Indhumathi Raman

Graph-Controlled Insertion-Deletion Systems Generating Language Classes Beyond Linearity

A regulated extension of an insertion-deletion system known as graph-controlled insertion-deletion (GCID) system has several components and each component contains some insertion-deletion rules. A rule is applied to a string in a component and the resultant string is moved to the target component specified in the rule. When resources are so limited (especially, when deletion is context-free) then GCID systems are not known to describe the class of recursively enumerable languages. Hence, it becomes interesting to find the descriptional complexity of such GCID systems of small sizes with respect to language classes below \(\mathrm {RE}\). To this end, we consider closure classes of linear languages. We show that whenever GCID systems describe \(\mathrm {LIN}\) with t components, we can extend this to GCID systems with just one more component to describe, for instance, 2-\(\mathrm {LIN}\) and with further addition of one more component, we can extend to GCID systems that describe the rational closure of \(\mathrm {LIN}\).

On the computational completeness of graph-controlled insertion–deletion systems with binary sizes

Abstract A graph-controlled insertion–deletion (GCID) system is a regulated extension of an insertion–deletion system. Such a system has several components and each component has some insertion–deletion rules. The transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. The language of the system is the set of all terminal strings collected in the final component. The parameters in the size ( k ; n , i ′ , i ″ ; m , j ′ , j ″ ) of a GCID system denote (from left to right) the maximum number of components, the maximal length of the insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; the last three parameters follow a similar representation with respect to deletion. In this paper, we discuss the computational completeness of the families of GCID systems of size ( k ; 1 , i ′ , i ″ ; 1 , j ′ , j ″ ) with k ∈ { 3 , 5 } and for (nearly) all values of i ′ , i ″ j ′ , j ″ ∈ { 0 , 1 } . All proofs are based on the simulation of type-0 grammars given in Special Geffert Normal Form (SGNF). The novelty in our proof presentation is that the context-free and the non-context-free rules of the given SGNF grammar are simulated by GCID systems of different sizes and finally we combine them by stitching and overlaying to characterize the recursive enumerable languages. This proof presentation greatly simplifies and unifies the proof of such characterization results. We also connect some of the obtained GCID simulations to the domain of insertion–deletion P systems.

Generative Power of Matrix Insertion-Deletion Systems with Context-Free Insertion or Deletion

Matrix insertion-deletion systems combine the idea of matrix control as established in regulated rewriting with that of insertion and deletion as opposed to replacements. We improve on and complement previous computational completeness results for such systems, showing for instance that matrix insertion-deletion systems with matrices of length two, insertion rules of type 1,i?ź1,i?ź1 and context-free deletions are computationally complete. We also show how to simulate Kleene stars of metalinear languages with several types of systems with very limited resources. We also generate non-semilinear languages using matrices of length three with context-free insertion and deletion rules.

On path-controlled insertion–deletion systems

A graph-controlled insertion–deletion system is a regulated extension of an insertion–deletion system. It has several components and each component contains some insertion–deletion rules. These components are the vertices of a directed control graph. A transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. This also describes the arcs of the control graph. Starting from an axiom in the initial component, strings thus move through the control graph. The language of the system is the set of all terminal strings collected in the final component. In this paper, we investigate a variant of the main question in this area: which combinations of size parameters (the maximum number of components, the maximal length of the insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; plus three similar restrictions with respect to deletion) are sufficient to maintain computational completeness of such restricted systems under the additional restriction that the (undirected) control graph is a path? Notice that these results also bear consequences for the domain of insertion–deletion P systems, improving on a number of previous results from the literature, concerning in particular the number of components (membranes) that are necessary for computational completeness results.

Computational Completeness of Path-Structured Graph-Controlled Insertion-Deletion Systems

A graph-controlled insertion-deletion (GCID) system is a regulated extension of an insertion-deletion system. It has several components and each component contains some insertion-deletion rules. These components are the vertices of a directed control graph. A rule is applied to a string in a component and the resultant string is moved to the target component specified in the rule, describing the arcs of the control graph. We investigate which combinations of size parameters (the maximum number of components, the maximal length of the insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; a similar three restrictions with respect to deletion) are sufficient to maintain the computational completeness of such restricted systems with the additional restriction that the control graph is a path, thus, these results also hold for ins-del P systems.

On Succinct Description of Certain Context-Free Languages by Ins-Del and Matrix Ins-Del Systems

In this paper, we introduce some basic measures for insertion-deletion system and matrix insertion-deletion system. These measures are based on the number of variables, the number of productions and the number of symbols in a grammar. We show that with respect to these measures, both the systems are more succinct over context-free grammars in representing certain families of context-free languages.

Minimizing Rules and Nonterminals in Semi-conditional Grammars: Non-trivial for the Simple Case

A simple semi-conditional (SSC) grammar is a form of regulated rewriting system where the derivations are controlled either by a permitting string alone or by a forbidden string alone and is specified in the rule. The maximum length i (j, resp.) of the permitting (forbidden, resp.) strings serves as a measure of descriptional complexity known as the degree of such grammars. In addition to the degree, the numbers of nonterminals and of conditional rules are also counted into the descriptional complexity measures of these grammars. We improve on some previously obtained results on computational completeness of SSC grammars by minimizing the number of nonterminals and/or the number of conditional rules for a given degree (i, j). More specifically, we prove that every recursively enumerable language is generated by an SSC grammar of (i) degree (2, 1) with at most eight conditional rules and nine nonterminals, (ii) degree (3, 1) with at most seven conditional rules and eight nonterminals and (iii) degree (3, 1) with at most nine conditional rules and seven nonterminals.

Computational Completeness of Simple Semi-conditional Insertion-Deletion Systems.

Insertion-deletion (or ins-del for short) systems are well studied in formal language theory, especially regarding their computational completeness. The need for many variants on ins-del systems was raised by the computational completeness result of ins-del system with (optimal) size (1, 1, 1; 1, 1, 1). Several regulations like graph-control, matrix and semi-conditional have been imposed on ins-del systems. Typically, computational completeness are obtained as trade-off results, reducing the size, say, to (1, 1, 0, 1, 1, 0) at the expense of increasing other measures of descriptional complexity. In this paper, we study simple semi-conditional ins-del systems, where an ins-del rule can be applied only in the presence or absence of substrings of the derivation string. We show that simple semi-conditional ins-del system, with maximum permitting string length 2 and maximum forbidden string length 1 and sizes (2, 0, 0; 2, 0, 0), (1, 1, 0; 2, 0, 0), or (1, 1, 0; 1, 1, 1), are computationally complete. We also describe RE by a simple semi-conditional ins-del system of size (1, 1, 0; 1, 1, 0) and with maximum permitting and forbidden string lengths 3 and 1, respectively. The obtained results complement the existing results available in the literature.

Certain height-balanced subtrees of hypercubes

ABSTRACT A height-balanced tree is a desired data structure for performing operations such as search, insert and delete, on high-dimensional external data storage. Its preference is due to the fact that it always maintains logarithmic height even in worst cases. It is a rooted binary tree in which for every vertex the difference (denoted as balance factor) in the heights of the subtrees, rooted at the left and the right child of the vertex, is at most one. In this paper, we consider two subclasses of height-balanced trees and . A tree in is such that all the vertices up to (a predetermined) level t has balance factor one and the remaining vertices have balance factor zero. A tree in is such that all the vertices at alternate levels up to t has balance factor one and the remaining vertices have balance factor zero. We prove that every tree in the classes and is a subtree of the hypercube.

On counting and embedding a subclass of height-balanced trees

A height-balanced tree is a rooted binary tree in which, for every vertex V, the difference in the heights of the subtrees rooted at the left and right child of V (called the balance factor of V) is at most one. In this paper, we consider height-balanced trees in which the balance factor of every vertex beyond a level is 0. We prove that there are 22t-1 such trees and embed them into a generalized join of hypercubes.