Church subsequently modified his methods to include use of Herbrand – Gödel recursion and then proved ( 1936 ) that the Entscheidungsproblem is unsolvable: There is no generalized " effective calculation " ( method, algorithm ) that can determine whether or not a formula in either the recursive-or λ-calculus is " valid " ( more precisely: no method to show that a well formed formula has a " normal form ").

This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

The original algorithm was described only for natural numbers and geometric lengths ( real numbers ), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials in one variable.

The Euclidean algorithm has been generalized further to other mathematical structures, such as knots and multivariate polynomials.

generalized the algorithm to digital waveguide synthesis, which could also be used to model acoustic waves in tubes and on drum membranes.

This generalized version of the game is NP-complete ; it is unlikely that any algorithm more efficient than a brute-force search exists that can find solutions for arbitrary generalized FreeCell configurations.

replaced this with a more efficient idea that later generalized as the " subgroup algorithm " ( in which form it works just as well for permutations and rotations ).

The Rete algorithm provides a generalized logical description of an implementation of functionality responsible for matching data tuples (" facts ") against productions (" rules ") in a pattern-matching production system ( a category of rule engine ).

Its original version, due to Gary L. Miller, is deterministic, but the determinism relies on the unproven generalized Riemann hypothesis ; Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm.

One drawback is that, like Freenet, DHTs only directly support exact-match search, rather than keyword search, although Freenet's routing algorithm can be generalized to any key type where a closeness operation can be defined.

A generalized algorithm to correct for the pressure level difference between the finger and brachial sites within an individual patient was developed and this correction worked under all circumstances that it was tested, even when it was not designed for it, since it applied general physiological principles.

Bruun's algorithm is a fast Fourier transform ( FFT ) algorithm based on an unusual recursive polynomial-factorization approach, proposed for powers of two by G. Bruun in 1978 and generalized to arbitrary even composite sizes by H. Murakami in 1996.

The Bruun factorization, and thus the Bruun FFT algorithm, was generalized to handle arbitrary even composite lengths, i. e. dividing the polynomial degree by an arbitrary radix ( factor ), as follows.

Lloyd's Method I algorithm, originally described in 1957, can be generalized in a straighforward way for application to vector data.

The generalized Euclidean algorithm identifies the greatest common right or left divisor of two Hurwitz quaternions, where the " size " of the remainder is measured by the norm.

While Peterson's original formulation worked with only two processes, the algorithm can be generalized for more than two.

A generalized Feistel algorithm can be used to create strong permutations on small domains of size not a power of two ( see Format-Preserving Encryption ).

This is further generalized by DNA sequence alignment algorithms such as the Smith – Waterman algorithm, which make an operation's cost depend on where it is applied.

In 1992, Deutsch and Jozsa produced a deterministic algorithm which was generalized to a function which takes bits for its input.

" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.

Since algorithms are platform-independent ( i. e. a given algorithm can be implemented in an arbitrary programming language on an arbitrary computer running an arbitrary operating system ), there are significant drawbacks to using an empirical approach to gauge the comparative performance of a given set of algorithms.

When the condition number is exactly one, then the algorithm may find an approximation of the solution with an arbitrary precision.

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

Bruun's algorithm applies to arbitrary even composite sizes.

The choice of which element to remove from the input is arbitrary, and can be made using almost any choice algorithm.

An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.

Given unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church – Turing thesis.

It is important to realize that the word problem is in fact solvable for many groups G. For example, polycyclic groups have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable ; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the Todd – Coxeter algorithm and the Knuth – Bendix completion algorithm.

In object-oriented graphics, the image is described indirectly by an object endowed with a self-rendering method — a procedure which assigns colors to the image pixels by an arbitrary algorithm.

This is asymptotically the fastest known single-source shortest-path algorithm for arbitrary directed graphs with unbounded nonnegative weights.

The Markov chain is started from an arbitrary initial value and the algorithm is run for many iterations until this initial state is " forgotten ".

" Informally, hypothesis boosting problem asks whether an efficient learning algorithm that outputs an hypothesis whose performance is only slightly better than random guessing a weak learner implies the existence of an efficient algorithm that outputs an hypothesis of arbitrary accuracy a strong learner.

Conversely, an algorithm to test for solvability in arbitrary integers could be used to test a given equation for solvability in natural numbers by applying that supposed algorithm to the equation obtained from the given equation by replacing each unknown by the sum of the squares of four new unknowns.

Using this method, he showed how to solve the Hamiltonian cycle problem in arbitrary n-vertex graphs by a Monte Carlo algorithm in time O ( 1. 657 < sup > n </ sup >); for bipartite graphs this algorithm can be further improved to time O ( 1. 414 < sup > n </ sup >).

This Bailey – Borwein – Plouffe formula permits one to calculate binary or hexadecimal digits of pi beginning at an arbitrary position, by means of a simple algorithm.

Furthermore, a function does need not be described by a formula, expression, or algorithm, nor need it deal with numbers at all: the domain and codomain of a function may be arbitrary sets.

The Blahut – Arimoto algorithm, co-invented by Richard Blahut, is an elegant iterative technique for numerically obtaining rate – distortion functions of arbitrary finite input / output alphabet sources and much work has been done to extend it to more general problem instances.

Run-time efficiency is a topic of great interest in computer science: A program can take seconds, hours or even years to finish executing, depending on which algorithm it implements ( see also performance analysis, which is the analysis of an algorithm's run-time in practice ).

This point should be taken in consideration for implementations with a different number of rounds, as even though it increases security against an exhaustive attack, it weakens the security guaranteed by the algorithm.

Huffman coding is the most known algorithm for deriving prefix codes, so prefix codes are also widely referred to as " Huffman codes ", even when the code was not produced by a Huffman algorithm.

( membership problem is even polynomially decidable-see CYK algorithm and Earley's Algorithm )

The condition number may also be infinite, in which case the algorithm will not reliably find a solution to the problem, not even a weak approximation of it ( and not even its order of magnitude ) with any reasonable and provable accuracy.

It was noted by Biham and Shamir that DES is surprisingly resistant to differential cryptanalysis, in the sense that even small modifications to the algorithm would make it much more susceptible.

Since NC contains NL, it is also unknown whether a space-efficient algorithm for computing the GCD exists, even for nondeterministic Turing machines.

Using the Euclidean algorithm is a simple method that can even be performed without a calculator.

The " trick " that allows lossless compression algorithms, used on the type of data they were designed for, to consistently compress such files to a shorter form is that the files the algorithms are designed to act on all have some form of easily modeled redundancy that the algorithm is designed to remove, and thus belong to the subset of files that that algorithm can make shorter, whereas other files would not get compressed or even get bigger.

On the other hand, it has also been proven that there is no algorithm to determine whether a file is incompressible in the sense of Kolmogorov complexity ; hence, given any particular file, even if it appears random, it's possible that it may be significantly compressed, even including the size of the decompressor.

But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating ( T is a partial function ), so this fails to produce the required bijection.

But if the delay of the output ( relative to the input ) is bounded during a process that operates over an unlimited time, then that signal processing algorithm is real-time, even if the throughput delay may be very long.

This makes 2-4 trees an important tool for understanding the logic behind red – black trees, and this is why many introductory algorithm texts introduce 2-4 trees just before red – black trees, even though 2-4 trees are not often used in practice.

There are also search methods designed for quantum computers, like Grover's algorithm, that are theoretically faster than linear or brute-force search even without the help of data structures or heuristics.

Similarly, even when a sensible description of a particular " average case " ( which will probably only be applicable for some uses of the algorithm ) is possible, they tend to result in more difficult to analyse equations.

For example, when analysing an algorithm, it may be possible to find the longest possible path through the algorithm ( by considering the maximum number of loops, for instance ) even if it is not possible to determine the exact input that would generate this path ( indeed, such an input may not exist ).

In protest against legislation that prohibits publication of copy protection circumvention code in countries that implement the WIPO Copyright Treaty ( such as the United States ' Digital Millennium Copyright Act ), some have devised clever ways of distributing descriptions of the DeCSS algorithm, such as through steganography, through various Internet protocols, on t-shirts and in dramatic readings, as MIDI files, as a series of haiku poems, and even as a so-called illegal prime number.