Recursion on Abstract Structures

  title={Recursion on Abstract Structures},
  author={Peter G. Hinman},
  booktitle={Handbook of Computability Theory},
  • P. Hinman
  • Published in
    Handbook of Computability…
  • 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. 
7 Citations
First and Second Order Recursion on Abstract Data Types
This paper compares two scheme-based models of computation on abstract many-sorted algebras A: Feferman's system ACP(A) of "abstract computational procedures" based on a least fixed point operator,Expand
Gandy's Theorem for Abstract Structures without the Equality Test
It is proved that Gandy’s theorem holds for abstract structures without the equality test, providing a useful tool for dealing with inductive definitions using Σ-formulas over continuous data types. Expand
Fixed points on abstract structures without the equality test
It is proved that Gandy theorem holds for abstract structures and provides a useful tool for dealing with recursive definitions using Sigma-formulas. Expand
Theses for Computation and Recursion on Concrete and Abstract Structures
The main aim of this article is to examine proposed theses for computation and recursion on concrete and abstract structures. What is generally referred to as Church’s Thesis or the Church-TuringExpand
Recent Advances in S-Definability over Continuous Data Types
  • M. Korovina
  • Computer Science
  • Ershov Memorial Conference
  • 2003
The purpose of this paper is to survey the recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals and illustrate how computability can be expressed in the language of Σ-formulas. Expand
Fixed Points on the Real Numbers without the Equality Test
  • M. Korovina
  • Computer Science, Mathematics
  • Electron. Notes Theor. Comput. Sci.
  • 2002
It is proved that Gandy theorem holds for the reals without the equality test, which provides a useful tool for dealing with recursive definitions using σ-formulas. Expand
On Gupta-Belnap Revision Theories of Truth, Kripkean fixed points, and the next stable set
  • P. Welch
  • Mathematics, Computer Science
  • Bull. Symb. Log.
  • 2001
A simplified account of varied revision sequences is given —as a generalised algorithmic theory of truth —which enables something of a unification with the Kripkean theory oftruth using supervaluation schemes. Expand


Deterministic and Nondeterministic Computation and Horn Programs, on Abstract Data Types
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
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
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
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
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