Unambiguous 1-Uniform Morphisms

A morphism h is unambiguous with respect to a word w if there is no other morphism g that maps w to the same image as h. In the present paper we study the question of whether, for any given word, there exists an unambiguous 1-uniform morphism, i.e., a morphism that maps every letter in the word to an image of length 1.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Restricted ambiguity of erasing morphisms

A morphism h is called ambiguous for a string s if there is another morphism that maps s to the same image as h; otherwise, it is called unambiguous. In this paper, we examine some fundamental problems on the ambiguity of erasing morphisms. We provide a detailed analysis of so-called ambiguity partitions, and our main result uses this concept to characterise those strings that have a morphism of strongly restricted ambiguity. Furthermore, we demonstrate that there are strings for which the set of unambiguous morphisms, depending on the size of the target alphabet of these morphisms, is empty, finite or infinite. Finally, we show that the problem of the existence of unambiguous erasing morphisms is equivalent to some basic decision problems for nonerasing multi-pattern languages

Weakly Unambiguous Morphisms

A nonerasing morphism sigma is said to be weakly unambiguous with respect to a word w if sigma is the only nonerasing morphism that can map w to sigma(w), i.e., there does not exist any other nonerasing morphism tau satisfying tau(w) = sigma(w). In the present paper, we wish to characterise those words with respect to which there exists such a morphism. This question is nontrivial if we consider so-called length-increasing morphisms, which map a word to an image that is strictly longer than the word. Our main result is a compact characterisation that holds for all morphisms with ternary or larger target alphabets. We also comprehensively describe those words that have a weakly unambiguous length-increasing morphism with a unary target alphabet, but we have to leave the problem open for binary alphabets, where we can merely give some non-characteristic conditions

Flexibility of Crab Chemosensory Hairs Enables Flicking Antennules to Sniff

The first step in smelling is capture of odorant molecules from the surrounding fluid. We used lateral flagella of olfactory antennules of crabs Callinectes sapidus to study the physical process of odor capture by antennae bearing dense tufts of hair-like chemosensory sensilla (aesthetascs). Fluid flow around and through aesthetasc arrays on dynamically scaled models of lateral flagella of C. sapidus was measured by particle image velocimetry to determine how antennules sample the surrounding water when they flick. Models enabled separate evaluation of the effects of flicking speed, aesthetasc spacing, and antennule orientation. We found that crab antennules, like those of other malacostracan crustaceans, take a discrete water sample during each flick by having a rapid downstroke, during which water flows into the aesthetasc array, and a slow recovery stroke, when water is trapped in the array and odorants have time to diffuse to aesthetascs. However, unlike antennules of crustaceans with sparse aesthetasc arrays, crabs enhance sniffing via additional mechanisms: 1) Aesthetascs are flexible and splay as a result of the hydrodynamic drag during downstrokes, then clump together during return strokes; and 2) antennules flick with aesthetascs on the upstream side of the stalk during downstrokes, but are hidden downstream during return strokes. Aiming aesthetascs into ambient flow maintains sniffing. When gaps between aesthetascs are wide, changes in antennule speed are more effective at altering flow through the array than when gaps are narrow. Nonetheless, if crabs had fixed gap widths, their ability to take discrete samples of their odorant environment would be diminished

Morphic Primitivity and Alphabet Reductions

An alphabet reduction is a 1-uniform morphism that maps a word to an image that contains a smaller number of dfferent letters. In the present paper we investigate the effect of alphabet reductions on morphically primitive words, i. e., words that are not a fixed point of a nontrivial morphism. Our first main result answers a question on the existence of unambiguous alphabet reductions for such words, and our second main result establishes whether alphabet reductions can be given that preserve morphic primitivity. In addition to this, we study Billaud's Conjecture - which features a dfferent type of alphabet reduction, but is otherwise closely related to the main subject of our paper - and prove its correctness for a special case

Patterns with Bounded Treewidth

We show that any parameter of patterns that is an upper bound for the treewidth of appropriate encodings of patterns as relational structures, if restricted to a constant, allows the membership problem for pattern languages to be solved in polynomial time. Furthermore, we identify a new such parameter, called the scope coincidence degree

Unique Decipherability in Formal Languages

We consider several language-theoretic aspects of various notions of unique decipherability (or unique factorization) in formal languages. Given a language L at some position within the Chomsky hierarchy, we investigate the language of words UD(L) in L^* that have unique factorization over L. We also consider similar notions for weaker forms of unique decipherability, such as numerically decipherable words ND(L), multiset decipherable words MSD(L) and set decipherable words SD(L). Although these notions of unique factorization have been considered before, it appears that the languages of words having these properties have not been positioned in the Chomsky hierarchy up until now. We show that UF(L), ND(L), MSD(L) and SD(L) need not be context-free if L is context-free. In fact ND(L) and MSD(L) need not be context-free even if L is finite, although UD(L) and SD(L) are regular in this case. We show that if L is context-sensitive, then so are UD(L), ND(L), MSD(L) and SD(L). We also prove that the membership problem (resp., emptiness problem) for these classes is PSPACE-complete (resp., undecidable). We finally determine upper and lower bounds on the length of the shortest word of L^* not having the various forms of unique decipherability into elements of L

Finding Shuffle Words That Represent Optimal Scheduling of Shared Memory Access

In the present paper, we introduce and study the problem of computing, for any given finite set of words, a shuffle word with a minimum so-called scope coincidence degree. The scope coincidence degree is the maximum number of different symbols that parenthesise any position in the shuffle word. This problem is motivated by an application of a new automaton model and can be regarded as the problem of scheduling shared memory accesses of some parallel processes in a way that minimises the number of memory cells required. We investigate the complexity of this problem and show that it can be solved in polynomial time

Unambiguous Morphic Images of Strings

We study a fundamental combinatorial problem on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism T with T(i) ≠ σ(i) for some symbol i in α and, nevertheless, T(α) = σ(α). As a consequence of its elementary nature, this question shows a variety of connections to those topics in discrete mathematics which are based on finite strings and morphisms such as pattern languages, equality sets and, thus, the Post Correspondence Problem. Our studies demonstrate that the existence of unambiguous morphic images essen- tially depends on the structure of α: We introduce a partition of the set of all finite strings into those that are decomposable (referred to as prolix) in a particular manner and those that are indecomposable (called succinct). This partition, that is also known to be of major importance for the research on pattern languages and on finite fixed points of morphisms, allows to formulate our main result according to which a string α can be mapped by an injective morphism onto an unambiguous image if and only if α is succinct