Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-59/3/b/solution

For each fixed input length, take the complete truth table of the language. For every accepted string , form the minterm
An OR of these minterms equals the indicator function, since precisely at . This is the truth-table upper bound for circuit size.
Generate the negated input wires once and share them. With accepted strings, use at most binary AND gates and binary OR gates, besides the negations. Empty truth tables use a constant-zero Boolean circuit; length zero is handled by a constant Boolean circuit. Thus
This is an existence bound for a nonuniform circuit family, even when the language is undecidable. It supplies no algorithm for computing the truth tables of an arbitrary language.

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