Search NKS | Online
131 - 140 of 205 for Exists
Indeed, whenever computational irreducibility exists in a system it means that in effect there can be no way to predict how the system will behave except by going through almost as many steps of computation as the evolution of the system itself.
Knowing that universal systems exist already tells one that this must be true at least in some situations.
And if one asks whether any initial conditions exist that lead, for example, to a pattern that does not die out, then this too will in general be undecidable—though in a sense this is just an immediate consequence of the fact that given a particular initial condition one cannot tell whether or not the pattern it produces will ever die out.
In satisfiability one sets up a collection of rows of black, white and gray squares, then asks whether there exists any sequence of black and white squares that satisfies the constraint that on every row the color of at least one square agrees with the color of the corresponding square in the sequence.
In fact, my suspicion is that such a difference will exist in almost any system whose behavior seems to us complex.
And in fact particularly compared to what I do in this book the vast majority of mathematics practiced today still seems to follow remarkably closely the traditions of arithmetic and geometry that already existed even in Babylonian times.
But if we require something that follows too many of the details of us as humans there is already evidence that it does not exist.
It is suspected but not established that there exist at least some problems in NP that cannot be solved in polynomial time, potentially indicating that for an appropriate system it might be impossible to do cryptanalysis in any time polynomial in n .
But in number theory there are questions that were formulated more than 2000 years ago (such as whether any odd perfect numbers exist) that have still not been answered.
The argument is based on showing that an algebraic function always exists for which the coefficients in its power series correspond to any given nested sequence when reduced modulo some p .