Matrix Grammars Research Articles

On Eliminating the λ-Rules from Simple Matrix Grammars

It is proved that the λ-rules increase the generative capacity of context-free simple matrix grammars but can be eliminated from right-linear simple matrix grammars and from linear simple matrix grammars without diminishing their generative capacity.

On the power of two-dimensional grammars which are not length consistent

The shape regulated 2-dimensional matrix grammars of type 3 (SR2DM3 grammars), and a subclass of them, the depth increasing SR2DM3 grammars, are introduced. For each grammar in the latter class, we associate a 1-dimensional language, called its column language. It is shown that an e -free language is regular iff it is a column language of a depth increasing SR2DM3 grammar which is length consistent. The existence of a hierarchy between the regular and context-sensitive languages which are column languages of non-length consistent depth increasing SR2DM3 grammars is derived by formulating a machine characterization of these languages. Results concerning the structure of SR2DM3 grammars are also exhibited.

On the family of finite index matrix languages

It is proved that: 1. (1) M ƒ λ is a substitution closed full AFL closed also under doubling ( D( L) = { xx | x ϵ L}). 2. (2) M ƒ = M ƒ λ = M ac ƒ = M ac ƒ λ = M left ƒ = M left ƒ λ = PM ƒ = PM ƒ λ! 3. (3) LM ƒ ⊂ M ƒ, strict inclusion. (The families M - M left λ are as in Salomaa's book [18], LM is the family or simple matrix languages of Ibarra [11] and PM, PM λ are the families of λ-free, respectively, arbitrary, parallel matrix languages [4]. For an arbitrary family of languages, L we denote by L ƒ the family of finite index languages in L , the index being considered with respect to the class of grammars generating languages in L .) 4. (4) The emptiness and the finiteness problems for M ƒ are algorithmically solvable, whereas many other decision problems about M ƒ are unsolvable. 5. (5) The composition does not increase the generative capacity of (context-free), arbitrary or of finite index, matrix grammars. A conjecture with many significant implications is formulated in the last part of the paper.

On the effect of the finite index restriction on several families of grammars

In this paper we compare the effect of the finite index restriction on the generating power of the following classes of language-generating devices: ET 0 L systems, E 0 L systems, context-free grammars, context-free programmed grammars, ordered grammars, matrix grammars, and indian parallel grammars.

On context-free matrix forms

The notion of a grammar form is extended to context-free matrix grammars yielding so-called matrix forms. Their reduction and completeness properties are studied and some normal-form results and complete matrix forms are presented. The closure properties of language families ℒ( F ) obtained from a matrix form F are also examined. The type of interpretation applied here corresponds to strict interpretations of ordinary context-free grammar forms.

On the generative capacity of simple matrix grammars of finite index

On the generative capacity of simple matrix grammars of finite index

Sequential/parallel matrix picture languages

A new system, that of matrix grammars, for two-dimensional picture processing, is introduced. A hierarchy, induced on Chomsky's is found. Language operations such as union, catenation (row and column), Kleene's closure (row and column), and homomorphisms are investigated. It is found that the smallest class of these languages may serve as the class of arrays, which is defined as the smallest class of arrays closed under union, catenation (row and column) and Kleene's closure (row and column). Eight possible ways of defining a matrix language are discussed and it is suggested that one of them may lead to a normal form of matrix grammars. The method is advantageous over others on several points. Perhaps the most interesting of all is that it provides a compromise between purely sequential methods, which take too much time for large arrays and purely parallel methods, which usually take too much hardware for large arrays.

A language for two-dimensional digital picture processing

A new system, that of matrix grammars, for two-dimensional pattern processing, is introduced. A hierarchy, induced on Chomsky's is found. Language operations such as union, catenation (row and column), Kleene's closure (row and column), and homomorphisms are investigated. It is found that the smallest class of these languages may serve as the class of regular arrays, which is defined as the smallest class of arrays closed under union, catenation (row and column) and Kleene's closure (row and column). Eight possible ways of defining a matrix language are discussed and it is suggested that one of them may lead to a normal form for matrix grammars. The method is advantageous over others on several points. Perhaps the most interesting of all is that it provides a compromise between sequential methods which take too much time for large arrays and parallel methods which usually take too much hardware for large arrays.

A language for two-dimensional digital picture processing

A new system, that of matrix grammars, for two-dimensional pattern processing, is introduced. A hierarchy, induced on Chomsky's is found. Language operations such as union, catenation (row and column), Kleene's closure (row and column), and homomorphisms are investigated. It is found that the smallest class of these languages may serve as the class of regular arrays, which is defined as the smallest class of arrays closed under union, catenation (row and column) and Kleene's closure (row and column). Eight possible ways of defining a matrix language are discussed and it is suggested that one of them may lead to a normal form for matrix grammars. The method is advantageous over others on several points. Perhaps the most interesting of all is that it provides a compromise between sequential methods which take too much time for large arrays and parallel methods which usually take too much hardware for large arrays.

Sequential/Parallel Matrix Array Languages

Abstract A new system, that of matrix grammars, for two-dimensional pattern processing is introduced. Two hierarchies induced on Chomsky's are found and are compared with each other. Language operations such as union, catenation (row and column), Kleene's closure (row and column) and homomorphisms are investigated. It is found that the smallest class of these languages may serve as the class of regular arrays which is defined as the smallest class of arrays closed under union, catenation (row and column) and Kleene's closure (row and column). Eight possible ways of defining a matrix language are discussed and it is suggested that one of them may lead to a normal form of matrix grammers. The method is advantageous over others in several points. Perhaps the most interesting of all is that it provides a compromise between purely sequential methods which take too much time for large arrays and purely parallel methods which usually take too much hardware for large arrays.

On the complexity of regulated context-free rewriting

Some complexity measures which are well-known for context-free languages are generalized in order to classify matrix languages and programmed languages. It is shown that the complexity of some context-free languages decreases if they are generated by matrix grammars or programmed grammars. An arithmetic characterization is given for infinite languages generated by two matrices. The number of matrices (as a complexity measure) is shown to be independent from any other complexity measure regarded in this paper.

Simple matrix languages with a leftmost restriction

It is shown that every type 1 language can be generated by an ε -free simple matrix grammar (of order 2), if a certain leftmost restriction is imposed on the application of matrices. Dropping the ε -free restriction, every type 0 language can be obtained. This strengthens a result previously established for matrix grammars.

On matrix languages

The matrix grammar is a well-known concept of a grammar with restricted use of productions. By weakening the matrix restrictions imposed on context-free grammars several new types of grammars are introduced; some of them are proved to possess the same generative power as ordinary matrix grammars; for some other types inclusion properties are obtained.

Some remarks on state grammars and matrix grammars

The context-free matrix grammar and the state grammar without any restriction in applying productions are considered. It turned out that these grammars are equivalent in the generative power. Another type of state grammar called the state grammar with unconditional transfer is introduced, and it is shown that each context-free matrix language is a homomorphic image of the intersection of a state language with unconditional transfer and a regular set.

Periodic representation of equal matrix grammars

A substantial amount of research in formal grammars recently has been to impose constraints on the use of rules in addition to the restrictions on the form of the rules of grammars. Time-variant grammars and equal matrix grammars are amoung them. It has been shown here that for each equal matrix grammar, there is a procedure to systematically construct the almost right linear periodic grammar equivalent to it.

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.

Matrix grammars with a leftmost restriction

The family of languages generated by matrix grammars with context-free (context-free λ-free) core productions and with a leftmost restriction on derivations equals the family of recursively enumerable (context-sensitive) languages.

Simple matrix languages

Simple matrix languages and right-linear simple matrix languages are defined as subfamilies of matrix languages by putting restrictions on the form and length (degree) of the rewriting rules associated with matrix grammars. For each n ⩾ 1, let L ( n ) [ ℛ ( n ) ] be the class of simple matrix languages [right-linear simple matrix languages] of degree n , and let L = ⋃ n ⩾ 1 L ( n ) [ ℛ = ⋃ n ⩾ 1 ℛ ( n ) ] . It is shown that L ( 1 ) [ ℛ ( 1 ) ] coincides with the class of context-free languages [regular sets] and that L is a proper subset of the family of languages accepted by deterministic linear bounded automata. It is proved that L ( n ) [ ℛ ( n ) ] forms a hierarchy of classes of languages in L [ ℛ ] . The closure properties and decision problems associated with L ( n ) , L , ℛ ( n ) , and ℛ are thoroughly investigated. Let L B [ ℛ B ] be the bounded languages in L [ ℛ ] . It is shown that L B = ℛ B and that most of the positive closure and decision results which are true for bounded context-free languages are carried over in L B . A characterization of L B as the smallest family of languages which contains the bounded context-free languages and which is closed under the operations of union and intersection is proved.