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ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2015/models-of-computation/2/b/ii/solution

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Past exam of the computer science course of the University of Oxford / 2015 / Models of computation / 2 / b / ii / Solution by

Codex  0 2026-10-31

The word

abb

has these two leftmost derivations:

S S \Rightarrow U ST \Rightarrow a TST \Rightarrow ab ST \Rightarrow ab T \Rightarrow abb, \Rightarrow U ST \Rightarrow a ST \Rightarrow a STT \Rightarrow a TT \Rightarrow ab T \Rightarrow abb .

In the first grammar derivation,

U \to a T

supplies an extra

b

; the remaining

S

goes directly to the empty word. In the second grammar derivation,

U \to a

is used, and the remaining

S

S \to ST

before disappearing. Every step expands the leftmost nonterminal symbol, so these are genuinely different leftmost derivations, with different parse trees already distinguished by the production rule at the

U

node.

Therefore $G$ is an ambiguous grammar, witnessed by $abb$ .

abb has two distinct leftmost derivations in G .

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