Super-recursive algorithm

In computability theory, super-recursive algorithms are posited as a generalization of hypercomputation: hypothetical algorithms that are more powerful, that is, compute more than Turing machines.

The term was introduced by Mark Burgin, whose book Super-recursive algorithms develops their theory and presents several mathematical models.

Burgin argues that super-recursive algorithms can be used to disprove the Church-Turing thesis. This point of view has been criticized within the mathematical community and is not widely accepted.

Definition
Burgin (2005: 13) uses the term recursive algorithms for algorithms that can be implemented on Turing machines, and uses the word algorithm in a more general sense. Then a super-recursive class of algorithms is "a class of algorithms in which it is possible to compute functions not computable by any Turing machine" (Burgin 2005: 107)

Super-recursive algorithms are also related to algorithmic schemes, another novel concept from Burgin, which are more general than super-recursive algorithms. Burgin argues (2005: 115) that it is necessary to make a clear distinction between super-recursive algorithms and those algorithmic schemes that are not algorithms. Under this distinction, some types of hypercomputation are obtained by super-recursive algorithms.

Relation to the Church–Turing thesis
The Church–Turing thesis in recursion theory relies on a particular definition of the term algorithm. Based on his personal definitions that are more general than the one commonly used in recursion theory, Burgin argues that super-recursive algorithms disprove the Church–Turing thesis. He furthermore claims to prove that super-recursive algorithms could hypothetically provide even greater efficiency gains than using quantum algorithms.

Burgin's interpretation of super-recursive algorithms has encountered opposition in the mathematical community. One critic is logician Martin Davis, who argues that Burgin's claims have been well understood "for decades". Davis states,
 * "The present criticism is not about the mathematical discussion of these matters but only about the misleading claims regarding physical systems of the present and future."(Davis 2006: 128)

Davis disputes Burgin's claims that sets at level $$\Delta^0_2$$ of the arithmetical hierarchy can be called computable, saying
 * "It is generally understood that for a computational result to be useful one must be able to at least recognize that it is indeed the result sought." (Davis 2006: 128)