Using bits as flags does not make a binary number into a unary number because the unary number does not have a zero value. But if you read and understood my message then you probably knew what I meant. So there are three aspects of a bit in a binary number. Two are the bit states and the third is the position of the bit. But, what I am wondering about is whether the relation between the bit positions which complicates the use of a binary number in a comparative computation means that certain significant mathematical functions cannot be used on a binary number without a great deal of elaboration. I believe this may be the basis for problems in np and this knowledge might be used to create new kinds of systems that have something in common with neural networks and so on. Jim Bromer
On Tue, May 19, 2015 at 9:56 PM, Jim Bromer <[email protected]> wrote: > I wrote that the reason computers, physics and computational > mathematics have advanced so far is because the n-ary numerical > representation, where n>1, is an effective compression of the unary > system and because computational mathematics are able to use those > representations without decompressing them (into unary form for > example). I was thinking about it today and I realized that all the > n-ary numbers are able to represent every number uniquely so I > wondered what was lost going from unary to binary. A one-to-one > correspondence between any equal parts of two or more numbers is > possible with unary representations but it is not with binary > representations. So, for example, if you want to subtract an equal > amount from two n-ary numbers you won't know -for all possible cases- > how many columns of digits will be involved. Therefore you cannot use > that information as an absolute in a mathematical abstraction. This > means that many great functions and mathematical methods, some of > which are quite ancient, cannot be efficiently used for > multiplicities. So, for another example, even if you carefully laid > out a sophisticated vector system that included individuals and > relational information as well, you wouldn't be able to get very far > with it because of the complications that are created by the lack of a > one-to-one correspondence at the base representations. You would have > to do a great many operations to put all the information in the > appropriate form to use it for a complex analysis of a situation and > if you tried to change one small thing it could require a complete > recalculation. If you want to be able to keep track of corresponding > bits you have to use a flag system in which each columnar bit is meant > to categorically represent something. I don't know what the > mathematical name for this kind of representation is but it is very > inefficient for complicated systems because it is essentially a unary > system. What happens with a logical formula is that multiple > operations create a distribution of these corresponding (bit) > references by recombining them. In some ways logic is like a neural > network. The summary or result of logical functions are represented in > a system that is distributed and combined. NNs are usually thought of > as fields and the result of a logical formula as a line of bits but > there is something in common. Once the polynomial time solution for > Boolean Satisfiability is worked out a more insightful examination of > the similarities and differences of these systems may lead to novel > learning and representational systems. > Everything is finally starting to become very clear to me. I don't > mean that I have it all figured out - that would be absurd - but I do > mean that I can finally start to see the path. I can't describe it > very well but I can see it. > Jim Bromer ------------------------------------------- AGI Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/21088071-f452e424 Modify Your Subscription: https://www.listbox.com/member/?member_id=21088071&id_secret=21088071-58d57657 Powered by Listbox: http://www.listbox.com
