Bar Recursion

	Bar Recursion Bar recursion is a generalized form of recursion developed by C. Spector in his 1962 paper. It is related to bar induction in the same fashion that primitive recursion is related to ordinary induction, or transfinite recursion is related to transfinite induction. Technical definition Let V, R, and O be types, and ''i'' be any natural number, representing a sequence of parameters taken from ''V''. Then the function sequence ''f'' of functions ''f''''n'' from V''i''+''n'' → R to O is defined by bar recursion from the functions ''L''''n'' : R → O and ''B'' with ''B''''n'' : ((V''i''+''n'' → R) x (V''n'' → R)) → O if: * ''f''''n''((λα:V''i''+''n'')''r'') = ''L''''n''(''r'') for any ''r'' long enough that ''L''''n''+''k'' on any extension of ''r'' equals ''L''''n''. Assuming ''L'' is a continuous sequence, there must be such ''r'', because a continuous function can use only finitely much data. * ''f''''n''(''p'') = ''B''''n''(''p'', (λ''x'':V)''f''''n''+1(cat(''p'', '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Bar Induction Bar induction is a reasoning principle used in intuitionistic mathematics, introduced by L. E. J. Brouwer. Bar induction's main use is the intuitionistic derivation of the fan theorem, a key result used in the derivation of the uniform continuity theorem. It is also useful in giving constructive alternatives to other classical results. The goal of the principle is to prove properties for all infinite sequences of natural numbers (called choice sequences in intuitionistic terminology), by inductively reducing them to properties of finite lists. Bar induction can also be used to prove properties about all choice sequences in a spread (a special kind of set). Definition Given a choice sequence x_0,x_1,x_2,x_3,\ldots, any finite sequence of elements x_0,x_1,x_2,x_3,\ldots,x_i of this sequence is called an ''initial segment'' of this choice sequence. There are three forms of bar induction currently in the literature, each one places certain restrictions on a pair of predicates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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]
picture info	Mathematical Induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Transfinite Recursion Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then $P(\alpha)$ is also true. Then transfinite induction tells us that $P$ is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that $P(0)$ is true. * Successor case: Prove that for any successor ordinal $\alpha+1$ , $P(\alpha+1)$ follows from $P(\alpha)$ (and, if necessary, $P(\beta)$ for all $\beta < \alpha$ ). * Limit case: Prove that for any [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Transfinite Induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then $P(\alpha)$ is also true. Then transfinite induction tells us that $P$ is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that $P(0)$ is true. * Successor case: Prove that for any successor ordinal $\alpha+1$ , $P(\alpha+1)$ follows from $P(\alpha)$ (and, if necessary, $P(\beta)$ for all $\beta < \alpha$ ). * Limit case: Prove that for any [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Type Theory In mathematics, logic, and computer science, a type theory is the formal system, formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a Foundations of mathematics, foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed lambda calculus, typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most proof assistant, computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of constructions, Calculus of Inductive Constructions. History Type theory was created to avoid a paradox in a mathematical foundation based on naive set theory and formal logic. Russell's paradox, which was discovered by Bertrand Russell, existed because a set could be defined using "all possible sets", which included itself. Between 1902 and 1908, Bertrand Russe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Concatenation In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenation theory, also called string theory, string concatenation is a primitive notion. Syntax In many programming languages, string concatenation is a binary infix operator. The + (plus) operator is often overloaded to denote concatenation for string arguments: "Hello, " + "World" has the value "Hello, World". In other languages there is a separate operator, particularly to specify implicit type conversion to string, as opposed to more complicated behavior for generic plus. Examples include . in Edinburgh IMP, Perl, and PHP, .. in Lua, and & in Ada, AppleScript, and Visual Basic. Other syntax exists, like , , in PL/I and Oracle Database SQL. In a few languages, notably C, C++, and Python, there is string literal concatenation, m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Dependent Choice In mathematics, the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis."The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." The axiom of dependent choice is stated on p. 86. Formal statement A homogeneous relation R on X is called a total relation if for every a \in X, there exists some b \in X such that a\,R~b is true. The axiom of dependent choice can be stated as follows: For every nonempty set X and every total relation R on X, there exists a sequence (x_n)_ in X such that :x_n\, R~x_ for all n \in \N. ''x''0 may be taken to be any desired element of ''X''. If the set X above is restricted to be the set of all real numbers, then the resulting axiom is den ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]