Proof Size Comparison
A second question about proof complexity is whether a method is more efficient than another. Since the proof size depends on the formula, it is possible that one method can produce a short proof of a formula and only long proofs of another formula, while a second method can have exactly the opposite behavior. The assumptions of measuring the size of the proofs relative to the size of the formula and considering only the shortest proofs are also used in this context.
When comparing two proof methods, two outcomes are possible:
- for every proof of a formula produced using the first method, there is a proof of comparable size of the same formula produced by the second method;
- there exists a formula such that the first method can produce a short proof while all proofs obtained by the second method are consistently larger.
Several proofs of the second kind involve contradictory formulae expressing the negation of the pigeonhole principle, namely that pigeons can fit holes with no hole containing two or more pigeons.
Read more about this topic: Proof Complexity
Famous quotes containing the words proof, size and/or comparison:
“Talk shows are proof that conversation is dead.”
—Mason Cooley (b. 1927)
“Our brains are no longer conditioned for reverence and awe. We cannot imagine a Second Coming that would not be cut down to size by the televised evening news, or a Last Judgment not subject to pages of holier-than-Thou second- guessing in The New York Review of Books.”
—John Updike (b. 1932)
“I have travelled a good deal in Concord; and everywhere, in shops, and offices, and fields, the inhabitants have appeared to me to be doing penance in a thousand remarkable ways.... The twelve labors of Hercules were trifling in comparison with those which my neighbors have undertaken; for they were only twelve, and had an end; but I could never see that these men slew or captured any monster or finished any labor.”
—Henry David Thoreau (18171862)