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This paper discusses a mathematical concept of language that models both artificial and natural languages and thus provides a framework for a unified language processing methodology. This concept of a language is regarded as a communication tool that allows language users to develop knowledges, while interacting with their universe of discourse, and to communicate with each other, while exchanging knowledges. Criteria for consistent usage of a language are established using a Galois connection between language syntax and language semantics. Solutions to ambiguity, paraphrase, attitude, and other problems concerning the relationship between syntax and semantics are addressed. A general schema for language specification is introduced and algorithms that perform language generation and language analysis are discussed as universal tools defined by the specification schema. Language transformations performed by various kinds of translators are examined and correctness criteria of these translators are defined using the language Galois connection.The paper is structured as follows: Section 1 introduces the framework and justifies the necessity of a unified methodology for language processing. Section 2 presents the mathematical concept of a language. Section 3 illustrates the mathematical concept of a language with three kinds of language structures: natural language, logical language, and programming language. Section 4 discusses the algebraic mechanism of language specification that unifies the methodology for language processing tool development. Section 5 formalizes the criterion for the consistency of the language usage, defines the architecture of a unified language processing system, and shows how the consistency criteria for language usage can be employed as correctness criteria for the algorithms performing various language transformations. [Copyright &y& Elsevier]

Abstract: The longest common subsequence (LCS) problem is one of the classical and well-studied problems in computer science. The computation of the LCS is a frequent task in DNA sequence analysis, and has applications to genetics and molecular biology. In this paper we introduce new variants of LCS problem and present efficient algorithms to solve them. In particular we introduce the notion of gap constraints in the LCS problems. For the LCS problem with fixed gap, we first present a naive algorithm runs in time, where is the total number of ordered pairs of positions at which the two strings match and is the fixed gap constraint. We then improve the running time to using some novel techniques. Furthermore, we present an algorithm that is independent of and runs in time. Using these techniques, we also present a new algorithm to solve the original LCS problem. Additionally, we modify our algorithms to handle elastic and rigid gaps. We also apply the notion of rigidness to the original LCS problem and modify the traditional dynamic programming solution to handle the rigidness presenting a algorithm to solve the problem. Finally, we also improve the solution to Rigid Fixed Gap LCS to . Notably, in all of the above cases, we assume that the two given strings are of equal length i.e. . But our results can be easily extended to handle two strings of different length. [Copyright &y& Elsevier]