Talk:Linear grammar

It's pretty clear this article was copied almost verbatim from http://www.cs.umanitoba.ca/~cs303/2005SS/Regular-Grammar-SS.pdf, though the anon author seemed to take steps to make it less readable than in the original. I opted to cut out all but the first paragraph, leaving it to others to either rewrite the part I removed or to decide that it wasn't appropriate to this entry. 128.197.81.229 22:16, 27 January 2006 (UTC)

Linear and regular grammars

I merged Right_Linear_Grammar into this, but I think that somebody expert and native English speaker (I am not) should draw some precise connections with Regular_grammar. For example, what is the requirement for a linear grammar to have a corresponding a regular grammar? Also, a union of a right-linear grammar and a left-linear grammar can be said to be a linear grammar?

Nb93 14:46, 27 August 2006 (UTC)

Beware of terminology, we make unions of sets, not grammars. The union of the set of all left-linear grammars and the set of all right-linear grammars has no special meaning itself (except it's members only generate regular languages).

On the other hand, the mixture of left and right linear grammars is a (more general) linear grammar.

Languages accepted by left-linear and right-linear grammars are regular languages and the union of these two sets of grammars gives a set of all regular grammars.

The set of regular grammars is thus a subset of the set of linear grammars. And the sets of left-regular (also called left-linear) and right-regular (right-linear) are subsets of the set of regular grammars.

pavlix, 81.0.198.173 (talk) 16:33, 8 June 2008 (UTC)

I now explain this at lengh both in this article and in the regular grammar article. My worry is that both articles are now too specialist oriented to merit separate inclusion in a general encyclopedia. Rp (talk) 13:09, 24 July 2008 (UTC)

Incorrect argument needs to be replaced

The article says:

Since nondeterministic context-free languages cannot be accepted in linear time, linear languages cannot be accepted in linear time in the general case.

This is an incorrect argument. If some red stones are heavy, and not all heavy stones can be lifted, it doesn't follow that not all red stones can be lifted. A different argument is required to show that linear languages cannot all be accepted in linear time. Rp (talk) 21:52, 30 July 2026 (UTC)

Agreed, this statement should be removed. How about referencing one or both of these papers instead?

• Hervé Gallaire (1969), Recognition time of context-free languages by on-line Turing machines, Information and Control 15, 288–295
• Joel I. Seiferas (1986), A simplified lower bound for context-free-language recognition, Information and Control 69, 255–260

Gallaire established and Seiferas simplified the proof of a Ω(n²/log n) lower bound for the on-line recognition problem for linear grammars, where "on-line" that the algorithm has to decide for each prefix of a word whether it is in the language before reading the next character. However, Seiferas' paper mentions that the argument does not apply to random access machines / ordinary ("off-line") recognition, and I do not know whether there is a super-linear lower bound for linear grammars in that case. Noamz (talk) 23:34, 30 July 2026 (UTC)