|
μ Operator
In computability theory, the μ-operator, minimization operator, or unbounded search operator searches for the least natural number with a given property. Adding the μ-operator to the five primitive recursive operators makes it possible to define all computable functions. Definition Suppose that R(''y'', ''x''1, ..., ''x''''k'') is a fixed (''k''+1)-ary relation on the natural numbers. The μ-operator "μ''y''", in either the unbounded or bounded form, is a "number theoretic function" defined from the natural numbers to the natural numbers. However, "μ''y''" contains a '' predicate'' over the natural numbers that delivers ''true'' when the predicate is satisfied and ''false'' when it is not. The ''bounded'' μ-operator appears earlier in Kleene (1952) ''Chapter IX Primitive Recursive Functions, §45 Predicates, prime factor representation'' as: :"\mu y_ R(y). \ \ \mbox \ y |
Recursion 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov's Principle
Markov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are nevertheless provable in a constructive context as well. History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function. In computability theory In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then for some input it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. In intuitionistic logic In predicate logic, a predicate ''P'' over some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marvin L
Marvin may refer to: __NOTOC__ Geography ;In the United States * Marvyn, Alabama, also spelled Marvin, an unincorporated community * Marvin, Missouri, an unincorporated community * Marvin, North Carolina, a village * Marvin, South Dakota, a town * Robley, Virginia, also known as Marvin * Lake Marvin, a lake in Georgia ;Elsewhere * Marvin Islands, Nunavut, Canada People and fictional characters * Marvin (given name), including a list of people and fictional characters * Marvin (surname), including a list of people and fictional characters Arts and entertainment * ''Marvin the Album'', an album by the Australian group Frente! * "Marvin (Patches)", a song by Titãs * "Marvin" (Marvin the Paranoid Android song), a song by Marvin the Paranoid Android (1981) * ''Marvin'' (film), a 2017 French film * ''Marvin'' (comic), a newspaper comic strip Other uses * Marvin (robot), developed by the University of Kaiserslautern Robotics Research Lab in Germany See also * Marven Gardens, a ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. Being p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Subtraction
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] |
Mu Recursive Function
In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as in a formal one. If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines (this is one of the theorems that supports the Church–Turing thesis). The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other equivalent classes of functions are the functions of lambda calculus and the functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Primitive Recursive Function
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] |
Peano Axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book, ''The principles of arithmetic presented by a new method'' ( la, Arithmeti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Abstract Machine Models Of The Unbounded
Alternative or alternate may refer to: Arts, entertainment and media * Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki'' * ''The Alternative'' (film), a 1978 Australian television film * ''The Alternative'', a radio show hosted by Tony Evans * ''120 Minutes'' (2004 TV program), an alternative rock music video program formerly known as ''The Alternative'' *'' The American Spectator'', an American magazine formerly known as ''The Alternative: An American Spectator'' * Alternative comedy, a range of styles used by comedians and writers in the 1980s * Alternative comics, a genre of comic strips and books * Alternative media, media practices falling outside the mainstreams of corporate communication * Alternative reality, in fiction * Alternative title, the use of a secondary title for a work when it is distributed or sold in other countries Music * ''Alternative'' (album), a B-sides album by Pet Shop Boys * ''The Alternative'' (album), an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counter Machine
A counter machine is an abstract machine used in a formal logic and theoretical computer science to model computation. It is the most primitive of the four types of register machines. A counter machine comprises a set of one or more unbounded ''registers'', each of which can hold a single non-negative integer, and a list of (usually sequential) arithmetic and control instructions for the machine to follow. The counter machine is typically used in the process of designing parallel algorithms in relation to the mutual exclusion principle. When used in this manner, the counter machine is used to model the discrete time-steps of a computational system in relation to memory accesses. By modeling computations in relation to the memory accesses for each respective computational step, parallel algorithms may be designed in such a matter to avoid interlocking, the simultaneous writing operation by two (or more) threads to the same memory address. Basic features For a given counter machin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing-equivalent
Turing equivalence may refer to: * As related to Turing completeness In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Tur ..., Turing equivalence means having computational power equivalent to a universal Turing machine * Turing degree equivalence (of sets), having the same level of unsolvability See also * Turing machine equivalents * Turing test (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
