Abstract
Context-free languages are highly important in computer language processing technology as well as in formal language theory. The Pumping Lemma for Context-Free Languages states a property that is valid for all context-free languages, which makes it a tool for showing the existence of non-context-free languages. This paper presents a formalization, extending the previously formalized Lemma, of the fact that several well-known languages are not context-free. Moreover, we build on those results to construct a formal proof of the well-known property that context-free languages are not closed under intersection. All the formalization has been mechanized in the Coq proof assistant.
Highlights
A context-free grammar G is a four-tuple (V, Σ, P, S) where V is the vocabulary, Σ is the set of terminal symbols or alphabet ( N = V \Σ is the set of non-terminal symbols), P is the set of rules of the form X → β, where X ∈ N and β ∈ V ∗, and S is the start symbol of the grammar, S ∈ N
This is accomplished by means of the Pumping Lemma for Context-Free Languages, which can be used to prove that a given language is not context-free
Formalization of simplification enabled Chomsky normalization, which in turn enabled the formalization of the Pumping Lemma and the results presented here
Summary
We use the formalization of the Pumping Lemma previously obtained by the authors [7] in the Coq proof assistant [2]. The results presented here are important for various reasons They are applications of the previous formalization of the Pumping Lemma. They are the first ever languages to be proved not to be context-free using a computerized theorem prover. The present work brings mathematical formalization into a new area of application They give rise to interesting considerations about building formal proofs from text proofs.
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