On the Generative Capacity of Contextual Grammars with Strictly Locally Testable Selection Languages

Abstract

We continue the research on the generative capacity of contextual grammars where contexts are adjoined around whole words (externally) or around subwords (internally) which belong to special regular selection languages. All languages generated by contextual grammars where all selection languages are elements of a certain subregular language family form again a language family. We investigate contextual grammars with strictly locally testable selection languages and compare those families to families which are based on finite, monoidal, nilpotent, combinational, definite, suffixclosed, ordered, commutative, circular, non-counting, power-separating, or union-free languages.

Merging two Hierarchies of Internal Contextual Grammars with Subregular Selection

“…The only exceptions are the relations of the family I C (SUF) to the families I C (ORD) and I C (NC) where it is not known whether they are incomparable or whether I C (SUF) is a subset of the other and the relation of the family I C (REG Z n+1 ) to I C (RL V n ) for n ≥ 1 where it is not known whether they are incomparable or whether I C (REG Z n+1 ) is a subset of I C (RL V n ). We note here that in [4,8,9,10,16,25,5] a slightly different definition was used than in [27,11] and the present paper. This difference consists in the alphabet of the selection languages.…”

Relations of contextual grammars with strictly locally testable selection languages

“…The inclusion follows from the proper inclusion ORD ⊂ NC (see [24]) and Lemma 2.11. The properness follows from Lemma 3.3 with the witness language L SLT 2,¬ORD which belongs to the class EC(SLT 2 ), and hence, also to the class EC(NC ) (see [4]), but not to the class EC(ORD).…”

“…From [18], we know the proper inclusion SLT ⊂ NC . In [4], the proper inclusions COMB ⊂ SLT 1 and DEF ⊂ SLT as well as the incomparability of each family SLT k for k ≥ 1 with the families FIN , NIL, and DEF were mentioned but not proved. This will be done in the sequel.…”

“…Many incomparability results have been published in [3,4,28]. The only open questions are whether the class EC(ORD) is incomparable to the classes EC(SLT ) and EC(SLT k ) for k ≥ 2.…”

“…The study of contextual grammars with selection in special regular sets was started in [3] and continued in [4,7,8,28]. For the families defined by syntactic restrictions, in the external case as well as in the internal case, infinite hierarchies with respect to all parameters mentioned above were obtained.…”