In 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.
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:
In programming language theory, lazy evaluation, or call-by-need, is an evaluation strategy which delays the evaluation of an expression until its value is needed and which also avoids repeated evaluations.
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree. The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".
In computer science, algorithmic efficiency is a property of an algorithm which relates to the amount of computational resources used by the algorithm. Algorithmic efficiency can be thought of as analogous to engineering productivity for a repeating or continuous process.
Computability 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.
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
Hashcash is a proof-of-work system used to limit email spam and denial-of-service attacks. Hashcash was proposed in 1997 by Adam Back and described more formally in Back's 2002 paper "Hashcash - A Denial of Service Counter-Measure". In Hashcash the client has to concatenate a random number with a string several times and hash this new string. It then has to do so over and over until a hash beginning with a certain number of zeros is found.
The Penny Black Project is a Microsoft Research project that tries to find effective and practical ways of fighting spam. Because identifying spams consumes a recipient's time, the idea is to make the sender of emails "pay" a certain amount for sending them. The currency or the mode of payment could be CPU cycles, Turing tests or memory cycles. Such a payment would limit spammers' ability to send out large quantities of emails quickly.
In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls to pure functions and returning the cached result when the same inputs occur again. Memoization has also been used in other contexts, such as in simple mutually recursive descent parsing. It is a type of caching, distinct from other forms of caching such as buffering and page replacement. In the context of some logic programming languages, memoization is also known as tabling.
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.
In computer science, a problem is said to have overlapping subproblems if the problem can be broken down into subproblems which are reused several times or a recursive algorithm for the problem solves the same subproblem over and over rather than always generating new subproblems.
In computer science, corecursion is a type of operation that is dual to recursion. Whereas recursion works analytically, starting on data further from a base case and breaking it down into smaller data and repeating until one reaches a base case, corecursion works synthetically, starting from a base case and building it up, iteratively producing data further removed from a base case. Put simply, corecursive algorithms use the data that they themselves produce, bit by bit, as they become available, and needed, to produce further bits of data. A similar but distinct concept is generative recursion, which may lack a definite "direction" inherent in corecursion and recursion.
Proof of work (PoW) is a form of cryptographic proof in which one party proves to others that a certain amount of a specific computational effort has been expended. Verifiers can subsequently confirm this expenditure with minimal effort on their part. The concept was invented by Moni Naor and Cynthia Dwork in 1993 as a way to deter denial-of-service attacks and other service abuses such as spam on a network by requiring some work from a service requester, usually meaning processing time by a computer. The term "proof of work" was first coined and formalized in a 1999 paper by Markus Jakobsson and Ari Juels. The concept was adapted to digital tokens by Hal Finney in 2004 through the idea of "reusable proof of work" using the 160-bit secure hash algorithm 1 (SHA-1).
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.
The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.
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.
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.
Cynthia Dwork is an American computer scientist best known for her contributions to cryptography, distributed computing, and algorithmic fairness. She is one of the inventors of differential privacy and proof-of-work.
In cryptography, a memory-hard function (MHF) is a function that costs a significant amount of memory to efficiently evaluate. It differs from a memory-bound function, which incurs cost by slowing down computation through memory latency. MHFs have found use in key stretching and proof of work as their increased memory requirements significantly reduce the computational efficiency advantage of custom hardware over general-purpose hardware compared to non-MHFs.
