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M-rate 0L systems are interactionless Lindenmayer systems together with a function assigning to every string a set of multisets of productions that may be applied simultaneously to the string. Some questions that have been left open in the forerunner papers are examined, and the computational power of deterministic M-rate 0L systems is investigated, where also tabled and extended variants are taken into consideration.
We study the derivational complexity of context-free and context-sensitive grammars by counting the maximal number of non-regular and non-context-free rules used in a derivation, respectively. The degree of non-regularity/non-context-freeness of a language is the minimum degree of non-regularity/non-context-freeness of context-free/context-sensitive grammars generating it. A language has finite degree of non-regularity iff it is regular. We give a condition for deciding whether the degree of non-regularity of a given unambiguous context-free grammar is finite. The problem becomes undecidable for arbitrary linear context-free grammars. The degree of non-regularity of unambiguous context-free grammars generating non-regular languages as well as that of grammars generating deterministic context-free languages that are not regular is of order Omega(n). Context-free non-regular languages of sublinear degree of non-regularity are presented. A language has finite degree of non-context-freeness if it is context-free. Context-sensitive grammars with a quadratic degree of non-context-freeness are more powerful than those of a linear degree.
This paper is part of the investigation of some operations on words and languages with motivations coming from DNA biochemistry, namely three variants of hairpin completion and three variants of hairpin reduction. Since not all the hairpin completions or reductions of semilinear languages remain semilinear, we study sufficient conditions for semilinear languages to preserve their semilinearity property after applying the non-iterated hairpin completion or hairpin reduction. A similar approach is then applied to the iterated variants of these operations. Along these lines, we define the hairpin reduction root of a language and show that the hairpin reduction root of a semilinear language is not necessarily semilinear except the universal language. A few open problems are finally discussed.