The Work of Anil Nerode: A Retrospective

Remmel, Jeffrey B.; Crossley, John N.

doi:10.1007/978-1-4612-0325-4_1

Cited by 3 publications

(2 citation statements)

References 121 publications

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“…The effective versions of various branches of mathematics have been extensively developed beginning with Fröhlich and Shepherdson [3], Malcev [12], Rabin [15], Nerode and his collaborators (see [16]) in algebra and Grzegorczyk [5], Lacombe [11] and others in analysis. In line with these investigations, we begin with a computable language L. An effective structure for the language should have a computable domain and we should be able to calculate the relevant functions and relations on this domain: Definition 1.4 A structure A for a language L is computable if its domain A is a computable subset of ω and its functions and relations are uniformly computable, i.e., there is a computable enumeration a i of A such that the atomic diagram of (A, a i ) is computable.…”

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“…If one begins simply with languages and structures, then one is essentially in the realm of general effective mathematics. The effective versions of various branches of mathematics have been extensively developed beginning with Frohlich and Shepherdson [3], Malcev [12], Rabin [15], Nerode and his collaborators (see [16]) in algebra and Grzegorczyk [5], Lacombe [11] and others in analysis. In line with these investigations, we begin with a computable language 2'.…”

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Computably categorical structures and expansions by constants

Cholak¹,

Гончаров²,

Khoussainov

et al. 1999

J. symb. log.

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Effective model theory is the subject that analyzes the typical notions and results of model theory to determine their effective content and counterparts. The subject has been developed both in the former Soviet Union and in the west with various names (recursive model theory, constructive model theory, etc.) and divergent terminology. (We use “effective model theory” as the most general and descriptive designation. Harizanov [6] is an excellent introduction to the subject as is Millar [13].) The basic subjects of model theory include languages, structures, theories, models and various types of maps between these objects. There are many ways to introduce considerations of effectiveness into the area. The two most prominent derive from starting, on the one hand, with the notion of a theory and its models or, on the other, with just structures.If one begins with theories, then a natural version of effectiveness is to consider decidable theories (i.e., ones with a decidable (equivalently, computable or recursive) set of theorems). When one moves to models and wants them to be effective, one might start with the requirement that the model (of any theory) have a decidable theory (i.e., Th (), the set of sentences true in , is decidable). Typically, however, one wants to be able to talk about the elements of the model as well as its theory in the given language. Thus one naturally considers the model as a structure for the language expanded by adding a constant ai, for each element ai of . Of course, one requires that the mapping from the constants to the corresponding elements of be effective (computable). We are thus lead to the following basic definition:A structure or model is decidable if there is a computable enumeration ai of A, the domain of , such that Th(, ai,) is decidable. (Of course, ai, is interpreted as ai, for each i Є ω.)

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Computably categorical structures and expansions by constants

Cholak¹,

Гончаров²,

Khoussainov

et al. 1999

J. symb. log.

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On hyper‐torre isols

Barback¹

2006

Mathematical Logic Qtrly

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Key wordsRegressive isol, hyper-torre isol, hereditarily odd-even isol. MSC (2000) 03D50In this paper we present a contribution to a classical result of E. Ellentuck in the theory of regressive isols. E. Ellentuck introduced the concept of a hyper-torre isol, established their existence for regressive isols, and then proved that associated with these isols a special kind of semi-ring of isols is a model of the true universal-recursive statements of arithmetic. This result took on an added significance when it was later shown that for regressive isols, the property of being hyper-torre is equivalent to being hereditarily odd-even. In this paper we present a simplification to the original proof for establishing that equivalence.

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Chapter 11 A bibliography of recursive algebra and recursive model theory

Kalantari

1998

Handbook of Recursive Mathematics - Volume 1: Recursive Model Theory

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The Work of Anil Nerode: A Retrospective

Cited by 3 publications

References 121 publications

Computably categorical structures and expansions by constants

Computably categorical structures and expansions by constants

On hyper‐torre isols

Chapter 11 A bibliography of recursive algebra and recursive model theory

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