The theory of constructive (recursive) models follows from works of
Froehlich, Shepherdson, Mal'tsev, Kuznetsov, Rabin, and Vaught in
the 50s. Within the framework of this theory, algorithmic
properties of abstract models are investigated by constructing
representations on the set of natural numbers and studying
relations between algorithmic and structural properties of these
models. This book is a very readable exposition of the modern
theory of constructive models and describes methods and approaches
developed by representatives of the Siberian school of algebra and
logic and some other researchers (in particular, Nerode and his
colleagues). The main themes are the existence of recursive models
and applications to fields, algebras, and ordered sets (Ershov),
the existence of decidable prime models (Goncharov, Harrington),
the existence of decidable saturated models (Morley), the existence
of decidable homogeneous models (Goncharov and Peretyat'kin),
properties of the Ehrenfeucht theories (Millar, Ash, and Reed), the
theory of algorithmic dimension and conditions of autostability
(Goncharov, Ash, Shore, Khusainov, Ventsov, and others), and the
theory of computable classes of models with various properties.
Future perspectives of the theory of constructive models are also
discussed. Most of the results in the book are presented in
monograph form for the first time. The theory of constructive
models serves as a basis for recursive mathematics. It is also
useful in computer science, in particular, in the study of
programming languages, higher level languages of specification,
abstract data types, and problems of synthesis and verification of
programs. Therefore, the book will be usefulfor not only
specialists in mathematical logic and the theory of algorithms but
also for scientists interested in the mathematical fundamentals of
computer science. The authors are eminent specialists in
mathematical logic. They have established fundamental results on
elementary theories, model theory, the theory of algorithms, field
theory, group theory, applied logic, computable numberings, the
theory of constructive models, and the theoretical computer
science.
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