A short survey around the pumping lemma for context-free languages

TL;DR

The survey consolidates foundational aspects of context-free languages and grammars, emphasizing the Pumping Lemma and the Chomsky Normal Form framework. It links derivation processes to derivation trees, provides a direct Pumping Lemma proof, and demonstrates practical uses with canonical non-CFL examples. The discussion clarifies CFL closure properties, showing both robust operations (union, concatenation, star, substitution) and limits (intersection/complement), and establishes decidability results via the pumping bound. Overall, the work equips readers with a coherent toolkit for analyzing CFLs and distinguishing them from more powerful language classes.

Abstract

Following a seminar the present author gave to an Automata Theory course to computer science students, it will be presented, in a very synthetic and mostly selfcontained way, the principal properties of context free languages (CFL), with particular attention given to the Pumping Lemma (PL), and of grammars which generate them(CFG). We refer to Chomsky and Schutzenberger for the first works about it. What is known in literature as the Iteration Theorem here will be referred to as the Ogden's Lemma in a fully justified way. All definitions not strictly connected with the notion of context freeness will be omitted (we will give precise references for all of them). The symbology used is substantially the classical one, but we will replace some symbols to avoid confusion with those used in logic