Chomsky–Schützenberger representation theorem

In formal language theory, the Chomsky–Schützenberger representation theorem is a theorem derived by Noam Chomsky and Marcel-Paul Schützenberger in 1959^[1] about representing a given context-free language in terms of two simpler languages. These two simpler languages, namely a regular language and a Dyck language, are combined by means of an intersection and a homomorphism.

The theorem Proofs of this theorem are found in several textbooks, e.g. Autebert, Berstel & Boasson (1997) or Davis, Sigal & Weyuker (1994).

A few notions from formal language theory are in order.

A context-free language is regular, if it can be described by a regular expression, or, equivalently, if it is accepted by a finite automaton.

A homomorphism is based on a function ${\displaystyle h}$ which maps symbols from an alphabet ${\displaystyle \Gamma }$ to words over another alphabet ${\displaystyle \Sigma }$; If the domain of this function is extended to words over ${\displaystyle \Gamma }$ in the natural way, by letting ${\displaystyle h(xy)=h(x)h(y)}$ for all words ${\displaystyle x}$ and ${\displaystyle y}$, this yields a homomorphism ${\displaystyle h:\Gamma ^{*}\to \Sigma ^{*}}$.

A matched alphabet ${\displaystyle T\cup {\overline {T}}}$ is an alphabet with two equal-sized sets; it is convenient to think of it as a set of parentheses types, where ${\displaystyle T}$ contains the opening parenthesis symbols, whereas the symbols in ${\displaystyle {\overline {T}}}$ contains the closing parenthesis symbols. For a matched alphabet ${\displaystyle T\cup {\overline {T}}}$, the typed Dyck language ${\displaystyle D_{T}}$ is given by

${\displaystyle D_{T}=\{\,w\in (T\cup {\overline {T}})^{*}\mid w{\text{ is a correctly nested sequence of parentheses}}\,\}.}$

For example, the following is a valid sentence in the 3-typed Dyck language:

( [ [ ] { } ] ( ) { ( ) } )

A language L over the alphabet ${\displaystyle \Sigma }$ is context-free if and only if there exists

• a matched alphabet ${\displaystyle T\cup {\overline {T}}}$
• a regular language ${\displaystyle R}$ over ${\displaystyle T\cup {\overline {T}}}$,
• and a homomorphism ${\displaystyle h:(T\cup {\overline {T}})^{*}\to \Sigma ^{*}}$

such that ${\displaystyle L=h(D_{T}\cap R)}$.

We can interpret this as saying that any CFG language can be generated by first generating a typed Dyck language, filtering it by a regular grammar, and finally converting each bracket into a word in the CFG language.

• Autebert, Jean-Michel; Berstel, Jean; Boasson, Luc (1997). "Context-Free Languages and Push-Down Automata" (PDF). In G. Rozenberg and A. Salomaa, eds., Handbook of Formal Languages, Vol. 1: Word, Language, Grammar (pp. 111–174). Berlin: Springer-Verlag. ISBN 3-540-60420-0.
• Davis, Martin D.; Sigal, Ron; Weyuker, Elaine J. (1994). Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science (2nd ed.). Elsevier Science. p. 306. ISBN 0-12-206382-1.