Computability theoryComputability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory.
Computable functionComputable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines.
Turing machineA Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. The machine operates on an infinite memory tape divided into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states.
Halting problemIn computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is undecidable, meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine.
Ackermann functionIn computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function.
Model of computationIn computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology.
Turing completenessIn computability theory, a system of data-manipulation rules (such as a model of computation, a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing). This means that this system is able to recognize or decide other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set.
Universal Turing machineIn computer science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Common sense might say that a universal machine is impossible, but Turing proves that it is possible. He suggested that we may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q 1: q 2 . ....
Primitive recursive functionIn computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive.
Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
