Post Machine Explained | Post Correspondence Problem | TOA Concept with Examples

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Welcome to another powerful episode on Theory of Automata (TOA). In today’s video, we’re diving into one of the most important theoretical models in computer science — the Post Machine, also known as the Post Correspondence Problem (PCP).

🔍 What is a Post Machine?
A Post Machine is a string manipulation model developed by Emil Post that performs computations using a finite set of string pairs. It is not a real machine, but a theoretical model just like Turing Machine. It's used to study the undecidability and non-computability of problems.

It helps us answer: Can a certain computation always be decided by an algorithm?

💡 Purpose of the Post Machine
To model algorithms using a simple string mechanism

To demonstrate problems that are undecidable

To understand formal language theory and recursive enumerability

To study Turing-equivalence (Post machine is equivalent in power to Turing Machine)

🔠 Components of a Post Machine
Alphabet (Σ): A finite set of symbols used to build strings

Set of Instruction Pairs: Ordered list of actions to perform

Initial Input String: The starting string in the computation

String Manipulation Rules: Add/remove/check symbols from the left or right

🧠 What is the Post Correspondence Problem (PCP)?
The Post Correspondence Problem is an undecidable problem where you're given two lists of strings:

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List A: a₁, a₂, ..., aₙ
List B: b₁, b₂, ..., bₙ
You must find a sequence of indices i₁, i₂, ..., iₖ such that:

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aᵢ₁aᵢ₂...aᵢₖ = bᵢ₁bᵢ₂...bᵢₖ
✅ If such a sequence exists → the instance has a solution
❌ If not → it's unsolvable

🔧 Why Is PCP Important in TOA?
It’s one of the first problems proven to be undecidable

Shows limits of computation

Helps us understand why not everything can be computed

Used in Turing Machine reductions

Basis for many proofs of undecidability

🧪 Example of PCP:
Given:

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A = [ab, a, bba]
B = [a, bab, ba]
Try sequences like:

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1 → A: ab B: a ❌
1 2 → A: aba B: abab ❌
2 1 3 → A: aabbaa B: bababba ✅
Found a match → Solution exists!

📌 Key Points to Remember
PCP is undecidable

Not every PCP instance has a solution

You can try trial and error, but there's no algorithm to always find a solution

Post Machine ≈ Turing Machine in computational power

🏁 Applications and Relevance
Foundations of computability theory

Proof tool in formal language & automata theory

Inspiration for problem reduction techniques

Related to context-free and recursively enumerable languages

🧑‍🏫 What You’ll Learn in This Video:
✅ What is a Post Machine?
✅ What is the Post Correspondence Problem (PCP)?
✅ Why PCP is undecidable
✅ Real solved examples of PCP
✅ How PCP is used in TOA and Turing Machine proofs

🎓 Ideal For:
🎯 BSCS Students
🎯 Punjab University Exams
🎯 CS401 VU Students
🎯 GATE / NET / Competitive Exams
🎯 Anyone learning Computability & Automata Theory

📖 Related Topics on My Channel:
Turing Machine

Undecidability

Recursive Enumerable Languages

CFG vs Unrestricted Grammar

Halting Problem

Pumping Lemma

🔖 Hashtags for Search Reach:
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#PostMachine #PostCorrespondenceProblem #TOA #TheoryOfAutomata #Undecidability #Computability #EmilPost #TuringMachine #FormalLanguages #CS401 #UniversityLecture #AutomataTheory #StringManipulation #PCPExamples #TOATutorial