Prompt Title: Computability Theory: Computable Programs

The correct answer is (A) It can be solved using a Turing Machine.

Computability theory is concerned with the study of which problems can be solved algorithmically or computationally. A program P is considered computable if it can be solved using a Turing Machine, which is a mathematical model of a hypothetical machine that can simulate any computer algorithm.

Option (B) "It can be described with a mathematical formula" is not necessarily true as there may be programs that are not easily describable with a mathematical formula but can still be solved using a Turing Machine.

Option (C) "It can be solved by a Java program" is not necessarily true as there may be problems that are not solvable by a Java program but can still be solved using a Turing Machine.

A Java program can simulate a Turing Machine, and Java is indeed considered to be a Turing-complete language, meaning that it can solve any problem that can be solved by a Turing Machine. Therefore, any problem that can be solved by a Turing Machine can also be solved by a Java program.

However, the reverse is not true. There are problems that can be solved by a Turing Machine that cannot be solved by a Java program. For example, a Java program cannot access or modify arbitrary memory locations, whereas a Turing Machine can access and modify an arbitrary number of memory locations. This means that there are problems that can be solved by a Turing Machine using unbounded memory, but cannot be solved by a Java program that is limited by the amount of memory allocated to it.

Additionally, Turing Machines can manipulate and process infinite streams of input and output, whereas Java programs can only process finite input and output. This means that there are problems that can be solved by a Turing Machine using infinite input and output, but cannot be solved by a Java program that is limited by the amount of input and output it can handle.

In summary, while Java is Turing-complete and can solve any problem that can be solved by a Turing Machine, there are certain types of problems that can be solved by a Turing Machine using unbounded memory or infinite input/output that cannot be solved by a Java program that is limited by memory and input/output constraints.