While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved.
Modern theoretical computer science research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed, as shown below:
An algorithm is an effective method expressed as a finite list[3] of well-defined instructions[4] for calculating a function.[5] Starting from an initial state and initial input (perhaps empty),[6] the instructions describe a computation that, when executed, proceeds through a finite[7] number of well-defined successive states, eventually producing "output"[8] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.[9]
Automata Theory is the study of self-operating virtual machines to help in the logical understanding of input and output process, without or with intermediate stage(s) of computation (or any function/process).
Computational complexity theory is a branch of the theory of computation that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm.
A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity measures are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do.
Computational geometry is a branch of computer science devoted to the study of algorithms that can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry.
The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (CAD/CAM), but many problems in computational geometry are classical in nature, and may come from mathematical visualization.
Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples including the samples that have never been previously seen by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples.
Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in practice by any adversary. It is theoretically possible to break such a system, but it is infeasible to do so by any known practical means. These schemes are therefore termed computationally secure; theoretical advances, e.g., improvements in integer factorization algorithms, and faster computing technology require these solutions to be continually adapted. There exist information-theoretically secure schemes that provably cannot be broken even with unlimited computing power—an example is the one-time pad—but these schemes are more difficult to implement than the best theoretically breakable but computationally secure mechanisms.
Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks. For example, databases use B-tree indexes for small percentages of data retrieval and compilers and databases use dynamic hash tables as look up tables.
Data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Usually, efficient data structures are key to designing efficient algorithms. Some formal design methods and programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design. Storing and retrieving can be carried out on data stored in both main memory and in secondary memory.
Distributed computing studies distributed systems. A distributed system is a software system in which components located on networked computers communicate and coordinate their actions by passing messages.[15] The components interact with each other in order to achieve a common goal. Three significant characteristics of distributed systems are: concurrency of components, lack of a global clock, and independent failure of components.[15] Examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications, and blockchain networks like Bitcoin.
A computer program that runs in a distributed system is called a distributed program, and distributed programming is the process of writing such programs.[16] There are many alternatives for the message passing mechanism, including RPC-like connectors and message queues. An important goal and challenge of distributed systems is location transparency.
Information-based complexity (IBC) studies optimal algorithms and computational complexity for continuous problems. IBC has studied continuous problems as path integration, partial differential equations, systems of ordinary differential equations, nonlinear equations, integral equations, fixed points, and very-high-dimensional integration.
Formal methods are a particular kind of mathematics based techniques for the specification, development and verification of software and hardware systems.[17] The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design.[18]
Machine learning is a scientific discipline that deals with the construction and study of algorithms that can learn from data.[27] Such algorithms operate by building a model based on inputs[28]:2 and using that to make predictions or decisions, rather than following only explicitly programmed instructions.
Machine learning can be considered a subfield of computer science and statistics. It has strong ties to artificial intelligence and optimization, which deliver methods, theory and application domains to the field. Machine learning is employed in a range of computing tasks where designing and programming explicit, rule-based algorithms is infeasible. Example applications include spam filtering, optical character recognition (OCR),[29]search engines and computer vision. Machine learning is sometimes conflated with data mining,[30] although that focuses more on exploratory data analysis.[31] Machine learning and pattern recognition "can be viewed as two facets of the same field."[28]:vii
Dually, one can view processes occurring in nature as information processing. Such processes include self-assembly, developmental processes, gene regulation networks, protein–protein interaction networks, biological transport (active transport, passive transport) networks, and gene assembly in unicellular organisms. Efforts to understand biological systems also include engineering of semi-synthetic organisms, and understanding the universe itself from the point of view of information processing. Indeed, the idea was even advanced that information is more fundamental than matter or energy. The Zuse-Fredkin thesis, dating back to the 1960s, states that the entire universe is a huge cellular automaton which continuously updates its rules.[34][35] Recently it has been suggested that the whole universe is a quantum computer that computes its own behaviour.[36]
The universe/nature as computational mechanism is addressed by,[37] exploring nature with help the ideas of computability, and [38] studying natural processes as computations (information processing).
Parallel computing is a form of computation in which many calculations are carried out simultaneously,[40] operating on the principle that large problems can often be divided into smaller ones, which are then solved "in parallel". There are several different forms of parallel computing: bit-level, instruction level, data, and task parallelism. Parallelism has been employed for many years, mainly in high-performance computing, but interest in it has grown lately due to the physical constraints preventing frequency scaling.[41] As power consumption (and consequently heat generation) by computers has become a concern in recent years,[42] parallel computing has become the dominant paradigm in computer architecture, mainly in the form of multi-core processors.[43]
Parallel computer programs are more difficult to write than sequential ones,[44] because concurrency introduces several new classes of potential software bugs, of which race conditions are the most common. Communication and synchronization between the different subtasks are typically some of the greatest obstacles to getting good parallel program performance.
The maximum possible speed-up of a single program as a result of parallelization is known as Amdahl's law.
Programming language theory is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features. It falls within the discipline of theoretical computer science, both depending on and affecting mathematics, software engineering, and linguistics. It is an active research area, with numerous dedicated academic journals.
In programming language theory, semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages. It does so by evaluating the meaning of syntactically legal strings defined by a specific programming language, showing the computation involved. In such a case that the evaluation would be of syntactically illegal strings, the result would be non-computation. Semantics describes the processes a computer follows when executing a program in that specific language. This can be shown by describing the relationship between the input and output of a program, or an explanation of how the program will execute on a certain platform, hence creating a model of computation.
Experiments have been carried out in which quantum computational operations were executed on a very small number of qubits.[50] Both practical and theoretical research continues, and many national governments and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis.[51]
Computer algebra, also called symbolic computation or algebraic computation is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although, properly speaking, computer algebra should be a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes exact computation with expressions containing variables that have not any given value and are thus manipulated as symbols (therefore the name of symbolic computation).
Software applications that perform symbolic calculations are called computer algebra systems, with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the implementation), a dedicated memory manager, a user interface for the input/output of mathematical expressions, a large set of routines to perform usual operations, like simplification of expressions, differentiation using chain rule, polynomial factorization, indefinite integration, etc.
↑ Cook, Stephen A. (1971). "The complexity of theorem-proving procedures". Proceedings of the third annual ACM symposium on Theory of computing - STOC '71. pp.151–158. doi:10.1145/800157.805047. ISBN978-1-4503-7464-4.
↑ "Any classical mathematical algorithm, for example, can be described in a finite number of English words". Rogers, Hartley Jr. (1967). Theory of Recursive Functions and Effective Computability. McGraw-Hill. Page 2.
↑ Well defined with respect to the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1967, p.2).
↑ "an algorithm is a procedure for computing a function (with respect to some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1967, p.1).
↑ "An algorithm has zero or more inputs, i.e., quantities which are given to it initially before the algorithm begins" (Knuth 1973:5).
↑ "A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method'" (Knuth 1973:5).
↑ "An algorithm has one or more outputs, i.e. quantities which have a specified relation to the inputs" (Knuth 1973:5).
↑ Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without the use of continuous methods or analog devices . . . carried forward deterministically, without resort to random methods or devices, e.g., dice" (Rogers 1967, p.2).
↑ Rivest, Ronald L. (1990). "Cryptology". In J. Van Leeuwen (ed.). Handbook of Theoretical Computer Science. Vol.1. Elsevier.
↑ Bellare, Mihir; Rogaway, Phillip (21 September 2005). "Introduction". Introduction to Modern Cryptography. p.10.
1 2 Coulouris, George; Jean Dollimore; Tim Kindberg; Gordon Blair (2011). Distributed Systems: Concepts and Design (5thed.). Boston: Addison-Wesley. ISBN978-0-132-14301-1.
↑ Ghosh, Sukumar (2007). Distributed Systems – An Algorithmic Approach. Chapman & Hall/CRC. p.10. ISBN978-1-58488-564-1.
↑ Rando Allikmets, Wyeth W. Wasserman, Amy Hutchinson, Philip Smallwood, Jeremy Nathans, Peter K. Rogan, Thomas D. Schneider, Michael Dean (1998) Organization of the ABCR gene: analysis of promoter and splice junction sequences, Gene215:1, 111–122
↑ Burnham, K. P. and Anderson D. R. (2002) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Second Edition (Springer Science, New York) ISBN978-0-387-95364-9.
↑ Wernick, Yang, Brankov, Yourganov and Strother, Machine Learning in Medical Imaging, IEEE Signal Processing Magazine, vol. 27, no. 4, July 2010, pp. 25–38
↑ Mannila, Heikki (1996). Data mining: machine learning, statistics, and databases. Int'l Conf. Scientific and Statistical Database Management. IEEE Computer Society.
↑ Friedman, Jerome H. (1998). "Data Mining and Statistics: What's the connection?". Computing Science and Statistics. 29 (1): 3–9.
↑ S.V. Adve et al. (November 2008). "Parallel Computing Research at Illinois: The UPCRC Agenda"Archived 2008-12-09 at the Wayback Machine (PDF). Parallel@Illinois, University of Illinois at Urbana-Champaign. "The main techniques for these performance benefits– increased clock frequency and smarter but increasingly complex architectures– are now hitting the so-called power wall. The computer industry has accepted that future performance increases must largely come from increasing the number of processors (or cores) on a die, rather than making a single core go faster."
↑ Asanovic et al. Old [conventional wisdom]: Power is free, but transistors are expensive. New [conventional wisdom] is [that] power is expensive, but transistors are "free".
↑ Asanovic, Krste et al. (December 18, 2006). "The Landscape of Parallel Computing Research: A View from Berkeley" (PDF). University of California, Berkeley. Technical Report No. UCB/EECS-2006-183. "Old [conventional wisdom]: Increasing clock frequency is the primary method of improving processor performance. New [conventional wisdom]: Increasing parallelism is the primary method of improving processor performance... Even representatives from Intel, a company generally associated with the 'higher clock-speed is better' position, warned that traditional approaches to maximizing performance through maximizing clock speed have been pushed to their limit."
↑ Manin, Yu. I. (1980). Vychislimoe i nevychislimoe[Computable and Noncomputable] (in Russian). Sov.Radio. pp.13–15. Archived from the original on 10 May 2013. Retrieved 4 March 2013.
↑ Finkelstein, David (1968). "Space-Time Structure in High Energy Interactions". In Gudehus, T.; Kaiser, G. (eds.). Fundamental Interactions at High Energy. New York: Gordon & Breach.
Computer science is the study of computation, information, and automation. Computer science spans theoretical disciplines to applied disciplines.
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem.
Discrete mathematics is the study of mathematical structures that can be considered "discrete" rather than "continuous". Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics".
A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of both particles and waves, and quantum computing leverages this behavior using specialized hardware. Classical physics cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations exponentially faster than any modern "classical" computer. Theoretically a large-scale quantum computer could break some widely used encryption schemes and aid physicists in performing physical simulations; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications.
Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. One well known subject classification system for computer science is the ACM Computing Classification System devised by the Association for Computing Machinery.
In computer science, computational learning theory is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms.
The Gödel Prize is an annual prize for outstanding papers in the area of theoretical computer science, given jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery Special Interest Group on Algorithms and Computational Theory. The award is named in honor of Kurt Gödel. Gödel's connection to theoretical computer science is that he was the first to mention the "P versus NP" question, in a 1956 letter to John von Neumann in which Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time.
Shlomi Dolev is a Rita Altura Trust Chair Professor in Computer Science at Ben-Gurion University of the Negev (BGU) and the head of the BGU Negev Hi-Tech Faculty Startup Accelerator.
Leslie Gabriel Valiant is a British American computer scientist and computational theorist. He was born to a chemical engineer father and a translator mother. He is currently the T. Jefferson Coolidge Professor of Computer Science and Applied Mathematics at Harvard University. Valiant was awarded the Turing Award in 2010, having been described by the A.C.M. as a heroic figure in theoretical computer science and a role model for his courage and creativity in addressing some of the deepest unsolved problems in science; in particular for his "striking combination of depth and breadth".
Yuri Gurevich, Professor Emeritus at the University of Michigan, is an American computer scientist and mathematician and the inventor of abstract state machines.
Michael John Fischer is an American computer scientist who works in the fields of distributed computing, parallel computing, cryptography, algorithms and data structures, and computational complexity.
Informatics is the study of computational systems. According to the ACM Europe Council and Informatics Europe, informatics is synonymous with computer science and computing as a profession, in which the central notion is transformation of information. In some cases, the term "informatics" may also be used with different meanings, e.g. in the context of social computing, or in context of library science.
Information Processing Letters is a peer-reviewed scientific journal in the field of computer science, published by Elsevier. The aim of the journal is to enable fast dissemination of results in the field of information processing in the form of short papers. Submissions are limited to nine double-spaced pages.
Natural computing, also called natural computation, is a terminology introduced to encompass three classes of methods: 1) those that take inspiration from nature for the development of novel problem-solving techniques; 2) those that are based on the use of computers to synthesize natural phenomena; and 3) those that employ natural materials to compute. The main fields of research that compose these three branches are artificial neural networks, evolutionary algorithms, swarm intelligence, artificial immune systems, fractal geometry, artificial life, DNA computing, and quantum computing, among others.
Dana Angluin is a professor emeritus of computer science at Yale University. She is known for foundational work in computational learning theory and distributed computing.
The TUM School of Computation, Information and Technology (CIT) is a school of the Technical University of Munich, established in 2022 by the merger of three former departments. As of 2022, it is structured into the Department of Mathematics, the Department of Computer Engineering, the Department of Computer Science, and the Department of Electrical Engineering.
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