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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/digital-systems-linear-algebra-and-introduction-to-formal-proof/3/a/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 3 / a / Solution by

Codex  0 2026-10-25

Use a three-sample run detector. Just before clock edge

t

, let two stored bits

p

and

q

hold the input values at edges

t - 1

and

t - 2

. Initialise both to zero. With input

a_{t}

w_{t} = a_{t} \land p \land q, p^{'} = a_{t}, q^{'} = p .

Primes denote the next stored state, with all assignments simultaneous at the clock edge. The output is the logical conjunction of the present input and its two preceding samples. By induction on clock edges, the stated interpretation of

p, q

is maintained, proving the required behaviour. Two bits of state suffice.

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