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Primitive Recursion
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the ''n''th prime are all primitive recursive. In fact, for showing that a computable function is primitive recursive, it suffices to show that its time complexity is bounded above by a primitive recursive function of the input size. It is hence not that easy to devise a computable function that is ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arity
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In logic and philosophy, it is also called adicity and degree. In linguistics, it is usually named valency. Examples The term "arity" is rarely employed in everyday usage. For example, rather than saying "the arity of the addition operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation". In general, the naming of functions or operators with a given arity follows a convention similar to the one used for ''n''-based numeral systems such as binary and hexadecimal. One combines a Latin prefix with the -ary ending; for example: * A nullary function takes no arguments. ** Example: f()=2 * A unary function takes one argument. ** Example: f(x)=2x * A binary function takes two arguments. ** Examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primality Test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is ''trial division'': given an input number, ''n'', check whether it is evenly divisible by any prime number between 2 and (i.e. that the division leaves no remainder). If so, then ''n'' is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For example, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases: b^n = \underbrace_. The exponent is usually shown as a superscript to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' (raised) to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th". Starting from the basic fact stated above that, for any positive integer n, b^n is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular: \begin b^ & = \underbrace_ \\[1ex] & = \underbrace_ \times \underbrace_ \\[1ex] & = b^n \times b^m \end In other words, when multiplying a base raised to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the refutations of P. Definition ''Classical negation'' is an operation on one logical value, typically the value of a proposition, that produces a value of ''true'' when its operand is false, and a value of ''false'' when its operand is true. Thus if statement is true, then \neg P (pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S , assuming that R abbreviates "it is raining" and S abbreviates "it is snowing". In classical logic, disjunction is given a truth functional semantics according to which a formula \phi \lor \psi is true unless both \phi and \psi are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an ''inclusive'' interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this operator is typically written as \wedge or . A \land B is true if and only if A is true and B is true, otherwise it is false. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English "and". * In programming languages, the short-circuit and control structure. * In set theory, intersection. * In lattice theory, logical conjunction ( greatest lower bound). * In predicate logic, universal quantification. Notation And is usually denoted by an infix operator: in mathematics and logic, it is denoted by \wedge, or ; in electronics, ; and in programming languages, &, &&, or and. In Jan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If, And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...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 , one has \mathbf_(x)=1 if x\in A, and \mathbf_(x)=0 otherwise, where \mathbf_A is a common notation for the indicator function. Other common notations are I_A, and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, :\mathbf_(x)= \in A For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition The indicator function of a subset of a set is a function \mathbf_A \colon X \to \ defined as \mathbf_A(x) := \begin 1 ~&\text~ x \in A~, \\ 0 ~&\text~ x \notin A~. \end The Iverson bracket provides the equivalent notation, \in A/math> or to be used instead of \mathbf_(x)\,. The function \mathbf_A is sometimes denoted , , , or even just . Nota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth Value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false". Classical logic In classical logic, with its intended semantics, the truth values are ''true'' (denoted by ''1'' or the verum ⊤), and '' untrue'' or '' false'' (denoted by ''0'' or the falsum ⊥); that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantics of l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monus
In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. The monus operator may be denoted with the − symbol because the natural numbers are a CMM under subtraction; it is also denoted with the \mathop symbol to distinguish it from the standard subtraction operator. Notation Definition Let (M, +, 0) be a commutative monoid. Define a binary relation \leq on this monoid as follows: for any two elements a and b, define a \leq b if there exists an element c such that a + c = b. It is easy to check that \leq is reflexive and that it is transitive. M is called ''naturally ordered'' if the \leq relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements a and b, a unique smallest element c exists such that a \leq b + c, then is called a ''commutative monoid with monus'' and the ''monus'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
For Loop
In computer science a for-loop or for loop is a control flow statement for specifying iteration. Specifically, a for loop functions by running a section of code repeatedly until a certain condition has been satisfied. For-loops have two parts: a header and a body. The header defines the iteration and the body is the code that is executed once per iteration. The header often declares an explicit loop counter or loop variable. This allows the body to know which iteration is being executed. For-loops are typically used when the number of iterations is known before entering the loop. For-loops can be thought of as shorthands for while-loops which increment and test a loop variable. Various keywords are used to indicate the usage of a for loop: descendants of ALGOL use "for", while descendants of Fortran use "do". There are other possibilities, for example COBOL which uses "PERFORM VARYING". The name ''for-loop'' comes from the word for. For is used as the keyword in many progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
