Problem Detail: So, it’s known that PCP is undecidable even when we fix the number of tiles to $n geq 7$. I’m wondering, can anything similar be said for when there is a fixed word length? To be precise, here’s the problem:
Given fixed $m$ and $n$, with $n geq 7$, and words $u_1, ldots u_n$ and $v_1 ldots v_n$ such that $|u_i| leq m$ and $|v_i| leq m$, is there an index sequence $i_1, ldots i_k$ such that $u_{i_1} cdots u_{i_k} = v_{i_1} cdots v_{i_k}$.
For what values of $m$, if any, is this known to be undecidable? Note that this is similar to this question, but none of the 8 linked papers seemed by their titles to answer my question, and I haven’t fully read all 8 of them yet.
Asked By : jmite
Answered By : Raphael
For all $m geq 3$, the problem is undecidable. Proof by reduction from the word problem of unrestricted grammars:
- Take an arbitrary formal grammar. W.l.o.g. all left and right sides of rules have length at most $3$. This can be seen by translating any grammar into an equivalent TM and then converting back.
- Map the resulting grammar to PCP instances; no tile is longer than the longest left or right side of a rule. That is, with step 1, all tiles have length $geq 3$.
Best Answer from StackOverflow
Question Source : http://cs.stackexchange.com/questions/28787 Ask a Question Download Related Notes/Documents
