[permalink] [id link]
In computability theory, Rogers ' equivalence theorem characterizes the Gödel numberings, or effective numberings of the set of computable functions.
from
Wikipedia
Some Related Sentences
computability and theory
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.
In computability theory, the Church – Turing thesis ( also known as the Turing-Church thesis, the Church – Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis ) is a combined hypothesis (" thesis ") about the nature of functions whose values are effectively calculable ; or, in more modern terms, functions whose values are algorithmically computable.
Proofs in computability theory often invoke the Church – Turing thesis in an informal way to establish the computability of functions while avoiding the ( often very long ) details which would be involved in a rigorous, formal proof.
Closely related fields in theoretical computer science are analysis of algorithms and computability theory.
In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kind of problems can, in principle, be solved algorithmically.
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters.
Research in computability theory has typically focused on decision problems.
Automata theory and formal language theory are closely related to computability.
Although not very successful in that respect, the lambda calculus found early successes in the area of computability theory, such as a negative answer to Hilbert's Entscheidungsproblem.
It is still used in the area of computability theory, although Turing machines are arguably the preferred model for computation.
* Shawn Hedman, A first course in logic: an introduction to model theory, proof theory, computability, and complexity, Oxford University Press, 2004, ISBN 0-19-852981-3.
Covers logics in close relation with computability theory and complexity theory
In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems.
* Post correspondence problem, an important problem in computability theory
In the theory of formal languages in computability theory, a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of times, with the resulting string remaining in that language.
In computability theory, primitive recursive functions are a class of functions that form an important building block on the way to a full formalization of computability.
computability and Rogers
In computability theory, a Turing reduction from a problem A to a problem B, is a reduction which solves A, assuming B is already known ( Rogers 1967, Soare 1987 ).
In computability theory the s < sub > mn </ sub > theorem, ( also called the translation lemma, parameter theorem, or parameterization theorem ) is a basic result about programming languages ( and, more generally, Gödel numberings of the computable functions ) ( Soare 1987, Rogers 1967 ).
computability and equivalence
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.
computability and theorem
In computability theory, Rice's theorem states that, for any non-trivial property of partial functions, there is no general and effective method to decide whether an algorithm computes a partial function with that property.
Another way of stating Rice's theorem that is more useful in computability theory follows.
Another important step in computability theory was Rice's theorem, which states that for all non-trivial properties of partial functions, it is undecidable whether a Turing machine computes a partial function with that property.
The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic, especially in computability theory.
In the context of second-order arithmetic, results such as Post's theorem establish a close link between the complexity of a formula and the ( non ) computability of the set it defines.
The diagonal lemma is closely related to Kleene's recursion theorem in computability theory, and their respective proofs are similar.
In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.
The same definition of recursive function can be given, in computability theory, by applying Kleene's recursion theorem.
In computability theory the Myhill isomorphism theorem, named after John Myhill, provides a characterization for two numberings to induce the same notion of computability on a set.
In computability theory the utm theorem, or Universal Turing machine theorem, is a basic result about Gödel numberings of the set of computable functions.
computability and Gödel
In computability theory, the term " Gödel numbering " is used in settings more general than the one described above.
Important numberings are the Gödel numbering of the terms in first-order predicate calculus and numberings of the set of computable functions which can be used to apply results of computability theory on the set of computable functions itself.
In computability theory complete numberings are generalizations of Gödel numbering first introduced by A. I.
computability and numberings
Computably isomorphic numberings induce the same notion of computability on a set.
computability and effective
Rosser 1939 addresses the notion of " effective computability " as follows: " Clearly the existence of CC and RC ( Church's and Rosser's proofs ) presupposes a precise definition of " effective ".
The development ... leads to ... an identification of computability < sup >†</ sup > with effective calculability.
Post strongly disagreed with Church's " identification " of effective computability with the λ-calculus and recursion, stating:
In it he stated another notion of " effective computability " with the introduction of his a-machines ( now known as the Turing machine abstract computational model ).
An attempt to understand the notion of " effective computability " better led Robin Gandy ( Turing's student and friend ) in 1980 to analyze machine computation ( as opposed to human-computation acted out by a Turing machine ).
Other formalisms ( besides recursion, the λ-calculus, and the Turing machine ) have been proposed for describing effective calculability / computability.
Because all these different attempts at formalizing the concept of " effective calculability / computability " have yielded equivalent results, it is now generally assumed that the Church – Turing thesis is correct.
The fundamental results the researchers obtained established Turing computability as the correct formalization of the informal idea of effective calculation.
Indeed, the proof that a logical system or theory is undecidable will use the formal definition of computability to show that an appropriate set is not a decidable set, and then invoke Church's thesis to show that the theory or logical system is not decidable by any effective method ( Enderton 2001, pp. 206ff.
Note that the effective computability of these functions does not imply that they can be efficiently computed ( i. e. computed within a reasonable amount of time ).
Theory of recursive functions and effective computability.
Theory of recursive functions and effective computability.
0.177 seconds.