Undecidability of the Post Correspondence Problem
Neso Academy
27 min, 46 sec
The video lecture explains and proves the undecidability of the Post Correspondence Problem (PCP) by reducing a known undecidable problem, the acceptance problem of a Turing machine, to PCP.
Summary
- Review of the Post Correspondence Problem (PCP) and its undecidability mentioned in the previous lecture.
- Introduction of proof by reduction to establish the undecidability of PCP.
- Using the acceptance problem of a Turing machine, which is known to be undecidable, to prove the undecidability of PCP.
- Detailed walkthrough of converting a Turing machine's acceptance problem into an equivalent PCP using a step-by-step approach.
- Successful demonstration of a solution to the constructed PCP, thereby proving PCP's undecidability.
Chapter 1
The instructor recaps the undecidability of the Post Correspondence Problem (PCP) and introduces the aim of the lecture, which is to prove PCP's undecidability.
- The previous lecture discussed PCP and mentioned its undecidability.
- The current lecture's objective is to prove the undecidability of PCP.
- The approach involves reducing a known undecidable problem to PCP.
Chapter 2
The instructor explains the proof by reduction technique, where an already proven undecidable problem is reduced to PCP to prove its undecidability.
- Proof by reduction technique is introduced.
- The undecidable acceptance problem of a Turing machine is chosen for reduction to PCP.
- If the Turing machine's problem can be converted to PCP, it proves PCP is undecidable.
Chapter 3
An example Turing machine is presented to illustrate the reduction process.
- The Turing machine with states q0, q1, and q2 is described.
- Inputs 'a' and 'b', tape symbols 'a', 'b', 'x', and the blank symbol are introduced.
- The operation of the Turing machine is explained with examples.
Chapter 4
The lecturer begins demonstrating the conversion of the Turing machine into a PCP form step by step.
- The initial tape configuration of the Turing machine is represented.
- A step-by-step method to create dominoes for PCP is initiated.
- The first few steps involve creating dominoes corresponding to the Turing machine's states and transitions.
Chapter 5
The steps to create dominoes for the Turing machine's transitions are detailed.
- Dominoes for the right transition of the Turing machine are created.
- Left transitions require considering other symbols that might be on the tape.
- Different dominoes are formed for all possible symbols that could appear during a left transition.
Chapter 6
Additional steps for creating dominoes for the Turing machine configuration are explained.
- Dominoes for all possible tape symbols are created.
- Dominoes for tape symbols after reaching the accepting state are formed.
- A special domino for the hash symbol is established.
Chapter 7
The final steps to complete the set of dominoes for the PCP are shown.
- A domino for the blank symbol is added to the set.
- The final domino representing the accepting state of the Turing machine is created.
- The complete set of dominoes representing the Turing machine is now ready.
Chapter 8
The instructor demonstrates how to find a solution to the PCP using the set of dominoes.
- The starting domino for solving PCP is selected.
- Subsequent dominoes are chosen to match the top and bottom sequences.
- The process continues until the top and bottom sequences are completely matched.
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