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Formal language in mathematics and computer science
science, a recursive (or decidable) language is a recursive subset of the Kleene closure of an alphabet. Equivalently, a formal language is recursive if there
Recursive_language
Formal language
In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable,
Recursively enumerable language
Recursively_enumerable_language
Relationship between language and human evolution
recursive elements of language such as spatial prepositions The merger of this ability with their parents' non-recursive language to create recursive
Origin_of_language
Process of repeating items in a self-similar way
references can occur. A process that exhibits recursion is recursive. Video feedback displays recursive images, as does an infinity mirror. In mathematics and
Recursion
Process where AI repeatedly improves itself
Recursive self-improvement (RSI) is a process in which an artificial intelligence (AI) system iteratively designs a more capable successor. This may trigger
Recursive_self-improvement
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Use of functions that call themselves
solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own
Recursion_(computer_science)
Hierarchy of classes of formal grammars
every context-free language is context-sensitive, every context-sensitive language is recursive and every recursive language is recursively enumerable. These
Chomsky_hierarchy
Computational learning model
primitive recursive function of the current step number, and the learner encodes a language guess as a program that enumerates the language i.e. the class
Language identification in the limit
Language_identification_in_the_limit
Acronym whose expansion includes a copy of itself
A recursive acronym is an acronym that refers to itself, and appears most frequently in computer programming. The term was first used in print in 1979
Recursive_acronym
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
Island or lake that is itself within an island or lake
A recursive island or lake, also known as a nested island or lake, is an island that lies inside a lake, or vice versa. For the purposes of defining recursion
Recursive_islands_and_lakes
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
Type of a context-free grammar
characterized as precisely those that can be parsed by a predictive parser – a recursive descent parser without backtracking – and these can be readily written
LL_grammar
Set with algorithmic membership test
}}\,x\in S,\\0&{\text{if }}\,x\not \in S.\end{cases}}} Every recursive language is computable. Every finite or cofinite subset of the natural numbers
Computable_set
Text-string-oriented programming language
unlike SNOBOL4 patterns, are not recursive, which gives a distinct computational advantage to SNOBOL4 patterns. (Recursive expressions did appear in Perl
SNOBOL
Structured system of communication
exceedingly complex meanings. It is distinguished by the property of recursivity: for example, a noun phrase can contain another noun phrase (as in "[[the
Language
Language defined by context-sensitive grammar
Formal Languages by A. Salomaa, page 14, Example 2.5). An example of recursive language that is not context-sensitive is any recursive language whose decision
Context-sensitive_language
Procedural programming language
B is a programming language developed at Bell Labs circa 1969 by Ken Thompson and Dennis Ritchie. B was designed for recursive, non-numeric, machine-independent
B_(programming_language)
American AI company (founded 2024)
coding agent harness maintained by Earendil Works. Built around a Recursive Language Model (RLM) architecture and a Continual Harness, it uses a persistent
Prime_Intellect
Subroutine call performed as final action of a procedure
target of a tail is the same subroutine, the subroutine is said to be tail recursive, which is a special case of direct recursion. Tail recursion (or tail-end
Tail_call
Model of computational complexity
that compute them. A related notion is the circuit complexity of a recursive language that is decided by a uniform family of circuits C 1 , C 2 , … {\displaystyle
Circuit_complexity
Computational problems no algorithm can solve
an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language. There are uncountably many undecidable problems
List_of_undecidable_problems
Top-down parser utilizing recursion
computer science, a recursive descent parser is a kind of top-down parser built from a set of mutually recursive procedures (or a non-recursive equivalent) where
Recursive_descent_parser
Programming language
LOOP is a simple register language designed to precisely capture the primitive recursive functions. The language is derived from the counter-machine model
LOOP_(programming_language)
General purpose functional programming language
function will return a function that will accept the second, and so on. Recursive functions require letrec so the function name is in scope within its body
ML_(programming_language)
Type of artificial intelligence model
parameters. Tiny Recursive Model by Samsung AI Labs Montréal "Advanced" (on-device) Apple Foundation Models by Apple Traditional AI language systems need
Small_language_model
Statically typed functional programming language
factorial(x: Int) -> Int { // The public function calls the private tail recursive function factorial_loop(x, 1) } fn factorial_loop(x: Int, accumulator:
Gleam_(programming_language)
decision problems that can be solved in time bounded by an elementary recursive function. Equivalently, these are the problems that can be solved in time
ELEMENTARY
Mathematical function that can be computed by a program
being Turing machines, register machines, lambda calculus and general recursive functions. Although these four are of a very different nature, they provide
Computable_function
Computer programming standard meant to be interoperable across programming languages
code ISO/IEC 10967 ISO/IEC 11404 Language interoperability Pivot language Recursive transcompiling Specification language Transcompiler "Information technology
Language-independent specification
Language-independent_specification
Mathematical logic concept
a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable
Computably_enumerable_set
Muran language
to be used as a phone in only this language. An extremely limited clause structure, not allowing for nested recursive sentences like "Mary said that John
Pirahã_language
Programming language with hardware abstraction
lower level than most other languages. C supports constructs such as expression evaluation, parameterized and recursive functions, data types and structures
High-level programming language
High-level_programming_language
Programming language family
Scheme is a statically scoped and properly tail-recursive dialect of the Lisp programming language invented by Guy L. Steele, Jr. and Gerald Jay Sussman
Lisp_(programming_language)
Large language family mostly of Southeast Asia and the Pacific
probable Sino-Tibetan homeland. Sagart also groups the Austronesian languages in a recursive-like fashion, placing Kra-Dai as a sister branch of Malayo-Polynesian
Austronesian_languages
general recursive fixed-point queries, which compute transitive closures. In standard SQL:1999 hierarchical queries are implemented by way of recursive common
Hierarchical and recursive queries in SQL
Hierarchical_and_recursive_queries_in_SQL
Processing of natural language by a computer
Natural language processing (NLP) is the processing of natural language information by a computer. NLP is a subfield of computer science and is closely
Natural_language_processing
Quickly growing function
examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function
Ackermann_function
Recursive visual effect
Droste effect (Dutch pronunciation: [ˈdrɔstə]) is the effect of a picture recursively appearing within itself, in a place where a similar picture would realistically
Droste_effect
Turing machine that halts for any input
is a member of a formal language. The class of languages that can be decided by such machines is the set of recursive languages. Given an arbitrary Turing
Decider_(Turing_machine)
Indo-Aryan language spoken in Pakistan
have noted their existence in Multani and have variously called them "recursives" or "injectives", while Grierson incorrectly treated them as "double consonants"
Saraiki_language
and that every recursive language is a unique solution of some equation. For a one-letter alphabet, Leiss discovered the first language equation with a
Language_equation
American AI entrepreneur (born 1983)
Amodei has also called for a slowdown of AI development, especially recursive self-improvement by democratic states, while engaging China in arms control-analogous
Dario_Amodei
Process in which a first language is being acquired
principle called recursion. Evidence suggests that every individual has three recursive mechanisms that allow sentences to go indeterminately. These three mechanisms
Language_acquisition
Complexity class consisting of all recursive languages
solvable by a Turing machine, which is the set of all recursive languages (also called decidable languages). R is equivalent to the set of all total computable
R_(complexity)
Problem in mathematics and theoretical computer science
However, the language S(x) may not even be a recursive language, since there are uncountably many such x, but only countably many recursive languages. A function
Semi-membership
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
Dialect of Lisp
programming and associated techniques such as recursive algorithms. It was also one of the first programming languages to support first-class continuations. It
Scheme_(programming_language)
Randomized polynomial time class of computational complexity theory
class R, although this name is more commonly used for the class of recursive languages. If the correct answer is YES and the algorithm is run n times with
RP_(complexity)
Hypothetical event
and recursive self-improvement". Substantial dangers are associated with an intelligence explosion singularity originating from a recursively self-improving
Technological_singularity
Ability to solve a problem by an effective procedure
widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally
Computability
Yes-or-no question that cannot ever be solved by a computer
called decidable or effectively solvable if the formalized set of A is a recursive set. Otherwise, A is called undecidable. A problem is called partially
Undecidable_problem
Family of higher-order functions
function that analyzes a recursive data structure and, through use of a given combining operation, recombines the results of recursively processing its constituent
Fold_(higher-order_function)
expressions. Other classes of languages that are closed under alternation include context-free languages and recursive languages. The vertical bar notation
Alternation (formal language theory)
Alternation_(formal_language_theory)
Sequence of program instructions invokable by other software
complex problems. Recursive languages provide a new copy of local variables on each call. If the programmer desires the recursive callable to use the
Function (computer programming)
Function_(computer_programming)
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
Microsoft programming language
let printList2 lst = List.iter (printfn "%d") lst /// Iteration using a recursive function and pattern matching let rec printList3 lst = match lst with
F Sharp (programming language)
F_Sharp_(programming_language)
Study of computable functions and Turing degrees
methods, and formal languages. The study of which mathematical constructions can be effectively performed is sometimes called recursive mathematics. Computability
Computability_theory
Defining elements of a set in terms of other elements in the set
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements
Recursive_definition
Constructed alien languages
complicated nature of center embedding in a sentence: The lack of large scale recursive center embedding has been claimed to be an issue of human short-term memory
Heptapod_languages
is an extremely simple E program: println("Hello, world!") Here is a recursive function for computing the factorial of a number, written in E. Functions
E_(programming_language)
Software programming optimization technique
gains), such as in simple mutually recursive descent parsing. In the context of some logic programming languages, memoization is also known as tabling
Memoization
Type of grammar for describing formal languages
closer to how string recognition tends to be done in practice, e.g. by a recursive descent parser. Unlike CFGs, PEGs cannot be ambiguous; a string has exactly
Parsing_expression_grammar
Data type defined by combining other types
datatype is recursive, the entire sum of products is wrapped in a recursive type, and each constructor also rolls the datatype into the recursive type. For
Algebraic_data_type
Mathematical-logic system
is M; this means a recursive function definition cannot be written with let. The letrec construction would allow writing recursive function definitions
Lambda_calculus
Scripting language created in 1994
parser in 1997 and formed the base of PHP 3, changing the language's name to the recursive acronym PHP: Hypertext Preprocessor. Public testing of PHP
PHP
Digital piracy and destructive book scanning operation
Panama aims to develop a dataset of books for training Anthropic's large language models. Prior to Project Panama, Anthropic used pirated books and was sued
Project_Panama
Functional programming language
0 = 0; fib 1 = 1; fib n = fib (n-2) + fib (n-1) if n>1; Better (tail-recursive and linear-time) version: fib n = fibs (0,1) n with fibs (a,b) n = if
Pure_(programming_language)
Relational database programming language
understanding and maintenance, and it is more expressive, in particular for recursive queries. HTSQL: URL based query method IBM Business System 12 (IBM BS12):
SQL
Programming paradigm based on applying and composing functions
allow recursive functions. Lisp first introduced many paradigmatic features of functional programming, though early Lisps were multi-paradigm languages, and
Functional_programming
KEE Kent Recursive Calculator (KRC) KiXtart Klerer-May System Knowledge Interchange Format (KIF) Kodu Kojo Kotlin KRL KRL (KUKA Robot Language) KRYPTON
List_of_programming_languages
Allowable structure and composition of phrases and sentences of a metalanguage
certain formal metalanguages used for describing recursive languages (formally called context-free languages) that have terminals, nonterminals, and metasymbols
Metasyntax
Chinese text-to-video model
Beijing-based technology company Kuaishou. Kling generates videos from natural language descriptions, called prompts. Kling AI was developed by Kuaishou, a Chinese
Kling_AI
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)
anonymous routines, a recursive typing system with higher-order functions, etc.; not only the context-free part, but the full language syntax and semantics
History of programming languages
History_of_programming_languages
Programming language
means 'recursive'. *) match integers with | [] -> 0 (* Yield 0 if integers is the empty list []. *) | first :: rest -> first + sum rest;; (* Recursive call
OCaml
Higher-order function which returns some fixed point of the input function
the lambda calculus and in functional programming languages, and provide a means to allow for recursive definitions. Applied to a non-constant function
Fixed-point_combinator
Public recursive DNS resolver based in Switzerland
Quad9 is a global public recursive DNS resolver that blocks access to domains associated with malware and phishing. Quad9 is operated by the Quad9 Foundation
Quad9
Data type that refers to itself in its definition
programming, a recursive data type is a data type whose definition contains values of the same type. It is also known as a recursively defined, inductively
Recursive_data_type
General-purpose programming language
types of recursive functions), e.g. def formatApples(x: Int) = "I ate %d apples".format(x) or (with a return type declared for a recursive function)
Scala_(programming_language)
Statistical model of language
Archived from the original on 30 October 2020. Retrieved 25 February 2019. "Recursive Deep Models for Semantic Compositionality Over a Sentiment Treebank".
Language_model
2026 software security vulnerability
5 that allow remote code execution (RCE). PHP is a popular programming language on which many website and web application codebases are built; these vulnerabilities
PHP2Shell
Specification of a mathematical group by generators and relations
then call a subset U of FS recursive (respectively recursively enumerable) if f(U) is recursive (respectively recursively enumerable). If S is indexed
Presentation_of_a_group
Complex animal communication
Brecht, Katharina F.; Johnston, Melissa; Nieder, Andreas (2022-11-04). "Recursive sequence generation in crows". Science Advances. 8 (44) eabq3356. Bibcode:2022SciA
Animal_language
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
Programming style in which control is passed explicitly
349 (1975), which sets out the first version of the Scheme programming language. John C. Reynolds gives a detailed account of the many discoveries of continuations
Continuation-passing_style
Prototype-based programming language
:= Object clone // creates a new, empty object named "A" A simple non-recursive factorial function, in Io: factorial := method(n, if(n == 0, return 1)
Io_(programming_language)
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)
A recursive transition network ("RTN") is a graph theoretical schematic used to represent the rules of a context-free grammar. RTNs have application to
Recursive_transition_network
General-purpose functional programming language
(* corners *) Note that a type synonym cannot be recursive; datatypes are necessary to define recursive constructors. (This is not at issue in this example
Standard_ML
Thesis on the nature of computability
with Jacques Herbrand, formalized the definition of the class of general recursive functions: the smallest class of functions (with arbitrarily many arguments)
Church–Turing_thesis
General-purpose programming language
declarations of objects with construct scope Recursive allocatable components – as an alternative to recursive pointers in derived types The Final Draft
Fortran
3-volume treatise on mathematics, 1910–1913
theory specifies the rules of syntax (rules of grammar) usually as a recursive definition that starts with "0" and specifies how to build acceptable
Principia_Mathematica
October 2026 in information security
infrastructure of LiteLLM, an open-source gateway for accessing multiple large language model providers, was compromised in a software supply chain attack. Attackers
Black October (information security)
Black_October_(information_security)
Multi-paradigm system programming language
functional programming features such as function literals, closures, recursively-immutable objects and the use of higher-order functions. There are two
D_(programming_language)
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
Markup language and file format
sequence using the iterator design pattern. This allows for writing of recursive descent parsers in which the structure of the code performing the parsing
XML
Programming paradigm based on objects
between open recursion and encapsulated state. Researchers have used recursive types and co-algebraic data types to incorporate essential features of
Object-oriented_programming
Programming language
All assignments and function arguments create copies. This means that recursive data structures and cycles are impossible. Like C, functions can modify
Sawzall (programming language)
Sawzall_(programming_language)
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RECURSIVE LANGUAGE
RECURSIVE LANGUAGE
RECURSIVE LANGUAGE
RECURSIVE LANGUAGE
RECURSIVE LANGUAGE
RECURSIVE LANGUAGE
RECURSIVE LANGUAGE
RECURSIVE LANGUAGE
RECURSIVE LANGUAGE
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