CS355 – The Mathematics and Theory of Computer Science

Reductions

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.

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:

ab

a

c

abc

a

ca

ab

a

ca

b

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.:

c

abc

ab

a

a

ca

ca

b

ab

a

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

ba

acc

a

ca

ab

abc

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.

PCP

• An instance of the PCP is a collection P of dominos:

and a match is a sequence i1, i2, …,il, where: ti1, ti2, …,til = bi1, bi2, …,bil. The problem is to determine whether P has a match. Let:

PCP = {<P> | P is an instance of the PCP with a match}

,,...,,2

2

1

1

k

k

b

t

b

t

b

tP

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.