CODEWITHSUNDEEP
algorithmcomplexity-theorycomputation-theoryturing-machinesformal-languages
HEMANTH M
HEMANTH M

Reputation: 1

Understanding Unary PCP Reduction to a Matching Problem (UPCP)

this the question

I'm trying to understand the relationship between the Post Correspondence Problem (PCP) and a related problem called Unary PCP (UPCP). In UPCP, we are given dominoes of the form (d_i = [1a_i1b_i]) for positive integer sequences (a_i) and (b_i), and the goal is to determine if there exists a match between the upper and lower tiles.

I have a few questions regarding this setup:

  1. If there exists an (i) such that (a_i = b_i), does this guarantee that the given instance (\langle P \rangle) of PCP is in UPCP? Intuitively, it seems like this should be the case, but I'm not sure how to formally prove it.

  2. I understand that PCP can be reduced to UPCP ((PCP \leq_m UPCP)) by setting (a_i = b_i) for all (i). However, I'm struggling to find a reduction in the opposite direction ((UPCP \leq_m PCP)). Can someone provide some insights or hints on how to approach this reduction?

  3. There is a condition where if there exist indices (i, j) such that (a_i < a_j) and (b_j < b_i), then the instance (\langle P \rangle) is in UPCP. However, I'm unsure how to prove this condition. Does anyone have a proof or a counterexample for this?

  4. Lastly, I'd like to confirm if my understanding is correct: UPCP is decidable because we can check if there exists an (i) such that (a_i = b_i) in a finite number of steps. Is this reasoning correct?

I appreciate any insights or explanations that can help clarify these concepts. Thank you!

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