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 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. or .

Mathematics

Notation

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

A ''matched alphabet'' $T \cup \overline T$ is an alphabet with two equal-sized sets; it is convenient to think of it as a set of parentheses types, where $T$ contains the opening parenthesis symbols, whereas the symbols in $\overline T$ contains the closing parenthesis symbols. For a matched alphabet $T \cup \overline T$, the '' typed Dyck language'' $D_T$ is given by
:$D_T = \.$
For example, the following is a valid sentence in the 3-typed Dyck language:

( [ ">[_.html" ;"title="[ ">[ ( ) )

Theorem

A language ''L'' over the alphabet $\Sigma$ is context-free if and only if there exists
:* a matched alphabet $T \cup \overline T$
:* a regular language $R$ over $T \cup \overline T$,
:* and a homomorphism $h : (T \cup \overline T)^* \to \Sigma^*$
:such that $L = h(D_T \cap R)$.

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.

References

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{{DEFAULTSORT:Chomsky-Schutzenberger representation theorem
Noam Chomsky
Formal languages
Theorems in discrete mathematics