Empowered by artificial intelligence technologies, computers today can engage in convincing conversations with people, compose songs, paint paintings, play chess and go, and diagnose diseases, to name just a few examples of their technological prowess.

These successes could be taken to indicate that computation has no limits. To see if that’s the case, it’s important to understand what makes a computer powerful.

There are two aspects to a computer’s power: the number of operations its hardware can execute per second and the efficiency of the algorithms it runs. The hardware speed is limited by the laws of physics. Algorithms – basically sets of instructions – are written by humans and translated into a sequence of operations that computer hardware can execute. Even if a computer’s speed could reach the physical limit, computational hurdles remain due to the limits of algorithms.

These hurdles include problems that are impossible for computers to solve and problems that are theoretically solvable but in practice are beyond the capabilities of even the most powerful versions of today’s computers imaginable. Mathematicians and computer scientists attempt to determine whether a problem is solvable by trying them out on an imaginary machine.

## An imaginary computing machine

The modern notion of an algorithm, known as a Turing machine, was formulated in 1936 by British mathematician Alan Turing. It’s an imaginary device that imitates how arithmetic calculations are carried out with a pencil on paper. The Turing machine is the template all computers today are based on.

To accommodate computations that would need more paper if done manually, the supply of imaginary paper in a Turing machine is assumed to be unlimited. This is equivalent to an imaginary limitless ribbon, or “tape,” of squares, each of which is either blank or contains one symbol.

The machine is controlled by a finite set of rules and starts on an initial sequence of symbols on the tape. The operations the machine can carry out are moving to a neighboring square, erasing a symbol and writing a symbol on a blank square. The machine computes by carrying out a sequence of these operations. When the machine finishes, or “halts,” the symbols remaining on the tape are the output or result.

Computing is often about decisions with yes or no answers. By analogy, a medical test (type of problem) checks if a patient’s specimen (an instance of the problem) has a certain disease indicator (yes or no answer). The instance, represented in a Turing machine in digital form, is the initial sequence of symbols.

A problem is considered “solvable” if a Turing machine can be designed that halts for every instance whether positive or negative and correctly determines which answer the instance yields.

## Not every problem can be solved

Many problems are solvable using…