Difference Between Decidability and Computability

Both computabilities, as well as decidability, tells us about the existence of an algorithm for a particular problem or function, while solving a particular problem it becomes really important for us to figure out if the problem is solvable or not, it is important both in terms of efficiency and execution. 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 but there are subtle differences between them. In this article, we will discuss briefly the differences between computability and decidability.

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.

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 for every input of the problem defined on the domain.

Difference Between Decidability and Computability:

S. No.	Computability	Decidability
1.	Computability talks about if 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.
2.	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.
3.	If the given set is computable it is also decidable.	However decidable sets are not always computable.
4.	Computability is required to decide if the problem is solvable.	Decidability is required to decide if the problem is computable.
5.	Computability is dependent or defined on a particular domain as well as a range.	Decidability is solely dependent or defined on a particular domain.
6.	Computability of a problem/function is a bit critical to apply.	Decidability of a problem/function is much simpler to apply.
7.	Computability is a characteristic concept where we try to find out if we are able to compute every input of a particular problem.	Decidability is a generalized concept where we try to find out if there is the Turing machine that accepts and halts for every input of the problem defined on the domain.
8.	Computability determines the solvability of a problem in finite time.	Decidability determines after certain steps of an algorithm if the answer can be achieved.