Context-free Array Research Articles

Relations between Control Mechanisms for Sequential Grammars1

We extend and refine previous results within the general framework for regulated rewriting based on the applicability of rules in sequential grammars [3]. Besides the well-known control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules, and priority relations on rules we also consider the new variant of activation and blocking of rules as investigated in [1, 2, 4]. Moreover, we exhibit special results for strings and multisets as well as for arrays in the general variant defined on Cayley grids of finitely presented groups. Especially we prove that array grammars defined on Cayley grids of finitely presented groups using #-context-free array productions together with control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules, priority relations on rules, or activation and blocking of rules have the same computational power as such array grammars using arbitrary array productions.

Equi-triangular array grammars with equal matrix type of rules

Generation of two-dimensional languages of picture arrays, rectangular, triangular or hexagonal in shape, has been considered in several studies motivated by problems in picture processing and pattern recognition. Picture arrays in the form of equi-triangular arrays (et-arrays) composed of equilateral triangles and their generation by context-free type of array grammars have been considered recently. Here generation of picture languages of equi-triangular arrays based on equal matrix grammar type of rules is introduced and investigated. The rules in the matrix are regular type of rules except for initial rules. It has been proved that these equi-triangular array grammars with equal matrix grammar type of rules can generate picture languages of et-arrays that cannot be generated by context-free equi-triangular array grammars. Also the family of equi-triangular array grammars with equal matrix type of rules is shown to properly include the family of regular picture languages of et-arrays.

Flat splicing context free array grammar systems generating picture arrays

In the area of DNA computing, a biologically motivated operation called splicing on linear words as well as circular words was introduced by Head 1987, 1992. Motivated by this operation, the concept of flat splicing on words was introduced by Berstel et al. 2012. This operation was extended by Subramanian et al. 2015a to picture arrays. Based on flat splicing on picture arrays, a new model of picture generation known as flat splicing regular array grammar system FSRAGS was also proposed by Subramanian et al. 2015b. Here we extend this model by considering a well-known two-phase context-free grammar in the components of the grammar system with the components deriving picture arrays in parallel using the rules and with two different components communicating using the array flat splicing operations. We call the resulting model as flat splicing context-free array grammar system FSCFAGS and obtain certain theoretical results on the generative power of FSCFAGS. As an application we construct a FSCFAGS and a corresponding algorithm to generate a certain class of floor design, also called a kolam pattern.

On the power of permitting features in cooperating context-free array grammar systems

Among many grammar models for picture description, cooperating context-free distributed array grammar systems utilize the mechanism of cooperation between component array grammars. Here, we endow these grammar systems with the feature of permitting symbols, and examine the generative capacity of the resulting permitting cooperating context-free distributed array grammar systems with certain other kinds of isometric and non-isometric array grammars. An advantage is that the number of components is reduced when this permitting feature is used in a cooperating context-free distributed array grammar system.

Grammar-Based Specification and Parsing of Binary File Formats

The capability to validate and view or play binary file formats, as well as to convert binary file formats to standard or current file formats, is critically important to the preservation of digital data and records. This paper describes the extension of context-free grammars from strings to binary files. Binary files are arrays of data types, such as long and short integers, floating-point numbers and pointers, as well as characters. The concept of an attribute grammar is extended to these context-free array grammars. This attribute grammar has been used to define a number of chunk-based and directory-based binary file formats. A parser generator has been used with some of these grammars to generate syntax checkers (recognizers) for validating binary file formats. Among the potential benefits of an attribute grammar-based approach to specification and parsing of binary file formats is that attribute grammars not only support format validation, but support generation of error messages during validation of format, validation of semantic constraints, attribute value extraction (characterization), generation of viewers or players for file formats, and conversion to current or standard file formats. The significance of these results is that with these extensions to core computer science concepts, traditional parser/compiler technologies can potentially be used as a part of a general, cost effective curation strategy for binary file formats.

COOPERATING ARRAY GRAMMAR SYSTEMS

The aim of this paper is to elaborate the power of cooperation in generating pictures by array grammars. As it is expected, the generative capacity of cooperating array grammar systems (with a fixed number, with a number greater than a given threshold, or with the maximal number of derivation steps in each component when it is enabled) is strictly greater than that of context-free array grammars. Yet the same result is also obtained in the case of systems with regular components, which contradicts the corresponding result for string grammar systems. In fact, some more results for array grammar systems are obtained which either contradict the results for the corresponding string grammar systems or are not even known for these string grammar systems. Various non-context-free sets of arrays which can be generated in a simple way by cooperating array grammar systems are presented and show the power of the mechanism of cooperation for picture descritpion.

ON PARALLEL RECOGNITION OF TWO CLASSES OF 2-D ARRAY PATTERNS

We investigate the parallel complexity of recognition problems for context-free and regular array (image) sets. We show that the sequential time complexity of the recognition of an n × n image is O(n5). The space required for these recognition problems is O(n5). We prove that there are log 2n time parallel algorithms with BM (n4) and n2 BM (n) processors for the recognition of context-free and regular array sets, respectively, where BM (n) is the number of processors sufficient to multiply two boolean n × n matrices in logarithmic time. We develop also a methodology for processing images using composition systems.

Graph grammars with path-controlled embedding

In this paper we define graph grammars, called the node-replacement graph grammars with path-controlled embedding (nPCE grammars), which use a sequence of edges instead of a single edge to embed a newly replaced graph into the host graph. Then we show that there exists a class of nPCE grammars generating a class of graph languages which is equal to the class of context-free array languages under some stipulations. It means that the path-controlled embedding brings no significant contexts into the production rules. Then, we characterize the above-mentioned class of nPCE grammars by making use of the partial path group.

GENERATING RECTANGLES USING TWO-DIMENSIONAL GRAMMARS WITH TIME AND SPACE COMPLEXITY ANALYSES

A two-dimensional grammar for generating all possible rectangles is presented and illustrated by examples. The time and space complexity analyses of this grammar together with a parallel context-free array grammar and a free grammar are also presented. Generating pictures using two-dimensional grammars appear to be a fertile field for further study. The study of two-dimensional grammars has useful applications in region filling. pattern recognition. robotics, pictorial information system design and related areas.

CONTEXT-SENSITIVITY OF TWO-DIMENSIONAL REGULAR ARRAY GRAMMARS

Regular array grammars (RAGs) are the lowest subclass in the Chomsky-like hierarchy of isometric array grammars. The left-hand side of each rewriting rule of RAGs has one nonterminal symbol and at most one "#" (a blank symbol). Therefore, the rewriting rules cannot sense contexts of non-# symbols. However, they can sense # as a kind of context. In this paper, we investigate this #-sensing ability. and study the language generating power of RAGs. Making good use of this ability, We show a method for RAGs to sense the contexts of local shapes of a host array in a derivation. Using this method, we give RAGs which generate the sets of all solid upright rectangles and all solid squares. On the other hand. it is proved that there is no context-free array grammar (and thus no RAG) which generates the set of all hollow upright rectangles.

PUSHDOWN RECOGNIZERS FOR ARRAY PATTERN

We investigate the factors that make it difficult to generalize pushdown automata for one-dimensional strings to two-dimensional arrays. Then we resolve the problems and construct two-dimensional pushdown array automata (PDAA). The relationship between isometric context-free array languages and pushdown array automata is established. Several examples of array automata are presented, and a pushdown array automaton is tested on VAX8650/VMS using PASCAL.

GRAMMARS ON THE HEXAGONAL ARRAY

In this paper, we will define array grammars on the hexagonal grid. One of the advantages obtained from using the hexagonal grid is that each point has only one kind of neighbor. It is shown that the class of the context-free array grammars on the hexagonal grid includes the class of the rectangular ones even if their languages are restricted to the pictures connected on the rectangular grid. It is also shown that, in the monotonic case, they own the same class of languages jointly.

Estimating the Size of Context-Free Tiling Languages

The problem of counting polyominoes motivates this paper. We will develop a general question for study that has counting polyominoes as a special case. We generalize in two ways. Polyominoes are shapes on the tiling made of square tiles. We will consider shapes on other tilings. The set of all polyominoes can be generated by a context-free array grammar, but the size of this set is estimated by counting the words of certain subsets and supersets that are generated by more convenient grammars. Our general question is the problem of counting the words of a context-free array language on a periodic tiling.Counting polyominoes is a difficult problem that has not been completely solved yet. There are various techniques for roughly estimating the number of polyominoes of a given size. We will extend some of these techniques to our general question.

Several properties of array languages

The languages generated by connected and disconnected array grammars are studied. We define substitutions of array languages, and we study closure properties of classes of array languages under substitutions. A characterization by means of substitutions for connected context-free array languages and a Kleene-like theorem for regular array languages are given.

The complexity of some decision problems about two-dimensional array grammars

The complexity of several decision problems concerning two-dimensional isometric array grammars (IAG) is studied. Because of the two-dimensional and the “isometric” properties of IAG, many decision problems become very hard to solve even for regular array grammars (RAG), the lowest subclass of IAG in the Chomsky-like hierarchy. In this paper, it is shown that the membership problems for RAGs and for context-free array grammars (CFAG) are both NP-complete. The emptiness problem and the equivalence problem for RAGs are shown to be undecidable.

Hierarchical Structures and Complexities of Parallel Isometric Languages

The relationship between parallel isometric array languages and sequential isometric array languages is examined. Their hierarchical structures are investigated and a hierarchy is established by introducing parallel context-free array languages (PCFAL), derivation bounded array languages (DBAL), linear array languages (LAL), and extended regular afray languages (ERAL). It is interesting to find that some fundamental aspects that hold in one-dimensional string languages do not hold in their two-dimensional counterparts. Some parsing techniques are also explored. It is shown that while parallel parsing grammars may be simpler to write and parallel processing usually takes less time than sequential ones, the nature of parallel parsing is very complicated. Finally, several future research topics concerned with parallel isometric array languages including their complexities, hierarchical structures, and application to pattern recognition are discussed.

Parallel context-free array grammar normal forms

Several possible normal forms for parallel context-free array grammars (PCFAG) are examined. It is shown that PCFAG's do not have a 2-point Chomsky normal form. In fact, they do not have normal forms of any kind at all. One interesting implication of this result is that the algorithms for constructing parallel context-free array language (PCFAL) recognizers may be more difficult and complicated than their sequential counterparts.

Parallel Σ-erasing array acceptors

A parallel Σ-erasing array acceptor (PΣ-EAA) is introduced. It is proved that the class accepted by PΣ-EAA's is exactly the context-free array languages.

A chomsky hierarchy of isotonic array grammars and languages

An array is a two-dimensional generalization of a string. Both sides of each rewriting rule of an isotonic array grammar have the same shape. In this paper we complete the Chomsky hierarchy of isotonic array grammars by introducing isotonic context-free array grammars. We obtain Chomsky and Greibach normal forms for context-free array grammars. We show that the one-dimensional (vertical or horizontal) isotonic regular and context-free array languages coincide and are closely related to the regular string languages. We also obtain a finite state recognizer characterization of regular array languages.

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.