Programmed grammars -- A new device for generating formal languages

A new form of grammar, which is called a programmed grammar, is defined and some of its properties are described. Programmed grammars are a generalization of phrase structure grammars where each production has a label, a core consisting of an ordinary phrase structure production, and two associated sets of production labels. If a production can be applied to an intermediate string in a derivation, it is applied as far to the left as possible, and the next production to be used is selected from the first set of labels. If the production cannot be applied to the intermediate string, the next production is selected from the other set of labels. The generative power of programmed grammars with various types of production cores is investigated. The additional machinery of programmed grammars does not add any additional generative power if the cores are arbitrary, context sensitive, linear, or one-sided linear. However, programmed grammars whose cores contain a single symbol on the lefthand side and an arbitrary string on the righthand side can generate all recursively enumerable languages. The class of languages generated by grammars of this type with the additional restriction that the righthand side of a production core cannot be the null string is properly contained with the context sensitive languages and properly contains the context free languages. For grammars of this type, called cfpg´s, the emptiness and finiteness problems are undecidable. In addition every recursively enumerable language can be obtained from a cfpg language by a homomorphism. The subset of cfpg´s for which the two sets of production labels are the same for each production is also considered. The class of languages generated by these grammars is properly contained within the cfpg languages and properly contains the context free languages. Furthermore, the emptiness problem is decidable for these grammars. This paper is an extended abstract based on the author´s doctoral thesis.