Rani Siromoney

Abstract families of matrices and picture languages

A linguistic model to generate matrices (arrays of terminals) is presented. The sixclosure axioms which define abstract families of languages (AFL) are extended to matrices to obtain abstract families of matrices (AFM). Corresponding automata to recognize matrices are defined and equivalence established. Immediate application is shown in picture processing by generating a wide class of interesting pictures. Illustrations include generation of kolam patterns, wall paper designs and geometric patterns replicated several times. Simple transformations of a picture such as reflection, translation, halftum, and turning upside down are very naturally obtained.

Picture languages with array rewriting rules

Generative models of picture languages with array rewriting rules are presented. The rewriting rules are regular, context-free or context-sensitive with arrays of terminals in the place of strings of terminals. Derivations are restricted by the condition for row and column catenation. The grammars describe a wide variety of pictures and are more powerful than the matrix grammars for digital pictures introduced in our earlier paper. A distinct hierarchy is shown to exist between the different classes introduced. The models are closed under reflection (about base and rightmost vertical), halfturn, quarter-turn, transpose, and conjugation. Further closure properties such as union, product, star and homomorphism are examined. The models can be applied to generate several interesting patterns of kolam and to describe the repetitive patterns of two-dimensional crystallography. Each letter of the alphabet of different sizes can be generated by a context-free array grammar.

On equal matrix languages

A generative grammar called equal matrix grammar which generates a class which meets both context-sensitive and context-free languages is defined and the formal power series generated by context-free grammars is extended to grammars of this system. The Parikh mapping of this family of languages is shown to be semilinear. The Boolean and closure properties of a certain subfamily are examined. For this subfamily, the generative power of equal matrix grammar is higher than that of context-free grammars. For certain inherently ambiguous context-free languages, including that of Parikh, unambiguous grammars of this class exist. The application of equal matrix grammar to the generation of Tamil kernel sentences is given in the appendix.

Array Grammars and Kolam

Kolam designs of South Indian folk art are treated as examples of two-dimensional picture languages, Many of the complicated kolain patterns are seen to be generable by context-free array grammars. Two methods of kolam generation are discussed. One method is to generate the kolam patterns by array grammars with a finite number of primitives as terminals. The other is to generate labeled dots by array grammars and then to give a finite number of simple instructions to draw the kolam pattern on the framework of labeled dots.

Circular DNA and Splicing Systems

Circular strings representing DNA molecules and certain recombinant behaviour are formalized. Various actions of splicing schemes on linear and circular DNA molecules are examined. It is shown that there is a difference in the regularity result of Culik and Harju [1] between the linear and circular strings. A consequence of this result is that a conjecture of Head [4] that the circular string language of a splicing system under an action on circular strings is regular, when the set of initial circular strings is regular, is disproved.

Extended controlled table L-arrays

A generative two-dimensional rectangular array model which allows for growth along the edges is proposed. The growth takes place in parallel, restricted by tables, and growth along the four different edges is controlled by a control set. Special classes of these models where the distinction between terminals and nonterminals is removed, provide for rectangular developmental arrays; the hierarchy within these classes is studied. The generative power of the new model is compared with that of earlier array models.

Parallel/sequential rectangular arrays with tables

Parallel/sequential generative models with tables are introduced and properties studied. These have higher generative capacity than our earlier matrix models which extend the essence of restricted parallelism to two dimensional grammars. The effects of appearance checking, minimal table and control are examined. The Szilard language of the vertical grammars is shown to be regular.

Space-filling curves and infinite graphs

Two-dimensional geometric patterns in the form of the space filling curves of Peano and Hilbert are represented as a string, an array and a graph. The sequence of patterns is such that each curve is a prefix of the succeeding one and hence we get prefix preserving WDOL to generate the corresponding string representation. The limit language is generable by a CDOL system and consists of a single infinite word corresponding to the Peano curve and two infinite words for the Hilbert curve. The property that in the sequence each curve is a prefix of the succeeding one helps in the definition of the infinite arrays and infinite graphs representing the infinite curve patterns and we construct a Context-free Parentheses Kolam array grammar and a coding of a graph DOL system to generate the sequence as well as the limit.

Circular contextual insertions/deletions with applications to biomolecular computation

Insertions and deletions of small circular DNA strands into long linear DNA strands are phenomena that happen frequently in nature and thus constitute an attractive paradigm for biomolecular computing. The paper presents a new model for DNA-based computation that involves circular as well as linear molecules, and that uses the operations of insertion and deletion. After introducing the formal model, we investigate its properties and prove in particular that the circular insertion/deletion systems are capable of universal computation. We also give the results of an experimental laboratory implementation of our model. This shows that rewriting systems of the circular insertion/deletion type are viable alternatives in DNA computation.

Parallel context-free languages

Parallel context-free languages are generated by context-free grammars in which every occurrence of a nonterminal in a line of derivation is rewritten simultaneously by the same rule. It is shown that the intersection of parallel context-free languages and context-free languages is the class of derivationbounded languages. Also, the parallel context-free languages form a proper subclass of the context-sensitive languages and are closed under union, product, Kleene closure, and homomorphism. An intercalation theorem is proved for parallel context-free languages.