New class of languages extends semi-linearity beyond context-free sets
Semi-Linearity and Generalized Stratified Semi-Linearity beyond Context-Free Languages
Formal Languages and Automata TheoryComputational Complexity
Summary
This work expands a famous idea about context-free languages, called semi-linearity, to a broader family of languages recognized by certain memory-limited machines. The authors study these ‘k-storable’ languages and show they also have a structured, predictable pattern in their word compositions, generalizing known results from simpler languages. They do this by using a new, abstract mathematical description that doesn't depend on machine models. This suggests that k-storable languages naturally generalize context-free languages in a meaningful way.
What this means in practice
- •For language designers: Create new formal languages with predictable word count patterns beyond context-free limits using k-storable language properties.
- •For compiler developers: Design parsers for extended language classes that maintain structured word composition, aiding syntax analysis beyond context-free grammars.
A theory result. No direct application yet.
Authors
Tomoyuki Yamakami
Abstract
Parikh's famous theorem relates to the semi-linearity of context-free languages. We seek for expanding the scope of the semi-linear property from context-free languages to languages beyond them. Of those languages, this work focuses on the languages that are particularly recognized by one-way nondeterministic depth-k storage automata under a natural restriction of frozen blank sensitivity. Those languages (which are called $k$-storable languages) have been already proven to satisfy a natural generalization of the well-known pumping lemmas for regular and context-free languages. The semi-linearity of k-storable languages generalizes Parikh's theorem on the semi-linearity of context-free languages. As for bounded languages, Ginsburg and Spanier's well-known theorem characterizes context-free languages in terms of stratified semi-linearity. This notion is further generalized and is used to characterize the k-storable languages as well. Such a generalization is interesting in its own right. This helps us take an abstract treatment of k-storable languages without relying on "machines". Our results strengthen the belief that those languages are "natural" generalizations of context-free languages.