of length l in the language Q.Moreover, we give the exact o...

Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions amight be incomputable.

We survey the relations between four classes of real numbers: Liouville numbers, computable reals, Borel absolutely-normal numbers and Martin-Löf random reals. Expansions of reals play an important role in our analysis. The paper refers to the original material and does not repeat proofs. A characterisation of Liouville numbers in terms of their ex...

The lecture surveys approaches using finite automata to define several notions of (automata-theoretic) randomness.It focuses on the one hand on automata-theoretic randomness of infinite sequences in connection with automata-independent notions like disjunctivity and Borel normality.On the other hand it considers the scale of relaxations of randomne...

In this paper, we consider factor complexity/topological entropy of infinite binary sequences. In particular, we show that for any real number \(\alpha \) with \(0 \leqslant \alpha \leqslant 1\), there is a subset of the Cantor space with Hausdorff dimension \(\alpha \), such that each one of its elements has factor complexity \(\alpha \). This res...

In this paper we derive several results which generalise the constructive dimension of (sets of) infinite strings to the case of exact dimension. We start with proving a martingale characterisation of exact Hausdorff dimension. Then using semi-computable super-martingales we introduce the notion of exact constructive dimension of (sets of) infinite...

The Kolmogorov complexity function of an infinite word ξ maps a natural number to the complexity K(ξ | n) of the n -length prefix of ξ . We investigate the maximally achievable complexity function if ξ is taken from a constructively describable set of infinite words. Here we are interested in linear upper bounds where the slope is the Hausdorff dim...

The Kolmogorov complexity function of an infinite word ξ maps a natural number to the complexity K(ξ | n) of the n -length prefix of ξ . We investigate the maximally achievable complexity function if ξ is taken from a constructively describable set of infinite words. Here we are interested in linear upper bounds where the slope is the Hausdorff dim...

The space of one-sided infinite words plays a crucial rôle 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 i...

In this paper we derive several results which generalise the constructive dimension of (sets of) infinite strings to the case of exact dimension. We start with proving a martingale characterisation of exact Hausdorff dimension. Then using semi-computable super-martingales we introduce the notion of exact constructive dimension of (sets of) infinite...

In this paper we investigate several variants of P automata having infinite runs on finite inputs. By imposing specific conditions on the infinite evolution of the systems, it is easy to find ways for going beyond Turing if we are watching the behavior of the systems on infinite runs. As specific variants we introduce a new halting variant for P au...

The space of one-sided infinite words plays a crucial rôle 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 i...

This paper deals with the calculation of the Hausdorff measure of regular ω-languages, that is, subsets of the Cantor space definable by finite automata. Using methods for decomposing regular ω-languages into disjoint unions of parts of simple structure we derive two sufficient conditions under which ω-languages with a closure definable by a finite...

A quasiperiod of a word or an infinite string is a word which covers every part of the string. A word or an infinite string is referred to as quasiperiodic if it has a quasiperiod. It is obvious that a quasiperiodic infinite string cannot have every word as a subword (factor). Therefore, the question arises how large the set of subwords of a quasi...

In this paper we define and study finite state complexity of finite strings and infinite sequences and study connections of these complexity notions to randomness and normality. We show that the finite state complexity does not only depend on the codes for finite transducers, but also on how the codes are mapped to transducers. As a consequence we...

The subword complexity of an infinite word ξ is a function f(ξ,n) returning the number of finite subwords (factors, infixes) of length n of ξ. In the present paper we investigate infinite words for which the set of subwords occurring infinitely often is a regular language. Among these infinite words we characterise those which are eventually recurr...

Exact constructive dimension as a generalisation of Lutz’s [10,11] approach to constructive dimension was recently introduced in [19]. It was shown that it is in the same way closely related to a priori complexity, a variant of Kolmogorov complexity, of infinite sequences as their constructive dimension is related to asymptotic Kolmogorov complexit...

The present paper generalises results by Tadaki [12] and Calude et al. [1] on oscillation-free partially random infinite strings. Moreover, it shows that oscillation-free partial Chaitin randomness can be separated from oscillation-free partial strong Martin-Löf randomness by $\Pi_{1}^{0}$ -definable sets of infinite strings.

The present paper generalises results by Lutz and Ryabko. We prove a martingale characterisation of exact Hausdorff dimension. On this base we introduce the notion of exact constructive dimension of (sets of) infinite strings. Furthermore, we generalise Ryabko’s result on the Hausdorff dimension of the set of strings having asymptotic Kolmogorov co...

The main topic of the present work are universal machines for plain and prefix-free description complexity and their domains. It is characterised when an r.e. set W is the domain of a universal plain machine in terms of the description complexity of the spectrum function s_W mapping each non-negative integer n to the number of all strings of length...

The present paper generalises results by Tadaki [12] and Calude et al. [1] on oscillation-free partially random infinite strings. Moreover, it shows that oscillation-free partial Chaitin randomness can be separated from scillation-free partial strong Martin-L\"of randomness by Π^0_1-definable sets of infinite strings.

In this paper we prove that any Turing machine that uses only a finite computational space for every input cannot solve an uncomputable problem even when it runs in accelerated mode. We also propose two ways to define the language accepted by an accelerated Turing machine. Accordingly, the classes of languages accepted by accelerated Turing machine...

The space of one-sided infinite words plays a crucial rôle 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 i...

Infinite words are often considered as limits of finite words. As topological methods have been proved to be useful in the theory of ω-languages it seems to be providing to include finite and infinite words into one (topological) space. The attempts so far (see [3, Section 2.4]) have their drawbacks. Therefore, in the present paper we investigate t...

We provide an exact estimate on the maximal subword complexity for quasiperiodic infinite words. To this end we give a representation of the set of finite and of infinite words having a certain quasiperiod q via a finite language derived from q. It is shown that this language is a suffix code having a bounded delay of decipherability. Our estimate...

Kuratowski observed that, starting from a subset M of a topological space and applying the closure operator and the interior operator arbitrarily often, one can generate at most seven different sets. We show that there are forty nine different types of sets w.r.t. the inclusion relationships between their generated sets. All these types really occu...

In this note we prove that any Turing machine which uses only a finite computational space for every input cannot solve an uncomputable problem even in case it runs in accelerated mode.

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...

We study computably enumerable (c.e.) prefix codes that are capable of coding all positive integers in an optimal way up to a fixed constant: these codes will be called universal. We prove various characterisations of these codes, including the following one: a c.e. prefix code is universal if and only if it contains the domain of a universal self-...

It has been shown (see Part I [L. Staiger, Electron. Notes Theor. Comput. Sci. 221, 287–297 (2008; Zbl 1262.03070)]) that there are strongly Martin-Löf-ε-random ω-words that behave in terms of complexity like random ω-words. That is, in particular, the a priori complexity of these ε-random ω-words is bounded from below and above by linear functions...

In this paper we discuss three notions of partial randomness or ε-randomness. ε-randomness should display all features of randomness in a scaled down manner. However, as Reimann and Stephan [J. Reimann, and F. Stephan, On hierarchies of randomness tests, in: Mathematical Logic in Asia, Proceedings of the 9th Asian Logic Conference, Novosibirsk, Wor...

The present work clarifies the relation between domains of universal machines and r.e. prefix-free supersets of such sets. One such characterisation can be obtained in terms of the spectrum function s W (n) mapping n to the number of all strings of length n in the set W. An r.e. prefix-free set W is the superset of the domain of a universal machine...

Abstract. The present work clarifies the relation between domains of universal machines and r.e. prefix-free supersets of such sets. One such characterisation can be obtained in terms of the spectrum function sW(n) mapping n to the number of all strings of length n in the set W. An r.e. prefix-free set W is the superset of the domain of a universal...

Generalised Łukasiewicz languages are simply described languages having good information-theoretic properties. An especially desirable property is the one of being a prefix code. This paper addresses the question under which conditions a generalised Łukasiewicz language is a prefix code. Moreover, an upper bound on the delay of decipherability of a...

We present a brief survey of results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry.Special emphasis is placed on bounds on the complexity of strings in constructively given subsets of the Cantor space. Fi...

The paper presents an elementary approach for the calculation of the entropy of a class of languages. This approach is based on the consideration of roots of a real polynomial and is also suitable for calculating the Bernoulli measure. The class of languages we consider here is a generalisation of the Łukasiewicz language.

Finite automata are used for the encoding and compression of images. For black-and-white images, for instance, using the quad-tree representation, the black points correspond to ω-words defining the corresponding paths in the tree that lead to them. If the ω-language consisting of the set of all these words is accepted by a deterministic finite aut...

We explore the borderline between decidability and undecidability of the following question: “Let C be a class of codes. Given a machine ${\mathfrak{M}}$ of type X, is it decidable whether the language $L({{\mathfrak{M}}})$ lies in C or not?” for codes in general, ω-codes, codes of finite and bounded deciphering delay, prefix, suffix and bi(pre)fix...

Introduction Surprisingly, there are very few results of the following kind: Let C be a class of codes. "Given a device D of some fixed type, is it decidable whether the language L(D) dened by D is in class C or not?" The monograph [2] contains almost no information concerning this question. Known results are surveyed in Sections 3 and 9 of [12] b...

Generalised Łukasiewicz languages are simply described languages having good information-theoretic properties. An especially desirable property is the one of being a prefix code. This paper addresses the question under which conditions a generalised Łukasiewicz language is a prefix code. Moreover, an upper bound on the delay of decipherability of a...

Kraft’s inequality is a classical theorem in Information Theory which establishes the existence of prefix codes for certain (admissible) length distributions. We prove the following generalisation of Kraft’s theorem: For every admissible infinite length distribution one can construct a maximal prefix codes whose codewords satisfy this length distri...

If is a random sequence, then the sequence is clearly not random; however, seems to be “about half random”. L. Staiger [Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 (1993) 159–194 and A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory Comput. Syst. 31 (1998) 215–229] and K. Tadaki [A gene...

The paper investigates fixed points and attractors of infinite iterated function systems in Cantor space. By means of the theory of formal languages simple examples of the non-coincidence of fixed point and attractor (closure of the fixed point) are given.

Imagine a sequence in which the first letter comes from a binary alphabet, the second letter can be chosen on an alphabet with 10 elements, the third letter can be chosen on an alphabet with 3 elements and so on. Such sequences occur in various physical contexts, in which the coding of experimental outcome varies with scale. When can such a sequenc...

We use formal language theory to estimate the Hausdorff measure of sets of a certain shape in Cantor space. These sets are closely related to infinite iterated function systems in fractal geometry. Our results are used to provide a series of simple examples for the non-coincidence of limit sets and attractors for infinite iterated function systems...

An infinite sequence (ω-word) is referred to as disjunctive provided it contains every finite word as infix (factor). As Jürgensen and Thierrin [JT83] observed the set of disjunctive ω-words, D, has a trivial syntactic monoid but is not accepted by a finite automaton. In this paper we derive some topological properties of the set of disjunctive !-w...

We present results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry. Special emphasis is laid on bounds on the complexity of strings in constructively given subsets of the Cantor space. Finally, we compare t...

We use means of formal language theory to estimate the Hausdorff measure of sets of a certain shape in Cantor space. These sets are closely related to infinite iterated function systems in fractal geometry. Our results are used to provide a series of simple examples for the noncoincidence of limit sets and attractors for infinite iterated function...

The present paper proposes a generalisation of the notion of disjunctive (or rich) sequence, that is, of an infinite sequence of letters having each finite sequence as a subword. Our aim is to give a reasonable notion of disjunc-tiveness relative to a given set of sequences F. We show that a definition like "every subword which occurs at infinitely...

We derive the coincidence of Lutz's constructive dimension and Kolmogorov complexity for sets of infinite strings from Levin's early result on the existence of an optimal left computable cylindrical semi-measure M via simple calculations.

The paper presents an elementary approach for the calculation of the entropy of a class of context-free languages. This approach is based on the consideration of roots of a real polynomial and is also suitable for calculating the Bernoulli measure.

Lempel and Ziv (1976) proposed a computable string production-complexity. In this paper, our emphasis is on providing the rigorous development, where possible, for the theoretical aspects of a more recent and contrasting measure of string complexity. We derive expressions for complexity bounds subject to certain constraints. We derive an analytic a...

The paper investigates fixed points and attractors of infinite iterated function systems in Cantor space. By means of the theory of formal languages simple examples of the non-coincidence of fixed point and attractor (closure of the fixed point) are given.

The paper investigates fixed points and attractors of infinite iterated function systems in Cantor space. By means of the theory of formal languages simple examples of the non-coincidence of fixed point and attractor (closure of the fixed point) are given.

We use formal language theory to estimate the Hausdorff measure of sets of a certain shape in Cantor space. These sets are closely related to infinite iterated function systems in fractal geometry. Our results are used to provide a series of simple examples for the non-coincidence of limit sets and attractors for infinite iterated function systems.

We introduce ω -P automata based on the model of P systems with membrane channels (see [8]) using only communication rules. We show that ω -P automata with only two membranes can simulate the computational power of usual (non-deterministic) ω -Turing machines. A very restricted variant of ω -P automata allows for the simulation of ω -finite automat...

The set of random sequences is large in the sense of measure, but small in the sense of category. This is the case when we regard the set of infinite sequences over a finite alphabet as a subset of the usual Cantor space. In this note we will show that the above result depends on the topology chosen. To this end we will use a relativization of the...

We show how weighted finite automata define topologies on the set of all ω-words over a finite alphabet X. Moreover, we give a characterization of these topologies in terms of topologies on X ω induced by languages U⊆X * .

We consider for a real number α the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way which depends only on the relative informat...

The paper presents an elementary approach for the calculation of the entropy of a class of context-free languages. This approach is based on the consideration of roots of a real polynomial and is also suitable for calculating the Bernoulli measure.

We consider for a real number α the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way that depends only on the relative informati...

We consider disjunctive sequences, that is, infinite sequences (ω-words) having all finite words as infixes. It is shown that the set of all disjunctive sequences can be described in an easy way using recursive languages and, besides being a set of measure one, is a residual set in Cantor space. Moreover, we consider the subword complexity of seque...

Landwebers’s paper [La69] and the subsequent ones [SW74, TY83] pro- ved a strong relationship between acceptance conditions imposed on finite automata on ω-words and the first classes of the Borel hierarchy in the Cantor space of all ω-words, (Xω,ρ), over a finite alphabet X . In Theorem 5 of [SW74] it is shown that an ω-language accepted by a fi...

Finite automata are used for the encoding and compression of images. For black-and-white images, for instance, using the quad-tree representation, the black points correspond to ω-words defining the corresponding paths in the tree that lead to them. If the ω-language consisting of the set of all these words is accepted by a deterministic finite aut...

Valuations—morphisms from (Σ*, ·, 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 (...

We consider infinite games where a gambler plays a coin-tossing game against an adversary. The gambler puts stakes on heads or tails, and the adversary tosses a fair coin, but has to choose his outcome according to a previously given law known to the gambler. In other words, the adversary is not allowed to play all infinite heads-tails-sequences, b...

Using a specific model of communicating deterministic Turing machines we prove that the class of ω-languages accepted by deterministic Turing machines do not only depend on the output acceptance condition but also on the way in which the device is allowed to read the input. For the three modes of reading an input ω-word studied so far in the litera...

Using a specific model of communicating deterministic Tur- ing machines we prove that the class of -languages accepted by deterministic Turing machines via complete non-oscillating (complete oscillating) runs on the input coincides with the class of -definable ( -definable, respectively) -languages.

We investigate the relationship between the classes of -languages accepted by Turing machines according to two types of acceptance: 1) Machines of the first type are allowed to read only a finite part of the infinite input. 2) Machines of the second type have the additional possibility to reject by not reading the whole infinite input. It is shown...

We explore the borderline between decidability and undecidability of the following question: "Let C be a class of codes. Given a machine M of type X, is it decidable whether the language L(M) lies in C or not?" for codes in general, ω-codes, codes of finite and bounded deciphering delay, prefix, suffix and bi(pre)fix codes, and for finite automata...

Contents 1 Notation 4 2 Turing machines 5 2.1 Acceptance without state conditions . . . . . . . . . . 6 2.2 State-acceptance conditions for Turing machines . . . 8 3 Classes of !-languages accepted by deterministic Turing machines 9 3.1 !-languages accepted according to Type 1 . . . . . . . 10 3.2 Composition and Decomposition Theorem . . . . . . ....

We consider for a real number a the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way which depends only on the relative informat...

In several previous papers we have shown how to calculate Hausdorff dimension and measure for certain classes of regular ω-languages (cf. [MS94], [St89], and [St93]). In this note we show that the results obtained in the papers [MS94] and [St93] can be used to give an effective procedure for the calculation of the Hausdorff measure for arbitrary re...

This paper was written during my visit to the CDMTCS, Auckland, August 1998

Valuations — morphisms from (σ*, , e) to ((0, ∞), , 1) —are a generalization of Bernoulli morphisms introduced in [7]. Here, we show how to generalize the notion of entropy (of a language) in order to obtain new formulae to determine the Hausdorff dimension of fractal sets (also in Euclidean spaces) especially defined via regular Ω-languages. In th...

Rich ω-words are one-sided infinite strings which have every finite word as a subword (infix). Infix-regular ω-words are one-sided infinite strings for which the infix set of a suffix is a regular language. We show that for a regular -language F (a set of predicates definable in Büchi's restricted monadic second order arithmetic) the following cond...

this paper were presented at the International Workshop "Algebraic and Combinatorial Coding Theory", Varna, Bulgaria, Sept. 1988 1 2 L. Staiger A linear code of length n is simply a subspace of GF (q)

This paper links the concepts of Kolmogorov complexity (in complexity theory) and Hausdorff dimension (in fractal geometry) for a class of recursive (computable) ω -languages. It is shown that the complexity of an infinite string contained in a Σ 2 -definable set of strings is upper bounded by the Hausdorff dimension of this set and that this upper...

Rich omega-words are one-sided infinite strings which have every finite word as a subword (infix). Infix-regular omega-words are one-sided infinite string; for which the infix set of a suffix is a regular language. We show that for a regular omega-language F (a set of predicates definable in Buchi's restricted monadic second order arithmetic) the f...

This paper deals with another topological property of ω-power languages. It was observed in [St76, 80b] that finite-state (or regular) ω-languages which are nowhere dense in Cantor-space lack some subword (finite pattern). Here we generalize this result to finite-state ω-languages nowhere dense in an ω-power language.

In this paper we investigate several questions related to syntactic congruences and to minimal automata associated with ω-languages. In particular we investigate 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 refineme...

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 an accepting state is visited at least once or at least once but only finitely often.

For ω-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...

Following Jürgensen and Thierrin [21] we say that an infinite sequence is disjunctive if it contains any (finite) word, or, equivalently, if any word appears in the sequence infinitely many times. “Disjunctivity” is a natural qualitative property; it is weaker, than the property of “normality” (introduced by Borel [1]; see, for instance, Kuipers, N...