Theory Of Context-free Languages Research Articles

Infinitary Axiomatization of the Equational Theory of Context-Free Languages

We give a natural complete infinitary axiomatization of the equational theory of the context-free languages, answering a question of Leis (1992).

A mechanisation of some context-free language theory in HOL4

We describe the mechanisation of some foundational results in the theory of context-free languages (CFLs), using the HOL4 system. We focus on pushdown automata (PDAs). We show that two standard acceptance criteria for PDAs (“accept-by-empty-stack” and “accept-by-final-state”) are equivalent in power. We are then able to show that the pushdown automata (PDAs) and context-free grammars (CFGs) accept the same languages by showing that each can emulate the other. With both of these models to hand, we can then show a number of basic, but important results. For example, we prove the basic closure properties of the context-free languages such as union and concatenation. Along the way, we also discuss the varying extent to which textbook proofs (we follow Hopcroft and Ullman) and our mechanisations diverge: sometimes elegant textbook proofs remain elegant in HOL; sometimes the required mechanisation effort blows up unconscionably.

Inverse monoids, trees and context-free languages

This paper is concerned with a study of inverse monoids presented by a set X X subject to relations of the form e i = f i {e_i} = {f_i} , i ∈ I i \in I , where e i {e_i} and f i {f_i} are Dyck words, i.e. idempotents of the free inverse monoid on X X . Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on X X . In the finitely presented case the word problem is solved by using Rabin’s theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored.

Formes de langages et de grammaires

This paper is devoted to the study of context-free languages over infinite alphabets. This work can be viewed as a new attempt to study families of grammars, replacing the usual "grammar forms" and giving a new point of view on these questions. A language over an infinite alphabet or I-language appears as being a model for a family of usual languages; an interpretation is an homomorphism from the infinite alphabet to any finite alphabet. Using this notion of interpretation we can associate to each family of I-languages an image, called its shadow, which is a family of usual languages. The closure properties of families, generalizing to infinite alphabets the family of context-free languages, lead to define rational transductions between infinite alphabets or I-transductions, and then, families of I-languages closed under I-transductions, or I-cones. We study here relations between the closure properties of a family of I-languages and these of its shadow. As a result, we obtain that any union closed rational cone of context-free languages, principal or not, is the shadow of a principal I-cone. This work leads to new results about the classical theory of context-free languages. For instance, we prove that any principal rational cone of context-free languages can be generated by a context-free language, whose grammar has only 6 variables. This work also leads to more general considerations about the adequacy of some generating devices to the generated languages. It appears that the context-free grammars are fair, in a sense that we define, for generating context-free languages but that non-expansive context-free grammars are not for generating non-expansive context-free languages. This point of view raises a number of questions.

Formal languages and enumeration

We present a theory of generating functions in countably many non-commuting variables. This generalizes the theory of context free languages. Applications are given to compositions of a number, rooted planar tree, dissected polygons, and the theory of simple random walks.

N. Chomsky and M. P. Schützenberger. The algebraic theory of context-free languages. Computer programming and formal systems, edited by P. Braffort and D. Hirschberg, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1963, pp. 118–161.

N. Chomsky and M. P. Schützenberger. The algebraic theory of context-free languages. Computer programming and formal systems, edited by P. Braffort and D. Hirschberg, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1963, pp. 118–161.