Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes...

if it is not computable. A subset S {\displaystyle S} of the natural numbers is computable if there exists a total computable function f {\displaystyle...

recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in...

the recursive numbers, effective numbers, computable reals, or recursive reals. The concept of a computable real number was introduced by Émile Borel...

"On Non-Computable Functions". One of the most interesting aspects of the busy beaver game is that, if it were possible to compute the functions Σ(n) and...

of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by...

ideas leads to the author's definition of a computable function, and to an identification of computability with effective calculability. It is not difficult...

with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability...

important property of logspace computability is that, if functions f , g {\displaystyle f,g} are logspace computable, then so is their composition g...

exponential function, and the function which returns the nth prime are all primitive recursive. In fact, for showing that a computable function is primitive...

often in discussions of computability since it demonstrates that some functions are mathematically definable but not computable. A key part of the formal...

Logic for Computable Functions (LCF) is an interactive automated theorem prover developed at Stanford and Edinburgh by Robin Milner and collaborators in...

computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in...

pairing function) are computably enumerable sets. The preimage of a computably enumerable set under a partial computable function is a computably enumerable...

total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates...

numbering of the computable functions in terms of the smn theorem and the UTM theorem. The theorem states that a partial computable function u of two variables...

partial function computable by a partial Turing machine be extended (that is, have its domain enlarged) to become a total computable function? Is it possible...

function may refer to: Recursive function (programming), a function which references itself General recursive function, a computable partial function...

In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions...

acceptable definition of a computable function defines also the same functions. General recursive functions are partial functions from integers to integers...

recognize. The domain of any universal computable function is a computably enumerable set but never a computable set. The domain is always Turing equivalent...

a Turing machine. Hypercomputers compute functions that a Turing machine cannot and which are, hence, not computable in the Church–Turing sense. Technically...

usual for such a proof, computable means computable by any model of computation that is Turing complete. In fact computability can itself be defined via...

arbitrary function with domain ω and only countably many computable functions. A specific example of a set with an enumeration but not a computable enumeration...

computability theory, a function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is sequentially computable if, for every computable sequence...

total computable functions such that the index set of P {\displaystyle P} is decidable with a promise that the input is the index of a total computable function...

Turing-equivalent if every function it can compute is also Turing-computable; i.e., it computes precisely the same class of functions as do Turing machines...

computable function that represents an increase in computational resources, one can find a resource bound such that the set of functions computable within...

can be given either in terms of effective methods or in terms of computable functions. These are generally considered equivalent per Church's thesis. Indeed...

numbering of partial computable functions. Let φ e {\displaystyle \varphi _{e}} be a computable enumeration of all partial computable functions, and W e {\displaystyle...