A Remark on Code Sets and Context-Free Languages

Given two nonempty sets of nonempty codewords S = {s
1
,...,s
n
} and T = {t
1
,...,t
m
}, it is decidable whether there are i
1
,...,l
k
and j
1
,...,j
k
such that s
i1
...s
ik
= t
j1
...t
jk
, and furthermore, the set of all such solutions is a one counter language.