How long will it take for a program to print a truth table for n propositional symbols?
(Symbols: P1, P2, ..., Pn)
Can't seem to crack this question, not quite sure how to calculate this instance.
Automata and Computability
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It will take time proportional to at least n*2^n. Each of the propositional symbols can assume one of two values - true or false. The portion of the truth table that lists all possible assignments of n such variables must have at least 2 * 2 * … * 2 (n times) = 2^n rows of n entries each; and that's not even counting the subexpressions that make up the rest of the table. This lower bound is tight since we can imagine the proposition P1 and P2 and … and Pn and the following procedure taking time Theta(n*2^n) to write out the answer:
If you have more complicated propositions then you should probably take the number of subexpressions as another independent variable since that could have an asymptotically relevant effect (using n propositional symbols, you can have arbitrarily many unique subexpressions that must be given their own columns in a complete truth table).