Recursion

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Recursion is a technique to define a function, or process of repeating objects, in a self-similar way. In computer science it is a method or algorithm where the solution to a problem depends on solutions to smaller instances of the same problem.

Sample

In the Tower of Hanoi puzzle to move N disks from peg A to C can be reduced to three sub problems:

  1. Moving N-1 disks from peg A to intermediate peg B
  2. Move the largest Disk N from peg A to target C
  3. Finally move the N-1 parked disks from B to C
Tower of Hanoi.gif

Recursive Definitions

A recursive definition of an object refers inductive terms of itself. A function set need to specify the function for some discrete values like zero, one or empty (base case), and to reduce all other cases by divide and conquer toward the base case. Recurrence relation is an equation that recursively defines a sequence of symbols or numbers [2].

Some more or less computer chess related examples ...

PGN Syntax

The BNF-like Syntax of the Portable Game Notation by Steven Edwards contains some recursive definitions, most tail recursion for iteration of games, and tag-pairs and elements inside a game [3] :

<PGN-database> ::= <PGN-game> <PGN-database>
                   <empty>

<tag-section> ::= <tag-pair> <tag-section>
                  <empty>

<element-sequence> ::= <element> <element-sequence>
                       <recursive-variation> <element-sequence>
                       <empty>

<recursive-variation> ::= ( <element-sequence> )

Index of PV-Array

An index of a Triangular PV-Table may be defined recursively that way ...

index(0) = 0
index(ply+1) = index(ply) + N - ply

...which can be trivially transformed from tail recursion to iteration, finally applying a summation:

index(ply) = ½ ply (2N + 1 - ply )

Population Count

The recursive definition of population count (number of one-bits in a computer word i. e. a bitboard) can be transformed to iteration as well, but lacks an arithmetical sum-formula:

popcnt(0) = 0
popcnt(n) = popcnt(n ÷ 2) + (n mod 2)

Infix Expression

Formal syntax of programming languages likely contain recursive definitions, i. e. parsing of arithmetical (and/or boolean) infix notation therefor implies indirect recursion, as demonstrated in following example with constants and elementary arithmetic only, which might even evaluated at compile time:

Terminal and none terminal symbols of a variant of BNF below are embedded in ' ' resp. < >.

<expression> ::= <term>   [ {'+' | '-'} <term> ]...
      <term> ::= <factor> [ {'*' | '/'} <factor> ]...
    <factor> ::= <constant> | '(' <expression> ')'

Implementation

In procedual programming, recursion is done by a procedure aka subroutine, method or function, which calls itself if no base case is determined, utilizing the processor stack. One has to take care the nesting is not too deep or infinite, which yields in a Stack overflow and abnormal program termination or crashing.

Recursive Data types

Recursive data types contain references (i. e. pointer in C) to objects of the same type to build directed graphs, such as linked lists or trees.

Recursive Algorithms

Searching

Tree-like data structures which are traversed in depth-first manner can be implemented with recursion using a stack of nodes. Minimax and alpha-beta are typical examples of indirect recursive routines, where Min calls Max and Max calls Min, and Negamax turns the indirect recursion to a direct one. While tail recursion or primitive recursion can easily turned into iterative solutions, it is more complicated for not primitive recursion. However, recursion can turned to a non-recursive version based on the use of continuations, see Iterative Search.

Knuth and Moore already introduced an iterative solution of Alpha-Beta in 1975 [4] :

It is interesting to convert this recursive procedure to an iterative (non-recursive) form by a sequence of mechanical transformations, and to apply simple optimizations which preserve program correctness. The resulting procedure is surprisingly simple, but not as easy to prove correct as the recursive form.

So called recursive pruning, especially null move pruning, or extensions refers to the fact that these events may occur multiple times inside a variation or path of the (recursive) search process.

Sorting

A well-known sorting algorithm is Quicksort developed in 1960 by C. A. R. Hoare. It recursively partitions and sorts two sub-lists from a list, whose elements are either less and greater a chosen pivot element. However, for move ordering to sort move lists in alpha-beta, most chess programmer use a selection sort to pick a move with best assigned score, since the effort to sort other moves may needless in case of a beta-cutoff.

Sorting quicksort anim.gif

Visualization of the quicksort algorithm [5]

See also

Recursive vs. Iterative Search

Publications

Forum Posts

External Links

Recursion

Recursive Functions

Primitive recursive function from Wikipedia
Leaf subroutine from Wikipedia
Tail call from Wikipedia

Ackermann Function

McCarthy 91-Function

Introduced by Zohar Manna, Amir Pnueli and John McCarthy in 1970.

Tak

Self-reference

Droste cacao 100gr blikje, foto 02.JPG
Escher and the Droste effect - Universiteit Leiden » M. C. Escher
Serpiente alquimica.jpg
MagrittePipe.jpg

Fractals

Hilbert curve from Wikipedia » David Hilbert
Koch snowflake from Wikipedia » Helge von Koch
Von Koch curve.gif
Peano curve from Wikipedia » Giuseppe Peano
Sierpinski triangle from Wikipedia » Wacław Sierpiński

Misc

Chick Corea, Bill Connors, Stanley Clarke, Lenny White
Chick Corea, Al Di Meola, Stanley Clarke, Lenny White

References

  1. Tetrahedron in Bottrop, North Rhine-Westphalia, Germany, part of The Industrial Heritage Trail, Image by Gerd Isenberg, April 08, 2017. The design is reminiscent of the Sierpinski tetrix : placing four half-size tetrahedra corner to corner and adding an octahedron in the middle, a full-size tetrahedron is formed; this process can be repeated recursively to form larger and larger tetrahedra, from Tetrahedron in Bottrop from Wikipedia, Category:Tetraeder Bottrop - Wikimedia Commons, Halden im Ruhrgebiet (German)
  2. Hofstadter sequence from Wikipedia
  3. Standard: Portable Game Notation Specification and Implementation Guide by Steven Edwards, 18: Formal syntax
  4. Donald Knuth, Ronald W. Moore (1975). An analysis of alpha-beta pruning. Artificial Intelligence, Vol. 6, No. 4
  5. Quicksort from Wikipedia
  6. Metamathematics from Wikipedia
  7. Results of the KDD cup 2009

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