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Turing machines and unrestricted grammars are two different formalisms that define the RE languages. Some RE languages are decidable, but not all are.

We can define the decidable languages with Turing machines by saying that a language is decidable iff there is a TM for the language that halts and accepts all strings in the language and halts and rejects all strings not in the language. My question is this: is there an analogous definition of decidable languages based on unrestricted grammars rather than Turing machines?

Raphael
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templatetypedef
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3 Answers3

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A language is decidable, iff it is semi-decidable and its complement is semi-decidable. Moreover, a language is recursive-enumerable iff it is semi-decidable and thus you can find an unrestricted Grammar. Therfore:

A language $L$ is decidable iff there is both an unrestricted Grammar $G$ with $L(G) = L$ and an unrestricted Grammar $\bar G$ with $L(\bar G) = \bar L$.

Simon S
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There can not be a useful class of grammars for $\mathrm{R}$ (the set of recursive languages), since

  • every useful class of grammars is enumerable, and
  • $\mathrm{R}$ is not semi-decidable or, equivalently, not enumerable.

The first is obviously not a rigorous theorem (and can't be), it's just judgemental conjecture. The set of all grammars is enumerable, and any restriction that is not decidable is likely not very useful¹ in itself; in particular it won't be a syntactic restriction (like Chomsky's).

The second is formally true, see also here.

  1. Of course, people have defined such restrictions, and those classes have their uses, but it is even hard to see whether a given grammar falls into simpler subclasses.
Community
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Raphael
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2

This question is addressed in a paper by Henning Fernau from 1994. Henning states:

As an example, we consider the family of recursive languages. It is an open question whether there is a 'natural' grammatical characterization of this language class. As we will show in the following, any grammar family characterizing the recursive languages must have some strange properties.

We direct the reader who is curious to learn about those strange properties to the paper.

Yuval Filmus
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