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 Incoming links: Failure of the regular concatenation construction for a nonregular grammar

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 4F / d / Solution Created 2026-09-24 Updated 2026-09-25

Take

G = ({a, b, c}, {S, A, B}, {S \to A B, A \to a, B \to b}, S)

and

G^{'} = ({a, b, c}, {T}, {T \to c}, T) .

These are variable based and have disjoint variable sets. They satisfy

L (G) L (G^{'}) = {ab c} .

In

H^{reg}

, however, both terminal productions of

G

are replaced:

A \to a T, B \to b T .

Consequently its only terminal derivation is

S \Rightarrow A B \Rightarrow a TB \Rightarrow a T b T \Rightarrow a c b c,

so

L (H^{reg}) = {a c b c} \neq = {ab c} .

This realizes the failure of the regular concatenation construction for a nonregular grammar: two different terminal productions introduce two copies of

T

.

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