Recursion on Abstract Structures

@inproceedings{Hinman1999RecursionOA,
  title={Recursion on Abstract Structures},
  author={Peter G. Hinman},
  booktitle={Handbook of Computability Theory},
  year={1999}
}
  • P. Hinman
  • Published in
    Handbook of Computability…
    1999
  • Computer Science
We develop and compare two models for computability over an abstract structure. The first, characterized in term of generalized register machines, provides a good theory for a large class of first-order structures. The second, defined in terms of minimal solutions for functional equations, is more versatile and handles many common second-order examples. 
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References

SHOWING 1-10 OF 45 REFERENCES
Deterministic and Nondeterministic Computation and Horn Programs, on Abstract Data Types
TLDR
The notion of “semicomputability” is investigated, intended to generalize the notion of recursive enumerability of relations to abstract structures and leads to the formulation of a “Generalized Church-Turing Thesis” for definability of Relations on abstract structures. Expand
Computable Functions on Stream Algebras
TLDR
It is shown how models of deterministic parallel computation on A can be adapted to provide new models of computation on stream algebras over A, including simultaneous primitive recursion schemes with and without the least number operator. Expand
Computations in Higher Types
Abstract.- The computation domain.- Recursion on ?.- Connection with Kleene recursion in higher types.- Recursion in normal lists on ?.- Kleene recursion in normal objects of type n+2, n>0.-Expand
Examples of Semicomputable Sets of Real and Complex Numbers
We investigate the concept of semicomputability of relations on abstract structures. We consider three possible definitions of this concept, which all reduce to the classical notion of recursiveExpand
Computability over arbitrary fields
TLDR
The proposed definition of computability over arbitrary fields is based on the Shepherdson - Sturgis2 concept of an unlimited register machine. Expand
The Formal Language of Recursion
TLDR
This is the first of a sequence of papers in which this approach takes recursion to be a fundamental (primitive) process for constructing algorithms, not a derived notion which must be reduced to others—e.g. iteration or application and abstraction. Expand
Computability: An Introduction to Recursive Function Theory
Preface Prologue, prerequisites and notation 1. Computable functions 2. Generating computable functions 3. Other approaches to computability: Church's thesis 4. Numbering computable functions 5.Expand
On the Solvability of Algorithmic Problems
TLDR
This chapter discusses the basic concepts of a generalized Galois theory to make the paradigm of Galois available for the discussion of the solvability of algorithmic problems. Expand
On a theory of computation and complexity over the real numbers: $NP$- completeness, recursive functions and universal machines
We present a model for computation over the reals or an arbitrary (ordered) ring R. In this general setting, we obtain universal machines, partial recursive functions, as well as JVP-completeExpand
Recursion in Higher Types
Publisher Summary Recursion in higher types is an extension of the theory of recursive functions on the integers. This chapter presents an exposition of the basic notions and facts of this theory. ItExpand
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