The Maximal Complexity of Quasiperiodic Infinite Words

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. It turns out its star root Pq* is a suffix code having a bounded delay of decipherability. This allows us to calculate the maximal subword (or factor) complexity of quasiperiodic infinite strings having quasiperiod q and further to derive that maximally complex quasiperiodic infinite strings have quasiperiods aba or aabaa. It is shown that, for every length l≥3, a word of the form anban (or anbban if l is even) generates the most complex infinite string having this word as quasiperiod. We give the exact ordering of the lengths l with respect to the achievable complexity among all words of length l.

Separating the Words of a Language by Counting Factors

For a given language L, we study the languages X such that for all distinct words u, v ∈ L, there exists a word x ∈ X that appears a different number of times as a factor in u and in v. In particular, we are interested in the following question: For which languages L does there exist a finite language X satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.

An adaptive parallel algorithm for finite language decomposition

The computationally hard problem of finite language decomposition is investigated. A finite language L is decomposable if there are two languages L1 and L2 such that L = L1L2. Otherwise, L is prime. The main contribution of the paper is an adaptive parallel algorithm for finding all decompositions L1L2 of L. The algorithm is based on an exhaustive search and incorporates several original methods for pruning the search space. Moreover, the algorithm is adaptive since it changes its behavior based on the runtime acquired data related to its performance. Comprehensive computational experiments on more than 4000 benchmark languages generated over alphabets of various sizes have been carried out. The experiments showed that by using the power of parallel computing the decompositions of languages containing more than 200000 words can be found. Decompositions of languages of that size have not been reported in the literature so far.

Polynomial-time tests for difference terms in idempotent varieties

We consider the following practical question: given a finite algebra [Formula: see text] in a finite language, can we efficiently decide whether the variety generated by [Formula: see text] has a difference term? We answer this question (positively) in the idempotent case and then describe algorithms for constructing difference term operations.

Hegel: Metacritics, Philosophical Language, and Memory

Hegel has a metacritical standpoint that can be related to but not reduced to the Herderian metacritique. Hegel’s philosophical language must not be understood in terms of the opposition between an ‘absolute’ and a ‘finite’ language; rather, it must be understood in terms of the opposition between abstract and concrete language. At a theoretical level, concrete language cannot be understood without assuming an organic function of memory. At the practical level, the difference between abstract and concrete language will be understood as the difference between everyday language and a philosophical one. Hegel justifies this last difference by following Humboldtian standpoints.

Arcs, hypercubes, and graphs as quotients of projective Fraïssé limits

We establish some basic properties of quotients of projective Fraïssé limits and exhibit some classes of compact metric spaces that are the quotient of a projective Fraïssé limit of a projective Fraïssé family in a finite language. We prove the result for the arcs directly, and by applying some closure properties we obtain all hypercubes and graphs as well.

Inductive Synthesis of Cover-Grammars with the Help of Ant Colony Optimization

A cover-grammar of a finite language is a context-free grammar that accepts all words in the language and possibly other words that are longer than any word in the language. In this paper, we describe an efficient algorithm aided by Ant Colony System that, for a given finite language, synthesizes (constructs) a small cover-grammar of the language. We also check its ability to solve a grammatical inference task through the series of experiments.

Separating the Classes of Recursively Enumerable Languages Based on Machine Size

In the late nineteen sixties it was observed that the r.e. languages form an infinite proper hierarchy [Formula: see text] based on the size of the Turing machines that accept them. We examine the fundamental position of the finite languages and their complements in the hierarchy. We show that for every finite language L one has that L, [Formula: see text] for some [Formula: see text] where m is the length of the longest word in L, c is the cardinality of L, and [Formula: see text]. If [Formula: see text], then [Formula: see text] for some [Formula: see text]. We also prove that for every n, there is a finite language Ln with [Formula: see text] such that [Formula: see text] but Ln, [Formula: see text] for some [Formula: see text]. Several further results are shown that how the hierarchy can be separated by increasing chains of finite languages. The proofs make use of several auxiliary results for Turing machines with advice.