I was really surprised when I discovered that converting the cross-products, used to determine the partial products, of a binary multiplication into the form of Boolean logic was logically simple. It turns out that the methodology of the multiplication problem, when you are given the multiplicands is simple just as the multiplication algorithm is simple. However, if you were to try to work backwards to use the partial products to try to determine the multiplicands the problem is much more difficult. It is simple as long as poorly formed partial products have been filtered out beforehand. The formalization of the problem, (expressed in pure Boolean form), is not quite so simple if the algorithm is supposed to detect or avoid poorly formed partial products.
It took me a long time to figure this out so I won't be calling any of the other boys in this group slow anytime soon. So anyway, if you wanted to use the multiplication algorithm as a factorization algorithm you would have to work the algorithm backwards in order to determine the multiplicands given the product, or to determine the multiplicands given a system of partial products. The use of the partial products as the given isn't simple because if a bit in a partial product = 0 it could be because the corresponding bits in one or both of the multiplicands that produced the cross product were 0. That shows that there are three ways to explain a bit in a cross product that equals 0. On the other hand if the bit in the cross product was 1 then we would know that both bits were 1, which shows that there should be a reasonably "easy" way to define the Boolean Logic of the partial products so that it would only allow correctly formed partial products to get past the Boolean Formula. This reasoning shows how complicated turning an algorithm into a true Boolean Formula can be. First we start off with the standard multiplication algorithm, ok. But if you want to use that algorithm in an unconventional way, you have to make sure that the implicit relations are well-formed in the sense that the unconventional usage will not present the given values in an unacceptable poorly-formed way. So even if you want to consider the problem in purely abstract way -using only Boolean variables- you have to be prepared for unexpected effects when your claim jumps the conventional rails. If you want to take the conversion of a multiplication function into a pure (variable) Boolean form and then use it to detect a factorization given a solution to the derived Boolean Formula, the multiplication method would first have to be expressed as a reversible function. Jim Bromer ------------------------------------------- AGI Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/21088071-c97d2393 Modify Your Subscription: https://www.listbox.com/member/?member_id=21088071&id_secret=21088071-2484a968 Powered by Listbox: http://www.listbox.com
