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Blum Axioms
In computational complexity theory the Blum axioms or Blum complexity axioms are axioms that specify desirable properties of complexity measures on the set of computable functions. The axioms were first defined by Manuel Blum in 1967. Importantly, Blum's speedup theorem and the Gap theorem hold for any complexity measure satisfying these axioms. The most well-known measures satisfying these axioms are those of time (i.e., running time) and space (i.e., memory usage). Definitions A Blum complexity measure is a pair (\varphi, \Phi) with \varphi a numbering of the partial computable functions \mathbf^ and a computable function :\Phi: \mathbb \to \mathbf^ which satisfies the following Blum axioms. We write \varphi_i for the ''i''-th partial computable function under the Gödel numbering \varphi, and \Phi_i for the partial computable function \Phi(i). * the domains of \varphi_i and \Phi_i are identical. * the set \ is recursive. Examples * (\varphi, \Phi) is a complexity mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable 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 their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel Numbering
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. Kurt Gödel developed the concept for the proof of his incompleteness theorems. A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects. Simplified overview Gödel noted that each statement within a system can be represented by a natural number (its ''Gödel number''). The significance of this was that propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator function of is the function \mathbf_A defined by \mathbf_\!(x) = 1 if x \in A, and \mathbf_\!(x) = 0 otherwise. Other common notations are and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, \mathbf_(x) = \left x\in A\ \right For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition Given an arbitrary set , the indicator function of a subset of is the function \mathbf_A \colon X \mapsto \ defined by \operatorname\mathbf_A\!( x ) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \,. \end The Iverson bracket provides the equivalent notation \left x\in A\ \right/math> or that can be used instead of \mathbf_\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean-valued Function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = ), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information. In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses, it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression. In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
In computational complexity theory, a complexity class is a set (mathematics), set of computational problems "of related resource-based computational complexity, complexity". The two most commonly analyzed resources are time complexity, time and space complexity, memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time complexity, time or space complexity, memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P (complexity), P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. Counting problem (complexity), counting problems and function problems) and using other models of computation (e.g. probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Computable Function
Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precise definition of the concept of algorithm, every formal definition of computability must refer to a specific model of computation. Many such models of computation have been proposed, the major ones being Turing machines, register machines, lambda calculus and general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for every model of computation that has ever been proposed, the computable functions for such a model are computable for the above four models of computation. The Church–Turing thesis is the unprovable assertion that every notion of computability that can be imagined can compute only functions that are computable in the above sense. Befo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursive Language
In mathematics, logic and computer science, a recursive (or ''decidable'') language is a recursive subset of the Kleene closure of an alphabet. Equivalently, a formal language is recursive if there exists a Turing machine that decides the formal language. In theoretical computer science, such always-halting Turing machines are called total Turing machines or algorithms. The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply ''decidable''. The class of all recursive languages is often called R, although this name is also used for the class RP. This type of language was not defined in the Chomsky hierarchy. All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of A Function
In mathematics, the domain of a function is the Set (mathematics), set of inputs accepted by the Function (mathematics), function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be". More precisely, given a function f\colon X\to Y, the domain of is . In modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both sets of real numbers, the function can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function f\colon X\to Y, the set is called the ''codomain'': the set to which all outputs must belong. The set of specific outputs the function assigns to elements of is called its ''Range of a function, range'' or ''Image (mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Computable Function
Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precise definition of the concept of algorithm, every formal definition of computability must refer to a specific model of computation. Many such models of computation have been proposed, the major ones being Turing machines, register machines, lambda calculus and general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for every model of computation that has ever been proposed, the computable functions for such a model are computable for the above four models of computation. The Church–Turing thesis is the unprovable assertion that every notion of computability that can be imagined can compute only functions that are computable in the above sense. Befo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a "logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. Non- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numbering (computability Theory)
There are many different numbering schemes for assigning nominal numbers to entities. These generally require an agreed set of rules, or a central coordinator. The schemes can be considered to be examples of a primary key of a database management system table, whose table definitions require a database design. In computability theory, the simplest numbering scheme is the assignment of natural numbers to a set of objects such as functions, rational numbers, graphs, or words in some formal language. A numbering can be used to transfer the idea of computability and related concepts, which are originally defined on the natural numbers using computable functions, to these different types of objects. A simple extension is to assign cardinal numbers to physical objects according to the choice of some base of reference and of measurement units for counting or measuring these objects within a given precision. In such case, numbering is a kind of classification, i.e. assigning a numeric pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gap Theorem
:''See also Gap theorem (other) for other gap theorems in mathematics.'' In computational complexity theory, the Gap Theorem, also known as the Borodin–Trakhtenbrot Gap Theorem, is a major theorem about the complexity of computable functions. It essentially states that there are arbitrarily large computable gaps in the hierarchy of complexity classes. For any computable function that represents an increase in computational resources, one can find a resource bound such that the set of functions computable within the expanded resource bound is the same as the set computable within the original bound. The theorem was proved independently by Boris Trakhtenbrot and Allan Borodin. Although Trakhtenbrot's derivation preceded Borodin's by several years, it was not known nor recognized in the West until after Borodin's work was published. Gap theorem The general form of the theorem is as follows. :Suppose is an abstract (Blum) complexity measure. For any total computable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
