62 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    DNA Computing: Modelling in Formal Languages and Combinatorics on Words, and Complexity Estimation

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    DNA computing, an essential area of unconventional computing research, encodes problems using DNA molecules and solves them using biological processes. This thesis contributes to the theoretical research in DNA computing by modelling biological processes as computations and by studying formal language and combinatorics on words concepts motivated by DNA processes. It also contributes to the experimental research in DNA computing by a scaling comparison between DNA computing and other models of computation. First, for theoretical DNA computing research, we propose a new word operation inspired by a DNA wet lab protocol called cross-pairing polymerase chain reaction (XPCR). We define and study a word operation called word blending that models and generalizes an unexpected outcome of XPCR. The input words are uwx and ywv that share a non-empty overlap w, and the output is the word uwv. Closure properties of the Chomsky families of languages under this operation and its iterated version, the existence of a solution to equations involving this operation, and its state complexity are studied. To follow the XPCR experimental requirement closely, a new word operation called conjugate word blending is defined, where the subwords x and y are required to be identical. Closure properties of the Chomsky families of languages under this operation and the XPCR experiments that motivate and implement it are presented. Second, we generalize the sequence of Fibonacci words inspired by biological concepts on DNA. The sequence of Fibonacci words is an infinite sequence of words obtained from two initial letters f(1) = a and f(2)= b, by the recursive definition f(n+2) = f(n+1)*f(n), for all positive integers n, where * denotes word concatenation. After we propose a unified terminology for different types of Fibonacci words and corresponding results in the extensive literature on the topic, we define and explore involutive Fibonacci words motivated by ideas stemming from theoretical studies of DNA computing. The relationship between different involutive Fibonacci words and their borderedness and primitivity are studied. Third, we analyze the practicability of DNA computing experiments since DNA computing and other unconventional computing methods that solve computationally challenging problems often have the limitation that the space of potential solutions grows exponentially with their sizes. For such problems, DNA computing algorithms may achieve a linear time complexity with an exponential space complexity as a trade-off. Using the subset sum problem as the benchmark problem, we present a scaling comparison of the DNA computing (DNA-C) approach with the network biocomputing (NB-C) and the electronic computing (E-C) approaches, where the volume, computing time, and energy required, relative to the input size, are compared. Our analysis shows that E-C uses a tiny volume compared to that required by DNA-C and NB-C, at the cost of the E-C computing time being outperformed first by DNA-C and then by NB-C. In addition, NB-C appears to be more energy efficient than DNA-C for some input sets, and E-C is always an order of magnitude less energy efficient than DNA-C

    Universal groups of cellular automata

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    We prove that the group of reversible cellular automata (RCA), on any alphabet A, contains a subgroup generated by three involutions which contains an iso-morphic copy of every finitely generated group of RCA on any alphabet B. This result follows from a case study of groups of RCA generated by symbol permutations and par-tial shifts (equivalently, partitioned cellular automata) with respect to a fixed Cartesian product decomposition of the alphabet. For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group. For non-prime non-four alphabets, it contains copies of all finitely generated groups of RCA. We also prove this property for the group generated by RCA of biradius one on any full shift with large enough alphabet, and also for some perfect finitely generated groups of RCA

    On the Properties and Structure of Bordered Words and Generalizations

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    Combinatorics on words is a field of mathematics and theoretical computer science that is concerned with sequences of symbols called words, or strings. One class of words that are ubiquitous in combinatorics on words, and theoretical computer science more broadly, are the bordered words. The word w has a border u if u is a non-empty proper prefix and suffix of w. The word w is said to be bordered if it has a border. Otherwise w is said to be unbordered. This thesis is primarily concerned with variations and generalizations of bordered and unbordered words. In Chapter 1 we introduce the field of combinatorics on words and give a brief overview of the literature on borders relevant to this thesis. In Chapter 2 we give necessary definitions, and we present a more in-depth literature review on results on borders relevant to this thesis. In Chapter 3 we complete the characterization due to Harju and Nowotka of binary words with the maximum number of unbordered conjugates. We also show that for every number, up to this maximum, there exists a binary word with that number of unbordered conjugates. In Chapter 4 we give results on pairs of words that almost commute and anti-commute. Two words x and y almost commute if xy and yx differ in exactly two places, and they anti-commute if xy and yx differ in all places. We characterize and count the number of pairs of words that almost and anti-commute. We also characterize and count variations of almost-commuting words. Finally we conclude with some asymptotic results related to the number of almost-commuting pairs of words. In Chapter 5 we count the number of length-n bordered words with a unique border. We also show that the probability that a length-n word has a unique border tends to a constant. In Chapter 6 we present results on factorizations of words related to borders, called block palindromes. A block palindrome is a factorization of a word into blocks that turns into a palindrome if each identical block is replaced by a distinct character. Each block is a border of a central block. We call the number of blocks in a block palindrome the width of the block palindrome. The largest block palindrome of a word is the block palindrome of the word with the maximum width. We count all length-n words that have a width-t largest block palindrome. We also show that the expected width of a largest block palindrome tends to a constant. Finally we conclude with some results on another extremal variation of block palindromes, the smallest block palindrome. In Chapter 7 we present the main results of the thesis. Roughly speaking, a word is said to be closed if it contains a non-empty proper border that occurs exactly twice in the word. A word is said to be privileged if it is of length ≤ 1 or if it contains a non-empty proper privileged border that occurs exactly twice in the word. We give new and improved bounds on the number of length-n closed and privileged words over a k-letter alphabet. In Chapter 8 we work with a generalization of bordered words to pairs of words. The main result of this chapter is a characterization and enumeration result for this generalization of bordered words to multiple dimensions. In Chapter 9 we conclude by summarizing the results of this thesis and presenting avenues for future research

    On some one-sided dynamics of cellular automata

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    A dynamical system consists of a space of all possible world states and a transformation of said space. Cellular automata are dynamical systems where the space is a set of one- or two-way infinite symbol sequences and the transformation is defined by a homogenous local rule. In the setting of cellular automata, the geometry of the underlying space allows one to define one-sided variants of some dynamical properties; this thesis considers some such one-sided dynamics of cellular automata. One main topic are the dynamical concepts of expansivity and that of pseudo-orbit tracing property. Expansivity is a strong form of sensitivity to the initial conditions while pseudo-orbit tracing property is a type of approximability. For cellular automata we define one-sided variants of both of these concepts. We give some examples of cellular automata with these properties and prove, for example, that right-expansive cellular automata are chain-mixing. We also show that left-sided pseudo-orbit tracing property together with right-sided expansivity imply that a cellular automaton has the pseudo-orbit tracing property. Another main topic is conjugacy. Two dynamical systems are conjugate if, in a dynamical sense, they are the same system. We show that for one-sided cellular automata conjugacy is undecidable. In fact the result is stronger and shows that the relations of being a factor or a susbsystem are undecidable, too

    Cellular automata with complicated dynamics

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    A subshift is a collection of bi-infinite sequences (configurations) of symbols where some finite patterns of symbols are forbidden to occur. A cellular automaton is a transformation that changes each configuration of a subshift into another one by using a finite look-up table that tells how any symbol occurring at any possible context is to be changed. A cellular automaton can be applied repeatedly on the configurations of the subshift, thus making it a dynamical system. This thesis focuses on cellular automata with complex dynamical behavior, with some different definitions of the word “complex”. First we consider a naturally occurring class of cellular automata that we call multiplication automata and we present a case study with the point of view of symbolic, topological and measurable dynamics. We also present an application of these automata to a generalized version of Mahler’s problem. For different notions of complex behavior one may also ask whether a given subshift or class of subshifts has a cellular automaton that presents this behavior. We show that in the class of full shifts the Lyapunov exponents of a given reversible cellular automaton are uncomputable. This means that in the dynamics of reversible cellular automata the long term maximal propagation speed of a perturbation made in an initial configuration cannot be determined in general from short term observations. In the last part we construct, on all mixing sofic shifts, diffusive glider cellular automata that can decompose any finite configuration into two distinct components that shift into opposing direction under repeated action of the automaton. This implies that every mixing sofic shift has a reversible cellular automaton all of whose directions are sensitive in the sense of the definition of Sablik. We contrast this by presenting a family of synchronizing subshifts on which all reversible cellular automata always have a nonsensitive direction

    On the k-Abelian Equivalence Relation of Finite Words

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    This thesis is devoted to the so-called k-abelian equivalence relation of sequences of symbols, that is, words. This equivalence relation is a generalization of the abelian equivalence of words. Two words are abelian equivalent if one is a permutation of the other. For any positive integer k, two words are called k-abelian equivalent if each word of length at most k occurs equally many times as a factor in the two words. The k-abelian equivalence defines an equivalence relation, even a congruence, of finite words. A hierarchy of equivalence classes in between the equality relation and the abelian equivalence of words is thus obtained. Most of the literature on the k-abelian equivalence deals with infinite words. In this thesis we consider several aspects of the equivalence relations, the main objective being to build a fairly comprehensive picture on the structure of the k-abelian equivalence classes themselves. The main part of the thesis deals with the structural aspects of k-abelian equivalence classes. We also consider aspects of k-abelian equivalence in infinite words. We survey known characterizations of the k-abelian equivalence of finite words from the literature and also introduce novel characterizations. For the analysis of structural properties of the equivalence relation, the main tool is the characterization by the rewriting rule called the k-switching. Using this rule it is straightforward to show that the language comprised of the lexicographically least elements of the k-abelian equivalence classes is regular. Further word-combinatorial analysis of the lexicographically least elements leads us to describe the deterministic finite automata recognizing this language. Using tools from formal language theory combined with our analysis, we give an optimal expression for the asymptotic growth rate of the number of k-abelian equivalence classes of length n over an m-letter alphabet. Explicit formulae are computed for small values of k and m, and these sequences appear in Sloane’s Online Encyclopedia of Integer Sequences. Due to the fact that the k-abelian equivalence relation is a congruence of the free monoid, we study equations over the k-abelian equivalence classes. The main result in this setting is that any system of equations of k-abelian equivalence classes is equivalent to one of its finite subsystems, i.e., the monoid defined by the k-abelian equivalence relation possesses the compactness property. Concerning infinite words, we mainly consider the (k-)abelian complexity function. We complete a classification of the asymptotic abelian complexities of pure morphic binary words. In other words, given a morphism which has an infinite binary fixed point, the limit superior asymptotic abelian complexity of the fixed point can be computed (in principle). We also give a new proof of the fact that the k-abelian complexity of a Sturmian word is n + 1 for length n 2k. In fact, we consider several aspects of the k-abelian equivalence relation in Sturmian words using a dynamical interpretation of these words. We reprove the fact that any Sturmian word contains arbitrarily large k-abelian repetitions. The methods used allow to analyze the situation in more detail, and this leads us to define the so-called k-abelian critical exponent which measures the ratio of the exponent and the length of the root of a k-abelian repetition. This notion is connected to a deep number theoretic object called the Lagrange spectrum
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