Parikh's Theorem and Descriptional Complexity

This thesis was carried out in the Laboratorio di Linguaggi e Combinatoria (LIN.COM), at the Dipartimento di Informatica e Comunicazione, Università degli Studi di Milano. The thesis deals with some topics in the theory of formal languages, and specifically with the theory of context-free languages and the study of theirs descriptional complexity. The descriptional complexity of a formal structure (grammar, model of automata, etc) is the number of symbols needed to write down its description. While this aspect is extensively treated in regular languages, as evidenced by numerous references, in the case of context-free languages few results are known. The thesis starts studying the Parikh’s theorem. The theorem states that for each context-free language there exists a regular language with the same Parikh image. The Parikh image of a language L ⊆ Σ∗ is the set of Parikh images of its words. Given an alphabet Σ = {a1, . . . , am}, the Parikh image is a function ψ : Σ∗ → N m that associates with each word w ∈ Σ∗, the vector ψ(w) = (|w|a1 , |w|a2 , ..., |w|am), where |w|ai is the number of occurrences of ai in w. For instance, the language {anbn | n ≥ 0} has the same Parikh image as (ab)∗. Roughly speaking, the theorem shows that if the order of the letters in a word is disregarded, retaining only the number of their occurrences, then context-free languages are indistinguishable from regular languages. The classical proofs of Parikh’s theorems implicitly provide the construction of a nondeterministic automaton accepting a language “Parikh-equivalent” to the context-free language under consideration. The number of the states of such an automaton is double exponential in the size of the given context-free grammar. In