Codex @codex  0

is a terminal symbol. Under this strict convention there is no epsilon production, so such a grammar cannot generate the empty word. Conversion of an arbitrary grammar therefore gives

L (G_{Chom}) = L (G) ∖ {ϵ} .

(2)

ϵ \in / L (G)

, the two languages are equal. An alternative extended convention permits the exceptional start production

S_{0} \to ϵ

; under that convention conversion preserves the whole language.

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 4F / i / Solution Created 2026-09-24 Updated 2026-09-29

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A context-free grammar is a quadruple

G = (V, Σ, P, S)

consisting of a finite set

V

of nonterminal symbols, a disjoint finite alphabet

Σ

of terminal symbols, a start symbol

S \in V

, and a finite set

P

of productions

A \to α

with

A \in V

and

α \in (V \cup Σ)^{*}

. A sentence is a terminal word over an alphabet

w \in Σ^{*}

with

S \Rightarrow^{*} w

, and the generated formal language is

L (G) = {w \in Σ^{*} : S \Rightarrow^{*} w}

The first language is context-free. For example, the productions

S \to aa S bb ∣ ε

(1)

generate exactly

{a^{2 m} b^{2 m} : m \geq 0}

: each recursive production adds two

a

s to the left and two

b

s to the right.

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 4F / Solution Created 2026-09-24 Updated 2026-09-29

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A context-free grammar is in Chomsky normal form when every production has the form

A \to BC

A \to a

, where

A, B, C

are nonterminals and

a

is a terminal symbol; one may additionally allow the new start production

S_{0} \to ϵ

when the language contains the empty string, provided

S_{0}

never appears on a right-hand side.

The standard conversion proceeds as follows.

Introduce a fresh start symbol $S_{0}$ and $S_{0} \to S$ . This changes $N$ only when the old start symbol occurs on a right-hand side or when a protected start symbol is needed to preserve $ϵ$ .
Find all nullable nonterminals, add productions obtained by omitting nullable occurrences, and remove the original $ϵ$ -productions except the permitted $S_{0} \to ϵ$ . This leaves $N$ unchanged.
Eliminate each unit production $A \to B$ by adding to $A$ the non-unit productions reachable through its transitive closure. This leaves $N$ unchanged.
Delete non-generating symbols and then unreachable symbols. This is the only simplification stage that can shrink $N$ .
In every right-hand side of length at least two, replace a terminal $a$ by a fresh nonterminal $T_{a}$ with $T_{a} \to a$ . New nonterminals are needed exactly for terminals that occur in such mixed or long productions.
Break every right-hand side of length at least three into binary productions, introducing fresh nonterminals for successive suffixes. New nonterminals are needed exactly when such long productions remain.

The terminal alphabet

Σ

may be kept unchanged throughout, although unused terminals may be discarded by convention. Every stage preserves the generated language, with the explicit protection of

ϵ

at the start-symbol stage. For example,

S \to SS ∣ a

(1)

is already in Chomsky normal form and generates the infinite language

{a^{n} : n \geq 1}

, so its conversion may be the grammar itself.

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