Chomsky–Schützenberger enumeration theorem

In formal language theory, the Chomsky–Schützenberger enumeration theorem is a theorem derived by Noam Chomsky and Marcel-Paul Schützenberger about the number of words of a given length generated by an unambiguous context-free grammar. The theorem provides an unexpected link between the theory of formal languages and abstract algebra.

Statement

In order to state the theorem, a few notions from algebra and formal language theory are needed.

Let ${\displaystyle \mathbb {N} }$ denote the set of nonnegative integers. A power series over ${\displaystyle \mathbb {N} }$ is an infinite series of the form

${\displaystyle f=f(x)=\sum _{k=0}^{\infty }a_{k}x^{k}=a_{0}+a_{1}x^{1}+a_{2}x^{2}+a_{3}x^{3}+\cdots }$

with coefficients ${\displaystyle a_{k}}$ in ${\displaystyle \mathbb {N} }$. The multiplication of two formal power series ${\displaystyle f}$ and ${\displaystyle g}$ is defined in the expected way as the convolution of the sequences ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$:

${\displaystyle f(x)\cdot g(x)=\sum _{k=0}^{\infty }\left(\sum _{i=0}^{k}a_{i}b_{k-i}\right)x^{k}.}$

In particular, we write ${\displaystyle f^{2}=f(x)\cdot f(x)}$, ${\displaystyle f^{3}=f(x)\cdot f(x)\cdot f(x)}$, and so on. In analogy to algebraic numbers, a power series ${\displaystyle f(x)}$ is called algebraic over ${\displaystyle \mathbb {Q} (x)}$, if there exists a finite set of polynomials ${\displaystyle p_{0}(x),p_{1}(x),p_{2}(x),\ldots ,p_{n}(x)}$ each with rational coefficients such that

${\displaystyle p_{0}(x)+p_{1}(x)\cdot f+p_{2}(x)\cdot f^{2}+\cdots +p_{n}(x)\cdot f^{n}=0.}$

A context-free grammar is said to be unambiguous if every string generated by the grammar admits a unique parse tree or, equivalently, only one leftmost derivation. Having established the necessary notions, the theorem is stated as follows.

Chomsky–Schützenberger theorem. If ${\displaystyle L}$ is a context-free language admitting an unambiguous context-free grammar, and ${\displaystyle a_{k}:=|L\ \cap \Sigma ^{k}|}$ is the number of words of length ${\displaystyle k}$ in ${\displaystyle L}$, then ${\displaystyle G(x)=\sum _{k=0}^{\infty }a_{k}x^{k}}$ is a power series over ${\displaystyle \mathbb {N} }$ that is algebraic over ${\displaystyle \mathbb {Q} (x)}$.

Proofs of this theorem are given by Kuich & Salomaa (1985), and by Panholzer (2005).

Usage

Asymptotic estimates

The theorem can be used in analytic combinatorics to estimate the number of words of length n generated by a given unambiguous context-free grammar, as n grows large. The following example is given by Gruber, Lee & Shallit (2012): the unambiguous context-free grammar G over the alphabet {0,1} has start symbol S and the following rules

S → M | U

M → 0M1M | ε

U → 0S | 0M1U.

To obtain an algebraic representation of the power series ${\displaystyle G(x)}$ associated with a given context-free grammar G, one transforms the grammar into a system of equations. This is achieved by replacing each occurrence of a terminal symbol by x, each occurrence of ε by the integer '1', each occurrence of '→' by '=', and each occurrence of '|' by '+', respectively. The operation of concatenation at the right-hand-side of each rule corresponds to the multiplication operation in the equations thus obtained. This yields the following system of equations:

S = M + U

M = M²x² + 1

U = Sx + MUx²

In this system of equations, S, M, and U are functions of x, so one could also write ${\displaystyle S(x)}$, ${\displaystyle M(x)}$, and ${\displaystyle U(x)}$. The equation system can be resolved after S, resulting in a single algebraic equation:

${\displaystyle x(2x-1)S^{2}+(2x-1)S+1=0}$.

This quadratic equation has two solutions for S, one of which is the algebraic power series ${\displaystyle G(x)}$. By applying methods from complex analysis to this equation, the number ${\displaystyle a_{n}}$ of words of length n generated by G can be estimated, as n grows large. In this case, one obtains ${\displaystyle a_{n}\in O(2+\epsilon )^{n}}$ but ${\displaystyle a_{n}\notin O(2-\epsilon )^{n}}$ for each ${\displaystyle \epsilon >0}$.^[1]

The following example is from Bassino & Nicaud (2011):

$${\displaystyle \left\{{\begin{array}{l }{S\rightarrow XY}\\{T\rightarrow aT|TbT|YcY}\\{Y\rightarrow YaY|cY|abTaYYa|X}\\{X\rightarrow a|b|c}\end{array}}\Rightarrow \left\{{\begin{array}{l}s(z)=x(z)y(z)\\t(z)=zt(z)+zt(z)^{2}+zy(z)^{2}\\y(z)=zy(z)^{2}+zy(z)+z^{4}t(z)y(z)^{2}+x(z)\\x(z)=3z\end{array}}\right.\right.}$$

$${\displaystyle s(z)^{8}-27\left(z^{3}-z^{2}\right)s(z)^{5}+\ldots +59049z^{10}=0}$$

Inherent ambiguity

In classical formal language theory, the theorem can be used to prove that certain context-free languages are inherently ambiguous. For example, the Goldstine language ${\displaystyle L_{G}}$ over the alphabet ${\displaystyle \{a,b\}}$ consists of the words ${\displaystyle a^{n_{1}}ba^{n_{2}}b\cdots a^{n_{p}}b}$ with ${\displaystyle p\geq 1}$, ${\displaystyle n_{i}>0}$ for ${\displaystyle i\in \{1,2,\ldots ,p\}}$, and ${\displaystyle n_{j}\neq j}$ for some ${\displaystyle j\in \{1,2,\ldots ,p\}}$.

It is comparably easy to show that the language ${\displaystyle L_{G}}$ is context-free.^[2] The harder part is to show that there does not exist an unambiguous grammar that generates ${\displaystyle L_{G}}$. This can be proved as follows: If ${\displaystyle g_{k}}$ denotes the number of words of length ${\displaystyle k}$ in ${\displaystyle L_{G}}$, then for the associated power series holds ${\displaystyle G(x)=\sum _{k=0}^{\infty }g_{k}x^{k}={\frac {1-x}{1-2x}}-{\frac {1}{x}}\sum _{k\geq 1}x^{k(k+1)/2-1}}$. Using methods from complex analysis, one can prove that this function is not algebraic over ${\displaystyle \mathbb {Q} (x)}$. By the Chomsky-Schützenberger theorem, one can conclude that ${\displaystyle L_{G}}$ does not admit an unambiguous context-free grammar.^[3]

Notes

References