TOWARD UNDERSTANDING THE GENERATIVE CAPACITY OF ERASING RULES IN MATRIX GRAMMARS

This article presents approaches to the open problem of whether erasing rules can be eliminated in matrix grammars. The class of languages generated by non-erasing matrix grammars is characterized by the newly introduced linear Petri net grammars. Petri net grammars are known to be equivalent to arbitrary matrix grammars (without appearance checking). In linear Petri net grammars, the marking has to be linear in size with respect to the length of the sentential form. The characterization by linear Petri net grammars is then used to show that applying linear erasing to a Petri net language yields a language generated by a non-erasing matrix grammar. It is also shown that in Petri net grammars (with final markings and arbitrary labeling), erasing rules can be eliminated, which yields two reformulations of the problem of whether erasing rules in matrix grammars can be eliminated.

LEARNING EQUAL MATRIX GRAMMARS BASED ON CONTROL SETS

An equal matrix grammar is a parallel rewriting system. In this paper, we consider the problem of learning equal matrix grammars from examples. We introduce a learning method based on control sets and show two subclasses learnable in polynomial time with learning methods for regular sets. We also show that for any equal matrix language there exists an equal matrix grammar learnable efficiently from positive structural examples only.

SIROMONEY ARRAY GRAMMARS AND APPLICATIONS

The Siromoney matrix model is a simple and elegant model for describing two-dimensional digital picture languages. The notion of attaching indices to nonterminals in a generative grammar, introduced and investigated by Aho. is considered in the vertical phase of a Siromoney matrix grammar (SMG). The advantage of this study is that the new model retains the simplicity and elegance of SMG but increases the generative power and enables us to describe pictures not generable by SMG. Besides certain closure properties and hierarchy results. applications of these two-dimensional grammars to describe tilings, polyominoes, distorted patterns and parquet deformations are studied.