Automata and Computability

Question

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.

Answer 1

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:

fill up P1's column with 2^(n-1) TRUE and then 2^(n-1) FALSE
fill up P2's column with 2^(n-2) TRUE and then 2^(n-2) FALSE, alternating
…
fill up Pn's column with 1 TRUE and 1 FALSE, alternating
fill up the last column with a TRUE at the top and FALSE all the way down

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).