Central sets and substitutive dynamical systems

In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of \nats possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone-\v{C}ech compactification \beta \nats . This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone-\v{C}ech compactification of \nats.Comment: arXiv admin note: substantial text overlap with arXiv:1110.4225, arXiv:1301.511

On the dual post correspondence problem

The Dual Post Correspondence Problem asks whether, for a given word α, there exists a pair of distinct morphisms σ,τ, one of which needs to be non-periodic, such that σ(α) = τ(α) is satisfied. This problem is important for the research on equality sets, which are a vital concept in the theory of computation, as it helps to identify words that are in trivial equality sets only. Little is known about the Dual PCP for words α over larger than binary alphabets, especially for so-called ratio-primitive examples. In the present paper, we address this question in a way that simplifies the usual method, which means that we can reduce the intricacy of the word equations involved in dealing with the Dual PCP. Our approach yields large sets of words for which there exists a solution to the Dual PCP as well as examples of words over arbitrary alphabets for which such a solution does not exist

Hierarchies of primitive recursive wordsequence functions: Comparisons and decision problems

AbstractIn this paper we consider wordsequence functions, i.e., functions of the type ƒ: Σ∗′ → Σ∗‵ where Σ is a finite alphabet and r ⩾ 0, s > 0. By starting with finite sets of basic functions and by taking the closure with respect to composition, cylindrification and iteration, we give some characterizations of primitive recursive wordsequence functions. We define some hierarchies of length ω2 of these functions by bounding the number of successive compositions and the depth of the nested iterations in the definitions of the functions. In such a manner we obtain refinements of the Axt, Grzegorczyk and Meyer and Ritchie generalized hierarchies of length ω of primitive recursive wordfunctions defined by Von Henke, Indermark and Weihrauch (1972).We consider Loop programs on words (see Ausiello and Moscarini (1976)) by allowing more than one output register, and we prove that the class of functions computed by these programs coincides with the class of primitive recursive wordsequence functions. The hierarchies of functions induce some hierarchies of programs.For the case of functions ƒ: Σ∗′ → Σ∗, our hierarchies are compared with the Axt et al. generalized hierarchies.We also compare our hierarchies with storage hierarchies, and we analyze the power of the Loop programs as acceptors.Finally, we state some decidability results for the considered classes

Combinatorics of unique maximal factorization families (UMFFs)

Suppose a set W of strings contains exactly one rotation (cyclic shift) of every primitive string on some alphabet Σ. Then W is a circ-UMFF if and only if every word in Σ+ has a unique maximal factorization over W. The classic circ-UMFF is the set of Lyndon words based on lexicographic ordering (1958). Duval (1983) designed a linear sequential Lyndon factorization algorithm; a corresponding PRAM parallel algorithm was described by J. Daykin, Iliopoulos and Smyth (1994). Daykin and Daykin defined new circ-UMFFs based on various methods for totally ordering sets of strings (2003), and further described the structure of all circ-UMFFs (2008). Here we prove new combinatorial results for circ-UMFFs, and in particular for the case of Lyndon words. We introduce Acrobat and Flight Deck circ-UMFFs, and describe some of our results in terms of dictionaries. Applications of circ-UMFFs pertain to structured methods for concatenating and factoring strings over ordered alphabets, and those of Lyndon words are wide ranging and multidisciplinary