Narrow Results

We continue the study of pictorial languages as formalized in Ref. 1 (based on the operations of shifting and superposing elementary pictures). If the pixels are superposed without composing their colors, then we produce only recursive languages. When the colors of the superposed pixels can be composed, then any array grammar can be simulated (hence, all recursively enumerable languages can be obtained). Bidimensional pictorial frameworks with a nonrecursive membership problem are obtained in the restricted case when (1) we do not allow the superposition of nontransparent pixels, excepting the fact that (2) for each color there is a complementary color, which, superposed on the original one, leads to a transparent pixel.

We define a syntactic model for generating sets of images, where an image can be viewed as an array over finite alphabet. This model is called image grammar. Image grammar can be considered as a generalization of classical Chomsky grammar. Then we study some combinatorial and language theoretical properties such as reduction, pumping lemmas, complexity measure, we give a strict infinite hierarchy. We also characterize these families in terms of deterministic substitutions and Chomsky languages.

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.

Systolic pyramid automata accepting square arrays are defined. Homogeneous and semi-homogeneous pyramid automata are shown to have equal power though regular pyramid automata are more powerful. Languages accepted by these automata are compared with languages generated by array grammars and languages accepted by one-way 2-D cellular automata. Hexagonal pyramid automata are also considered and are shown to accept some languages generated by hexagonal array grammars.

In a "coordinate grammar", the rewriting rules replace sets of symbols having been given coordinates by sets of symbols whose coordinates are given functions of the coordinates of the original symbols. It was shown in 1972 that coordinate grammars are "too powerful"; even if the rules are all of finite-state types and the functions are all computable by finite transducers, the grammar has the power of a Turing machine. This paper shows that if we require the functions to be shift-invariant and the rules to be of bounded diameter, then such grammars do have a useful hierarchy of types; in fact, when we require that their sentential forms always remain connected, they turn out to be equivalent to "isometric grammars".

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.

Language theoretic public key cryptosystems for strings and pictures are discussed. Two methods of constructing public key cryptosystems for the safe transmission or storage of chain code pictures are presented; the first one encrypts a chain code picture as a string and the second one as a two-dimensional array.

We introduce a new model for generating finite, digitized, connected pictures called puzzle grammars and study its generative power by comparison with array grammars. We note how this model generalizes the classical Chomskian grammars and study the effect of direction-independent rewriting rules. We prove that regular control does not increase the power of basic puzzle grammars. We show that for basic and context-free puzzle grammars, the membership problem is NP-complete and the emptiness problem is undecidable.

Nivat et al. proposed a class of grammars called puzzle grammars. Such models are suitable for describing and generating connected arrays consisting of unit cells. In this paper, we introduce the stochastic version of puzzle grammars. Conditions for their consistency are given. Although the simplest of puzzle grammars, called basic puzzle grammar, generates a larger class of pictures than regular array grammars, the additional generative power is restricted and it requires considerable effort to write grammars for even pictures whose complexity is not high. We propose a parallel version of the puzzle grammar model which lends itself naturally to the generation of pictures. Several examples are given to illustrate the power of this model. Its stochastic version is presented along with an application to the clustering of syntactic patterns.

Two methods of generating isosceles right triangles by basic puzzle grammars are presented. The methods use two different sets of tiles to tile the triangular region. This is of interest in view of the fact that the set of isosceles right triangles is a nontrivial example of a two-dimensional language which cannot be generated by any regular array grammar.

In a coordinate grammar, the rewriting rules replace sets of symbols having given coordinates by sets of symbols whose coordinates are given functions of the coordinates of the original symbols. Usually, at each step of a derivation, only one rule is applied and only one instance of its left hand side is rewritten. This type is referred to sequential grammars. As a counterpart of this grammar, parallel coordinate grammars are defined as generalized parallel isometric grammars. In the parallel grammars, the rewriting rule are used in parallel in a derivation application. The paper discusses some properties of parallel coordinate grammars and examines a relationship between the sequential coordinate grammars and parallel ones.

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.

In a coordinate grammar, the rewriting rules replace sets of symbols having given coordinates by sets of symbols whose coordinates are given functions of the coordinates of the original symbols. Usually, at each step of a derivation, only one rule is applied and only one instance of its left hand side is rewritten. This type is referred to sequential grammars. As a counterpart of this grammar, parallel coordinate grammars are defined as generalized parallel isometric grammars. In the parallel grammars, the rewriting rule are used in parallel in a derivation application. The paper discusses some properties of parallel coordinate grammars and examines a relationship between the sequential coordinate grammars and parallel ones.

Nivat et al. proposed a class of grammars called puzzle grammars. Such models are suitable for describing and generating connected arrays consisting of unit cells. In this paper, we introduce the stochastic version of puzzle grammars. Conditions for their consistency are given. Although the simplest of puzzle grammars, called basic puzzle grammar, generates a larger class of pictures than regular array grammars, the additional generative power is restricted and it requires considerable effort to write grammars for even pictures whose complexity is not high. We propose a parallel version of the puzzle grammar model which lends itself naturally to the generation of pictures. Several examples are given to illustrate the power of this model. Its stochastic version is presented along with an application to the clustering of syntactic patterns.

We define a syntactic model for generating sets of images, where an image can be viewed as an array over finite alphabet. This model is called image grammar. Image grammar can be considered as a generalization of classical Chomsky grammar. Then we study some combinatorial and language theoretical properties such as reduction, pumping lemmas, complexity measure, closure properties and decidability results. In terms of complexity measure, we give a strict infinite hierarchy. We also characterize these families in terms of deterministic substitutions and Chomsky languages.

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.

Systolic pyramid automata accepting square arrays are defined. Homogeneous and semihomogeneous pyramid automata are shown to have equal power though regular pyramid automata are more powerful. Languages accepted by these automata are compared with languages generated by array grammars and languages accepted by one-way 2-D cellular automata. Hexagonal pyramid automata are also considered and are shown to accept some languages generated by hexagonal may grammars.

In a “coordinate grammar”, the rewriting rules replace sets of symbols having been given coordinates by sets of symbols whose coordinates are given functions of the coordinates of the original symbols. It was shown in 1972 that coordinate grammars are “too powerful”; even if the rules are all of finite-state types and the functions are all computable by finite transducers, the grammar has the power of a Turing machine. This paper shows that if we require the functions to be shift-invariant and the rules to be of bounded diameter, then such grammars do have a useful hierarchy of types; in fact, when we require that their sentential forms always remain connected, they turn out to be equivalent to “isometric grammars”.

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.

Language theoretic public key cryptosystems for strings and pictures are discussed. Two methods of constructing public key cryptosystems for the safe transmission or storage of chain code pictures are presented; the first one encrypts a chain code picture as a string and the second one as a two-dimensional array.