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Theory of computer sciences
In the formal language theory of computer science, left recursion is a special case of recursion where a string is recognized as part of a language by
Left_recursion
Type of grammar for describing formal languages
parsing, left recursion must be eliminated. Direct recursion, be that left or right, is important in context-free grammars, because there recursion is the
Parsing_expression_grammar
Parsing technique
by Frost, Hafiz, and Callaghan, which do accommodate ambiguity and left recursion in polynomial time and which generate polynomial-sized representations
Top-down_parsing
Top-down parser that parses input from left to right
method, see removing left recursion. A simple example for left recursion removal: The following production rule has left recursion on E E -> E '+' T E
LL_parser
Type of parser
{\displaystyle {\texttt {'aaaa'}}} . Left recursion happens when a grammar production refers to itself as its left-most element, either directly or indirectly
Packrat_parser
Top-down parser utilizing recursion
contain left recursion. Any context-free grammar can be transformed into an equivalent grammar that has no left recursion, but removal of left recursion does
Recursive_descent_parser
Software programming optimization technique
contextual comparison is the key to accommodate indirect (or hidden) left-recursion. When performing a successful lookup in a memotable, instead of returning
Memoization
Analysing a string of symbols, according to the rules of a formal grammar
implementations of top-down parsing cannot accommodate direct and indirect left-recursion and may require exponential time and space complexity while parsing
Parsing
Higher-order function that combines several parsers
language Haskell that solve the long-standing problem of accommodating left recursion, and work as a complete top-down parsing tool in polynomial time and
Parser_combinator
Subroutine call performed as final action of a procedure
tail recursive, which is a special case of direct recursion. Tail recursion (or tail-end recursion) is particularly useful, and is often easy to optimize
Tail_call
Use of functions that call themselves
recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves
Recursion_(computer_science)
Process of repeating items in a self-similar way
Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines
Recursion
Parser generator program
SourceForge.[citation needed] ANTLR 4 deals with direct left recursion correctly, but not with left recursion in general, i.e., grammar rules x that refer to
ANTLR
Type of a context-free grammar
may or may not be LALR(1). LL grammars cannot have rules containing left recursion. Each LL(k) grammar that is ε-free can be transformed into an equivalent
LL_grammar
Computer science and linguistics concept relating to non-terminal production
Maynooth, Co. Kildare, Ireland. Moore, Robert C. (2000), "Removing Left Recursion from Context-free Grammars", Proceedings of the 1st North American Chapter
Recursive_grammar
grammar::peg Packrat Tcl Mixed All Free, BSD Grako Packrat + Cut + Left Recursion Python, C++ (beta) Separate All Free, BSD IronMeta Packrat C# Mixed
Comparison of parser generators
Comparison_of_parser_generators
Type of binary relation
and recursion on S gives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The
Well-founded_relation
In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random
Topological_recursion
Form for context-free grammars
sequence of nonterminal symbols. Observe that the grammar does not have left recursions. Every context-free grammar can be transformed into an equivalent grammar
Greibach_normal_form
Type of software bug
Notice that the function on the left must store in its stack exp number of integers, which will be multiplied when the recursion terminates and the function
Stack_overflow
Pattern defining an infinite sequence of numbers
(analysis of algorithms) Mathematical induction Orthogonal polynomials Recursion Recursion (computer science) Time scale calculus Jacobson, Nathan, Basic Algebra
Recurrence_relation
Alternate way to define a function in APL
separated by ⋄ or new-lines, wherein ⍺ denotes the left argument and ⍵ the right, and ∇ denotes recursion (function self-reference). For example, the function
Direct_function
Tool for analyzing divide-and-conquer algorithms
p(input x of size n): if n < some constant k: Solve x directly without recursion else: Create a subproblems of x, each having size n/b Call procedure p
Master theorem (analysis of algorithms)
Master_theorem_(analysis_of_algorithms)
International hacker group
Health website was taken down on Friday, 10 December around 1 AM. Lapsus$ left a message, "Contact us if you want your data back", along with their Telegram
Lapsus$
Type of algorithm in computer science
corecursion is a type of operation that is dual to (structural) recursion. Whereas recursion consumes a data structure by first handling the topmost layer
Corecursion
Recursive algorighm in linear algebra
Levinson recursion or Levinson–Durbin recursion is a procedure in linear algebra to recursively calculate the solution to an equation involving a Toeplitz
Levinson_recursion
The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable S = ∑ i = 1 N X i {\displaystyle
Panjer_recursion
Programming language
like other programming languages would overflow its stack attempting left recursion. META II uses a $ (zero or more) sequence operator. The expr parsing
META_II
In programming languages, the object or class the currently running code belongs to
recursion, and means that these methods can be overridden by derived classes or objects. By contrast, direct named recursion or anonymous recursion of
This_(computer_programming)
Divide and conquer sorting algorithm
( n / p ) ) {\displaystyle {\mathcal {O}}\left(p\log \left(n/p\right)\right)} . The expected recursion depth is O ( log ( ∑ i | S i | ) ) = O ( log
Merge_sort
Study of computable functions and Turing degrees
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated
Computability_theory
Family of higher-order functions
without reference to the recursive case (here, on its left i.e., in its first argument), then the recursion would stop. This means that while foldr recurses
Fold_(higher-order_function)
Higher-order function Y for which Y f = f (Y f)
imperative language. Used in this way, the Y combinator implements simple recursion. The lambda calculus does not allow a function to appear as a term in
Fixed-point_combinator
Divide and conquer sorting algorithm
obvious that when recursing on (lo..p) and (p+1..hi), because the left half of the recursion includes the returned index, it is the partition function's job
Quicksort
programming formalisms — such as restricted formal languages, type systems, or recursion schemes — whose expressive power coincides exactly with a given complexity
Implicit computational complexity
Implicit_computational_complexity
Proof method in mathematical logic
induction. Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical
Structural_induction
Function computable with bounded loops
composition h ∘ g 1 {\displaystyle h\circ g_{1}} is obtained. Primitive recursion operator ρ {\displaystyle \rho } : Given the k-ary function g ( x 1 ,
Primitive_recursive_function
Generalization of addition, multiplication, exponentiation, tetration, etc.
copies of }}a},\quad n\geq 2} It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann
Hyperoperation
Rewriting system and type of formal grammar
above to the earlier recursion, one gets: Axiom First recursion Second recursion Third recursion Fourth recursion Seventh recursion, scaled down ten times
L-system
Defining elements of a set in terms of other elements in the set
\end{aligned}}} This definition is valid for each natural number n, because the recursion eventually reaches the base case of 0. The definition may also be thought
Recursive_definition
Class of algorithms
approach (left) as well as iterative approach (right). Implementations in iterative approach are able to avoid the drawbacks of recursion, particularly
Tree_traversal
Web browser developed by Google
performed poorly on recursion-intensive benchmarks, such as those of Google, because the Mozilla team had not implemented recursion-tracing yet. Two weeks
Google_Chrome
Fractal composed of triangles
Sierpiński pyramid recursion (8 steps)
Sierpiński_triangle
Sierpiński–Menger type of fractal
1/3-scaled corner cubes and the central one from each cube left from the previous recursion (lighter) or by removing only corner cubes (heavier). In one
Mosely_snowflake
Programming paradigm based on objects
self, to refer to the current object. In languages that support open recursion, a method in an object can call other methods in the same object, including
Object-oriented_programming
Non-comparative lexicographical sorting algorithm
required by LSD. However, MSD sorts are more amenable to subdivision and recursion. Each bucket created by an MSD step can itself be radix sorted using the
Radix_sort
1957 technique for modelling problems of decision making under uncertainty
forward recursion or backward recursion algorithms, as outlined below. Stochastic dynamic programs can be solved to optimality by using backward recursion or
Stochastic dynamic programming
Stochastic_dynamic_programming
One of several equivalent definitions of a computable function
{\displaystyle a+b=R^{2}\left(U_{1}^{1},;S_{1}^{3}(S,U_{2}^{3})\right)} Fibonacci number McCarthy 91 function Recursion theory Recursion Recursion (computer science)
General_recursive_function
Algorithm for integer multiplication
products can be computed by recursive calls of the Karatsuba algorithm. The recursion can be applied until the numbers are so small that they can (or must)
Karatsuba_algorithm
Type of transfinite numbers
epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: ε 0 = ω ω ω ⋅ ⋅ ⋅ = sup { ω
Epsilon_number
Muran language
researchers, numerals and counting are based on recursion in the language, the absence of recursion in their language entails a lack of counting. That
Pirahã_language
Logic puzzle forming a picture in a grid
the next step of the solution even without contradictions and deeper recursion. However, finding such sets is usually as difficult as finding contradictions
Nonogram
Computation model defining an abstract machine
Kleene and J. B. Rosser by use of Church's lambda-calculus and Gödel's recursion theory (1934). Church's paper (published 15 April 1936) showed that the
Turing_machine
Coefficients coupled with angular momentum
factor is always an integer, so it remains the same when transposed to the left, and the inverse relation follows upon making the substitution m3 → −m3:
3-j_symbol
Polynomial sequence
_{5}(x)&=\left(2{\sqrt {15}}\,\pi ^{\frac {1}{4}}\right)^{-1}\,\left(4x^{5}-20x^{3}+15x\right)\,e^{-{\frac {1}{2}}x^{2}}.\end{aligned}}} Following recursion relations
Hermite_polynomials
Size of a mathematical ball
{d}{dR}}V_{n}(R)={\frac {n}{R}}V_{n}(R).} Thus, An − 1(R) inherits formulas and recursion relationships from Vn(R), such as A n − 1 ( R ) = 2 π n / 2 Γ ( n 2 )
Volume_of_an_n-ball
Limited form of tree data structure
inequality Merkle tree Optimal binary search tree Random binary tree Recursion (computer science) Red–black tree Rope (computer science) Self-balancing
Binary_tree
Formal system in mathematical logic
such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems that introduce polymorphic types (like
Simply_typed_lambda_calculus
Coefficients in angular momentum eigenstates of quantum systems
space inner product. From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. There also
Clebsch–Gordan_coefficients
Formula on random variables
Variance Propagation of error Markov chain central limit theorem Panjer recursion Inverse-variance weighting Donsker's theorem Paired difference test Klenke
Bienaymé's_identity
Degradation of AI models trained on synthetic data
Yarin; Papernot, Nicolas; Anderson, Ross (2023-05-31). "The Curse of Recursion: Training on Generated Data Makes Models Forget". arXiv:2305.17493 [cs
Model_collapse
Method in numerical analysis
_{1}(x)+a_{2}T_{2}(x)+\cdots +a_{n}T_{n}(x).} The coefficients in the recursion relation for the Chebyshev polynomials are α ( x ) = 2 x , β = − 1 , {\displaystyle
Clenshaw_algorithm
System to identify resources on a network
to excessive length. RD: 1 bit Recursion Desired, indicates if the client means a recursive query. RA: 1 bit Recursion Available, in a response, indicates
Domain_Name_System
Sequence of operations for a task
Undecidable, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An Unsolvable Problem
Algorithm
Technique for polynomial interpolation
algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences. It is similar to Aitken's algorithm
Neville's_algorithm
Quickly growing function
stack reflects the recursion depth. As the reduction according to the rules {r4, r5, r7} involves a smaller maximum depth of recursion, this computation
Ackermann_function
American mathematician and logician (1897 – 1954)
Post had been interested in astronomy, but at the age of twelve lost his left arm in a car accident. This loss was a significant obstacle to being a professional
Emil_Leon_Post
Representation theory
j(p)-j(q)=\left({1 \over p}-{1 \over q}\right)\prod _{n,m=1}^{\infty }(1-p^{n}q^{m})^{c_{nm}}} for the elliptic modular function j. Peterson gave a recursion formula
Weyl_character_formula
Finite or infinite ordered list of elements
using recursion. This is in contrast to the definition of sequences of elements as functions of their positions. To define a sequence by recursion, one
Sequence
Operations on ordinals that extend classical arithmetic
set that represents the result of the operation or by using transfinite recursion. In addition to these standard operations for ordinals, there are also
Ordinal_arithmetic
Algorithm that estimates unknowns from a series of measurements over time
above system is known as the inverse Wiener-Hopf factor. The backward recursion is the adjoint of the above forward system. The result of the backward
Kalman_filter
Algorithmic technique
(roughly) every N/2 inputs if the recursion stops at exactly n = N. By making N sufficiently large, the overhead of recursion can be made negligible (precisely
Pairwise_summation
Mathematical proposition equivalent to the axiom of choice
directly using transfinite recursion, still assuming the axiom of choice. For that, see for example Transfinite recursion theorem § Example: a basis construction
Zorn's_lemma
Term that does not contain any variables
contains no variables. Ground terms may be defined by logical recursion (formula-recursion): Elements of C {\displaystyle C} are ground terms; If f ∈ F
Ground_expression
Lithograph print by M. C. Escher
BBC television series Doctor Who. The storyline also relied heavily on recursion, a favorite theme in Escher's later and more famous works, and used ideas
Castrovalva_(M._C._Escher)
Rule system for formal languages
become enclosed by matching parentheses; and the third rule terminates the recursion. A second canonical example is two different kinds of matching nested
Context-free_grammar
Thesis on the nature of computability
functions (with arbitrarily many arguments) that is closed under composition, recursion, and minimization, and includes zero, successor, and all projections.
Church–Turing_thesis
Algorithm for finding max graph matchings
graphs and directly addresses these challenges. It first introduces a new recursion-free Blossom algorithm. In the traditional algorithm, a blossom is contracted
Blossom_algorithm
Simple Turing complete logic
combinator directly, Hxy = x(yy). This function can be used to achieve recursion. If β is the function that applies α to the self application of something
SKI_combinator_calculus
Spline function
internal knots. B-splines can be constructed by means of the Cox–de Boor recursion formula. We start with the B-splines of degree p = 0 {\displaystyle p=0}
B-spline
American theoretical physicist
Britto on a method calculating scattering amplitudes known as the BCFW recursion relations. Witten has been honored with numerous awards including a MacArthur
Edward_Witten
2010 film by Christopher Nolan
and the splinter suffix -ception, which refer to layering, nesting, or recursion, in reference to the movie's key element of a "dream within a dream".
Inception
Mathematical logic concept
complexity class containing all computably enumerable sets is RE. In recursion theory, the lattice of c.e. sets under inclusion is denoted E {\displaystyle
Computably_enumerable_set
Automated methods for the creation of mazes
of recursion – in the worst case, the routine may need to recur on every cell of the area being processed, which may exceed the maximum recursion stack
Maze_generation_algorithm
Algorithm to compare text strings using wildcard syntax
characters left to match.) Git/Rsync's wildmatch ABORT also covers invalid inputs. The new INN uwildmat does the same. Asterisk advancement in recursion. This
Matching_wildcards
Product of numbers from 1 to n
by a constant factor at each level of recursion, so the total time for these steps at all levels of recursion adds in a geometric series to O ( n log
Factorial
Algorithm that arranges lists in order
sorted; sometimes O(log n) additional memory is considered "in-place". Recursion: Some algorithms are either typically recursive or typically non-recursive
Sorting_algorithm
Type of matrix factorization
is −1. Matrix formulae to calculate rows and columns of LU factors by recursion are given in the remaining part of Banachiewicz's paper as Eq. (2.3) and
LU_decomposition
Binary tree variant
allowing tree traversal without recursion and the extra storage (proportional to the tree's depth) that recursion requires. "A binary tree is threaded
Threaded_binary_tree
Fractal named after mathematician Benoit Mandelbrot
connectedness locus has also been studied; here one considers the two-parameter recursion z ↦ z 3 + 3 k z + c {\displaystyle z\mapsto z^{3}+3kz+c} , whose two critical
Mandelbrot_set
Type of geomantic divination
interpreting a series of figures formed by a randomized process that involves recursion, followed by analyzing them, often augmented with astrological interpretations
Arabic_geomancy
Mathematical transform that expresses a function of time as a function of frequency
This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases n + 2 and n allowing to compute, e.g., the three-dimensional
Fourier_transform
Functions of an angle
previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation. Being defined as fractions
Trigonometric_functions
Functions whose execution you can pause
coroutines for state machines or concurrency is similar to using mutual recursion with tail calls, as in both cases the control changes to a different one
Coroutine
Homomorphism from an initial algebra into another algebra
nLab". Ki Yung Ahn; Sheard, Tim (2011). "A hierarchy of mendler style recursion combinators: taming inductive datatypes with negative occurrences". Proceedings
Catamorphism
Algorithm for the kth smallest element in an array
eliminating the tail recursion with a loop: function select(list, left, right, k) is loop if left = right then return list[left] pivotIndex := ... //
Quickselect
Numbers obtained by adding the two previous ones
steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization). Most identities involving Fibonacci numbers can be
Fibonacci_sequence
Programming language running on the Erlang virtual machine
Immutable data, with an emphasis, like other functional languages, on recursion and higher-order functions instead of side-effect-based looping Shared
Elixir_(programming_language)
One That I Want" from Grease.[citation needed] "recursion( see it )" shows a "Did you mean: recursion", linking to the same search over and over again
List_of_Google_Easter_eggs
Programming language that uses first order logic
called tail call optimization for deterministic predicates exhibiting tail recursion or, more generally, tail calls: A clause's stack frame is discarded before
Prolog
LEFT RECURSION
LEFT RECURSION
Boy/Male
Indian, Jain
Cleft
Girl/Female
Arabic, Muslim
Left Hand Side
Boy/Male
Arabic, Muslim
Wealthy; Towards Left; Easy
Biblical
his raiment; his left hand; his astonishment
Boy/Male
Biblical
His raiment; his left hand; his astonishment.
Girl/Female
Biblical
Requiring, lent, pit.
Girl/Female
Biblical
Left hand, shut.
Boy/Male
Egyptian
Left handed.
Boy/Male
Biblical
Asked, lent, a grave. Demanded, lent, ditch, death.
Boy/Male
Celtic, German, Irish, Polish
Famous Fighter; Left Handed; Renowned Fighter
Boy/Male
Indian
Leniecy
Surname or Lastname
English
English : from Middle English lofte ‘upper chamber’, ‘attic’, possibly bestowed on a household servant who worked in an upper chamber, or used in the same sense as Loftus.Danish : habitational name from a place called Loft.
Surname or Lastname
English
English : topographic name for someone who lived by a watercourse or road junction, Old English gelǣt, or a habitational name from Leat in Devon, or The Leete in Essex, named with this element.
Boy/Male
Native American
Left handed.
Surname or Lastname
English
English : from a short form of the medieval female personal name Lettice (see Leece 1).German : from Middle High German lette ‘clay’, ‘clayey soil’, hence a topographic name for someone who farmed on fertile clay soil.
Biblical
left hand; shut
Boy/Male
Muslim
Lent Lily
Biblical
requiring; lent; pit
Boy/Male
Indian
Lent Lily
Boy/Male
Muslim
Leniecy
LEFT RECURSION
LEFT RECURSION
Girl/Female
Indian
Great goddess.
Boy/Male
Sikh
Wisdom, Friend of the supreme
Female
Norse
Old Norse name VERÃANDI means "present; that which is happening; in the making." In mythology, this is the name of one of the three Norns, a goddess of destiny. The other two are Skuld ("future") and Urðr ("fate; that which happened").Â
Girl/Female
Muslim
Capable one
Boy/Male
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sikh, Tamil, Telugu
Blessings
Girl/Female
Hindu
Light victory wealthy
Female
Russian
(Саша) Unisex pet form of Russian Aleksandr and Aleksandra, both SASHA means "defender of mankind."
Surname or Lastname
English, German, and Dutch
English, German, and Dutch : patronymic from a pet form of Nicholas (German Nikolaus).Irish (County Limerick) : Anglicized form of Gaelic Mac Nioceais ‘son of Nicholas’, which was taken by some Limerick families named Woulfe.German (southern and Upper Rhine) : from a nickname from Middle High German nickes(e), nixe ‘water sprite’.
Girl/Female
Australian, British, Celtic, Danish, English, German, Irish, Japanese
Irish Woman; Blessed Prize
Boy/Male
Scottish
Supplanter.
LEFT RECURSION
LEFT RECURSION
LEFT RECURSION
LEFT RECURSION
LEFT RECURSION
n.
A piece made by splitting; as, a cleft of wood.
n.
A rise; a degree of elevation; as, the lift of a lock in canals.
imp. & p. p.
of Let
n.
That part of surrounding space toward which the left side of one's body is turned; as, the house is on the left when you face North.
n.
Alt. of Left-handiness
n.
A lift gate. See Lift gate, below.
a.
Clumsy; awkward; unlucky; insincere; sinister; malicious; as, a left-handed compliment.
n.
A court-leet; the district within the jurisdiction of a court-leet; the day on which a court-leet is held.
a.
Having the left hand or arm stronger and more dexterous than the right; using the left hand and arm with more dexterity than the right.
a.
Of or pertaining to that side of the body in man on which the muscular action of the limbs is usually weaker than on the other side; -- opposed to right, when used in reference to a part of the body; as, the left hand, or arm; the left ear. Also said of the corresponding side of the lower animals.
n.
Help; assistance, as by lifting; as, to give one a lift in a wagon.
n.
The greater part or bulk of anything; as, the heft of the crop was spoiled.
n.
The state or quality of being left-handed; awkwardness.
a.
Incised nearly to the midrib; as, a cleft leaf.
n.
Those members of a legislative assembly (as in France) who are in the opposition; the advanced republicans and extreme radicals. They have their seats at the left-hand side of the presiding officer. See Center, and Right.
n.
A gallery or raised apartment in a church, hall, etc.; as, an organ loft.
a.
Situated on the left; nearer the left hand than the right; as, the left-hand side; the left-hand road.
n.
The space or distance through which anything is lifted; as, a long lift.
obs. imp.
of Let, to allow.