"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 (e.g., approximate solutions versus precise ones)."
The study of the fundamental principles of computing, including computation models, formal languages, automata theory, and complexity theory.
Automata Theory: Theory of finite state machines, including deterministic and non-deterministic automata, regular expressions, and their relationship to formal languages.
Turing Machines: A theoretical model of computation that can simulate any computer algorithm, including the concept of decidability and undecidability.
Formal Languages: Study of formal grammar, syntax, and semantics of computer languages, including Chomsky hierarchy, context-free languages, and pushdown automata.
Computability Theory: Investigating the ability of a problem to be solved algorithmically, using concepts such as halting problem, Church-Turing thesis, and Rice's theorem.
Complexity Theory: Classifying problems according to their computational complexity, including big-O notation, P and NP complexity classes, and NP-complete problems.
Automata Theory: It involves exploring mathematical models for computing machines, known as automata, and their properties.
Complexity Theory: It encompasses the study of the computational complexity of problems and the resources, such as time and space, required by different algorithms to solve them.
Computability Theory: It involves the investigation of the limitations of computation, specifically identifying problems that are not solvable by any algorithm or computing machine.
Formal Language Theory: It examines the formalization of natural languages and their grammars, including the generation and parsing of sentences using formal rules.
Algorithm Design and Analysis: It involves the development and analysis of algorithms for solving computationally difficult problems, focusing on issues such as optimization, performance, and correctness.
Information Theory: It studies the quantification, transmission, and processing of information, including topics such as data compression, error correction, and encryption.
Cryptography: It deals with the design of algorithms and protocols for secure communication, including aspects such as confidentiality, integrity, and authenticity.
Distributed Computing: It focuses on the design and analysis of algorithms for coordinating and managing the behavior of multiple computers or devices, particularly in situations where resources are distributed or shared.
Computational Geometry: It examines algorithms and data structures for efficiently solving geometric problems, such as finding intersections, computing distances, and constructing shapes.
Quantum Computing: It involves developing algorithms and computing models that utilize the principles of quantum mechanics, potentially offering significant speedups for certain types of computations.
"The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory."
"What are the fundamental capabilities and limitations of computers?"
"In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation."
"The most commonly examined model is the Turing machine."
"Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to prove results, and because it represents what many consider the most powerful possible 'reasonable' model of computation."
"[T]he Turing machine represents what many consider the most powerful possible 'reasonable' model of computation (see Church–Turing thesis)."
"It might seem that the potentially infinite memory capacity is an unrealizable attribute..."
"...but any decidable problem solved by a Turing machine will always require only a finite amount of memory."
"So in principle, any problem that can be solved (decided) by a Turing machine can be solved by a computer that has a finite amount of memory."
"[...] how efficiently [problems] can be solved."
"[...] what problems can be solved on a model of computation..."
"In theoretical computer science and mathematics..."
"[...] approximate solutions versus precise ones."
"[...] what problems can be solved on a model of computation..."
"[The Turing machine] can be analyzed and used to prove results."
"There are several models in use..."
"...because it represents what many consider the most powerful possible 'reasonable' model of computation."
"[...] the field [theory of computation] 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..."
"...how efficiently they can be solved or to what degree (e.g., approximate solutions versus precise ones)."