Search results
Results from the WOW.Com Content Network
The set being enumerated is then called recursively enumerable (or computably enumerable in more contemporary language), referring to the use of recursion theory in formalizations of what it means for the map to be computable. In this sense, a subset of the natural numbers is computably enumerable if it is the range of a computable function. In ...
In computer programming, an enumerated type (also called enumeration, enum, or factor in the R programming language, and a categorical variable in statistics) is a data type consisting of a set of named values called elements, members, enumeral, or enumerators of the type.
If A and B are computably enumerable sets then A ∩ B, A ∪ B and A × B (with the ordered pair of natural numbers mapped to a single natural number with the Cantor pairing function) are computably enumerable sets. The preimage of a computably enumerable set under a partial computable function is a computably enumerable set.
A real number is computable if and only if the set of natural numbers it represents (when written in binary and viewed as a characteristic function) is computable. The set of computable real numbers (as well as every countable, densely ordered subset of computable reals without ends) is order-isomorphic to the set of rational numbers.
Harrington gave a further example of an automorphic property: that of the creative sets, the sets which are many-one equivalent to the halting problem. Besides the lattice of computably enumerable sets, automorphisms are also studied for the structure of the Turing degrees of all sets as well as for the structure of the Turing degrees of c.e. sets.
Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers. Determining whether a given initial point with rational coordinates is periodic, or whether it lies in the basin of attraction of a given open set, in a piecewise-linear iterated map in two dimensions ...
Matiyasevich's theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says: . Every computably enumerable set is Diophantine, and the converse.. A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever.
The set of all provable sentences in an effective axiomatic system is always a recursively enumerable set.If the system is suitably complex, like first-order arithmetic, then the set T of Gödel numbers of true sentences in the system will be a productive set, which means that whenever W is a recursively enumerable set of true sentences, there is at least one true sentence that is not in W.