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Rice–Shapiro Theorem
In computability theory, the Rice–Shapiro theorem is a generalization of Rice's theorem, and is named after Henry Gordon Rice and Norman Shapiro. Formal statement Let ''A'' be a set of partial-recursive unary functions on the domain of natural numbers such that the set Ix(A):=\ is recursively enumerable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ..., where \varphi_n denotes the n-th partial-recursive function in a Gödel numbering. Then for any unary partial-recursive function ''\psi'', we have: :\psi \in A \Leftrightarrow \exists a finite function \theta \subseteq \psi such that \theta \in A. In the given statement, a finite function is a function with a finite domain x_1,x_2,...,x_m and \theta \subseteq \psi means that for every x \in \ it holds that \psi(x) is defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rice's Theorem
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is ''non-trivial'' if it is neither true for every partial computable function, nor false for every partial computable function. Rice's theorem can also be put in terms of functions: for any non-trivial property of partial functions, no general and effective method can decide whether an algorithm computes a partial function with that property. Here, a property of partial functions is called ''trivial'' if it holds for all partial computable functions or for none, and an effective decision method is called ''general'' if it decides correctly for every algorithm. The theorem is named after Henry Gordon Rice, who proved it in his doctoral dissertation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Gordon Rice
Henry Gordon Rice (July 18, 1920 – April 14, 2003) was an American logician and mathematician best known as the author of Rice's theorem, which he proved in his doctoral dissertation of 1951 at Syracuse University with thesis advisor Paul C. Rosenbloom. Rice was also a Professor of Mathematics at the University of New Hampshire. After 1960 he was employed by Computer Sciences Corporation in El Segundo, California, El Segundo. Rice died on April 14, 2003, in Davis, California. References External links * 1920 births 2003 deaths Syracuse University alumni 20th-century American mathematicians 21st-century American mathematicians American logicians Mathematicians from New York (state) {{US-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Shapiro
Norman Zalmon Shapiro was an American mathematician, who was the co-author of the Rice–Shapiro theorem. Education Shapiro obtained a BS in Mathematics at University of Illinois in 1952. Shapiro spent the summer of 1954 at Bell Laboratories in Murray Hill, New Jersey where, in collaboration with Karel de Leeuw, Ed Moore, and Claude Shannon, he investigated the question of whether providing a Turing machine augmented with an oracle machine producing an infinite sequence of random events (like the tosses of a fair coin) would enable the machine to output a non-computable sequence. The well-known efficacy of Monte Carlo methods might have led one to think otherwise, but the result was negative. Stated precisely: : An infinite string, S, on a finite alphabet is computable if it can be output with probability one by a Turing machine augmented by an oracle machine giving an infinite sequence of equal-probability zeroes and ones. Moreover, the result continues to hold if the output pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the set of input numbers for which the algorithm halts is exactly ''S''. Or, equivalently, *There is an algorithm that enumerates the members of ''S''. That means that its output is simply a list of all the members of ''S'': ''s''1, ''s''2, ''s''3, ... . If ''S'' is infinite, this algorithm will run forever. The first condition suggests why the term ''semidecidable'' is sometimes used. More precisely, if a number is in the set, one can ''decide'' this by running the algorithm, but if the number is not in the set, the algorithm runs forever, and no information is returned. A set that is "completely decidable" is a computable set. The second condition suggests why ''computably enumerable'' is used. The abbreviations c.e. and r.e. are oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (topology)
In mathematics, a base (or basis) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under some cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In The Foundations Of Mathematics
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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