# What is 'K' in the Bounded Post Correspondence Problem?

I'm only just starting to learn about the PCP, and I'm especially interested in the Bounded version. From Wikipedia:

One of the most important variants of PCP is the bounded Post correspondence problem, which asks if we can find a match using no more than k tiles, including repeated tiles.

I am not sure I understand what Wiki means by a 'tile' in this context.

I do understand that a typical PCP problem takes as input two equal-length lists of nonempty strings, each of which contains some number of words. An example is this:

M = (abb, aa, aaa) and N = (bba, aaa, aa)

In terms of the Bounded PCP, what would the integer K be here? Is it the length an individual word can be (which in this case is maxed at 3)? Is it the length of a single list?

Hope this is clear. I appreciate any help profusely in advance.

• You've flipped around the problem. You've given as input a pair of lists, but it takes as input a list of pairs. – jmite Nov 14 '16 at 4:47

Suppose your list is M=(bba,ab,bba,a) N=(bb,aa,bb,baa) Then its equivalent tile representation is as follows Now your pcp problem is equivalent to finding a sequence of tiles(repetition allowed) such that concatenating all upper halves and concatenating all lower halves gives you same string. K here is nothing but number of tiles that can be used

Edit : As Yuval Filmus subtly mentions in his comment, there are many other interpretations of K. In an another formulation K can also be the maximum length of concatenated string, in which case the bounded PCP problem is Pspace complete. But usually many books refer to K as being number of tiles that are allowed.Please refer to Sipser book for more.

• There might be several different definitions of bounded, see for example this answer. In particular, the Wikipedia definition seems to be different. – Yuval Filmus Nov 14 '16 at 8:12
• I would say something like "finding a multiset of tiles" instead, to stress you can repeat the same tile. If you can't then the problem becomes decidable. – chi Nov 14 '16 at 8:50
• sequence would be much better representation to insist on the order . Anyways thanks for pointing out :) – MysticForce Nov 14 '16 at 9:29
• Thank you for the answer. For this version of the Bounded PCP, the question is whether we can find a matching solution using no more than k tiles, as opposed to the general PCP which allows an infinite supply of such tiles, right? – Fiery Phoenix Nov 14 '16 at 15:19
• @Fiery Phoenix Yes thats true – MysticForce Nov 15 '16 at 11:26