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Formal language
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
Recursively_enumerable_language
Mathematical logic concept
which enumerates any maximal recursively enumerable set dominates every general recursive function. There exists maximal recursively enumerable set of
Computably_enumerable_set
Hierarchy of classes of formal grammars
context-free language is context-sensitive, every context-sensitive language is recursive and every recursive language is recursively enumerable. These are
Chomsky_hierarchy
Ability to solve a problem by an effective procedure
recursively enumerable, but not recursive? And, furthermore, are there languages which are not even recursively enumerable? The halting problem is one of
Computability
Set with algorithmic membership test
function, or the empty set. Computably enumerable Decidability (logic) Recursively enumerable language Recursive language Recursion That is, under the Set-theoretic
Computable_set
Formal language in mathematics and computer science
class RP. This type of language was not defined in the Chomsky hierarchy. All recursive languages are also recursively enumerable. All regular, context-free
Recursive_language
Type of formal grammar
Recursively enumerable languages are closed under Kleene star, concatenation, union, and intersection, but not under set difference; see Recursively enumerable
Unrestricted_grammar
Model of computation in computer science
Therefore, all recursively enumerable languages are unambiguously recognizable. Conversely, every unambiguously recognizable language is recognizable
Unambiguous_Turing_machine
Measure of unsolvability
the language ⟨ ≤, = ⟩. A degree is called recursively enumerable (r.e.) or computably enumerable (c.e.) if it contains a recursively enumerable set.
Turing_degree
Ordered listing of items in collection
computable. The set being enumerated is then called recursively enumerable (or computably enumerable in more contemporary language), referring to the use
Enumeration
Process of repeating items in a self-similar way
reducible to non-recursively defined values: in this case F(0) = 0 and F(1) = 1. Applying the standard technique of proof by cases to recursively defined sets
Recursion
context-free languages and the recursively enumerable languages, and other families of formal languages studied in the scientific literature. A formal language is
Abstract_family_of_languages
of the production rules. Such sets are recursively enumerable languages and every recursively enumerable language is the restriction of some such set to
Post_canonical_system
Computational learning model
{\displaystyle \{s_{1},...,s_{n-1}\}} . It is shown that a class of recursively enumerable languages is learnable in the limit if it has finite elasticity. A bound
Language identification in the limit
Language_identification_in_the_limit
standard Turing machine and therefore accepts precisely the recursively enumerable languages. A multitrack Turing machine with n {\displaystyle n} -tapes
Multi-track_Turing_machine
On solvability of Diophantine equations
making the notion of recursive enumerability perfectly rigorous. It is evident that Diophantine sets are recursively enumerable (also known as semi-decidable)
Hilbert's_tenth_problem
Topics referred to by the same term
technological advancement Type-0 language or Recursively enumerable language in the Chomsky hierarchy of formal languages Type 0 string theory, a model of
Type_0
context free languages can be any recursively enumerable language. The quotient of two recursively enumerable languages is recursively enumerable. These closure
Quotient_of_a_formal_language
any recursively enumerable set of well-formed formulas of a first-order language is recursively axiomatizable, and even primitively recursively axiomatizable
Craig's_theorem
Type of Turing machine
machine can calculate any recursive function, decide any recursive language, and accept any recursively enumerable language. According to the Church–Turing
Universal_Turing_machine
Complexity class
In computability theory and computational complexity theory, RE (recursively enumerable) is the class of decision problems for which a 'yes' answer can
RE_(complexity)
Function computable with bounded loops
Peano arithmetic) is also recursively enumerable, as one can enumerate all the proofs of the theory. While all primitive recursive functions are provably
Primitive_recursive_function
particular by the families of regular languages, context-free languages and the recursively enumerable languages. The concept of a cone is a more abstract
Cone_(formal_languages)
Computation model defining an abstract machine
alphabet. A set of strings which can be enumerated in this manner is called a recursively enumerable language. The Turing machine can equivalently be
Turing_machine
Mathematical function that can be computed by a program
if and only if the word w is in the language. The term enumerable has the same etymology as in computably enumerable sets of natural numbers. The following
Computable_function
Computer science and linguistics concept relating to non-terminal production
Otherwise it is called a non-recursive grammar. For example, a grammar for a context-free language is left recursive if there exists a non-terminal
Recursive_grammar
Study of computable functions and Turing degrees
computably enumerable (c.e.) set, which is a set that can be enumerated by a Turing machine (other terms for computably enumerable include recursively enumerable
Computability_theory
automaton Chomsky hierarchy Context-sensitive language, context-sensitive grammar Recursively enumerable language Register machine Stack machine Petri net
List of computability and complexity topics
List_of_computability_and_complexity_topics
Specification of a mathematical group by generators and relations
call a subset U of FS recursive (respectively recursively enumerable) if f(U) is recursive (respectively recursively enumerable). If S is indexed as above
Presentation_of_a_group
Educational software
automaton pumping lemma for context-free language CYK parser LL parser SLR parser Topics on recursively enumerable language: Turing machine unrestricted grammar
JFLAP
Limitative results in mathematical logic
its set of theorems is recursively enumerable. This means that there is a computer program that, in principle, could enumerate all the theorems of the
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Kleene's recursion theorem Recursively enumerable set Recursively enumerable language Decidable language Undecidable language Rice's theorem Post's theorem
List of mathematical logic topics
List_of_mathematical_logic_topics
Control flow construct for executing code repeatedly
programming languages, such as C, C++, and Java, use that term for the three-part for loop, which is not an enumeration. Other programming languages, such as
Loop_(statement)
Type of formal grammar
homomorphism, and Kleene plus. Every recursively enumerable language L can be written as h(L) for some context-sensitive language L and some string homomorphism
Context-sensitive_grammar
Yes-or-no question that cannot ever be solved by a computer
partially decidable, semi-decidable, solvable, or provable if A is a recursively enumerable set. In computability theory, the halting problem is a decision
Undecidable_problem
Programming language family
itself naturally to recursion. Mathematical problems such as the enumeration of recursively defined sets are simple to express in this notation. For example
Lisp_(programming_language)
Automata that lists elements of some given set
recognizable languages are also recursively enumerable. Proof A Turing Recognizable language can be Enumerated by an Enumerator Consider a Turing Machine M
Enumerator_(computer_science)
Sequence of words formed by specific rules
problem for semigroups was recursively insoluble", and later devised the canonical system for the creation of formal languages. In 1907, Leonardo Torres
Formal_language
Overview of and topical guide to logic
Primitive recursive function Recursion (computer science) Recursive language Recursive set Recursively enumerable language Recursively enumerable set Reduction
Outline_of_logic
Hierarchy of complexity classes for formulas defining sets
{\displaystyle \psi } has only bounded quantifiers. These are exactly the recursively enumerable sets. The set of natural numbers that are indices for Turing machines
Arithmetical_hierarchy
Non-contradiction of a theory
recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable
Consistency
grammar Prefix grammar Pumping lemma Recursively enumerable language Regular expression Regular grammar Regular language S-attributed grammar Star height
List of formal language and literal string topics
List_of_formal_language_and_literal_string_topics
Extension of recursion theory to admissible ordinals beyond the natural numbers
} ) are α {\displaystyle \alpha } -recursively-enumerable. It's of note that α {\displaystyle \alpha } -recursive sets are members of L α + 1 {\displaystyle
Alpha_recursion_theory
Study of abstract machines and automata
closely related to formal language theory. In this context, automata are used as finite representations of formal languages that may be infinite. Automata
Automata_theory
Yes/no problem in computer science
provable if the set of inputs for which the answer is YES is a recursively enumerable set. Problems that are not decidable are undecidable, which means
Decision_problem
Use of functions that call themselves
case). Merge sort is a sorting algorithm that recursively sorts and merges divided subarrays (the recursive case) until each subarray consists of one element
Recursion_(computer_science)
Formalization of the natural numbers
Skolem arithmetic. The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function, including the
Primitive recursive arithmetic
Primitive_recursive_arithmetic
(recursively enumerable, but not recursive). This concludes the proof. Corollary The set of finitely satisfiable sentences is recursively enumerable.
Trakhtenbrot's_theorem
Topics referred to by the same term
Earth radius RE (complexity) (recursively enumerable), a complexity class of decision problems Recursively enumerable (r.e.), in computability theory
Re
Set of sentences in a formal language
theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms
Theory_(mathematical_logic)
Solution of some Diophantine equation
conjunction of Matiyasevich's result with the fact that most recursively enumerable languages are not decidable implies that a solution to Hilbert's tenth
Diophantine_set
Type of a context-free grammar
parsers or by recursive descent parsers, and many computer languages[clarification needed] are designed to be LL(1) for this reason. Languages based on grammars
LL_grammar
System for reasoning about vagueness
s : S → {\displaystyle \rightarrow } [0,1] of a set S is recursively enumerable if a recursive map h : S×N → {\displaystyle \rightarrow } Ü exists such
Fuzzy_logic
Set of all true first-order statements about the arithmetic of natural numbers
so Th( N {\displaystyle {\mathcal {N}}} ) is not decidable nor recursively enumerable. Th( N {\displaystyle {\mathcal {N}}} ) is closely related to the
True_arithmetic
Theorem in computability theory
operator Φ there is a recursively enumerable set F such that Φ(F) = F and F is the smallest set with this property. For any recursive operator Ψ there is
Kleene's_recursion_theorem
Type of automaton
context-sensitive languages, which is a proper superclass of the context-free languages, and a proper subclass of Turing-recognizable (i.e. recursively enumerable) languages
Pushdown_automaton
Type of Turing reduction
simply m-complete, iff B {\displaystyle B} is recursively enumerable and every recursively enumerable set A {\displaystyle A} is m-reducible to B {\displaystyle
Many-one_reduction
Concept in computability theory
partial function with domain A, then A is said to be B-recursively enumerable and B-computably enumerable. We say A {\displaystyle A} is Turing equivalent to
Turing_reduction
computing – Recursive descent parser – Recursion (computer science) – Recursive set – Recursively enumerable language – Recursively enumerable set – Reference
Index_of_computing_articles
Mathematical model for deduction or proof systems
then generalizing it. A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules
Formal_system
Russian mathematician (1934–2019)
affirmative answer to Post's problem regarding the existence of recursively enumerable Turing degrees between 0 and 0' . This result, now known as the
Albert_Muchnik
Computational problems no algorithm can solve
undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e., such undecidable languages may be recursively enumerable
List_of_undecidable_problems
Problem in computer science
when run on input x} represents the halting problem. This set is recursively enumerable, which means there is a computable function that lists all of the
Halting_problem
Axiomatic logical system
same language, and both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable
Robinson_arithmetic
Sequence of characters, data type
list) of data other than just characters. Depending on the programming language and precise data type used, a variable declared to be a string may either
String_(computer_science)
primitive recursive, showing that PR is strictly contained in R (Cooper 2004:88). On the other hand, we can "enumerate" any recursively enumerable set (see
PR_(complexity)
Fundamental theorem in mathematical logic
the set of logically valid formulas in second-order logic is not recursively enumerable. The same is true of all higher-order logics. It is possible to
Gödel's_completeness_theorem
Ability of a computing system to simulate Turing machines
enumerable. Also, since all functions in these languages are total, algorithms for recursively enumerable sets cannot be written in these languages,
Turing_completeness
Ordinals in mathematics and set theory
construction) prove their existence. If T {\displaystyle T} is a recursively enumerable set theory consistent with V=L, then the least α {\displaystyle
Large_countable_ordinal
Halting probability of a random computer program
sequence. Calude, Hertling, Khoussainov, and Wang showed that a recursively enumerable real number is an algorithmically random sequence if and only if
Chaitin's_constant
Academic subfield of computer science
Retrieved 6 January 2015. Henry Gordon Rice (1953). "Classes of Recursively Enumerable Sets and Their Decision Problems". Transactions of the American
Theory_of_computation
Limited form of tree data structure
always visit the current node; next, we recursively traverse the current node's left subtree, and then we recursively traverse the current node's right subtree
Binary_tree
Notation techniques for grammars in computer science
input for a parser generator. They describe precisely all recursively enumerable languages, which makes parsing impossible in general: it is an undecidable
Van_Wijngaarden_grammar
Structure of a formal language
constructing practical language translation tools. A recursive grammar is a grammar that contains production rules that are recursive. For example, a grammar
Formal_grammar
Programming language
as a map[string]interface{} (map of string to empty interface). This recursively describes data in the form of a dictionary with string keys and values
Go_(programming_language)
Summary of a mathematical proof
assumed to be effective, which means that the set of axioms must be recursively enumerable. This means that it is theoretically possible to write a finite-length
Proof sketch for Gödel's first incompleteness theorem
Proof_sketch_for_Gödel's_first_incompleteness_theorem
Operation in computability theory
4310/mrl.1999.v6.n6.a10. Retrieved 2008-07-13. Soare, R.I. (1987). Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably
Turing_jump
Higher-order function Y for which Y f = f (Y f)
the set of fixed-point combinators of untyped lambda calculus is recursively enumerable. The Y combinator can be expressed in the SKI-calculus as Y = S
Fixed-point_combinator
Technique used in mathematical logic
Rado graph. any two many-complete recursively enumerable sets are recursively isomorphic. We establish a language L {\displaystyle {\mathcal {L}}} and
Back-and-forth_method
Set of problems in computational complexity theory
problems are often synonymously referred to as languages, since strings of bits represent formal languages (a concept borrowed from linguistics); for example
Complexity_class
Lemma that defines a property of regular languages
theory of formal languages, the pumping lemma for regular languages is a lemma that describes an essential property of all regular languages. Informally,
Pumping lemma for regular languages
Pumping_lemma_for_regular_languages
Mathematical theory
M learns S if M learns every f in S. Basic results are that all recursively enumerable classes of functions are learnable while the class REC of all computable
Solomonoff's theory of inductive inference
Solomonoff's_theory_of_inductive_inference
Concept in mathematical logic and set theory
sets are exactly the ω 1 C K {\displaystyle \omega _{1}^{CK}} -recursively-enumerable subsets of ω {\displaystyle \omega } . [Bar75, p. 168] A function
Analytical_hierarchy
Abstract machine used to study decision problems
Robert I. (1987). "Fundamentals of Recursively Enumerable Sets and the Recursion Theorem". Recursively Enumerable Sets and Degrees. Perspectives in Mathematical
Oracle_machine
Finite-state machine
function composition. Clearly, this process may be recursively continued, giving the following recursive definition of δ ^ : Q × Σ ⋆ → Q {\displaystyle {\widehat
Deterministic finite automaton
Deterministic_finite_automaton
Axioms for the natural numbers
which consists of a recursively enumerable and even decidable set of axioms. For each formula φ(x, y1, ..., yk) in the language of Peano arithmetic,
Peano_axioms
American mathematician (1920–1983)
Decision problems about algebraic and logical systems as a whole and recursively enumerable degrees of unsolvability. 1968 Contributions to Math. Logic (Colloquium
William_Boone_(mathematician)
American mathematician
the function which enumerates the complement of any maximal recursively enumerable set must grow faster than any general recursive function. In 1963,
Stanley_Tennenbaum
Mathematical-logic system
again. As a concrete example, consider the factorial function F(n), recursively defined by F(n) = 1, if n = 0; else n × F(n − 1). In the lambda expression
Lambda_calculus
American mathematician and logician (1897 – 1954)
1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem
Emil_Leon_Post
Proprietary array programming language
factorial function can be implemented directly in Q as {prd 1+til x} or recursively as {$[x=0;1;x*.z.s[x-1]]} Note that in both cases the function implicitly
Q (programming language from Kx Systems)
Q_(programming_language_from_Kx_Systems)
In mathematics, a local language is a formal language for which membership of a word in the language can be determined by looking at the first and last
Local language (formal language)
Local_language_(formal_language)
Turing machine that halts for any input
{\displaystyle M_{1},M_{2},\ldots } of machines, there would be a recursively enumerable sequence T 1 , … T 2 , … {\displaystyle T_{1},\ldots T_{2},\ldots
Decider_(Turing_machine)
Proof method in mathematical logic
to prove that some proposition P(x) holds for all x of some sort of recursively defined structure, such as formulas, lists, or trees. A well-founded
Structural_induction
Problem in finite group theory
authors require the class K {\displaystyle K} to be definable by a recursively enumerable set of presentations. Throughout the history of the subject, computations
Word_problem_for_groups
Microsoft programming language
language that encompasses functional, imperative, and object-oriented programming methods. It is most often used as a cross-platform Common Language Infrastructure
F Sharp (programming language)
F_Sharp_(programming_language)
Formal grammar
straightforward to enumerate its language in increasing weight order. In particular, any nonterminal with infinite minimum weight produces the empty language. See:
Regular_tree_grammar
Branch of mathematical logic
the existence of separating sets for computationally inseparable recursively enumerable sets. In particular, one can simply write down two Σ 1 0 {\displaystyle
Reverse_mathematics
Mathematical logic concept
functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation, factorials or the Fibonacci
Gentzen's_consistency_proof
Language used to describe another language
and linguistics, a metalanguage is a language used to describe another language, often called the object language. Expressions in a metalanguage are often
Metalanguage
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RECURSIVELY ENUMERABLE-LANGUAGE