Solution

The correct option is 1) Only PCP in X has a solution.

Post correspondence Problem

• The Post Correspondence Problem (PCP) is a mathematical problem that asks whether it is possible to form a string by concatenating elements of two given lists A and B, such that the resulting string is equal to another string in the other list.
• In more detail, given two lists A = {n1, n2, . . . , nk} and B = {x1, x2, . . . , xk} of strings over the alphabet Σ, the task is to find a sequence of integers i1, i2, . . . , im (where m ≥ 1 and repetition of indices is allowed) such that ni1 ni2 . . . nim = xi1 xi2 . . . xim.
• 1) For example, if Σ = {0, 1}, A = {1, 10111, 10}, and B = {111, 10, 0}, there exists a solution, m = 4, i1 = 2, i2 = 1, i3 = 1, i4 = 3, such that ni1 ni2 ni3 ni4 = 1011110.
• However, not all instances of PCP have solutions.
• 2) For example, if Σ = {0, 1}, A = {10, 011, 101}, and B = {101, 11, 011}, there is no solution because it's not possible to form the strings from the elements of the lists.
• In general, PCP is a well-known problem in computational theory and is known to be undecidable, meaning that there is no algorithm that can determine whether a solution exists for all instances of the problem in a finite amount of time.