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Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e., such undecidable languages may be recursively enumerable. Many, if not most, undecidable problems in mathematics can be posed as word problems : determining when two distinct strings of symbols (encoding some mathematical ...
Thus every recursively enumerable set is in . The converse is true as well: for every formula φ ( n ) {\displaystyle \varphi (n)} in Σ 1 0 {\displaystyle \Sigma _{1}^{0}} with k existential quantifiers, we may enumerate the k {\displaystyle k} –tuples of natural numbers and run a Turing machine that goes through all of them until it finds ...
The predicates can be used to obtain Kleene's normal form theorem for computable functions (Soare 1987, p. 15; Kleene 1943, p. 52—53). This states there exists a fixed primitive recursive function such that a function : is computable if and only if there is a number such that for all , …, one has
In computer science, corecursion is a type of operation that is dual to recursion.Whereas recursion works analytically, starting on data further from a base case and breaking it down into smaller data and repeating until one reaches a base case, corecursion works synthetically, starting from a base case and building it up, iteratively producing data further removed from a base case.
A recursive operator is an enumeration operator that, when given the graph of a partial recursive function, always returns the graph of a partial recursive function. A fixed point of an enumeration operator Φ is a set F such that Φ(F) = F. The first enumeration theorem shows that fixed points can be effectively obtained if the enumeration ...
The main form of computability studied in the field was introduced by Turing in 1936. [9] A set of natural numbers is said to be a computable set (also called a decidable, recursive, or Turing computable set) if there is a Turing machine that, given a number n, halts with output 1 if n is in the set and halts with output 0 if n is not in
Sections 4.3 (The master method) and 4.4 (Proof of the master theorem), pp. 73–90. Michael T. Goodrich and Roberto Tamassia. Algorithm Design: Foundation, Analysis, and Internet Examples. Wiley, 2002. ISBN 0-471-38365-1. The master theorem (including the version of Case 2 included here, which is stronger than the one from CLRS) is on pp. 268 ...
Catmull–Clark surfaces are defined recursively, using the following refinement scheme. [1]Start with a mesh of an arbitrary polyhedron.All the vertices in this mesh shall be called original points.