"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."
Recursively enumerable languages are a type of language that can be recognized by a Turing machine. They are also known as Turing-recognizable languages.
Formal Languages: Formal languages are sets of strings constructed from a given alphabet and defined by a set of rules or grammar.
Recursive Languages: Recursive languages belong to a class of formal languages that can be recognized by a Turing machine.
Recursively Enumerable Languages: Recursively enumerable languages are a superset of recursive languages and are defined by a Turing machine that can recognize them in an infinite amount of time.
Turing Machines: Turing machines are theoretical computing devices used to model the behavior of algorithms. They consist of an infinite tape and a finite set of rules.
Church-Turing thesis: The Church-Turing thesis states that any computable function can be computed using a Turing machine.
Grammar: A grammar is a set of production rules that define a formal language by specifying valid strings that can be generated from a given set of symbols.
Chomsky Hierarchy: The Chomsky hierarchy is a classification of formal languages based on the type of grammar used to generate them.
Undecidability: Undecidability applies to problems for which there is no algorithm that can determine a yes or no answer in finite time.
Halting Problem: The halting problem is a classic example of an undecidable problem, which asks whether a given Turing machine will halt or continue running indefinitely.
Church's thesis: Church's thesis is an informal statement that any algorithmic process can be expressed in terms of lambda calculus.
Godel's incompleteness theorem: The incompleteness theorem states that any formal system that is powerful enough to express arithmetic will be either inconsistent or incomplete.
Post Correspondence Problem: The Post Correspondence problem is an example of an undecidable problem that asks whether two sets of strings can be concatenated to produce the same string.
Formal Logic: Formal logic is a branch of mathematics that deals with the validity of reasoning based on a set of rules or axioms.
Complexity Theory: Complexity theory is a branch of computer science that deals with the efficiency of algorithms and their scalability.
Automata Theory: Automata theory is a branch of computer science that deals with the study of sequential machines and the languages that they recognize.
Computability Theory: Computability theory is a branch of computer science that deals with the study of the limits of computation and the problems that cannot be solved by any algorithm.
Pumping lemma: The pumping lemma is a useful tool for proving that a given language is not a regular language.
Language Recognition: Language recognition is the ability of a machine or program to determine if a given string belongs to a specified language.
Regular Languages: Regular languages are a class of formal languages defined by regular expressions and are recognized by finite-state automata.
Finite-state Automata: Finite-state automata are theoretical computing machines used to recognize and manipulate regular languages.
Type 0 or Unrestricted: This category contains all possible formal languages that can be interpreted by Turing machines. These languages are unconstrained by any limitations, so they are also called undecidable languages.
Type 1 or Context-Sensitive: These are languages where the rules for producing symbols must adhere to a contextual structure or meaning. While they are not as powerful as unrestricted languages, they can still be interpreted by linear-bounded automata.
Type 2 or Context-Free: This category includes languages that are not constrained by context, and thus, the order in which symbols are produced is irrelevant. They are often used to represent programming languages and can also be interpreted by pushdown automata.
Type 3 or Regular: These languages are the simplest level of formal languages and have a very limited way of representing patterns. They can be parsed by finite automata and are used to represent formal grammars.
Type 4 or Recursively Enumerable: This category of languages is sometimes referred to as recursively inseparable languages. These are the most complex formal languages, and while they may not be recognizable by any formal grammar or automaton, they can be generated by a Turing machine. These languages encompass many of the most interesting and important problems in computer science, including the halting problem.
"Recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable)."
"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."