Three-letter Alphabet Research Articles

<i>m</i>-aperiodic words on three-letter alphabet

The work is devoted to the study of sets of aperiodic words over a finite alphabet. The set of aperiodic words can be considered as a dictionary of some finite formal language. The existence of infinite words in two-letter or three-letter alphabets that do not contain subwords that are third powers or, respectively, squares of other words was first discovered more than a hundred years ago. S.I. Adyan in 2010 constructed an example of an infinite sequence of irreducible words, each of which is the beginning of the next and does not contain word squares in a two-letter alphabet. S.E. Arshon established the existence of an n-digit asymmetric repetition-free sequence for an alphabet of at least three letters. In the monograph by S.I. Ad- yan proved that in an alphabet of two symbols there exist infinite 3-aperiodic sequences. In the works of other authors, generalizations of aperiodicity were considered, when not only the powers of some subwords were excluded. In the monograph by A.Yu. Olshansky proved the infinity of the set of 6-aperiodic words in a two-letter alphabet and obtained an estimate for the number of such words of any given length. The au- thor previously considered the case of a three-letter alphabet only in the case of 6-aperiodic words. In this article, we prove the infinity of the set of m-aperiodic words in the three-letter alphabet at m≥4and obtain an estimate for the set of such words. The results can be applied when encoding information in space communications.

The smallest Mealy automaton generating an indicable regular branch group

Abstract We construct a two-state Mealy automaton A over the three-letter alphabet generating a regular branch group G ⁢ ( A ) {G(A)} , which surjects onto the infinite cyclic group. Some algebraic and geometric properties of the group G ⁢ ( A ) {G(A)} are derived. In particular, this group has a nearly finitary subgroup of index two, is amenable, just non-solvable, has exponential growth, and its action on the corresponding regular rooted tree is self-replicating, contracting, but it does not have the congruence subgroup property. We also derive in detail an ascending finite L-presentation for the group G ⁢ ( A ) {G(A)} .

Toehold-Mediated Strand Displacement in Random Sequence Pools.

Toehold-mediated strand displacement (TMSD) has been used extensively for molecular sensing and computing in DNA-based molecular circuits. As these circuits grow in complexity, sequence similarity between components can lead to cross-talk, causing leak, altered kinetics, or even circuit failure. For small non-biological circuits, such unwanted interactions can be designed against. In environments containing a huge number of sequences, taking all possible interactions into account becomes infeasible. Therefore, a general understanding of the impact of sequence backgrounds on TMSD reactions is of great interest. Here, we investigate the impact of random DNA sequences on TMSD circuits. We begin by studying individual interfering strands and use the obtained data to build machine learning models that estimate kinetics. We then investigate the influence of pools of random strands and find that the kinetics are determined by only a small subpopulation of strongly interacting strands. Consequently, their behavior can be mimicked by a small collection of such strands. The equilibration of the circuit with the background sequences strongly influences this behavior, leading to up to 1 order of magnitude difference in reaction speed. Finally, we compare two established and one novel technique that speed up TMSD reactions in random sequence pools: a three-letter alphabet, protection of toeholds by intramolecular secondary structure, or by an additional blocking strand. While all of these techniques were useful, only the latter can be used without sequence constraints. We expect that our insights will be useful for the construction of TMSD circuits that are robust to molecular noise.

Periodic stationary solutions of the Nagumo lattice differential equation: existence regions and their number

The Nagumo lattice differential equation admits stationary solutions with arbitrary spatial period for sufficiently small diffusion rate. The continuation from the stationary solutions of the decoupled system (a system of isolated nodes) is used to determine their types; the solutions are labelled by words from a three-letter alphabet. Each stationary solution type can be assigned a parameter region in which the solution can be uniquely identified. Numerous symmetries present in the equation cause some of the regions to have identical or similar shape. With the help of combinatorial enumeration, we derive formulas determining the number of qualitatively different existence regions. We also discuss possible extensions to other systems with more general nonlinear terms and/or spatial structure.

Partial Information Decomposition and the Information Delta: A Geometric Unification Disentangling Non-Pairwise Information.

Information theory provides robust measures of multivariable interdependence, but classically does little to characterize the multivariable relationships it detects. The Partial Information Decomposition (PID) characterizes the mutual information between variables by decomposing it into unique, redundant, and synergistic components. This has been usefully applied, particularly in neuroscience, but there is currently no generally accepted method for its computation. Independently, the Information Delta framework characterizes non-pairwise dependencies in genetic datasets. This framework has developed an intuitive geometric interpretation for how discrete functions encode information, but lacks some important generalizations. This paper shows that the PID and Delta frameworks are largely equivalent. We equate their key expressions, allowing for results in one framework to apply towards open questions in the other. For example, we find that the approach of Bertschinger et al. is useful for the open Information Delta question of how to deal with linkage disequilibrium. We also show how PID solutions can be mapped onto the space of delta measures. Using Bertschinger et al. as an example solution, we identify a specific plane in delta-space on which this approach’s optimization is constrained, and compute it for all possible three-variable discrete functions of a three-letter alphabet. This yields a clear geometric picture of how a given solution decomposes information.

Kaleidoscope substitutions and pairwise well-formed modes: major-minor duality transformationally revisited

Pairwise well-formed modes are investigated as substitutions on words over a three-letter alphabet. Motivating examples are the major and minor modes, here described in terms of the substitutions and . Pairwise well-formedness is investigated through the inspection of the two-letter-mergings of a substiution, namely concatenations with letter insertions from the right and letter projections from the left. The appropriate generalizations of the two mentioned examples are the plain and twisted kaleidoscope substitutions. Four of their mergings turn out to be Sturmian morphisms with interesting interdependencies. The diatonic mergings are musically related to the medieval modes, which are fifth-generated. The syntonic mergings are third-generated and provide a common neutral anchor for the major and minor modes. The paper thus provides a new proposal in order to integrate the diatonic side of harmonic tonality with its triadic aspects, which we characterize as its syntonic side.

Subset Synchronization in Monotonic Automata

We study extremal and algorithmic questions of subset and careful synchronization in monotonic automata. We show that several synchronization problems that are hard in general automata can be solved in polynomial time in monotonic automata, even without knowing a linear order of the states preserved by the transitions. We provide asymptotically tight bounds on the maximum length of a shortest word synchronizing a subset of states in a monotonic automaton and a shortest word carefully synchronizing a partial monotonic automaton. We provide a complexity framework for dealing with problems for monotonic weakly acyclic automata over a three-letter alphabet, and use it to prove NP-completeness and inapproximability of problems such as {\sc Finite Automata Intersection} and the problem of computing the rank of a subset of states in this class. We also show that checking whether a monotonic partial automaton over a four-letter alphabet is carefully synchronizing is NP-hard. Finally, we give a simple necessary and sufficient condition when a strongly connected digraph with a selected subset of vertices can be transformed into a deterministic automaton where the corresponding subset of states is synchronizing.

Existence of Words over Tree-Letter Alphabet not Containing Squares with Replacement Errors

The paper concerns some issues related to the existence of periodic structures in words from formal languages. Squares, i.e., fragments of the form xx, where x is some word, and ∆-squares, i.e., fragments of the form xy, where the word x is different from the word y by not more than ∆ letters, are considered as periodic structures. We show the existence of arbitrarily long words over three-letter alphabet not Containing ∆-Squares with the period exceeding ∆. In particular, such words are constructed for all possible values ∆.

Remarks concerning the freeness problem over morphism and matrix semigroups

We study the freeness problem over morphism and matrix semigroups. We show that the freeness problem is undecidable for morphisms over a three-letter alphabet. We show that there is a commutative semiring R such that the freeness problem is undecidable for upper-triangular 2×2 matrices having entries in R.

Discrete segments of [formula omitted] constructed by synchronization of words

We study a natural and naive composition algorithm which takes three input words written on two-letter alphabets and synchronizes them into a word on a three-letter alphabet. We show that in the case where the three input words are compatible Christoffel words, the algorithm provides a synchronization of the letters which allows the geometrical interpretation of the input words to be inherited by the output word forming a 3D discrete line segment. A second approach is considered while applying our composition algorithm to words defined by stripes meeting at a corner of discrete planes. We show that, under certain conditions, the output of the algorithm corresponds to the normal vector of the plane.

Avoiding abelian squares in partial words

Erdős raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes. Several of our constructions are based on iterating morphisms. In the case of one hole, we prove that the minimal alphabet size is four, while in the case of more than one hole, we prove that it is five. We also investigate the number of partial words of length n with a fixed number of holes over a five-letter alphabet that avoid abelian squares and show that this number grows exponentially with n.

Generalized Davenport–Schinzel sequences and their 0–1 matrix counterparts

A generalized Davenport–Schinzel sequence is one over a finite alphabet whose subsequences are not isomorphic to a forbidden subsequence σ. What is the maximum length of such a σ-free sequence, as a function of its alphabet size n? Is the extremal function linear or nonlinear? And what characteristics of σ determine the answers to these questions? It is known that such sequences have length at most n ⋅ 2 ( α ( n ) ) O ( 1 ) , where α is the inverse-Ackermann function and the O ( 1 ) depends on σ. We resolve a number of open problems on the extremal properties of generalized Davenport–Schinzel sequences. Among our results: 1. We give a nearly complete characterization of linear and nonlinear σ ∈ { a , b , c } ⁎ over a three-letter alphabet. Specifically, the only repetition-free minimally nonlinear forbidden sequences are a b a b a and a b c a c b c . 2. We prove there are at least four minimally nonlinear forbidden sequences. 3. We prove that in many cases, doubling a forbidden sequence has no significant effect on its extremal function. For example, Nivaschʼs upper bounds on alternating sequences of the form ( a b ) t and ( a b ) t a , for t ⩾ 3 , can be extended to forbidden sequences of the form ( a a b b ) t and ( a a b b ) t a . 4. Finally, we show that the absence of simple subsequences in σ tells us nothing about σʼs extremal function. For example, for any t, there exists a σ t avoiding a b a b a whose extremal function is Ω ( n ⋅ 2 α t ( n ) ) . Most of our results are obtained by translating questions about generalized Davenport–Schinzel sequences into questions about the density of 0–1 matrices avoiding certain forbidden submatrices. We give new and often tight bounds on the extremal functions of numerous forbidden 0–1 matrices.

BS Seeker: precise mapping for bisulfite sequencing

BackgroundBisulfite sequencing using next generation sequencers yields genome-wide measurements of DNA methylation at single nucleotide resolution. Traditional aligners are not designed for mapping bisulfite-treated reads, where the unmethylated Cs are converted to Ts. We have developed BS Seeker, an approach that converts the genome to a three-letter alphabet and uses Bowtie to align bisulfite-treated reads to a reference genome. It uses sequence tags to reduce mapping ambiguity. Post-processing of the alignments removes non-unique and low-quality mappings.ResultsWe tested our aligner on synthetic data, a bisulfite-converted Arabidopsis library, and human libraries generated from two different experimental protocols. We evaluated the performance of our approach and compared it to other bisulfite aligners. The results demonstrate that among the aligners tested, BS Seeker is more versatile and faster. When mapping to the human genome, BS Seeker generates alignments significantly faster than RMAP and BSMAP. Furthermore, BS Seeker is the only alignment tool that can explicitly account for tags which are generated by certain library construction protocols.ConclusionsBS Seeker provides fast and accurate mapping of bisulfite-converted reads. It can work with BS reads generated from the two different experimental protocols, and is able to efficiently map reads to large mammalian genomes. The Python program is freely available at http://pellegrini.mcdb.ucla.edu/BS_Seeker/BS_Seeker.html.

A generalization of Thue freeness for partial words

This paper approaches the combinatorial problem of Thue freeness for partial words. Partial words are sequences over a finite alphabet that may contain a number of “holes”. First, we give an infinite word over a three-letter alphabet which avoids squares of length greater than two even after we replace an infinite number of positions with holes. Then, we give an infinite word over an eight-letter alphabet that avoids longer squares even after an arbitrary selection of its positions are replaced with holes, and show that the alphabet size is optimal. We find similar results for overlap-free partial words.

On the triplex substitution – combinatorial properties

If a substitution τ over a three-letter alphabet has a positively linear complexity, that is, P τ ( n ) = C 1 n + C 2 ( n ⩾ 1 ) with C 1 , C 2 ⩾ 0 , there are only 4 possibilities: P τ ( n ) = 3 , n + 2 , 2 n + 1 or 3 n. The first three cases have been studied by many authors, but the case 3 n remained unclear. This leads us to consider the triplex substitution σ : a ↦ a b , b ↦ a c b , c ↦ a c c . Studying the factor structure of its fixed point, which is quite different from the other cases, we show that it is of complexity 3 n. We remark that the triplex substitution is also a typical example of invertible substitution over a three-letter alphabet. To cite this article: B. Tan et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Periodicity Problem of Substitutions over Ternary Alphabets

In this paper, we characterize the substitutions over a three-letter alphabet which generate a ultimately periodic sequence.

On the number of repetition-free words

A well-known problem is studied of the number of repetition-free words of the two types: the square-free words over a three-letter alphabet and the cube-free words over a two-letter alphabet.

Statistical geometry approach to the study of functional effects of human nonsynonymous SNPs

The ability to predict the effect of nonsynonymous SNPs (nsSNPs) on protein function is important for the success of genetic disease association studies. Here we present a statistical geometry approach to nsSNP classification based on Delaunay tessellation, whereby the impact of nsSNPs on protein function is correlated with the change in the four-body statistical potential (DeltaQ) of the protein caused by the amino acid substitution. We observed that the DeltaQ of polymorphic proteins with disease-associated nsSNPs (daSNPs) was on average significantly lower than the DeltaQ of the proteins with neutral SNPs (ntSNPs). Clustering amino acid substitutions into conservative and nonconservative groups, and using a three-letter alphabet based on side-chain polarity showed significantly lower DeltaQ in nonconservative changes to daSNPs and when hydrophobic residues were substituted by charged or by polar residues. We also found that the daSNPs in the protein core caused much lower DeltaQ than surface daSNPs. This approach demonstrates a strong correlation between the computed DeltaQ and SNP classification. Integration of our approach with the existing models will help achieve a more precise recognition of nsSNPs that underlie polygenic diseases. All of the programs were written in Java and are available from the authors upon request.

Two-dimensional iterated morphisms and discrete planes

Iterated morphisms of the free monoid are very simple combinatorial objects which produce infinite sequences by replacing iteratively letters by words. The aim of this paper is to introduce a formalism for a notion of two-dimensional morphisms; we show that they can be iterated by using local rules, and that they generate two-dimensional patterns related to discrete approximations of irrational planes with algebraic parameters. We associate such a two-dimensional morphism with any usual Pisot unimodular one-dimensional iterated morphism over a three-letter alphabet.

Two-dimensional iterated morphisms and discrete planes

Iterated morphisms of the free monoid are very simple combinatorial objects which produce infinite sequences by replacing iteratively letters by words. The aim of this paper is to introduce a formalism for a notion of two-dimensional morphisms; we show that they can be iterated by using local rules, and that they generate two-dimensional patterns related to discrete approximations of irrational planes with algebraic parameters. We associate such a two-dimensional morphism with any usual Pisot unimodular one-dimensional iterated morphism over a three-letter alphabet.