CS355 – The Mathematics and Theory of Computer Science Reductions

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth Undecidable Problem Let us look at an undecidable problem concerning simple manipulation of strings. It is called the Post correspondence problem (PCP). Can describe the problem as a type of puzzle.

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth PCP We begin with a collection of dominos, each containing two strings, one on each side: for e.g. And a collection of dominos looks like:

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth PCP Goal: Make a list of these dominos (repeats allowed) so that the string we get on the top of the dominos is the same as the string we get on the bottom of them. This list is called a match. E.g.:

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth PCP Reading off the top and bottom strings gives us: –abcaaabc For some collection of dominos there may not be a match! E.g.: Top strings longer than bottom / No c in bottom

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth PCP THE PCP is to determine whether a collection of dominos has a match. Problem is unsolvable by algorithm. Let us state the problem precisely and then express it as a language.

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth PCP An instance of the PCP is a collection P of dominos: and a match is a sequence i 1, i 2, …,i l, where: t i1, t i2, …,t il = b i1, b i2, …,b il. The problem is to determine whether P has a match. Let: PCP = { | P is an instance of the PCP with a match}

Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth PCP PCP is an undecidable problem and there is a proof in Sipser if anyone is interested in looking it up. Let us look at a simpler undecidable problem.