Intercalation properties of context-free languages

Context-freedom of a language implies certain intercalation properties known as pumping or iteration lemmas. Although the question of a converse result for some of the properties has been studied, it is still not entirely clear how these properties are related, which are the stronger ones and which are weaker;Among the intercalation properties for context-free languages the better known are the general pumping conditions (generalized Ogden\u27s, Ogden\u27s and classic pumping conditions), Sokolowski-type conditions (Sokolowski\u27s and Extended Sokolowski\u27s conditions) and the Interchange condition. We present a rather systematic investigation of the relationships among these properties; it turns out that the three types of properties, namely pumping, Sokolowski-type and interchange, above are independent. However, the interchange condition is strictly stronger than the Sokolowski\u27s condition;Intercalation properties of some subclasses of context-free languages are also studied. We prove a pumping lemma and an Ogden\u27s lemma for nonterminal bounded languages and show that none of these two conditions is sufficient. We also investigate three of Igarashi\u27s pumping conditions for real-time deterministic context-free languages and show that these conditions are not sufficient either. Furthermore, we formulate linear analogues of the general pumping and interchange conditions and then compare them to the general context-free case. The results show that these conditions are also independent

Descriptional Succinctness of Some Grammatical Formalisms for Natrual Language

We investigate the problem of describing languages compactly in different grammatical formalisms for natural languages. In particular, the problem is studied from the point of view of some newly developed natural language formalisms like linear control grammars (LCGs) and tree adjoining grammars (TAGs); these formalisms not only generate non-context-free languages that capture a wide variety of syntactic phenomena found in natural language, but also have computationally efficient polynomial time recognition algorithms. We prove that the formalisms enjoy the property of unbounded succinctness over the family of context-grammars, i.e. they are, in general, able to provide more compact representations of natural languages as compared to standard context-free grammars

A Pumping Lemma Scheme for the Control Language Hierarchy

In [9] Weir introduced control grammars as a model for describing the syntactic structure of natural languages. Informally, a control grammar is a pair {G, C} where G is a context-free grammar whose productions are assigned labels from a finite set of labels II, and C (called the control set) is a set of strings over II. A derivation in a control grammar is similar to that in an ordinary context-free grammar except that the control set C is used to further constrain the set of valid derivations. In particular, if one views a derivation as a tree, then (in a manner to be described later) each edge in such a tree is given a label from II according to the production of G associated with the edge. The derivation tree is considered valid if certain paths in the tree correspond to strings which are in the control set C. The language generated by the control grammar is then the set of strings having at least one derivation tree in the sense just described

On non-recursive trade-offs between finite-turn pushdown automata

It is shown that between one-turn pushdown automata (1-turn PDAs) and deterministic finite automata (DFAs) there will be savings concerning the size of description not bounded by any recursive function, so-called non-recursive tradeoffs. Considering the number of turns of the stack height as a consumable resource of PDAs, we can show the existence of non-recursive trade-offs between PDAs performing k+ 1 turns and k turns for k >= 1. Furthermore, non-recursive trade-offs are shown between arbitrary PDAs and PDAs which perform only a finite number of turns. Finally, several decidability questions are shown to be undecidable and not semidecidable

Hypertrees: a study in language specification

A hierarchy of data structures, called the hypertree hierarchy, is presented which has strings and trees as its smallest two elements. A generalized frontiering operation is presented. Grammars, automata and regular expressions are extended to this hierarchy. These are shown to be equivalent and the resulting languages are called regular. This leads to a hierarchy of regular languages on the hypertree hierarchy. These are projected onto the set of strings by the frontier operation resulting in a true hierarchy of string languages. This hierarchy is called the (IO) algebraic language hierarchy. It has regular languages, context free languages and macro languages as its first three levels. It is contained in the set of context sensitive languages but is not equal to it. Other characterizations are presented

FORMULAE AND ASYMPTOTICS FOR COEFFICIENTS OF ALGEBRAIC FUNCTIONS

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