Post correspondence problem

Key takeaways

Post correspondence problem belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Post correspondence problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Post correspondence problem from memory before moving on to harder problems.

Reference excerpt

The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the halting problem and the Entscheidungsproblem it is often used in proofs of undecidability.

Definition of the problem Let A {\displaystyle A} be an alphabet with at least two symbols. The input of the problem consists of two finite lists α 1 , … , α N {\displaystyle \alpha _{1},\ldots ,\alpha _{N}} and β 1 , … , β N {\displaystyle \beta _{1},\ldots ,\beta _{N}} of words over A {\displaystyle A} . A solution to this problem is a sequence of indices ( i k ) 1 ≤ k ≤ K {\displaystyle (i_{k})_{1\leq k\leq K}} with K ≥ 1 {\displaystyle K\geq 1} and 1 ≤ i k ≤ N {\displaystyle 1\leq i_{k}\leq N} for all k {\displaystyle k} , such that

α i 1 … α i K = β i 1 … β i K . {\displaystyle \alpha _{i_{1}}\ldots \alpha _{i_{K}}=\beta _{i_{1}}\ldots \beta _{i_{K}}.}

The decision problem then is to decide whether such a solution exists or not.

Alternative definition

g : ( i 1 , … , i K ) ↦ α i 1 … α i K {\displaystyle g:(i_{1},\ldots ,i_{K})\mapsto \alpha _{i_{1}}\ldots \alpha _{i_{K}}}

h : ( i 1 , … , i K ) ↦ β i 1 … β i K . {\displaystyle h:(i_{1},\ldots ,i_{K})\mapsto \beta _{i_{1}}\ldots \beta _{i_{K}}.}

This gives rise to an equivalent alternative definition often found in the literature, according to which any two homomorphisms g , h {\displaystyle g,h} with a common domain and a common codomain form an instance of the Post correspondence problem, which now asks whether there exists a nonempty word w {\displaystyle w} in the domain such that

g ( w ) = h ( w ) {\displaystyle g(w)=h(w)} . Another definition describes this problem easily as a type of puzzle. We begin with a collection of dominos, each containing two strings, one on each side. An individual domino looks like

[ a a b ] {\displaystyle {\begin{bmatrix}a\\ab\end{bmatrix}}}

and a collection of dominos looks like

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Post correspondence problem

Start with the simplest possible case. Write down what Post correspondence problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Post correspondence problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Post correspondence problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Post correspondence problem

In research: Post correspondence problem appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Post correspondence problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Post correspondence problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computability theory, Theory of computation, Undecidable problems, so understanding it makes those chapters shorter.
In everyday life: Look for Post correspondence problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Post correspondence problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Post correspondence problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Post correspondence problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Post correspondence problem in simple terms?

The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the halting problem and the Entscheidungsproblem it is often used in proofs of undecidability.

Why does Post correspondence problem matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Post correspondence problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Post correspondence problem.