On regularity of context-free languages

AbstractThis paper considers conditions under which a context-free language is regular and conditions which imposed on (productions of) a rewriting system generating a context-free language will guarantee that the generated language is regular. In particular: 1.(1) necessary and sufficient conditions on productions of a unitary grammar are given that guarantee the generated language to be regular (a unitary grammar is a semi-Thue system in which the left-hand of each production is the empty word), and2.(2) it is proved that commutativity of a linear language implies its regularity. To obtain the former result, we give a generalization of the Myhill–Nerode characterization of the regular languages in terms of well-quasi orders, along with a generalization of Higman's well-quasi order result concerning the subsequence embedding relation on Σ*. In obtaining the latter results, we introduce the class of periodic languages, and demonstrate how they can be used to characterize the commutative regular languages. Here we also utilize the theory of well-quasi orders

Detecting palindromes, patterns, and borders in regular languages

Given a language L and a nondeterministic finite automaton M, we consider whether we can determine efficiently (in the size of M) if M accepts at least one word in L, or infinitely many words. Given that M accepts at least one word in L, we consider how long a shortest word can be. The languages L that we examine include the palindromes, the non-palindromes, the k-powers, the non-k-powers, the powers, the non-powers (also called primitive words), the words matching a general pattern, the bordered words, and the unbordered words.Comment: Full version of a paper submitted to LATA 2008. This is a new version with John Loftus added as a co-author and containing new results on unbordered word

Transduction of Automatic Sequences and Applications

We consider the implementation of the transduction of automatic sequences, and their generalizations, in the Walnut software for solving decision problems in combinatorics on words. We provide a number of applications, including (a) representations of n! as a sum of three squares (b) overlap-free Dyck words and (c) sums of Fibonacci representations. We also prove results about iterated running sums of the Thue-Morse sequence

On parse trees and Myhill–Nerode-type tools for handling graphs of bounded rank-width

AbstractRank-width is a structural graph measure introduced by Oum and Seymour and aimed at better handling of graphs of bounded clique-width. We propose a formal mathematical framework and tools for easy design of dynamic algorithms running directly on a rank-decomposition of a graph (on contrary to the usual approach which translates a rank-decomposition into a clique-width expression, with a possible exponential jump in the parameter). The main advantage of this framework is a fine control over the runtime dependency on the rank-width parameter. Our new approach is linked to a work of Courcelle and Kanté [7] who first proposed algebraic expressions with a so-called bilinear graph product as a better way of handling rank-decompositions, and to a parallel recent research of Bui-Xuan, Telle and Vatshelle

An introduction to finite automata and their connection to logic

This is a tutorial on finite automata. We present the standard material on determinization and minimization, as well as an account of the equivalence of finite automata and monadic second-order logic. We conclude with an introduction to the syntactic monoid, and as an application give a proof of the equivalence of first-order definability and aperiodicity