1. Algorithmic Problems in Algebraic Structures Undecidability Paul Bell Supervisor: Dr. Igor Potapov Department of Computer Science www.csc.liv.ac.uk/~pbell Impossible Objects There are some objects which look reasonable at a first glance but in fact cannot ever exist. These objects are present in both mathematics and the real world: Similar objects can appear in algorithmics. An algorithm is just a program to perform a particular procedure. Some algorithms are more complex than others and need more space and time to perform a task. For example to sort a list of numbers (5,3,2,8,13,4,...) is quite easy but to find factors of a large number can be very difficult. Here is a challenge: Try and write a program which can factorize (i.e. find prime numbers p 1,p 2,...,p N, whose product is..) the number: 135066410865995223349603216278805969938881475605667027524485143851526510604859533833940287150571909441798 207282164471551373680419703964191743046496589274256239341020864383202110372958725762358509643110564073501 508187510676594629205563685529475213500852879416377328533906109750544334999811150056977236890927563 This is an RSA challenge number (RSA-1024). If you can find the factors of this number, you can claim a real prize : $100,000! This shows how confident people are that this problem is difficult. Naïve algorithms may take hundreds of years to return the answer. Undecidable Problems In our work we instead deal with “Undecidable Problems”. These are in some sense impossible objects because it can be shown that there does not exist an algorithm which can ever solve them – even with the fastest computers and given thousands of years! There exist many undecidable problems in different areas. We are exploring the limits of computation in a formal sense. This shows problems which humans and computers cannot ever solve. These problems are not necessarily esoteric mathematical curiosities but can be present in the real world as shown below. Scalar Reachability Problem: Given a set of n matrices of dimension 4, is it possible to multiply them together in some arbitrary finite product and eventually get a “Scalar Matrix” (A matrix which scales all vectors by a constant amount)? By using a new encoding technique, we recently proved this is an undecidable problem! In some senses it is infinitely more difficult than factorization of a number. Current Research Borders of Undecidability We are exploring the border that exists between decidable and undecidable problems. Making small changes to a problem often moves it from decidable to undecidable. It is interesting to try and discover the reason for this change. Post's Correspondence Problem This is a famous problem shown to be undecidable by Emil Post in 1946. It can be easily stated as a sort of “game” in the following way: Given an infinite set of such “tiles”, is it possible to put a finite sequence of them next to each other in such a way that the word on the top and the bottom is equal? Does any such solution exist? This undecidable problem is very useful for showing undecidability in other systems via “reduction”. Iterative Function Systems Fractals generate highly complex images using a simple set of rules. In fact we recently showed that 2D iterative function systems can simulate a Turing machine! Thus the reachability question of a given point is undecidable. It is an open problem whether reachability in one dimensional affine maps of the form f i (x) = ax + b, is undecidable. We have shown undecidability results in several different systems. There exist many open problems however and we are currently working towards new techniques for proving their decidability status. We can graph a summary of our current undecidability results: ● Identity Problem – Given a set of matrices, does some product of them equal the identity matrix? ● Find strict lower and upper bounds for the above problems. Vector reachability asks if one vector can be mapped to another by multiplying it by matrices from a set. We showed this to be undecidable for just two matrices in dimension 15. This is currently the smallest known bound on any undecidable problem in linear algebra. Undecidability Results Open Problems To encode Post’s correspondence problem, we use a unique mapping from words to matrices shown to the right. Scalar Matrix