Abstract
If a full AFLℒ is not closed under substitution, thenℒ ô ℒ, the result of substituting members ofℒ intoℒ, is not substitution closed and henceℒ generates an infinite hierarchy of full AFL's. Ifℒ 1 andℒ 2 are two incomparable full AFL's, then the least full AFL containingℒ 1 andℒ 2 is not substitution closed. In particular, the substitution closure of any full AFL properly contained in the context-free languages is itself properly contained in the context-free languages. If any set of languages generates the context-free languages, one of its members must do so. The substitution closure of the one-way stack languages is properly contained in the nested stack languages. For eachn, there is a class of full context-free AFL's whose partial ordering under inclusion is isomorphic to the natural partial ordering onn-tuples of positive integers.
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This paper was completed while the author was in the Division of Engineering and Applied Physics of Harvard University. Research sponsored in part by the Air Force Cambridge Research Laboratories, Office of Aerospace Research, USAF, under contracts F-1962870C0023 and F-1962868C0029, and by the Air Force Office of Scientific Research, Office of Aerospace Research, USAF, under AFOSR Grant No. AF-AFOSR-1203-67A and the Division of Engineering and Applied Physics of Harvard University.
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Greibach, S.A. Chains of full AFL's. Math. Systems Theory 4, 231–242 (1970). https://doi.org/10.1007/BF01691106
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DOI: https://doi.org/10.1007/BF01691106