Computability

Computability is the ability to solve a problem using a clear, step-by-step method. It is an important idea in the field of computability theory, which is part of mathematical logic and the theory of computation in computer science. Whether a problem is computable depends on whether there is an algorithm that can solve it.

People often study models like Turing-computable problems, μ-recursive functions, and the lambda calculus. These models all have the same level of power when it comes to solving problems. Some areas look at weaker forms of computability, like those studied in automata theory, while others explore ideas even stronger than what a Turing machine can do, such as those in hypercomputation. Understanding computability helps us know what problems can truly be solved by computers and how they can be solved.

Problems

A big idea in computability is the idea of a computational problem. This is a job we can try to solve with a computer or a set of clear steps.

There are two main types of problems. A decision problem asks us to decide if something is true or false. For example, we might want to know if a number is prime. A function problem asks us to find a result from an input. For example, we might take a string of numbers and reverse them. Other types of problems include search problems and optimization problems. Computability theory looks at which problems we can solve with different ways of computing.

Formal models of computation

Main article: Model of computation

A model of computation is a way to explain how computers and other computing systems work. It helps us learn about the problems that can be solved and how to solve them step by step. One key model is the Turing machine. This model uses a long tape to read, write, and move symbols, showing what any computer can do with enough time and space.

Other models include lambda calculus, which uses special expressions and rules to change them. There is also combinatory logic, which uses different rules but can do similar things. Simpler models, like finite automata, have a set number of states and change based on inputs. Pushdown automata have a stack to help remember things. These models help scientists understand what can and cannot be computed.

Power of automata

With computational models, we can learn about their limits and see what kinds of problems they can solve.

Power of finite-state machines

A regular language is one that a finite-state machine can recognize. But some languages cannot be recognized this way. For example, a finite-state machine cannot recognize all strings with an equal number of 'a's and 'b's because it would need an infinite number of states to keep track.

Power of pushdown automata

A Context-free language can be recognized by a pushdown automaton. This model is more powerful than a finite-state machine. It can handle languages like the one with equal numbers of 'a's and 'b's. But there are still some languages, such as those involving prime numbers, that even pushdown automata cannot recognize.

Power of Turing machines

Turing machines are the most powerful computational models. They can recognize any context-free language and even more complex languages. Turing machines can simulate any other computational model, showing how flexible they are in solving problems.

Concurrency-based models

Many computational models use concurrency, like the parallel random-access machine and the Petri net. These models let many tasks run together, but they can only do the same kinds of calculations as a Turing machine. So, even though they work differently, they do not change what we think is computable.

Stronger models of computation

The Church–Turing thesis says that no computer can do more than a special kind of machine called a Turing machine. But computer scientists have imagined special machines called hypercomputers that go beyond normal computers.

Some ideas include machines that can do endless steps very quickly, called Zeno machines, and machines that can ask special questions to solve problems that normal machines cannot. Even these very strong machines still have their own limits and cannot solve every problem.