The Post Correspondence Problem

Summary

In this lecture, we explore the Post Correspondence Problem (PCP), an undecidable problem introduced by Emil Post in 1946. PCP involves finding a sequence of dominos, each with a top and bottom string, such that the concatenated strings from the top and bottom are identical. The lecture presents examples demonstrating the process of matching dominos and forming equal strings, illustrating how some instances can be solved while others cannot. Ultimately, the undecidability of PCP is highlighted, emphasizing the absence of a universal algorithm for determining the solvability of all PCP instances.

Takeaways

• 😀 The Post Correspondence Problem (PCP) is an undecidable decision problem introduced by Emil Post in 1946.
• 🤔 PCP involves finding a sequence of dominos that results in identical strings on the top and bottom.
• 🧩 Dominos consist of two parts: a top (numerator) and a bottom (denominator).
• 🔍 The main task in PCP is to arrange dominos so that the concatenated strings from the top and bottom match.
• 🔄 Dominos can be used multiple times in the sequence, allowing for greater flexibility in finding solutions.
• 📊 The lecture provides examples to illustrate how to approach solving a PCP using specific dominos.
• ⚠️ Not all PCP instances can be solved; some lead to infinite loops, showcasing the problem's undecidability.
• 💡 A finite sequence of dominos is required to achieve a matching top and bottom string.
• ❌ The undecidability of PCP means there is no algorithm that can universally determine if a solution exists for every instance.
• 🔗 The proof of the undecidability of PCP will be discussed in future lectures, highlighting its significance in computational theory.

Q & A

What is the Post Correspondence Problem (PCP)?

The Post Correspondence Problem (PCP) is an undecidable decision problem that involves finding a sequence of dominoes (tiles) such that the concatenated symbols on the top and bottom parts of the dominoes are identical.

Who introduced the Post Correspondence Problem?

The PCP was introduced by Emil Post in 1946.

What do the dominoes consist of in the PCP?

Each domino has a top part (numerator) and a bottom part (denominator), which can be represented by symbols.

What is the goal when arranging dominoes in the PCP?

The goal is to find a finite sequence of dominoes such that the concatenated symbols on the top are the same as those on the bottom.

Can the same domino be used multiple times in a solution?

Yes, any domino can be used any number of times in the solution.

Why is the PCP classified as an undecidable problem?

The PCP is classified as undecidable because there is no general algorithm that can determine whether any given PCP instance has a solution, as some problems can lead to infinite loops without a resolution.

What is an example of how to solve a PCP?

In one example, a sequence was formed using dominoes that resulted in the top string 'ABCA' and the bottom string 'ABCA', demonstrating a successful solution.

What happens if a PCP leads to an infinite sequence?

If a PCP leads to an infinite sequence, it means that there is no solution for that particular instance, highlighting the complexity of PCP.

How can PCP be represented besides using tiles?

PCP can also be represented using tables where the top and bottom parts of the dominoes are listed alongside their corresponding indices.

What is a key takeaway about the nature of PCP based on the examples given?

A key takeaway is that while some instances of PCP can be solved, others may not have a solution at all, illustrating the undecidable nature of the problem.