An Entertaining Example of Using the Concepts of Context-Free Grammar and Pushdown Automation

A formal-linguistic approach for solving an entertaining task is offered in this paper. The well-known task of the Hanoi towers is discussed in relation to some concepts of formal languages and grammars. A context-free grammar which generates an algorithm for solving this task is described. A deterministic pushdown automation which in its work imitates the work of monks in solving the task of the Hanoi towers is built.

Context-free Grammars for Triangular Arrays

We consider context-free grammars of the form G = {f → f^b1＋b2＋1g^a1＋a2, g → f^b1 g^a1＋1},where ai and bi are integers sub ject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T（n, k）}0≤k≤n satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let Tn （x） =∑k=0^n T（n, k）x^k. Based on the differential operator with respect to G, we define a sequence of linear operators Pn such that Tn＋1（x） = Pn（Tn（x））. Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Br？ndén, we obtain a necessary and sufficient condition for the operator Pn to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh.Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers.

Predicting RNA Secondary Structure Using Profile Stochastic Context-Free Grammars and Phylogenic Analysis

Stochastic context-free grammars （SCFGs） have been applied to predicting RNA secondary structure. The prediction of RNA secondary structure can be facilitated by incorporating with comparative sequence analysis. However, most of existing SCFG-based methods lack explicit phylogenic analysis of homologous RNA sequences, which is probably the reason why these methods are not ideal in practical application. Hence, we present a new SCFG-based method by integrating phylogenic analysis with the newly defined profile SCFG. The method can be summarized as： 1） we define a new profile SCFG, M, to depict consensus secondary structure of multiple RNA sequence alignment; 2） we introduce two distinct hidden Markov models, λ and λ＇, to perform phylogenic analysis of homologous RNA sequences. Here, λ＇ is for non-structural regions of the sequence and λ＇ is for structural regions of the sequence; 3） we merge λ and λ＇ into M to devise a combined model for prediction of RNA secondary structure. We tested our method on data sets constructed from the Rfam database. The sensitivity and specificity of our method are more accurate than those of the predictions by Pfold.

A Grammatical Approach of Multi-Localization of GUIs: Application to the Multi-Localization of the Checkers Game GUI

By proposing tools that help for the accomplishment of tasks in almost all sectors of activities, computer science has revolutionized the world in a general way. Nowadays, it addresses the peculiarities of peoples through their culture in order to produce increasingly easy-to-use software for end users: This is the aim of software localization. Localizing a software consists among other things, in adapting its GUI according to the end user culture. We propose in this paper a generic approach allowing accomplishing this adaptation, even for multi-user applications like gaming applications, collaborative editors, etc. Techniques of functional interpretations of abstracts structures parameterized by algebras, constitute the formal base of our approach.

LR(K) Parser Construction Using Bottom-up Formal Analysis

Design and construction of an error-free compiler is a difficult and challenging process. The main functionality of a compiler is to translate a source code to an executable machine code correctly and efficiently. In formal verification of software, semantics of a language has more meanings than the syntax. It means source program verification does not give guarantee the generated code is correct. This is because the compiler may lead to an incorrect target program due to bugs in itself. It means verification of a compiler is much more important than verification of a source program. In this paper, we present a new approach by linking context-free grammar and Z notation to construct LR(K) parser. This has several advantages because correctness of the compiler depends on describing rules that must be written in formal languages. First, we have defined grammar then language derivation procedure is given using right-most derivations. Verification of a given language is done by recursive procedures based on the words. Ambiguity of a language is checked and verified. The specification is analyzed and validated using Z/Eves tool. Formal proofs are presented using powerful techniques of reduction and rewriting available in Z/Eves.

Several variants of the Dumont differential system and permutation statistics

The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont(1979)and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme for the cycle structure of permutations. We then introduce two types of Jacobi pairs of differential equations. We present a general method to derive the solutions of these differential equations. As applications,we present some characterizations for several permutation statistics.