scholarly journals The Maximal Complexity of Quasiperiodic Infinite Words

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 306
Author(s):  
Ludwig Staiger
Keyword(s):  
Infinite Words ◽  
Bounded Delay ◽  
Finite Language ◽  
Infinite Strings ◽  
Infinite String

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

2021 ◽  
Vol 180 (4) ◽  
pp. 375-393
Author(s):  
Aleksi Saarela
Keyword(s):  
Regular Languages ◽  
Infinite Words ◽  
Language L ◽  
Finite Language

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.

FINITELY BASED WORDS

2000 ◽  
Vol 10 (04) ◽  
pp. 457-480 ◽  
Author(s):  
OLGA SAPIR
Keyword(s):  
Sufficient Conditions ◽  
Basis Property ◽  
Free Monoid ◽  
Finite Basis ◽  
Finitely Based ◽  
Finite Basis Property ◽  
Letter Alphabet ◽  
Finite Language ◽  
Rees Quotient ◽  
The Ideal

Let W be a finite language and let Wc be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in Wc. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

scholarly journals Isoscattering strings of concatenating graphs and networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Michał Ławniczak ◽  
Adam Sawicki ◽  
Małgorzata Białous ◽  
Leszek Sirko
Keyword(s):  
Trace Function ◽  
Quantum Graphs ◽  
Mathematical Approach ◽  
Large Graphs ◽  
Graphs And Networks ◽  
Scattering Matrices ◽  
Theoretical Predictions ◽  
Infinite Strings ◽  
Microwave Networks ◽  
Insight Into

AbstractWe identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs and networks for which the isoscattering properties are preserved for $$n \rightarrow \infty $$ n → ∞ . The theoretical predictions are confirmed experimentally using $$n=2$$ n = 2 units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the $$2n \times 2n $$ 2 n × 2 n scattering matrices $${\hat{S}}$$ S ^ of the systems to 2n diagonal elements, while the old measures of isoscattering require all $$(2n)^2$$ ( 2 n ) 2 entries. The studied problem generalizes a famous question of Mark Kac “Can one hear the shape of a drum?”, originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.

scholarly journals Computational and Proof Complexity of Partial String Avoidability

2021 ◽  
Vol 13 (1) ◽  
pp. 1-25
Author(s):  
Dmitry Itsykson ◽  
Alexander Okhotin ◽  
Vsevolod Oparin
Keyword(s):  
Lower Bounds ◽  
Circuit Complexity ◽  
Conjunctive Normal Form ◽  
Proof Complexity ◽  
Proof Systems ◽  
Finite Set ◽  
Propositional Proof Systems ◽  
Classical Proof ◽  
Exponential Size ◽  
Infinite String

The partial string avoidability problem is stated as follows: given a finite set of strings with possible “holes” (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this article establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form satisfiability problem, where each clause has infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting proof lines (such as clauses, inequalities, polynomials, etc.). First, it is proved that there is a particular formula that has a short refutation in Resolution with a shift rule but requires classical proofs of exponential size. At the same time, it is shown that exponential lower bounds for classical proof systems can be translated for their shifted versions. Finally, it is shown that superpolynomial lower bounds on the size of shifted proofs would separate NP from PSPACE; a connection to lower bounds on circuit complexity is also established.

Fuzzy regular languages over finite and infinite words

2006 ◽  
Vol 157 (11) ◽  
pp. 1532-1549 ◽  
Author(s):  
Werner Kuich ◽  
George Rahonis
Keyword(s):  
Regular Languages ◽  
Infinite Words

The spectrum of resplendency

1990 ◽  
Vol 55 (2) ◽  
pp. 626-636
Author(s):  
John T. Baldwin
Keyword(s):  
Homogeneous Model ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Uncountable Cardinal ◽  
Recursive Theory ◽  
Finite Language

AbstractLet T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

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