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Inherent difficulty of computational problems
theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage
Computational complexity theory
Computational_complexity_theory
Amount of resources to perform an algorithm
computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation
Computational_complexity
Academic subfield of computer science
three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question:
Theory_of_computation
Set of problems in computational complexity theory
In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly
Complexity_class
Computational complexity of quantum algorithms
Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational
Quantum_complexity_theory
Measurement of computational complexity
In computational complexity theory, asymptotic computational complexity is the use of asymptotic analysis for the estimation of the computational complexity
Asymptotic computational complexity
Asymptotic_computational_complexity
Subfield of computer science and mathematics
transmitted data. Computational complexity theory is a branch of the theory of computation that focuses on classifying computational problems according
Theoretical_computer_science
Estimate of time taken for running an algorithm
the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly
Time_complexity
American computer scientist (born 1981)
particularly computational complexity theory. At Cornell, he became interested in quantum computing and devoted himself to computational complexity and quantum
Scott_Aaronson
Feature of systems that defy description
important factor of complexity. In several scientific fields, "complexity" has a precise meaning: In computational complexity theory, the amounts of resources
Complexity
Mathematical model describing how an output of a function is computed given an input
science, and more specifically in computability theory and computational complexity theory, a model of computation is a model that describes how an output of
Model_of_computation
Implicit computational complexity (ICC) is a subfield of computational complexity theory that characterizes programs by constraints on the way in which
Implicit computational complexity
Implicit_computational_complexity
Classification of computer problems
Geometric complexity theory (GCT), is a research program in computational complexity theory proposed by Ketan Mulmuley and Milind Sohoni. The goal of
Geometric_complexity_theory
Computational complexity
problems in computer science In computational complexity theory, NL (Nondeterministic Logarithmic-space) is the complexity class containing decision problems
NL_(complexity)
Measure of algorithmic complexity
the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size
Kolmogorov_complexity
Aspect of computational complexity theory
In computational complexity theory, a computational resource is a resource used by some computational models in the solution of computational problems
Computational_resource
Proof checkable by a randomized algorithm
In computational complexity theory, a probabilistically checkable proof (PCP) is a type of proof that can be checked by a randomized algorithm using a
Probabilistically checkable proof
Probabilistically_checkable_proof
Conceptual framework
usage of the term complexity specifically refers to sociologic theories of society as a complex adaptive system, however, social complexity and its emergent
Social_complexity
American computer scientist (born 1969)
is an American theoretical computer scientist working in computational complexity theory and algorithms. Williams graduated from the Alabama School
Ryan Williams (computer scientist)
Ryan_Williams_(computer_scientist)
Algorithmic runtime requirements for common math procedures
list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity of performing
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
American computer scientist (1928–2022)
seminal paper which established the foundations for the field of computational complexity theory". Hartmanis was born in Latvia on July 5, 1928. He was a son
Juris_Hartmanis
Topics referred to by the same term
Complexity theory may refer to: Computational complexity theory, a field in theoretical computer science and mathematics Assembly theory, to quantify the
Complexity_theory
Branch of mathematical logic
Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic
Descriptive_complexity_theory
Concept in the philosophy of mathematics
possibility of avoiding unwieldy large numbers can be based on computational complexity theory, as in András Kornai's work on explicit finitism (which does
Ultrafinitism
Algorithmic runtime requirements for matrix multiplication
problems in computer science In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
(computational complexity theory, structural complexity theory) Cook's theorem (computational complexity theory) Fagin's theorem (computational complexity theory) Full
List_of_theorems
Algorithm characteristic in computations
In computational complexity theory, the average-case complexity of an algorithm is the amount of some computational resource (typically time) used by the
Average-case_complexity
Academic conference in computer science
targets research in computational complexity theory. This currently[when?] includes(but is not limited to the study of models of computation ranging from deterministic
Computational Complexity Conference
Computational_Complexity_Conference
Branch of computational complexity theory
computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their
Parameterized_complexity
Hypothesis in computational complexity theory
In computational complexity theory, a computational hardness assumption is the hypothesis that a particular problem cannot be solved efficiently (where
Computational hardness assumption
Computational_hardness_assumption
Subfield of mathematical topology
computer science, in particular, computational geometry and computational complexity theory. A primary concern of algorithmic topology, as its name suggests
Computational_topology
Computer memory needed by an algorithm
complexity theory – Inherent difficulty of computational problems Computational resource – Aspect of computational complexity theory Time complexity –
Space_complexity
American computer scientist
at the University of California, San Diego, specializing in computational complexity theory. Impagliazzo received a BA in mathematics from Wesleyan University
Russell_Impagliazzo
In computational complexity theory of computer science, the structural complexity theory or simply structural complexity is the study of complexity classes
Structural_complexity_theory
In computational complexity theory, the element distinctness problem or element uniqueness problem is the problem of determining whether all the elements
Element_distinctness_problem
Class of computational complexity
}{=}}PSPACE}}} More unsolved problems in computer science In computational complexity theory, PSPACE is the set of all decision problems that can be solved
PSPACE
Field in logic and theoretical computer science
specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources that
Proof_complexity
American-Canadian computer scientist, contributor to complexity theory
research in computational complexity theory, which has considerably improved our understanding of the inherent difficulty of computational problems and
Stephen_Cook
Model of computational complexity
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according
Circuit_complexity
Class of problems solvable in polynomial time
In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class. It contains all decision problems that can
P_(complexity)
Algorithm that employs a degree of randomness as part of its logic or procedure
Carlo algorithm repeatedly till a correct answer is obtained. Computational complexity theory models randomized algorithms as probabilistic Turing machines
Randomized_algorithm
Notion in combinatorial game theory
Combinatorial game theory measures game complexity in several ways: State-space complexity (the number of legal game positions from the initial position)
Game_complexity
Complexity class
In computational complexity theory, a computational problem H is called NP-hard if, for every problem L which can be solved in non-deterministic polynomial-time
NP-hardness
Indian American professor of computer science (born 1957)
of algorithms, together with work on computational complexity theory, cryptography, and algorithmic game theory. During the 1980s, he made seminal contributions
Vijay_Vazirani
American computer scientist
known for his work in computational complexity theory, computability theory, computational learning theory, and Ramsey theory. He is currently a professor
William_Gasarch
of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics
List_of_complexity_classes
computational complexity theory to prove a relation between graph reachability and complexity classes.[citation needed] A theoretical computational model
Configuration_graph
principle. Computational complexity theory deals with how hard computations are, in quantitative terms, both with upper bounds (algorithms whose complexity in
List of computability and complexity topics
List_of_computability_and_complexity_topics
Complexity class (logarithmic space)
In computational complexity theory, L (also known as LSPACE, LOGSPACE or DLOGSPACE) is the complexity class containing decision problems that can be solved
L_(complexity)
Branch of the discipline of sociology
science. In relevant literature, computational sociology is often related to the study of social complexity. Social complexity concepts such as complex systems
Computational_sociology
Complexity class used to classify decision problems
problems in computer science In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision
NP_(complexity)
Quantified formulas with real-number variables
In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of
Existential theory of the reals
Existential_theory_of_the_reals
Study of resources used by an algorithm
broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem
Analysis_of_algorithms
Topics referred to by the same term
it. Solomonoff–Kolmogorov–Chaitin complexity, the most widely used such measure. In computational complexity theory, although it would be a non-formal
Algorithmic_complexity
included Stephen Cook, founder of the theory of NP-completeness which laid the groundwork for computational complexity theory, and Geoffrey Hinton, the "Godfather
Computer science at the University of Toronto
Computer_science_at_the_University_of_Toronto
Problem a computer might be able to solve
The field of computational complexity theory addresses such questions by determining the amount of resources (computational complexity) solving a given
Computational_problem
The polynomial hierarchy is contained in probabilistic Turing machine in polynomial time
Toda's theorem is a result in computational complexity theory that was proven by Seinosuke Toda in his paper "PP is as Hard as the Polynomial-Time Hierarchy"
Toda's_theorem
Area of mathematics
scientific computation The mathematics of scientific computation, in particular numerical analysis, the theory of numerical methods Computational complexity Computer
Computational_mathematics
Abstract machine that models computation
In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two
Interactive_proof_system
Problem in computational complexity theory
\epsilon >0} ? More unsolved problems in computer science In computational complexity theory, the 3SUM problem asks if a given set of n {\displaystyle n}
3SUM
Computer system simulating intelligence
stochastic. A recent definition of the IEEE Computational Intelligence Societey describes CI as the theory, design, application and development of biologically
Computational_intelligence
Algorithm whose behavior and output may depend on the run
probabilistically, for instance using an analysis of its expected time. In computational complexity theory, nondeterminism is often modeled using an explicit mechanism
Nondeterministic_algorithm
Category of mathematical proof
fundamental limitations in the provability of formal systems. In computational complexity theory, techniques like relativization (the addition of an oracle)
Proof_of_impossibility
Measure in information theory
Logical depth is a measure of complexity for individual strings devised by Charles H. Bennett based on the computational complexity of an algorithm that can
Logical_depth
Existential second order logic captures NP
oldest result of descriptive complexity theory, a branch of computational complexity theory that characterizes complexity classes in terms of logic-based
Fagin's_theorem
Computational benchmark
conclusion to be valid, only very mild assumptions in the theory of computational complexity have to be invoked. In this sense, quantum random sampling
Quantum_supremacy
Abstract computation model
In computational complexity theory, an alternating Turing machine (ATM) is a non-deterministic Turing machine (NTM) with a rule for accepting computations
Alternating_Turing_machine
Interactive proof system in computational complexity theory
In computational complexity theory, an Arthur–Merlin protocol, introduced by Babai (1985), is an interactive proof system in which the verifier's coin
Arthur–Merlin_protocol
Randomized polynomial time class of computational complexity theory
In computational complexity theory, randomized polynomial time (RP) is the complexity class of decision problems for which a probabilistic Turing machine
RP_(complexity)
Complexity of sending information in a distributed algorithm
Note that, unlike in computational complexity theory, communication complexity is not concerned with the amount of computation performed by Alice or
Communication_complexity
Influence of local substructure of a graph on global properties
graph theory. Extremal graph theory is closely related to fields such as Ramsey theory, spectral graph theory, computational complexity theory, and additive
Extremal_graph_theory
Theorem in computational complexity theory
Mahaney's theorem is a theorem in computational complexity theory proven by Stephen Mahaney that states that If any sparse language is NP-hard, then P=NP
Mahaney's_theorem
Transformation of one computational problem to another
In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem. A sufficiently
Reduction_(complexity)
Yes/no problem in computer science
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a
Decision_problem
American computer scientist (1935–2011)
American computer scientist, a noted researcher in computational complexity theory and database theory, and a target of the Unabomber. Fischer was born
Patrick_C._Fischer
Concept in computational complexity theory
In computational complexity theory, BPL (Bounded-error Probabilistic Logarithmic-space), sometimes called BPLP (Bounded-error Probabilistic Logarithmic-space
BPL_(complexity)
Theory of machine learning
bounds, computational learning theory studies the time complexity and feasibility of learning. In computational learning theory, a computation is considered
Computational_learning_theory
Memory space for a non-deterministic Turing machine
In computational complexity theory, non-deterministic space or NSPACE is the computational resource describing the memory space for a non-deterministic
NSPACE
Sequence of words formed by specific rules
natural languages). In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as
Formal_language
Set of problem-solving methods
Computational thinking refers to the thought processes involved in formulating problems so their solutions can be represented as computational steps and
Computational_thinking
Graph used in computational complexity theory and graph theory
In graph theory and computational complexity theory, a Frankl–Rödl graph is a graph defined by connecting pairs of vertices of a hypercube that are at
Frankl–Rödl_graph
Model of computation
In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for combinational digital logic circuits. A formal
Boolean_circuit
1997 computer science textbook
Introduction to the Theory of Computation (ISBN 0-534-95097-3) is a textbook in theoretical computer science, written by Michael Sipser and first published
Introduction to the Theory of Computation
Introduction_to_the_Theory_of_Computation
Mathematics award
including: All mathematical aspects of computer science, including computational complexity theory, logic of programming languages, analysis of algorithms, cryptography
IMU_Abacus_Medal
Professor of computer science
computer science, especially computational complexity theory, and in recent years has been working on "geometric complexity theory", an approach to the P versus
Ketan_Mulmuley
Israeli computer scientist
of Jerusalem. He is known for his research in computational complexity theory and algorithmic game theory. Nisan did his undergraduate studies at the Hebrew
Noam_Nisan
Both deterministic and nondeterministic machines can solve more problems given more space
In computational complexity theory, the space hierarchy theorems are separation results that show that both deterministic and nondeterministic machines
Space_hierarchy_theorem
Class in computational complexity theory
}{=}}{\mathsf {P}}} More unsolved problems in computer science In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems
NC_(complexity)
Concept in computational complexity theory
Cobham and Jack Edmonds), asserts that computational problems can be feasibly computed on some computational device only if they can be computed in polynomial
Cobham's_thesis
Set of computational problems stated by Richard Karp (1973)
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
Model of computational complexity
In computational complexity theory, the decision tree model is the model of computation in which an algorithm can be considered to be a decision tree,
Decision_tree_model
Propositional proof complexity: past, present and future. Technical Report TR98-067, Electronic Colloquium on Computational Complexity. Nathan Segerlind
Propositional_proof_system
Topics referred to by the same term
function of "n" or the limiting behavior of a function, e.g. in computational complexity theory The nth tensor power of Serre's twisting sheaf O ( 1 ) {\displaystyle
O(n)
Basic framework of mathematics
mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently, parts of computer
Foundations_of_mathematics
In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the yes and no answers. Equivalently
Complement_(complexity)
Basic concept of graph theory
the minimum values of κ(u, v) and λ(u, v), respectively. In computational complexity theory, SL is the class of problems log-space reducible to the problem
Connectivity_(graph_theory)
When a finite set S of relations yields polynomial-time or NP-complete problems
In computational complexity theory, a branch of computer science, Schaefer's dichotomy theorem, proved by Thomas Jerome Schaefer, states necessary and
Schaefer's_dichotomy_theorem
Subfield of mathematical optimization
optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including
Combinatorial_optimization
Application of complexity science to economics
interactions between economic agents. The complexity science approach has also been applied as the primary field in computational economics. The "nearly archetypal
Complexity_economics
Mathematical game
games such as Chinese checkers and Halma. They determined the computational complexity of the one-player and two-player versions of this game, and special
Pebble_game
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