In mathematics and computer science, an algorithm is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes and deduce valid inferences.
In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:
A 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.
In computability theory, a system of data-manipulation rules is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine. This means that this system is able to recognize or decode other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all programming languages today are Turing-complete.
Punched tape or perforated paper tape is a form of data storage device that consists of a long strip of paper through which small holes are punched. It was developed from and was subsequently used alongside punched cards, the difference being that the tape is continuous.
In 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 ; which will be called "m-configurations". He then described the operation of such machine, as described below, and argued:
It is my contention that these operations include all those which are used in the computation of a number.
In mathematical logic and theoretical computer science, a register machine is a generic class of abstract machines, analogous to a Turing machine and thus Turing complete. Unlike a Turing machine that uses a tape and head, a register machine utilizes multiple uniquely addressed registers to store non-negative integers. There are several sub-classes of register machines, including counter machines, pointer machines, random-access machines (RAM), and Random-Access Stored-Program Machine (RASP), each varying in complexity. These machines, particularly in theoretical studies, help in understanding computational processes. The concept of register machines can also be applied to virtual machines in practical computer science, for educational purposes and reducing dependency on specific hardware architectures.
In computer science, random-access machine is a model of computation that describes an abstract machine in the general class of register machines. The RA-machine is very similar to the counter machine but with the added capability of 'indirect addressing' of its registers. The 'registers' are intuitively equivalent to main memory of a common computer, except for the additional ability of registers to store natural numbers of any size. Like the counter machine, the RA-machine contains the execution instructions in the finite-state portion of the machine.
Computable 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. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions.
A Post–Turing machine is a "program formulation" of a type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of computation. Post's model and Turing's model, though very similar to one another, were developed independently. Turing's paper was received for publication in May 1936, followed by Post's in October. A Post–Turing machine uses a binary alphabet, an infinite sequence of binary storage locations, and a primitive programming language with instructions for bi-directional movement among the storage locations and alteration of their contents one at a time. The names "Post–Turing program" and "Post–Turing machine" were used by Martin Davis in 1973–1974. Later in 1980, Davis used the name "Turing–Post program".
Turing's proof is a proof by Alan Turing, first published in November 1936 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem". It was the second proof of the negation of Hilbert's Entscheidungsproblem; that is, the conjecture that some purely mathematical yes–no questions can never be answered by computation; more technically, that some decision problems are "undecidable" in the sense that there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In Turing's own words: "what I shall prove is quite different from the well-known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K [Principia Mathematica]".
The following are examples to supplement the article Turing machine.
A Turing machine is a hypothetical computing device, first conceived by Alan Turing in 1936. Turing machines manipulate symbols on a potentially infinite strip of tape according to a finite table of rules, and they provide the theoretical underpinnings for the notion of a computer algorithm.
As presented by Hao Wang, his basic machine B is an extremely simple computational model equivalent to the Turing machine. It is "the first formulation of a Turing-machine theory in terms of computer-like models". With only 4 sequential instructions it is very similar to, but even simpler than, the 7 sequential instructions of the Post–Turing machine. In the same paper, Wang introduced a variety of equivalent machines, including what he called the W-machine, which is the B-machine with an "erase" instruction added to the instruction set.
Algorithm characterizations are attempts to formalize the word algorithm. Algorithm does not have a generally accepted formal definition. Researchers are actively working on this problem. This article will present some of the "characterizations" of the notion of "algorithm" in more detail.
In theoretical computer science the random-access stored-program (RASP) machine model is an abstract machine used for the purposes of algorithm development and algorithm complexity theory.
A counter machine or counter automaton is an abstract machine used in a formal logic and theoretical computer science to model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one or more unbounded registers, each of which can hold a single non-negative integer, and a list of arithmetic and control instructions for the machine to follow. The counter machine is typically used in the process of designing parallel algorithms in relation to the mutual exclusion principle. When used in this manner, the counter machine is used to model the discrete time-steps of a computational system in relation to memory accesses. By modeling computations in relation to the memory accesses for each respective computational step, parallel algorithms may be designed in such a matter to avoid interlocking, the simultaneous writing operation by two threads to the same memory address.
There are many variants of the counter machine, among them those of Hermes, Ershov, Péter, Minsky, Lambek, Shepherdson and Sturgis, and Schönhage. These are explained below.
The history of the Church–Turing thesis ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable. It is an important topic in modern mathematical theory and computer science, particularly associated with the work of Alonzo Church and Alan Turing.
In 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. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically definable but not computable.
