Algorithmic Problems in Algebraic Structures Undecidability Paul Bell Supervisor: Dr. Igor Potapov Department of Computer Science 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. For example consider the following fascinating pictures by M.C.Escher: In fact such objects can appear in algorithmics. An algorithm is really 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: This is an RSA challenge number (RSA-1024). If you can find the factors of this number, you can claim a legitimate prize : $100,000! This shows how confident people are that this problem is difficult. In fact it's relatively easy to write a program to find the factors but 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 and many researchers work towards proving decidability status for them. Doing so allows us to explore 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! We showed that in general, there cannot exist an algorithm to solve this 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. For example, in the scalar reachability problem (below), it is easily shown to be decidable in dimension 1 and we have shown it to be undecidable in dimension 4, but what about dimensions 2 and 3? There exist many such bounds between problems and it is interesting to try and find the reasons they become undecidable by making small changes to the system in question. Post's Correspondence Problem This is a famous problem shown to be undecidable by Emil Post in It can be easily stated as a sort of “game” in the following way: Given a finite set of types of such “tiles”, is it possible to put a sequence of them next to each other (by choosing the same tiles as many times as needed), in such a way that the word on the top and the bottom is equal? Does any such solution exist? With just seven tiles and a binary alphabet, it was shown that the problem is undecidable by Matiyasevich and Senizergues. It is however decidable if we use only two tiles. This 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 2 dimensional iterative function systems can simulate a Turing machine! Therefore the reach- ability 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. The “zero in the upper right corner problem” for a set of matrices asks if there exists any product of matrices that gives a matrix with a zero in the top right element. Vector reachability asks if a vector can be mapped to another vector 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 the unique mapping from words to matrices shown to the right. Scalar Matrix