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• Complexity_class abstract "In computational complexity theory, a complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form:the set of problems that can be solved by an abstract machine M using O(f(n)) of resource R, where n is the size of the input.Complexity classes are concerned with the rate of growth of the requirement in resources as the input n increases. It is an abstract measurement, and does not give time or space in requirements in terms of seconds or bytes, which would require knowledge of implementation specifics. The function inside the O(...) expression could be a constant, for algorithms which are unaffected by the size of n,or an expression involving a logarithm, an expression involving a power of n, i.e a polynomial expression, and many others. The O is read as \"order of..\". For the purposes of computational complexity theory, some of the details of the function can be ignored, for instance many possible polynomials can be grouped together as a class.The resource in question can either be time, essentially the number of primitve operations on an abstract machine, or (storage) space. For example, the class NP is the set of decision problems whose solutions can be determined by a non-deterministic Turing machine in polynomial time, while the class PSPACE is the set of decision problems that can be solved by a deterministic Turing machine in polynomial space.The simplest complexity classes are defined by the following factors: The type of computational problem: The most commonly used problems are decision problems. However, complexity classes can be defined based on function problems (an example is FP), counting problems (e.g. #P), optimization problems, promise problems, etc. The model of computation: The most common model of computation is the deterministic Turing machine, but many complexity classes are based on nondeterministic Turing machines, boolean circuits, quantum Turing machines, monotone circuits, etc. The resource (or resources) that are being bounded and the bounds: These two properties are usually stated together, such as \"polynomial time\", \"logarithmic space\", \"constant depth\", etc.Many complexity classes can be characterized in terms of the mathematical logic needed to express them; see descriptive complexity.Bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that \"the time complexities in any two reasonable and general models of computation are polynomially related\" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP.The Blum axioms can be used to define complexity classes without referring to a concrete computational model.".
• Complexity_class wikiPageExternalLink complexity_theory.html.
• Complexity_class wikiPageExternalLink Complexity_Zoo.
• Complexity_class wikiPageID "502426".
• Complexity_class wikiPageLength "17831".
• Complexity_class wikiPageOutDegree "149".
• Complexity_class wikiPageRevisionID "706871305".
• Complexity_class wikiPageWikiLink AC_(complexity).
• Complexity_class wikiPageWikiLink ALL_(complexity).
• Complexity_class wikiPageWikiLink Abstract_machine.
• Complexity_class wikiPageWikiLink Algorithm.
• Complexity_class wikiPageWikiLink Arthur–Merlin_protocol.
• Complexity_class wikiPageWikiLink BPP_(complexity).
• Complexity_class wikiPageWikiLink BQP.
• Complexity_class wikiPageWikiLink Big_O_notation.
• Complexity_class wikiPageWikiLink Blum_axioms.
• Complexity_class wikiPageWikiLink Boolean_circuit.
• Complexity_class wikiPageWikiLink Category:Complexity_classes.
• Complexity_class wikiPageWikiLink Category:Computational_complexity_theory.
• Complexity_class wikiPageWikiLink Category:Measures_of_complexity.
• Complexity_class wikiPageWikiLink Circuit_complexity.
• Complexity_class wikiPageWikiLink Co-NP.
• Complexity_class wikiPageWikiLink Cobhams_thesis.
• Complexity_class wikiPageWikiLink Complete_(complexity).
• Complexity_class wikiPageWikiLink Computability_theory.
• Complexity_class wikiPageWikiLink Computational_complexity_theory.
• Complexity_class wikiPageWikiLink Computational_model.
• Complexity_class wikiPageWikiLink Computational_problem.
• Complexity_class wikiPageWikiLink Computational_resource.
• Complexity_class wikiPageWikiLink Context-free_grammar.
• Complexity_class wikiPageWikiLink Context-sensitive_grammar.
• Complexity_class wikiPageWikiLink Counting_problem_(complexity).
• Complexity_class wikiPageWikiLink DSPACE.
• Complexity_class wikiPageWikiLink DTIME.
• Complexity_class wikiPageWikiLink David_S._Johnson.
• Complexity_class wikiPageWikiLink Decision_problem.
• Complexity_class wikiPageWikiLink Descriptive_complexity_theory.
• Complexity_class wikiPageWikiLink EXPSPACE.
• Complexity_class wikiPageWikiLink EXPTIME.
• Complexity_class wikiPageWikiLink FP_(complexity).
• Complexity_class wikiPageWikiLink Function_problem.
• Complexity_class wikiPageWikiLink IP_(complexity).
• Complexity_class wikiPageWikiLink Interactive_proof_system.
• Complexity_class wikiPageWikiLink L_(complexity).
• Complexity_class wikiPageWikiLink List_of_complexity_classes.
• Complexity_class wikiPageWikiLink List_of_undecidable_problems.
• Complexity_class wikiPageWikiLink Log-space_reduction.
• Complexity_class wikiPageWikiLink Logarithm.
• Complexity_class wikiPageWikiLink Logical_conjunction.
• Complexity_class wikiPageWikiLink Logical_connective.
• Complexity_class wikiPageWikiLink Logical_disjunction.
• Complexity_class wikiPageWikiLink Mathematical_logic.
• Complexity_class wikiPageWikiLink Michael_Garey.
• Complexity_class wikiPageWikiLink NC_(complexity).
• Complexity_class wikiPageWikiLink NEXPTIME.
• Complexity_class wikiPageWikiLink NL_(complexity).
• Complexity_class wikiPageWikiLink NP-completeness.
• Complexity_class wikiPageWikiLink NP-hardness.
• Complexity_class wikiPageWikiLink NP_(complexity).
• Complexity_class wikiPageWikiLink NSPACE.
• Complexity_class wikiPageWikiLink NTIME.
• Complexity_class wikiPageWikiLink Negation.
• Complexity_class wikiPageWikiLink Neil_Immerman.
• Complexity_class wikiPageWikiLink Non-deterministic_Turing_machine.
• Complexity_class wikiPageWikiLink Optimization_problem.
• Complexity_class wikiPageWikiLink PSPACE.
• Complexity_class wikiPageWikiLink P_(complexity).
• Complexity_class wikiPageWikiLink Polynomial.
• Complexity_class wikiPageWikiLink Polynomial-time_reduction.
• Complexity_class wikiPageWikiLink Probabilistic_Turing_machine.
• Complexity_class wikiPageWikiLink Promise_problem.
• Complexity_class wikiPageWikiLink QMA.
• Complexity_class wikiPageWikiLink Quantum_Turing_machine.
• Complexity_class wikiPageWikiLink RP_(complexity).
• Complexity_class wikiPageWikiLink Recursive_language.
• Complexity_class wikiPageWikiLink Recursively_enumerable_language.
• Complexity_class wikiPageWikiLink Regular_grammar.
• Complexity_class wikiPageWikiLink Savitchs_theorem.
• Complexity_class wikiPageWikiLink Sharp-P.
• Complexity_class wikiPageWikiLink Space_hierarchy_theorem.
• Complexity_class wikiPageWikiLink Subset.
• Complexity_class wikiPageWikiLink Time_complexity.
• Complexity_class wikiPageWikiLink Time_hierarchy_theorem.
• Complexity_class wikiPageWikiLink Turing_machine.
• Complexity_class wikiPageWikiLink Turing_reduction.
• Complexity_class wikiPageWikiLink ZPP_(complexity).
• Complexity_class wikiPageWikiLink File:DottedLine.png.
• Complexity_class wikiPageWikiLink File:SolidLine.png.
• Complexity_class wikiPageWikiLinkText "Calculability".
• Complexity_class wikiPageWikiLinkText "Complexity Classes".
• Complexity_class wikiPageWikiLinkText "Complexity class".
• Complexity_class wikiPageWikiLinkText "Relationships between complexity classes".
• Complexity_class wikiPageWikiLinkText "as hard".
• Complexity_class wikiPageWikiLinkText "class".
• Complexity_class wikiPageWikiLinkText "classes".
• Complexity_class wikiPageWikiLinkText "complexity class".
• Complexity_class wikiPageWikiLinkText "complexity classes".
• Complexity_class wikiPageWikiLinkText "computational complexity".
• Complexity_class wikiPageWikiLinkText "time-complexity class".
• Complexity_class wikiPageUsesTemplate Template:ComplexityClasses.

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