Full AFLs and nested iterated substitution
TL;DR: A superAFL is a family of languages closed under union with unitary sets, intersection with regular sets, and nested iterated substitution and containing at least one nonunitary set containing all context-free languages.
Abstract: A superAFL is a family of languages closed under union with unitary sets, intersection with regular sets, and nested iterated substitution and containing at least one nonunitary set Every superAFL is a full AFL containing all context-free languages If L is a full principal AFL, then Ŝ∞(L), the least superAFL containing L, is full principal If L is not substitution closed, the substitution closure of L is properly contained in Ŝ∞(L) The indexed languages form a superAFL which is not the least superAFL containing the one-way stack languages If L has a decidable emptiness problem, so does Ŝ∞(L) IfDs is an AFA, L = L(Ds) and Dw is the family of machines whose data structure is a pushdown store of tapes of Ds, then L(Ds) = Ŝ∞(L) if, and only if, Ds is nontrivial If Ds is uniformly erasable and L(Ds) has a decidable emptiness problem, then it is decidable if a member of Dw is finitely nested
Citations
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TL;DR: A relationship between parallel rewriting systems and two-way machines is investigated, finding restrictions on the “copying power” of these devices endow them with rich structuring and give insight into the issues of determinism, parallelism, and copying.
181 citations
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TL;DR: The syntactic structure of sets of ancestors and sets of descendants is considered, as well as that of unions of congruence classes, taken over (infinite) context-free languages or regular sets.
113 citations
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TL;DR: TOL languages form an infinite hierarchy with respect to “natural” complexity measures introduced in this paper, and are contained in the family of context-free programmed languages.
Abstract: We discuss a family of systems and languages (called TOL) which have originally arisen from the study of mathematical models for the development of some biological organisms. From a formal language theory point of view, a TOL system is a rewriting system where at each step of a derivation every symbol in a string is rewritten in a context-free way, but different rewriting steps may use different sets of production rules and the language consists of all strings derivable from the single fixed string (the axiom). The family of TOL languages (as well as its different subfamilies considered here) is not closed with respect to usually considered operations; it is “incomparable” with context-free languages, but it is contained in the family of context-free programmed languages. TOL languages form an infinite hierarchy with respect to “natural” complexity measures introduced in this paper.
84 citations
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TL;DR: 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.
Abstract: If a full AFLℒ is not closed under substitution, thenℒ o ℒ, the result of substituting members ofℒ intoℒ, is not substitution closed and henceℒ generates an infinite hierarchy of full AFL's. Ifℒ1 andℒ2 are two incomparable full AFL's, then the least full AFL containingℒ1 andℒ2 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.
79 citations
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TL;DR: This work defines several pushdown machines of which the control is recursive without parameters, or even iterative, and which work on a generalized pushdown as storage, and characterize the n-fold composition of total deterministic macro tree transducers by recursive push down machines with an iterated push down as storage.
73 citations
Cites background from "Full AFLs and nested iterated subst..."
..., the pushdown of pushdowns, is already considered in [28, 35, 36, 9, 17]....
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References
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TL;DR: A new type of grammar for generating formal languages, called an indexed grammar, is presented, and the class of languages generated by indexed grammars has closure properties and decidability results similar to those for context-free languages.
Abstract: A new type of grammar for generating formal languages, called an indexed grammar, is presented. An indexed grammar is an extension of a context-free grammar, and the class of languages generated by indexed grammars has closure properties and decidability results similar to those for context-free languages. The class of languages generated by indexed grammars properly includes all context-free languages and is a proper subset of the class of context-sensitive languages. Several subclasses of indexed grammars generate interesting classes of languages.
476 citations
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15 Oct 1968
TL;DR: Two new classes of grammars based on programming macros are studied, one of which is IO but not OI and the other OI but not IO, showing that neither class contains the other.
Abstract: Two new classes of grammars based on programming macros are studied. Both involve appending arguments to the intermediate symbols of a context-free grammar. They differ only in the order in which nested terms may be expanded: IO is expansion from the inside-out; OI from the outside-in. Both classes, in common with the context-free, have decidable emptiness and derivation problems, and both are closed under the operations of union, concatenation, Kleene closure (star), reversal, intersection with a regular set, and arbitrary homomorphism. OI languages are also closed under inverse homomorphism while IO languages are not. We exhibit two languages, one of which is IO but not OI and the other OI but not IO, showing that neither class contains the other. However, both trivially contain the class of context-free languages, and both are contained in the class of contextsensitive languages. Finally, the class of OI languages is identical to the class of indexed languages studied by Aho, and indeed many of the above. theorems about OI languages follow directly from the equivalence.
227 citations
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TL;DR: A number of operations which either preserve sets accepted by one-way stack automata or preserve setsaccepted by deterministic one- way stack Automata are presented.
Abstract: A number of operations which either preserve sets accepted by one-way stack automata or preserve sets accepted by deterministic one-way stack automata are presented. For example, sequential transduction preserves the former; set complementation, the latter. Several solvability questions are also considered.
223 citations
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TL;DR: The quasi-real-time one- way stack languages are not a full AFL but are a proper subAFL of the one-way stack languages, which properly include the stack languages.
122 citations