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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
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)
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
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
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
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
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 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
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
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
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
In mathematics, topological recursion is a recursive definition of invariants of spectral curves. It has applications in enumerative geometry, random
Topological_recursion
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
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
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
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
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
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)
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
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
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)
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
Higher-order function Y for which Y f = f (Y f)
or a for loop. Used in this way, the Y combinator implements general recursion. The lambda calculus does not have global names, for a function to refer
Fixed-point_combinator
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)
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
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
Fractal composed of triangles
Sierpiński pyramid recursion (8 steps)
Sierpiński_triangle
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
Method for aligning biological sequences
j {\displaystyle F_{0j}=d*j} F i 0 = d ∗ i {\displaystyle F_{i0}=d*i} Recursion, based on the principle of optimality: F i j = max ( F i − 1 , j − 1 +
Needleman–Wunsch_algorithm
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
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
Logical connective AND
problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic Algebraic
Logical_conjunction
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
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
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
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
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
Dialect of Lisp
iteration construct, do, but it is more idiomatic in Scheme to use tail recursion to express iteration. Standard-conforming Scheme implementations are required
Scheme_(programming_language)
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
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
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
Formula that visually represents itself when graphed
regarding the shape of the plot is stored within k {\displaystyle k} . Recursion – Process of repeating items in a self-similar way Strange loop – Cycles
Tupper's self-referential formula
Tupper's_self-referential_formula
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
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
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
O ( n ) {\displaystyle O(n)} operations per step of the recursion but only call the recursion on at most half the points each time. Let Sij be the i-th
Well-separated pair decomposition
Well-separated_pair_decomposition
Mathematical algorithm
{tr} A^{k}~;...} Observe A−1 = − Mn /c0 = (−1)n−1Mn/detA terminates the recursion at λ. This could be used to obtain the inverse or the determinant of A
Faddeev–LeVerrier_algorithm
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
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
General-purpose programming language
FORTRAN 77, many F77 compilers supported recursion as an option, and the Burroughs mainframes, designed with recursion built-in, did so by default. It became
Fortran
Term in probability theory
probability mass function satisfies the recurrence formula (the Panjer recursion) p k p k − 1 = a + b k , k = 1 , 2 , 3 , … {\displaystyle {\frac
(a,b,0) class of distributions
(a,b,0)_class_of_distributions
Integers occurring in the coefficients of the Taylor series of 1/cosh t
denotes the rising factorial. The Euler numbers can be defined by the recursion E 2 n = − ∑ k = 1 n ( 2 n 2 k ) E 2 ( n − k ) , {\displaystyle E_{2n}=-\sum
Euler_numbers
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
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
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
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
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
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 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
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
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
Mathematical function characterizing set membership
function. (See paragraph below about the use of the inverse in classical recursion theory.) Given a probability space ( Ω , F , P ) {\displaystyle \textstyle
Indicator_function
General-purpose programming language
commonly provide tail call optimization to allow for extensive use of recursion without stack overflow problems. Limitations in JVM bytecode complicate
Scala_(programming_language)
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
LEFT RECURSION
LEFT RECURSION
Biblical
requiring; lent; pit
Boy/Male
Indian
Leniecy
Boy/Male
Muslim
Leniecy
Girl/Female
Biblical
Requiring, lent, pit.
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
Indian
Lent Lily
Boy/Male
Egyptian
Left handed.
Boy/Male
Biblical
Asked, lent, a grave. Demanded, lent, ditch, death.
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
Biblical
his raiment; his left hand; his astonishment
Girl/Female
Biblical
Left hand, shut.
Boy/Male
Muslim
Lent Lily
Girl/Female
Arabic, Muslim
Left Hand Side
Boy/Male
Celtic, German, Irish, Polish
Famous Fighter; Left Handed; Renowned Fighter
Boy/Male
Indian, Jain
Cleft
Boy/Male
Arabic, Muslim
Wealthy; Towards Left; Easy
Boy/Male
Native American
Left handed.
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.
Boy/Male
Biblical
His raiment; his left hand; his astonishment.
LEFT RECURSION
LEFT RECURSION
Surname or Lastname
English (mainly Devon and Cornwall)
English (mainly Devon and Cornwall) : variant spelling of Mitchell.
Boy/Male
Arabic, Muslim
Servant of the Giver of Might and Glory
Boy/Male
Argentina, Bengali, Indian
Loved by Everyone
Male
Polish
Polish form of Latin Darius, DARIUSZ means "possesses a lot; wealthy."
Boy/Male
American, Australian, British, Danish, English, Finnish, German, Japanese, Swedish
Secret Lore
Boy/Male
American, Christian, English, French, Gaelic, German, Hindu, Indian, Irish
Small and Dark-skinned; Dark; Swarthy; Little Dark One
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Tamil, Telugu
Life; God
Boy/Male
Hebrew
God's helper.
Boy/Male
Arabic, Muslim
High; Towering; Lofty; Tall
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu
Full Moon; Moonlight
LEFT RECURSION
LEFT RECURSION
LEFT RECURSION
LEFT RECURSION
LEFT RECURSION
n.
The space or distance through which anything is lifted; as, a long lift.
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 court-leet; the district within the jurisdiction of a court-leet; the day on which a court-leet is held.
n.
The greater part or bulk of anything; as, the heft of the crop was spoiled.
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.
obs. imp.
of Let, to allow.
n.
A rise; a degree of elevation; as, the lift of a lock in canals.
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.
Alt. of Left-handiness
a.
Situated on the left; nearer the left hand than the right; as, the left-hand side; the left-hand road.
n.
A lift gate. See Lift gate, below.
a.
Incised nearly to the midrib; as, a cleft leaf.
n.
Help; assistance, as by lifting; as, to give one a lift in a wagon.
n.
A gallery or raised apartment in a church, hall, etc.; as, an organ loft.
a.
Clumsy; awkward; unlucky; insincere; sinister; malicious; as, a left-handed compliment.
n.
A piece made by splitting; as, a cleft of wood.
n.
The state or quality of being left-handed; awkwardness.
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.