Decidability and Undecidability Problems

2. Undecidable Problems

An undecidable language is a type of language that cannot be decided by any algorithm or computer program. In other words, we can say there is no way to construct a program that can determine whether a given input string is a member of the language or not. This is because undecidable languages are inherently complex and may require infinite steps to determine membership.
One classic example of an undecidable language is the Halting problem, which asks whether a given Turing machine will halt onto a  given input. Another example is the Post Correspondence Problem, which asks whether a list of pairs of strings can be arranged in a certain way.
Undecidable languages are an important concept in computer science, as they show that some problems cannot be solved by algorithms or computer programs, no matter how sophisticated they are.

A recursively enumerable language is a formal language that can be generated by an algorithm or a Turing machine. The language may be infinite, and not all strings can be verified to belong in the language.

Example of an Undecidable Problem

Here are some examples of undecidable problems:

• Given a positive integer n, determine whether n is a prime number.
• Given a regular language L and a string s, determine whether s belongs to L.
• Given a context-free grammar G and a string s, determine whether s can be derived from G.
• Given a finite automaton A and a string s, determine whether A accepts s.
• Given two finite sets, A and B, determine whether A is a subset of B.
• Given a graph G and two vertices s and t, determine whether there exists a path from s to t in G.

Difference Between Decidable and Undecidable Problems in TOC

Frequently Asked Questions

What is Decidability and Undecidability?

Decidability refers to problems that can be solved by an algorithm in finite time, while undecidability refers to problems for which no such algorithm exists.

What is the Decidability of a Language?

A language is decidable if a Turing machine exists that always halts and correctly determines whether a given string belongs to the language.

What is the Undecidability of a Language?

A language is undecidable if no Turing machine can always determine, in finite time, whether a given string belongs to the language.

Can undecidable problems be solved by an algorithm?

Undecidable problems cannot be solved by any algorithm because they lack a well-defined solution that can be computed using a finite sequence of steps. These problems are typically too complex or exhibit fundamental limitations in computation.

Why is Decidability important?

Decidability is important because it allows us to determine whether a problem can be solved algorithmically, providing insight into the nature of computation and helping us to identify the limits of what can be computed.

Conclusion

Cheers if you reached here!! 

In this article, we discussed Decidability and Undecidability problems. We saw an example of a decidable problem. 

Recommended Readings:

• Basics of TOC
• Chomsky Hierarchy
• Universal Turing Machine
• Finite State Machine
• Automata Theory
• Introduction to Automata
• Yarn vs NPM
• DFA minimization
• Check If A String Is Palindrome
• Arden's theorem