
- Automata Theory Tutorial
- Automata Theory - Home
- Automata Theory Introduction
- Deterministic Finite Automaton
- Non-deterministic Finite Automaton
- NDFA to DFA Conversion
- DFA Minimization
- Moore & Mealy Machines
- Classification of Grammars
- Introduction to Grammars
- Language Generated by Grammars
- Chomsky Grammar Classification
- Regular Grammar
- Regular Expressions
- Regular Sets
- Arden's Theorem
- Constructing FA from RE
- Pumping Lemma for Regular Grammar
- DFA Complement
- Context-Free Grammars
- Context-Free Grammar Introduction
- Ambiguity in Grammar
- CFL Closure Properties
- CFG Simplification
- Chomsky Normal Form
- Greibach Normal Form
- Pumping Lemma for CFG
- Pushdown Automata
- Pushdown Automata Introduction
- Pushdown Automata Acceptance
- PDA & Context Free Grammar
- PDA & Parsing
- Turing Machine
- Turing Machine Introduction
- Accepted & Decided Language
- Multi-tape Turing Machine
- Multi-Track Turing Machine
- Non-Deterministic Turing Machine
- Semi-Infinite Tape Turing Machine
- Linear Bounded Automata
- Decidability
- Language Decidability
- Undecidable Language
- Turing Machine Halting Problem
- Rice Theorem
- Post Correspondence Problem
- Automata Theory Useful Resources
- Automata Theory - Quick Guide
- Automata Theory - Useful Resources
- Automata Theory - Discussion
Language Decidability
A language is called Decidable or Recursive if there is a Turing machine which accepts and halts on every input string w. Every decidable language is Turing-Acceptable.

A decision problem P is decidable if the language L of all yes instances to P is decidable.
For a decidable language, for each input string, the TM halts either at the accept or the reject state as depicted in the following diagram −

Example 1
Find out whether the following problem is decidable or not −
Is a number ‘m’ prime?
Solution
Prime numbers = {2, 3, 5, 7, 11, 13, …………..}
Divide the number ‘m’ by all the numbers between ‘2’ and ‘√m’ starting from ‘2’.
If any of these numbers produce a remainder zero, then it goes to the “Rejected state”, otherwise it goes to the “Accepted state”. So, here the answer could be made by ‘Yes’ or ‘No’.
Hence, it is a decidable problem.
Example 2
Given a regular language L and string w, how can we check if w ∈ L?
Solution
Take the DFA that accepts L and check if w is accepted

Some more decidable problems are −
- Does DFA accept the empty language?
- Is L1 ∩ L2 = ∅ for regular sets?
Note −
If a language L is decidable, then its complement L' is also decidable
If a language is decidable, then there is an enumerator for it.