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A recursively enumerable language in mathematics, logic and computer science, is a type of formal language which is also called partially decidable or Turing-recognizable. It is known as a type-0 language in the Chomsky hierarchy of formal languages. The class of all recursively enumerable languages is called RE. Euclid, detail from The School of Athens by Raphael. ...
Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...
In mathematics, logic, and computer science, a formal language is a set of finite-length words (i. ...
The Chomsky hierarchy is a containment hierarchy of classes of formal grammars that generate formal languages. ...
RE (recursively enumerable) is the class of decision problems for which a yes answer can be verified by a Turing machine in a finite amount of time. ...
Definitions
There exist three equivalent major definitions for the concept of a recursively enumerable language. - A recursively enumerable formal 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 machine (or other computable function) which will enumerate all valid strings of the language. Note that, if the language is infinite, the enumerating algorithm provided can be chosen so that it avoids repetitions, since we can test whether the string produced for number n is "already" produced for a number which is less than n. If it already is produced, use the output for input n+1 instead (recursively), but again, test whether it is "new".
- A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) that will halt and accept when presented with any string in the language as input but may either halt and reject or loop forever when presented with a string not in the language. Contrast this to recursive languages, which require that the Turing machine halts in all cases.
All regular, context-free, context-sensitive and recursive languages are recursively enumerable. In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if There is an algorithm that, when given an input — typically an integer or a tuple of integers or a sequence of characters — eventually halts if it...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
A Specimen of typeset fonts and languages, by William Caslon, letter founder; from the 1728 Cyclopaedia. ...
An artistic representation of a Turing Machine . ...
In computability theory computable functions or Turing computable functions are the basic objects of study. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
In various branches of mathematics and computer science, strings are sequences of various simple objects (symbols, tokens, characters, etc. ...
A recursive language in mathematics, logic and computer science, is a type of formal language which is also called recursive, decidable or Turing-decidable. ...
A regular language is a formal language (i. ...
In linguistics and computer science, a context-free grammar (CFG) is a formal grammar in which every production rule is of the form V → w where V is a non-terminal symbol and w is a string consisting of terminals and/or non-terminals. ...
A context-sensitive language is a formal language that can be defined by a context-sensitive grammar. ...
A recursive language in mathematics, logic and computer science, is a type of formal language which is also called recursive, decidable or Turing-decidable. ...
RE, together with its complement co-RE, form the basis for the arithmetical hierarchy. RE may mean: Aer Arann: IATA airline designator RE (complexity), the set of recursively enumerable languages Relative effectiveness factor, (R.E. factor) a measurement of an explosives power Real Estate Recursively Enumerable set The complexity class of all recursively enumerable languages Regular expression, in computer science, a string that...
In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the yes and no answers. ...
Co-RE is the set of all languages that are complements of a language in RE. In a sense, Co-RE contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever. ...
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene hierarchy classifies the set of arithmetic formulas (or arithmetic sets) according to their degree of solvability. ...
Closure Properties Recursively enumerable languages are closed under the following operations. That is, if L and P are two recursively enumerable languages, then the following languages are recursively enumerable as well: In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...
Note that recursively enumerable languages are not closed under set difference or complementation. The set difference LP and the complement of L may or may not be recursively enumerable. In mathematical logic and computer science, the Kleene star (or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. ...
In formal language theory (and therefore in programming languages), concatenation is the operation of joining two character strings end to end. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
See also A recursive language in mathematics, logic and computer science, is a type of formal language which is also called recursive, decidable or Turing-decidable. ...
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