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Recursively Enumerable Language
In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language. Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable. The class of all recursively enumerable languages is called RE. Definitions There are three equivalent definitions of a recursively enumerable language: # A recursively enumerable language is a recursively enumerable subset in the set of all possible words over the alphabet of the language. # A recursively enumerable language is a formal language for which there exists a Turing mac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Literal String
string literal or anonymous string is a literal for a string value in the source code of a computer program. Modern programming languages commonly use a quoted sequence of characters, formally "bracketed delimiters", as in x = "foo", where , "foo" is a string literal with value foo. Methods such as escape sequences can be used to avoid the problem of delimiter collision (issues with brackets) and allow the delimiters to be embedded in a string. There are many alternate notations for specifying string literals especially in complicated cases. The exact notation depends on the programming language in question. Nevertheless, there are general guidelines that most modern programming languages follow. Syntax Bracketed delimiters Most modern programming languages use bracket delimiters (also balanced delimiters) to specify string literals. Double quotations are the most common quoting delimiters used: "Hi There!" An empty string is literally written by a pair of quotes with n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of Set (mathematics), sets is the set of all element (set theory), elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of Zero, zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the List of mathematical symbols, table of mathematical symbols. Binary union The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, : A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot ha ... [...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 formalizations 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, and in some it is written without an operator. This is implemented in different ways: * Overloading the plus sign + Example from C#: "Hello, " + "World" has the value "Hello, World". * Dedicated operator, such as . in PHP, & in Visual Basic, and , , in SQL. This has the advantage over reusing + that it allows implicit type conversion to string. * string literal concatenation, which means that adjacent strings are concatenated without any operator. Example from C: "Hello, " "World" has the value "Hello, World". In many scientific publications or standards the con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene Star
In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a Set (mathematics), set to generate a set of all finite-length strings that are composed of zero or more repetitions of members from . It was named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize Automata theory, automata for regular expressions. In mathematics, it is more commonly known as the free monoid construction. Definition Given a set V, define :V^=\ (the set consists only of the empty string), :V^=V, and define recursively the set :V^=\ for each i>0. V^i is called the i-th power of V, it is a shorthand for the Concatenation#Concatenation of sets of strings, concatenation of V by itself i times. That is, ''V^i'' can be understood to be the set of all strings that can be represented as the concatenation of i members from V. The definition of Kleene star on V is : V^*=\bigcup_V^i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set (mathematics), set is closed under an Operation (mathematics), operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest superset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set (mathematics), set equipped with one or several methods for producing elements of from other elements of .Operation (mathematics), Operations and (partial function, partial) multivar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entscheidungsproblem
In mathematics and computer science, the ; ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every Structure (mathematical logic), structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936. Completeness theorem By Gödel's completeness theorem, the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced using logical rules and axioms, so the ' can also be viewed as asking for an algorithm to decide whether a given statement is provable using the rules of logic. In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the ' is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mortality (computability Theory)
In computability theory, the mortality problem is a decision problem related to the halting problem. For Turing machines, the halting problem can be stated as follows: ''Given a Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ..., and an input, decide whether the machine halts when run on the given input.'' In contrast, the mortality problem for Turing machines asks whether ''all'' executions of the machine, starting from ''any configuration'', halt. In the statement above, a ''configuration'' specifies both the machine's state (not necessarily its initial state), its tape position and the contents of the tape. While we usually assume that in the starting configuration all but finitely many cells on the tape are blanks, in the mortality problem the tape can have arb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Post Correspondence Problem
The Post correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the halting problem In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for ... and the ''Entscheidungsproblem'' it is often used in proofs of undecidability. Definition of the problem Let A be an alphabet with at least two symbols. The input of the problem consists of two finite lists \alpha_, \ldots, \alpha_ and \beta_, \ldots, \beta_ of words over A. A solution to this problem is a sequence of indices (i_k)_ with K \ge 1 and 1 \le i_k \le N for all k, such that : \alpha_ \ldots \alpha_ = \beta_ \ldots \beta_. The decision problem then is to decide whether such a solution exists or not. Alternative definition :g: (i_1,\ldots,i_K) \mapsto \alpha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halting Problem
In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is ''Undecidable problem, undecidable'', meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically Definable set, definable but not Computable function, computable. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program that might determine whether programs halt, that a "pathological" program exists for which makes an incorrect determination. Specifically, is the program that, when called with some input, passes its own s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetical Hierarchy
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy was invented independently by Kleene (1943) and Mostowski (1946).P. G. Hinman, ''Recursion-Theoretic Hierarchies'' (p.89), Perspectives in Logic, 1978. Springer-Verlag Berlin Heidelberg, ISBN 3-540-07904-1. The arithmetical hierarchy is important in computability theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
