The set of equations ( 5 ), ( 6 ), and the starting equation ( 7 ) is of a recursive type well suited to programming on the digital computer.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive.

However, the recursion is bounded because in each recursive application either m decreases, or m remains the same and n decreases.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

This extreme growth can be exploited to show that f, which is obviously computable on a machine with infinite memory such as a Turing machine and so is a computable function, grows faster than any primitive recursive function and is therefore not primitive recursive.

Seed AI is a hypothesized type of strong artificial intelligence capable of recursive self-improvement.

A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line segment to within a small tolerance.

The recursive equation is best introduced in a slightly more general form

Although the above recursive formula can be used for computation it is

It is not feasible to carry out such a computation using the above recursive formulae, since at least ( a constant multiple of ) p < sup > 2 </ sup > arithmetic operations would be required.

Note that this is different from the recursive languages which can be decided by an always-halting Turing machine.

Note that the set of grammars corresponding to recursive languages is not a member of this hierarchy.

Every regular language is context-free, every context-free language, not containing the empty string, is context-sensitive and every context-sensitive language is recursive and every recursive language is recursively enumerable.

A recursive definition, sometimes also called an inductive definition, is one that defines a word in terms of itself, so to speak, albeit in a useful way.

* A recursive query is one for which the DNS server will fully answer the query ( or give an error ) by querying other name servers as needed.

Using the following observation, one can create a recursive algorithm that computes x < sup > n </ sup > for an integer n using squaring and multiplication:

As one illustration, trees, ferns, cells of the nervous system, blood and lung vasculature, and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques.

⟨ X | R ⟩ of a group H, one can recursively enumerate all homomorphisms h: H → G by first enumerating all mappings h < sup >†</ sup >: X → G. Not all of these mappings extend to homomorphisms, but, since h < sup >†</ sup >( R ), is finite, it is possible to distinguish between homomorphism and non-homomorphisms by using the solution to the word problem in G. " Weeding out " non-homomorphisms gives the required recursive enumeration: h < sub > 1 </ sub >, h < sub > 2 </ sub >, ..., h < sub > n </ sub >, ....

Act II opens with Vladimir singing a recursive round about a dog, which could illustrate the cyclical nature of the play's universe, and also point toward the play's debt to the carnivalesque, music hall traditions, and vaudeville comedy ( this is only one of a number of canine references and allusions in the play ).

Identificazione di una donna ( Identification of a Woman, 1982 ), filmed in Italy, deals one more time with the recursive subjects of his Italian trilogy.

Here's an example of a recursive one:

It is an open question to give supports for a Church thesis for fuzzy mathematics the proposed notion of recursive enumerability for fuzzy subsets is the adequate one.

The list of moves for a tower being carried from one peg onto another one, as produced by the recursive algorithm has many regularities.

In 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 each such procedure usually implements one of the production rules of the grammar.

Systems explored in the book include cellular automata in one, two, and three dimensions ; mobile automata ; Turing machines in 1 and 2 dimensions ; several varieties of substitution and network systems ; primitive recursive functions ; nested recursive functions ; combinators ; tag systems ; register machines ; reversal-addition ; and a number of other systems.

This usage may seem odd, but it is possible to prove that if a group has a presentation with recursively enumerable then it has another one with recursive.

Hence, one can easily formulate the solution for finding shortest paths in a recursive manner, which is what the Bellman-Ford algorithm or the Floyd-Warshall algorithm does.

This identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle.

In other words X is if it is defined by a formula allowed to ask questions about membership in Y. Alternatively one can view the sets as those sets that can be built starting with sets recursive in Y and alternatively projecting and taking the complements of these sets up to n times.

In recursive implementations of D & C algorithms, one must make sure that there is sufficient memory allocated for the recursion stack, otherwise the execution may fail because of stack overflow.

As one of the examples used to demonstrate such reasoning, Manna's book includes a non-recursive algorithm that simulates the original ( recursive ) 91 function.

A formal derivation of the non-recursive version from the recursive one was given in a 1980 article by Mitchell Wand, based on the use of continuations.

Important features of natural language syntax such as agreement and reference are not part of the context-free grammar, but the basic recursive structure of sentences, the way in which clauses nest inside other clauses, and the way in which lists of adjectives and adverbs are swallowed by nouns and verbs, is described exactly.

A theory about some topic is usually first-order logic together with: a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things.

This comes in contrast with the direct meaning of the notion of semantic consequence, that quantifies over all structures in a particular language, which is clearly not a recursive definition.

Pascal has its roots in the ALGOL 60 language, but also introduced concepts and mechanisms which ( on top of ALGOL's scalars and arrays ) enabled programmers to define their own complex ( structured ) datatypes, and also made it easier to build dynamic and recursive data structures such as lists, trees and graphs.

The primitive recursive functions are among the number-theoretic functions, which are functions from the natural numbers ( nonnegative integers )

In the equivalence of models of computability, a parallel is drawn between Turing machines which do not terminate for certain inputs and an undefined result for that input in the corresponding partial recursive function.

Both handle cyclic, recursive and shared structures, storage / retrieval of class and metaclass info and include mechanisms for " on the fly " object migration ( i. e. to convert instances which were written by an older version of a class with a different object layout ).

The " regular expressions " in current versions of AWK and Perl are in fact extensions of regular expressions in the traditional sense, but regular expressions, unlike SNOBOL4 patterns, cannot be recursive, which gives a distinct computational advantage to SNOBOL4 patterns.

This work, along with Gödel's work on general recursive functions, established that there are sets of simple instructions, which, when put together, are able to produce any computation.

It also supports global and local variables, which permits recursive functions and subroutines to be written.

A recursive acronym ( synonymous with metacronym, recursive initialism, and recursive backronym ) is an initialism that refers to itself in the expression for which it stands.