The Generative Power of Two-Level Grammars

TLDR: Two-level grammars as introduced by the author in [2] have been used to define most of the syntax of ALGOL 68 and experience has shown that although these Grammars have the same generative power as Turing machines, they are much more manageable.

Abstract: Two-level grammars as introduced by the author in [2] have been used to define most of the syntax of ALGOL 68 [3]. Although SINTZOFF showed in [1] that every recursively enumerable set is generated by some such grammar, its use to the description of the full syntax of ALGOL 68 had to wait until recently [4]. The even more ambitious project of using such a grammar to give an integrated definition of both syntax and semantics of ALGOL 68 has been undertaken by the author but is still far from completion. Experience in these and other applications, e.g., to prove program correctness [5], have shown that although these grammars have the same generative power as Turing machines, they are much more manageable. An obvious question is the reason of this manageability. Certainly, a judicious choice of metanotions is of considerable value and in the big tasks, like the description of full-scale languages, a large number of metanotions are used. On the other hand, one might ask whether the ability to generate every recursively enumerable set requires so many metanotions.