The Post Correspondence Problem
Neso Academy
14 min, 29 sec
The video is a lecture on the Post Correspondence Problem (PCP), an undecidable problem introduced by Emil Post in 1946.
Summary
- The lecturer introduces the Post Correspondence Problem and its undecidable nature, following the discussion of the halting problem in a previous lecture.
- Examples of PCP are provided, showing how to arrange dominos (tiles with a top and bottom part) to make the strings on top and bottom match.
- The first example is solved by selecting dominos in a sequence that matches the top and bottom strings, demonstrating a solution to the PCP.
- A second example is presented using a different representation (a table format) and is also solved by arranging a sequence of dominos.
- A third example shows a PCP that cannot be solved, illustrating the undecidable nature of the problem and leading to the conclusion that no general algorithm can solve all instances of PCP.
Chapter 1
The introductory part of the lecture explains the concept of undecidable problems and introduces the Post Correspondence Problem (PCP).
- The lecture starts by referencing the halting problem as an example of an undecidable problem.
- The Post Correspondence Problem (PCP) is introduced as another undecidable problem, first presented by Emil Post in 1946.
- The problem involves arranging dominos or tiles in a way that the top and bottom strings formed by the sequence are identical.
Chapter 2
The lecture proceeds with a detailed explanation of how to solve an example of the Post Correspondence Problem.
- A set of four dominos is presented as an example to solve the PCP.
- The lecturer explains that a sequence must be found where the concatenated top strings match the concatenated bottom strings.
- Dominos can be used any number of times to achieve a matching sequence, and the sequence must be finite.
- A solution to the example is demonstrated step by step, starting with a domino that has matching symbols on the top and bottom.
Chapter 3
A second example using a tabular representation of PCP is solved to demonstrate another approach to solving the problem.
- The lecturer presents a different representation of PCP using a table format where dominos correspond to rows in the table.
- The task is to match the top and bottom strings by selecting dominos in a sequence.
- A solution is found by carefully choosing dominos that complement the extra elements on the top or bottom strings.
Chapter 4
An unsolvable PCP example is used to explain why the problem is considered undecidable and why no general algorithm can solve all instances of PCP.
- A third example of PCP is provided where no solution is possible, showcasing the undecidable nature of the problem.
- Different possibilities are explored to solve the PCP, but each attempt leads to an infinite loop without a matching sequence.
- The example illustrates why a generalized algorithm for solving all PCP instances cannot exist, as some instances may lead to infinite loops without a solution.
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