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The notion of Lyndon word and Lyndon factorization has shown to have unexpected applications in theory as well in developing novel algorithms on words. A counterpart to these notions are those of inverse Lyndon word and inverse Lyndon factorization. Differently from the Lyndon words, the inverse Lyndon words may be bordered. The relationship between the two factorizations is related to the inverse lexicographic ordering, and has only been recently explored. More precisely, a main open question is how to get an inverse Lyndon factorization from a classical Lyndon factorization under the inverse lexicographic ordering, named CFLin. In this paper we reveal a strong connection between these two factorizations where the border plays a relevant role. More precisely, we show two main results. We say that a factorization has the border property if a nonempty border of a factor cannot be a prefix of the next factor. First we show that there exists a unique inverse Lyndon factorization having the border property. Then we show that this unique factorization with the border property is the so-called canonical inverse Lyndon factorization, named ICFL. By showing that ICFL is obtained by compacting factors of the Lyndon factorization over the inverse lexicographic ordering, we provide a linear time algorithm for computing ICFL from CFLin., Comment: 11 pages, version submitted to MFCS2024. arXiv admin note: text overlap with arXiv:2404.17969, arXiv:1911.01851

The notion of inverse Lyndon word is related to the classical notion of Lyndon word. More precisely, inverse Lyndon words are all and only the nonempty prefixes of the powers of the anti-Lyndon words, where an anti-Lyndon word with respect to a lexicographical order is a classical Lyndon word with respect to the inverse lexicographic order. Each word $w$ admits a factorization in inverse Lyndon words, named the canonical inverse Lyndon factorization $\ICFL(w)$, which maintains the main properties of the Lyndon factorization of $w$. Although there is a huge literature on the Lyndon factorization, the relation between the Lyndon factorization $\CFL_{in}$ with respect to the inverse order and the canonical inverse Lyndon factorization $\ICFL$ has not been thoroughly investigated. In this paper, we address this question and we show how to obtain one factorization from the other via the notion of grouping. This result naturally opens new insights in the investigation of the relationship between $\ICFL$ and other notions, e.g., variants of Burrows Wheeler Transform, as already done for the Lyndon factorization., Comment: arXiv admin note: substantial text overlap with arXiv:1911.01851

Feature embedding methods have been proposed in literature to represent sequences as numeric vectors to be used in some bioinformatics investigations, such as family classification and protein structure prediction. Recent theoretical results showed that the well-known Lyndon factorization preserves common factors in overlapping strings. Surprisingly, the fingerprint of a sequencing read, which is the sequence of lengths of consecutive factors in variants of the Lyndon factorization of the read, is effective in preserving sequence similarities, suggesting it as basis for the definition of novels representations of sequencing reads. We propose a novel feature embedding method for Next-Generation Sequencing (NGS) data using the notion of fingerprint. We provide a theoretical and experimental framework to estimate the behaviour of fingerprints and of the $k$-mers extracted from it, called $k$-fingers, as possible feature embeddings for sequencing reads. As a case study to assess the effectiveness of such embeddings, we use fingerprints to represent RNA-Seq reads and to assign them to the most likely gene from which they were originated as fragments of transcripts of the gene. We provide an implementation of the proposed method in the tool lyn2vec, which produces Lyndon-based feature embeddings of sequencing reads.

Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes, namely the factorization conjecture and the triangle conjecture, proposed by Perrin and Sch\"{u}tzemberger. The latter concerns finite codes $Y$ which are subsets of $a^* B a^*$, where $a$ is a letter and $B$ is an alphabet not containing $a$. A structural property of finite maximal codes has recently been shown by Zhang and Shum. It exhibits a relationship between finite maximal codes and factorizations of cyclic groups. With the aim of highlighting the links between this result and other older ones on maximal and factorizing codes, we give a simpler and a new proof of this result. As a consequence, we prove that for any finite maximal code $X \subseteq (B \cup \{a \})^*$ containing the word $a^{pq}$, where $p,q$ are prime numbers, $X \cap a^* B a^*$ satisfies the triangle conjecture. Let $n$ be a positive integer that is a product of at most two prime numbers. We also prove that it is decidable whether a finite code $Y \cup a^{n} \subseteq a^* B a^* \cup a^*$ is included in a finite maximal code and that, if this holds, $Y \cup a^{n}$ is included in a code that also satisfies the factorization conjecture.

Lyndon words have been largely investigated and showned to be a useful tool to prove interesting combinatorial properties of words. In this paper we state new properties of both Lyndon and inverse Lyndon factorizations of a word $w$, with the aim of exploring their use in some classical queries on $w$. The main property we prove is related to a classical query on words. We prove that there are relations between the length of the longest common extension (or longest common prefix) $lcp(x,y)$ of two different suffixes $x,y$ of a word $w$ and the maximum length $\mathcal{M}$ of two consecutive factors of the inverse Lyndon factorization of $w$. More precisely, $\mathcal{M}$ is an upper bound on the length of $lcp(x,y)$. This result is in some sense stronger than the compatibility property, proved by Mantaci, Restivo, Rosone and Sciortino for the Lyndon factorization and here for the inverse Lyndon factorization. Roughly, the compatibility property allows us to extend the mutual order between local suffixes of (inverse) Lyndon factors to the suffixes of the whole word. A main tool used in the proof of the above results is a property that we state for factors $m_i$ with nonempty borders in an inverse Lyndon factorization: a nonempty border of $m_i$ cannot be a prefix of the next factor $m_{i+1}$. The last property we prove shows that if two words share a common overlap, then their Lyndon factorizations can be used to capture the common overlap of the two words. The above results open to the study of new applications of Lyndon words and inverse Lyndon words in the field of string comparison., Comment: arXiv admin note: text overlap with arXiv:1705.10277

Motivated by applications to string processing, we introduce variants of the Lyndon factorization called inverse Lyndon factorizations. Their factors, named inverse Lyndon words, are in a class that strictly contains anti-Lyndon words, that is Lyndon words with respect to the inverse lexicographic order. The Lyndon factorization of a nonempty word w is unique but w may have several inverse Lyndon factorizations. We prove that any nonempty word w admits a canonical inverse Lyndon factorization, named ICFL(w), that maintains the main properties of the Lyndon factorization of w: it can be computed in linear time, it is uniquely determined, it preserves a compatibility property for sorting suffixes. In particular, the compatibility property of ICFL(w) is a consequence of another result: any factor in ICFL(w) is a concatenation of consecutive factors of the Lyndon factorization of w with respect to the inverse lexicographic order.

Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes. In this paper we discuss this conjectures and their relations with factorizations of cyclic groups.

A splicing system is a formal model of a recombinant behaviour of sets of double stranded DNA molecules when acted on by restriction enzymes and ligase. In this survey we will concentrate on a specific behaviour of a type of splicing systems, introduced by P\u{a}un and subsequently developed by many researchers in both linear and circular case of splicing definition. In particular, we will present recent results on this topic and how they stimulate new challenging investigations., Comment: Appeared in: Discrete Mathematics and Computer Science. Papers in Memoriam Alexandru Mateescu (1952-2005). The Publishing House of the Romanian Academy, 2014. arXiv admin note: text overlap with arXiv:1112.4897 by other authors

We introduce the notion of specular sets which are subsets of groups called here specular and which form a natural generalization of free groups. These sets are an abstract generalization of the natural codings of linear involutions. We prove several results concerning the subgroups generated by return words and by maximal bifix codes in these sets., Comment: arXiv admin note: substantial text overlap with arXiv:1405.3529

We investigate the relation between bifix codes and interval exchange transformations. We prove that the class of natural codings of regular interval echange transformations is closed under maximal bifix decoding., Comment: arXiv admin note: substantial text overlap with arXiv:1305.0127, arXiv:1308.5396

We prove in this note that, for an alphabet with three letters, the set of first return to a given word in a set satisfying the tree condition is a basis of the free group.

We introduce a class of sets of words which is a natural common generalization of Sturmian sets and of interval exchange sets. This class of sets consists of the uniformly recurrent tree sets, where the tree sets are defined by a condition on the possible extensions of bispecial factors. We prove that this class is closed under maximal bifix decoding. The proof uses the fact that the class is also closed under decoding with respect to return words., Comment: arXiv admin note: text overlap with arXiv:1305.0127

Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words defined by the property of the extension graph of each word in the set to be acyclic or connected or a tree. We prove that in a uniformly recurrent tree set, the sets of first return words are bases of the free group on the alphabet. Concerning acyclic sets, we prove as a main result that a set $F$ is acyclic if and only if any bifix code included in $F$ is a basis of the subgroup that it generates., Comment: arXiv admin note: substantial text overlap with arXiv:1305.0127, arXiv:1011.5369, Monatsh. Math. (2015)

We describe in this paper a connection between bifix codes, symbolic dynamical systems and free groups. This is in the spirit of the connection established previously for the symbolic systems corresponding to Sturmian words. We introduce a class of sets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namemly the class of tree sets. We prove as a main result that for a uniformly recurrent tree set $F$, a finite bifix code $X$ on the alphabet $A$ is $F$-maximal of $F$-degree $d$ if and only if it is the basis of a subgroup of index $d$ of the free group on $A$., Comment: arXiv admin note: text overlap with arXiv:1011.5369, arXiv:1305.0120

We give partial results on the factorization conjecture on codes proposed by Schutzenberger. We consider finite maximal codes C over the alphabet A = {a, b} with C \cap a^* = a^p, for a prime number p. Let P, S in Z , with S = S_0 + S_1, supp(S_0) \subset a^* and supp(S_1) \subset a^*b supp(S_0). We prove that if (P,S) is a factorization for C then (P,S) is positive, that is P,S have coefficients 0,1, and we characterize the structure of these codes. As a consequence, we prove that if C is a finite maximal code such that each word in C has at most 4 occurrences of b's and a^p is in C, then each factorization for C is a positive factorization. We also discuss the structure of these codes. The obtained results show once again relations between (positive) factorizations and factorizations of cyclic groups.

We prove new results concerning the relation between bifix codes, episturmian words and subgroups offree groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set $F$ of words ($F$-maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let $F$ be a Sturmian set of words, defined as the set of factors of a strict episturmian word. Our results express the fact that an $F$-maximal bifix code of degree $d$ behaves just as the set of words of $F$ of length $d$. An $F$-maximal bifix code of degree $d$ in a Sturmian set of words on an alphabet with $k$ letters has $(k-1)d+1$ elements. This generalizes the fact that a Sturmian set contains $(k-1)d+1$ words of length $d$. Moreover, given an infinite word $x$, if there is a finite maximal bifix code $X$ of degree $d$ such that $x$ has at most $d$ factors of length $d$ in $X$, then $x$ is ultimately periodic. Our main result states that any $F$-maximal bifix code of degree $d$ on the alphabet $A$ is the basis of a subgroup of index $d$ of the free group on~$A$., Comment: 70 pages + index

Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. Some unanswered questions are related to the computational power of such systems, and finding a characterization of the class of circular languages generated by circular splicing systems is still an open problem. In this paper we solve this problem for complete systems, which are special finite circular splicing systems. We show that a circular language L is generated by a complete system if and only if the set Lin(L) of all words corresponding to L is a pure unitary language generated by a set closed under the conjugacy relation. The class of pure unitary languages was introduced by A. Ehrenfeucht, D. Haussler, G. Rozenberg in 1983, as a subclass of the class of context-free languages, together with a characterization of regular pure unitary languages by means of a decidable property. As a direct consequence, we characterize (regular) circular languages generated by complete systems. We can also decide whether the language generated by a complete system is regular. Finally, we point out that complete systems have the same computational power as finite simple systems, an easy type of circular splicing system defined in the literature from the very beginning, when only one rule is allowed. From our results on complete systems, it follows that finite simple systems generate a class of context-free languages containing non-regular languages, showing the incorrectness of a longstanding result on simple systems.