Languages convex with respect to binary relations, and their closure properties

A language is prefix-convex if it satisfies the condition that, if a word w and its prefix u are in the language, then so is every prefix of w that has u as a prefix. Prefix-convex languages include prefix-closed languages at one end of the spectrum, and prefix-free languages, which include prefix codes, at the other. In a similar way, we define suffix-, bifix-, factor-, and subword-convex languages and their closed and free counterparts. This provides a common framework for diverse languages such as codes, factorial languages and ideals. We examine the relationships among these languages. We generalize these notions to arbitrary binary relations on the set of all words over a given alphabet, and study the closure properties of such languages

Problems Related to Shortest Strings in Formal Languages

In formal language theory, studying shortest strings in languages, and variations thereof, can be useful since these strings can serve as small witnesses for properties of the languages, and can also provide bounds for other problems involving languages. For example, the length of the shortest string accepted by a regular language provides a lower bound on the state complexity of the language. In Chapter 1, we introduce some relevant concepts and notation used in automata and language theory, and we show some basic results concerning the connection between the length of the shortest string and the nondeterministic state complexity of a regular language. Chapter 2 examines the effect of the intersection operation on the length of the shortest string in regular languages. A tight worst-case bound is given for the length of the shortest string in the intersection of two regular languages, and loose bounds are given for two variations on the problem. Chapter 3 discusses languages that are defined over a free group instead of a free monoid. We study the length of the shortest string in a regular language that becomes the empty string in the free group, and a variety of bounds are given for different cases. Chapter 4 mentions open problems and some interesting observations that were made while studying two of the problems: finding good bounds on the length of the shortest squarefree string accepted by a deterministic finite automaton, and finding an efficient way to check if a finite set of finite words generates the free monoid. Some of the results in this thesis have appeared in work that the author has participated in \cite{AngPigRamSha,AngShallit}

A Whole-Genome RNA Interference Screen Reveals a Role for Spry2 in Insulin Transcription and the Unfolded Protein Response.

Insulin production by the pancreatic β-cell is required for normal glucose homeostasis. While key transcription factors that bind to the insulin promoter are known, relatively little is known about the upstream regulators of insulin transcription. Using a whole-genome RNA interference screen, we uncovered 26 novel regulators of insulin transcription that regulate diverse processes including oxidative phosphorylation, vesicle traffic, and the unfolded protein response (UPR). We focused on Spry2-a gene implicated in human type 2 diabetes by genome-wide association studies but without a clear connection to glucose homeostasis. We showed that Spry2 is a novel UPR target and its upregulation is dependent on PERK. Knockdown of Spry2 resulted in reduced expression of Serca2, reduced endoplasmic reticulum calcium levels, and induction of the UPR. Spry2 deletion in the adult mouse β-cell caused hyperglycemia and hypoinsulinemia. Our study greatly expands the compendium of insulin promoter regulators and demonstrates a novel β-cell link between Spry2 and human diabetes