Shuffling and Unshuffling

We consider various shuffling and unshuffling operations on languages and words, and examine their closure properties. Although the main goal is to provide some good and novel exercises and examples for undergraduate formal language theory classes, we also provide some new results and some open problems

Decision Problems on Copying and Shuffling

We study decision problems of the form: given a regular or linear context-free language $L$, is there a word of a given fixed form in $L$, where given fixed forms are based on word operations copy, marked copy, shuffle and their combinations

Square-free Words with Square-free Self-shuffles

We answer a question of Harju: For every n ≥ 3 there is a square-free ternary word of length n with a square-free self-shuffle.http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p

Topological Sorting with Regular Constraints

We introduce the constrained topological sorting problem (CTS): given a regular language K and a directed acyclic graph G with labeled vertices, determine if G has a topological sort that forms a word in K. This natural problem applies to several settings, e.g., scheduling with costs or verifying concurrent programs. We consider the problem CTS[K] where the target language K is fixed, and study its complexity depending on K. We show that CTS[K] is tractable when K falls in several language families, e.g., unions of monomials, which can be used for pattern matching. However, we show that CTS[K] is NP-hard for K = (ab)^* and introduce a shuffle reduction technique to show hardness for more languages. We also study the special case of the constrained shuffle problem (CSh), where the input graph is a disjoint union of strings, and show that CSh[K] is additionally tractable when K is a group language or a union of district group monomials. We conjecture that a dichotomy should hold on the complexity of CTS[K] or CSh[K] depending on K, and substantiate this by proving a coarser dichotomy under a different problem phrasing which ensures that tractable languages are closed under common operators

Combinatorial Algorithms for Subsequence Matching: A Survey

In this paper we provide an overview of a series of recent results regarding algorithms for searching for subsequences in words or for the analysis of the sets of subsequences occurring in a word.Comment: This is a revised version of the paper with the same title which appeared in the Proceedings of NCMA 2022, EPTCS 367, 2022, pp. 11-27 (DOI: 10.4204/EPTCS.367.2). The revision consists in citing a series of relevant references which were not covered in the initial version, and commenting on how they relate to the results we survey. arXiv admin note: text overlap with arXiv:2206.1389