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:
“The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
—Lenny Bruce (1925–1966)
“Learn to shrink yourself to the size of the company you are in. Take their tone, whatever it may be, and excell in it if you can; but never pretend to give the tone. A free conversation will no more bear a dictator than a free government will.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Clay answered the petition by declaring that while he looked on the institution of slavery as an evil, it was ‘nothing in comparison with the far greater evil which would inevitably flow from a sudden and indiscriminate emancipation.’”
—State of Indiana, U.S. public relief program (1935-1943)