Observations on grammar and language families

In this report, we emphasize the differences of grammar families and their properties versus language families and their properties. To this end, we investigate grammar families from an abstract standpoint, developping a new framework of reasoning. In particular when considering decidability questions, special care must be taken when trying to use decidability results (which are, in the first place, properties of grammar families) in order to establish results (e.g. hierarchy results) on language families. We illustrate this by inspecting some theorems and their proofs in the field of regulated rewriting. In this way, we also correct the formulation of an important theorem of Hinz and Dassow. As an exercise, we show that there is no `effective\u27 grammatical characterization of the family of recursive languages. Moreover, we show how to prove the strictness of the Chomsky hierarchy using decidability properties only. Most of the material of this report will be published in `fundamenta informaticae\u27

Membership for limited ET0L languages is not decidable

In this paper, we show how to encode arbitrary enumerable set of numbers given by register machines within limited EPT0L systems and programmed grammars with unconditional transfer.This result has various consequences, e.g.the existence of nonrecursive sets generable by 1lET0L systems or by programmed grammars with unconditional transfer. Moreover, ordered grammars are strictly less powerful than 1lET0L systems

Accepting grammars and systems

We investigate several kinds of regulated rewriting (programmed, matrix, with regular control, ordered, and variants thereof) and of parallel rewriting mechanisms (Lindenmayer systems, uniformly limited Lindenmayer systems, limited Lindenmayer systems and scattered context grammars) as accepting devices, in contrast with the usual generating mode. In some cases, accepting mode turns out to be just as powerful as generating mode, e.g. within the grammars of the Chomsky hierarchy, within random context, regular control, L systems, uniformly limited L systems, scattered context. Most of these equivalences can be proved using a metatheorem on so-called context condition grammars. In case of matrix grammars and programmed grammars without appearance checking, a straightforward construction leads to the desired equivalence result. Interestingly, accepting devices are (strictly) more powerful than their generating counterparts in case of ordered grammars, programmed and matrix grammars with appearance checking (even programmed grammarsm with unconditional transfer), and 1lET0L systems. More precisely, if we admit erasing productions, we arrive at new characterizations of the recursivley enumerable languages, and if we do not admit them, we get new characterizations of the context-sensitive languages. Moreover, we supplement the published literature showing: - The emptiness and membership problems are recursivley solvable for generating ordered grammars, even if we admit erasing productions. - Uniformly limited propagating systems can be simulated by programmed grammars without erasing and without appearance checking, hence the emptiness and membership problems are recursively solvable for such systems. - We briefly discuss the degree of nondeterminism and the degree of synchronization for devices with limited parallelism