Difference Between Decidability and Computability
Last Updated :
23 Jul, 2025
While solving a particular problem it becomes really important for us to figure out if the problem is solvable or not. Determining both computability and decidability is thus very important to figure out if the problem can be solved in a finite amount of time.
Even though both of them sound quite similar there are some differences between them. In this article, we will discuss briefly the differences between computability and decidability.
What is Computability?
The computability of a problem can be defined as whether it can be solved infinite amount of time. It is connected with the idea that if there exists an algorithm to solve the problem.
Hence, we can define computability as the algorithmically computable problem or function. In another way, we can define the computability of a problem or a function by its ability to be calculated by a Turing machine, for the given input if the Turing machine halts and produces the output we can declare it as computable.
Advantages of Computability
- Computability can be used to solve different numbers of complex problems.
- Computability should be used to build a good theoretical base.
- Some problems in Machine Learning can be solved with the Computability.
Disadvantages of Computability
- There is no guarantee that the Computability will solve a problem within the given time.
- In some problems, it will be very difficult to implement Computability.
- Computability can not be implemented to real-world problems.
What is Decidability?
Decidability is a much simpler concept where we try to figure out for a given problem if there exists an algorithm or a Turing machine that will halt within the given domain.
The output to the decidable problem is either YES or NO. Decidability is a generalized concept where we try to find out if there is a Turing machine that accepts and halts every input of the problem defined on the domain.
Advantages of Decidability
- Any answer to a problem can be easily predicted with the help of Decidability.
- If you want to perform program verification, then Decidability will be needed.
- A Decidability Problem can easily be solved without errors.
Disadvantages of Decidability
- You can use Decidability in some problems.
- Decidability follows a complex algorithm that makes it difficult to understand.
- Some Decidability problems can increase the time complexity.
Difference Between Decidability and Computability
Computability | Decidability |
|---|
Computability talks about whether the problem can be calculated algorithmically or by a Turing machine. | Decidability talks about if there exists an algorithm to solve a problem or if the Turing machine halts. |
If a function or a problem is computable it will take input on the tape of the Turing machine and output on the same tape of the Turing machine. | The output to the decidable problem is either YES or NO. |
If the given set is computable it is also decidable. | However decidable sets are not always computable. |
Computability is required to decide if the problem is solvable. | Decidability is required to decide if the problem is computable. |
Computability is dependent or defined on a particular domain as well as a range. | Decidability is solely dependent or defined on a particular domain. |
Computability of a problem/function is a bit critical to apply. | The decidability of a problem/function is much simpler to apply. |
Computability is a characteristic concept where we try to find out if we can compute every input of a particular problem. | Decidability is a generalized concept where we try to find out if there is a Turing machine that accepts and halts every input of the problem defined on the domain. |
Computability determines the solvability of a problem in finite time. | Decidability determines after certain steps of an algorithm if the answer can be achieved. |
Conclusion
In conclusion, Decidability and computability are two theoretical concepts in the field of computer science. Decidability relates to the solvability of a problem, and by that, it means that an algorithm exists that returns a ‘yes’ or ‘no’ for any input given to it. While computability raises the question of whether a problem can be solved with any algorithm in a finite period.