Computable and Non-Computable Problems

Problems are classified as computable or non-computable based on whether they can be solved by an algorithm. Computable problems have a clear, step-by-step procedure that always lead to a correct solution while non-computable problems cannot be solved by any algorithm, no matter how much time or resources are given.

Computable Problems

A computable problem is a problem for which an algorithm can always give a correct solution for every input. These problems can be solved using a finite sequence of steps, and the algorithm always stops after producing the correct output. Examples include arithmetic calculations, searching, sorting, and many standard computational tasks.

• A computable problem can be solved by an algorithm.
• The algorithm always produces the correct result for every valid input.
• The algorithm halts (stops) after a finite number of steps.
• Examples of computable problems include addition of numbers, finding the greatest common divisor (GCD), sorting a list, and searching for an element in a dataset.
• The term algorithm can also be represented using a computer program or a Turing Machine

Examples of Computable Problems

These are four simple examples of the computable problem:

• Computing the greatest common divisor of a pair of integers.
• Computing the least common multiple of a pair of integers.
• Finding the shortest path between a pair of nodes in a finite graph.
• Determining whether a propositional formula is a tautology.

Non-Computable Problems

A non-computable problem is one for which no algorithm can always give a correct solution. The most famous example is the Halting Problem, which asks whether a program (or Turing machine) will stop or run forever for a given input. Since some programs may loop infinitely, no general algorithm can solve this problem for all cases.

• A non-computable problem cannot be solved by any algorithm.
• The Halting Problem is the most famous example of a non-computable problem.
• It determines whether a program halts or runs forever for a given input.
• Programs that loop infinitely make this problem undecidable.
• The term “computer program” can be replaced by “Turing machine” or “algorithm”.

Examples of Non-Computable Problems

Halting Problem: The problem of determining whether a program (or Turing machine) will halt or run forever for a given input is non-computable. No algorithm can solve this problem for all possible programs and inputs.

Proving Computability or Non-Computability

We can show that a problem is computable by describing a procedure and proving that the procedure always terminates and always produces the correct answer. It is enough to provide a convincing argument that such a procedure exists. Finding the actual procedure is not necessary (but often helps to make the argument more convincing).

• To show that a problem is not computable, we need to show that no algorithm exists that solves the problem.
• Since there are an infinite number of possible procedures, we cannot just list all possible procedures and show why each one does not solve the problem.
• Instead, we need to construct an argument showing that if there were such an algorithm, it would lead to a contradiction.
• The core of our argument is based on knowing the Halting Problem is non-computable.
• If a solution to some new problem P could be used to solve the Halting Problem, then we know that P is also non-computable.
• That is, no algorithm exists that can solve P since if such an algorithm exists, it could be used to also solve the Halting Problem, which we already know is impossible.
• The proof technique where we show that a solution for some problem P can be used to solve a different problem Q is known as a reduction.
• A problem Q is reducible to a problem P if a solution to P could be used to solve Q.
• This means that problem Q is no harder than problem P since a solution to problem Q leads to a solution to problem P.