Computable Rings and Fields

Stoltenberg-Hansen, Viggo; Tucker, John V.

doi:10.1016/s0049-237x(99)80028-7

Cited by 59 publications

(32 citation statements)

References 108 publications

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“…However, questions proliferate as one reflects on the number of algebras using the rational numbers ( [33]). For example, we do not know the answer to these simple questions.…”

Section: Discussionmentioning

confidence: 99%

“…Now the common rational arithmetics and field extensions are all computable algebras. Indeed, in the theory of computable rings and fields there is a wealth of constructions of computable algebras that start with the rationals and the finite fields: see the introduction and survey Stoltenberg-Hansen and Tucker [33]. Therefore, according to our general theory of algebraic specifications of computable data types they have various equational specifications under initial and final algebra semantics.…”

Section: Introductionmentioning

confidence: 99%

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Elementary Algebraic Specifications of the Rational Complex Numbers

Bergstra

Tucker

2006

Algebra, Meaning, and Computation

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From the range of techniques available for algebraic specifications we select a core set of features which we define to the elementary algebraic specifications. These include equational specifications with hidden functions and sorts and initial algebra semantics. We give an elementary equational specification of the field operations and conjugation operator on the rational complex numbers Q(i) and discuss some open problems.

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“…However, questions proliferate as one reflects on the number of algebras using the rational numbers ( [33]). For example, we do not know the answer to these simple questions.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Elementary Algebraic Specifications of the Rational Complex Numbers

Bergstra

Tucker

2006

Algebra, Meaning, and Computation

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“…The theory of algebraic specifications is based on theories of universal algebras (e.g., Wechler [20], Meinke and Tucker [14]), computable algebras (StoltenbergHansen and Tucker [17]), and term rewriting (Terese [19]). The theory of computable fields is surveyed in Stoltenberg-Hansen and Tucker [18].…”

Section: Technical Preliminaries On Algebraic Specificationsmentioning

confidence: 99%

Division Safe Calculation in Totalised Fields

Bergstra

Tucker

2007

Theory Comput Syst

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“…Hence every ideal of a computable commutative Noetherian ring is c.e. It can be shown that it is even computable [25]. An example of a computable commutative Noetherian ring with identity is the ring Q[x 1 , .…”

Section: Learning Classes Of Ring Ideals From Textmentioning

confidence: 99%

Inductive Inference Systems for Learning Classes of Algorithmically Generated Sets and Structures

Harizanov

2007

Induction, Algorithmic Learning Theory, and Philosophy

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Abstract. Computability theorists have extensively studied sets whose elements can be enumerated by Turing machines. These sets, also called computably enumerable sets, can be identified with their Gödel codes. Although each Turing machine has a unique Gödel code, different Turing machines can enumerate the same set. Thus, knowing a computably enumerable set means knowing one of its infinitely many Gödel codes. In the approach to learning theory stemming from E.M. Gold's seminal paper [9], an inductive inference learner for a computably enumerable set A is a system or a device, usually algorithmic, which when successively (one by one) fed data for A outputs a sequence of Gödel codes (one by one) that at certain point stabilize at codes correct for A. The convergence is called semantic or behaviorally correct, unless the same code for A is eventually output, in which case it is also called syntactic or explanatory. There are classes of sets that are semantically inferable, but not syntactically inferable.Here, we are also concerned with generalizing inductive inference from sets, which are collections of distinct elements that are mutually independent, to mathematical structures in which various elements may be interrelated. This study was recently initiated by F. Stephan and Yu. 30

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Computable Rings and Fields

Cited by 59 publications

References 108 publications

Elementary Algebraic Specifications of the Rational Complex Numbers

Elementary Algebraic Specifications of the Rational Complex Numbers

Division Safe Calculation in Totalised Fields

Inductive Inference Systems for Learning Classes of Algorithmically Generated Sets and Structures

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