Some Single and Combined Operations on Formal Languages: Algebraic Properties and Complexity

In this thesis, we consider several research questions related to language operations in the following areas of automata and formal language theory: reversibility of operations, generalizations of (comma-free) codes, generalizations of basic operations, language equations, and state complexity. Motivated by cryptography applications, we investigate several reversibility questions with respect to the parallel insertion and deletion operations. Among the results we obtained, the following result is of particular interest. For languages L1, L2 ⊆ Σ∗, if L2 satisfies the condition L2ΣL2 ∩ Σ+L2Σ+ = ∅, then any language L1 can be recovered after first parallel-inserting L2 into L1 and then parallel-deleting L2 from the result. This property reminds us of the definition of comma-free codes. Following this observation, we define the notions of comma codes and k-comma codes, and then generalize them to comma intercodes and k-comma intercodes, respectively. Besides proving all these new codes are indeed codes, we obtain some interesting properties, as well as several hierarchical results among the families of the new codes and some existing codes such as comma-free codes, infix codes, and bifix codes. Another topic considered in this thesis are some natural generalizations of basic language operations. We introduce block insertion on trajectories and block deletion on trajectories, which properly generalize several sequential as well as parallel binary language operations such as catenation, sequential insertion, k-insertion, parallel insertion, quotient, sequential deletion, k-deletion, etc. We obtain several closure properties of the families of regular and context-free languages under the new operations by using some relationships between these new operations and shuffle and deletion on trajectories. Also, we obtain several decidability results of language equation problems with respect to the new operations. Lastly, we study the state complexity of the following combined operations: L1L2∗, L1L2R, L1(L2 ∩ L3), L1(L2 ∪ L3), (L1L2)R, L1∗L2, L1RL2, (L1 ∩ L2)L3, (L1 ∪ L2)L3, L1L2 ∩ L3, and L1L2 ∪ L3 for regular languages L1, L2, and L3. These are all the combinations of two basic operations whose state complexities have not been studied in the literature

Variable-Length Coding of Two-Sided Asymptotically Mean Stationary Measures

We collect several observations that concern variable-length coding of two-sided infinite sequences in a probabilistic setting. Attention is paid to images and preimages of asymptotically mean stationary measures defined on subsets of these sequences. We point out sufficient conditions under which the variable-length coding and its inverse preserve asymptotic mean stationarity. Moreover, conditions for preservation of shift-invariant $\sigma$-fields and the finite-energy property are discussed and the block entropies for stationary means of coded processes are related in some cases. Subsequently, we apply certain of these results to construct a stationary nonergodic process with a desired linguistic interpretation.Comment: 20 pages. A few typos corrected after the journal publicatio

Writing with Discipline: A Call for Avoiding APA Style Guide Errors in Manuscript Preparation

The education community in the United States—as in many countries—is extremely large and diverse. Indeed, as documented by Mosteller, Nave, and Miech (2004), The United States has more than 3.6 million teachers in elementary and secondary education, more than 100,000 principals, and about 15,000 school districts, each with its own set of district administrators, school board members, and concerned citizens. The parents and family members of the 60 million students in elementary and secondary education represent another constituency, as do the policymakers and legislators in the 50 states (along with the District of Columbia) and at the federal level. Postsecondary education represents another 1 million faculty members, along with an enrollment of 15 million undergraduates and 1.8 million graduate students. (p. 29) Indeed, with the number of individuals involved in the educational system, educational research has the potential to play a pivotal role in improving the quality of education—from Kindergarten through primary, through secondary, through tertiary education. Yet, for educational research to play such a role, its findings must be disseminated to individuals (e.g., educators, administrators, stakeholders, policymakers) and groups (e.g., teacher associations) who can most effectively use them (Mosteller et al., 2004; Onwuegbuzie, Leech, & Whitmore, 2008). Unfortunately, research findings do not disseminate themselves, regardless of how statistically, practically, clinically, or economically significant they are for the field of education. Rather, it is educational researchers in general and practitioner-researchers in particular who must convey these findings

The Impossibility of Extending Random Dictatorship to Weak Preferences

Random dictatorship has been characterized as the only social decision scheme that satisfies efficiency and strategyproofness when individual preferences are strict. We show that no extension of random dictatorship to weak preferences satisfies these properties, even when significantly weakening the required degree of strategyproofness

Incorporating Punctuation Into the Sentence Grammar: A Lexicalized Tree Adjoining Grammar Perspective

Punctuation helps us to structure, and thus to understand, texts. Many uses of punctuation straddle the line between syntax and discourse, because they serve to combine multiple propositions within a single orthographic sentence. They allow us to insert discourse-level relations at the level of a single sentence. Just as people make use of information from punctuation in processing what they read, computers can use information from punctuation in processing texts automatically. Most current natural language processing systems fail to take punctuation into account at all, losing a valuable source of information about the text. Those which do mostly do so in a superficial way, again failing to fully exploit the information conveyed by punctuation. To be able to make use of such information in a computational system, we must first characterize its uses and find a suitable representation for encoding them. The work here focuses on extending a syntactic grammar to handle phenomena occurring within a single sentence which have punctuation as an integral component. Punctuation marks are treated as full-fledged lexical items in a Lexicalized Tree Adjoining Grammar, which is an extremely well-suited formalism for encoding punctuation in the sentence grammar. Each mark anchors its own elementary trees and imposes constraints on the surrounding lexical items. I have analyzed data representing a wide variety of constructions, and added treatments of them to the large English grammar which is part of the XTAG system. The advantages of using LTAG are that its elementary units are structured trees of a suitable size for stating the constraints we are interested in, and the derivation histories it produces contain information the discourse grammar will need about which elementary units have used and how they have been combined. I also consider in detail a few particularly interesting constructions where the sentence and discourse grammars meet-appositives, reported speech and uses of parentheses. My results confirm that punctuation can be used in analyzing sentences to increase the coverage of the grammar, reduce the ambiguity of certain word sequences and facilitate discourse-level processing of the texts