The final example is in prenex normal form. The symbol ‘3’ means

Software 249 The final example is in prenex normal form. The symbol ‘3’ means ‘there exists’ and R stands for relationship in this equation. All logical statements can be translated into this form using a purely mechanical process. There is even a website that will do this for you. It’s useful but I don't recommend it as entertainment! In the example above, something exists in relation to the existence of something else: one person who loves another. Give me a name and I can look up the person they love. This is simple. A computer can easily solve such problems. Indeed there are hundreds of websites doing this every day. Once you've solved one problem of this type, you have solved them all. We can rearrange Diophantine equations into many different prenex forms. The simplest form might be, ‘there exists an x which solves the following equation, x equals three? This would be written out as Ix, x=3 and is of the 4 class — ‘there exists. There are slightly more complex classes than our simple 4 relationship: VV ‘for all, there exists for all’ or the class V°V ‘for all, for all, there exists, for all: Each of these groups of equation is called a ‘reduction class. One way to think about a reduction class is as a problem in topology, ‘knots, to non-mathematicians. Imagine someone handed you a bunch of tangled cables - the sort of mess you get when they are thrown haphazardly into a drawer. You can tease them apart and rearrange them but you must not cut them or break any connection. Once you have done this you will be left with a series of cables on the desk. They are all separate, looped or in someway knotted together. Each cable has a fundamental topological arrangement: straight cables, granny knots, figure eight, and so on. You have reduced them to their simplest form, their logical classes. The same goes for logical statements. Once you have rearranged logical statements into their simplest form you can lay them out and group them together according to their complexity. Each group makes up a reduction class and you can ask whether that class as a whole is automatically decidable. It isa huge task to untangle and classify mathematical problems, and it took Robinson and her colleagues nearly forty years to succeed. It turns out problems with a form as simple as VAV (for all, there exists, for all) have no general purpose algorithm. Each must be examined individually and solved by something that is not a computer. This is a remarkable result as the logic boundary is set quite low. An 44, (exists, exists), class of problem is automatically solvable by a general HOUSE_OVERSIGHT_015939