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Author: Ralph McKenzie Publisher: Springer Science & Business Media ISBN: 1461245524 Category : Mathematics Languages : en Pages : 209
Book Description
A mathematically precise definition of the intuitive notion of "algorithm" was implicit in Kurt Godel's [1931] paper on formally undecidable propo sitions of arithmetic. During the 1930s, in the work of such mathemati cians as Alonzo Church, Stephen Kleene, Barkley Rosser and Alfred Tarski, Godel's idea evolved into the concept of a recursive function. Church pro posed the thesis, generally accepted today, that an effective algorithm is the same thing as a procedure whose output is a recursive function of the input (suitably coded as an integer). With these concepts, it became possible to prove that many familiar theories are undecidable (or non-recursive)-i. e. , that there does not exist an effective algorithm (recursive function) which would allow one to determine which sentences belong to the theory. It was clear from the beginning that any theory with a rich enough mathematical content must be undecidable. On the other hand, some theories with a substantial content are decidable. Examples of such decidabLe theories are the theory of Boolean algebras (Tarski [1949]), the theory of Abelian groups (Szmiele~ [1955]), and the theories of elementary arithmetic and geometry (Tarski [1951]' but Tarski discovered these results around 1930). The de termination of precise lines of division between the classes of decidable and undecidable theories became an important goal of research in this area. algebra we mean simply any structure (A, h(i E I)} consisting of By an a nonvoid set A and a system of finitary operations Ii over A.
Author: Ralph McKenzie Publisher: Springer Science & Business Media ISBN: 1461245524 Category : Mathematics Languages : en Pages : 209
Book Description
A mathematically precise definition of the intuitive notion of "algorithm" was implicit in Kurt Godel's [1931] paper on formally undecidable propo sitions of arithmetic. During the 1930s, in the work of such mathemati cians as Alonzo Church, Stephen Kleene, Barkley Rosser and Alfred Tarski, Godel's idea evolved into the concept of a recursive function. Church pro posed the thesis, generally accepted today, that an effective algorithm is the same thing as a procedure whose output is a recursive function of the input (suitably coded as an integer). With these concepts, it became possible to prove that many familiar theories are undecidable (or non-recursive)-i. e. , that there does not exist an effective algorithm (recursive function) which would allow one to determine which sentences belong to the theory. It was clear from the beginning that any theory with a rich enough mathematical content must be undecidable. On the other hand, some theories with a substantial content are decidable. Examples of such decidabLe theories are the theory of Boolean algebras (Tarski [1949]), the theory of Abelian groups (Szmiele~ [1955]), and the theories of elementary arithmetic and geometry (Tarski [1951]' but Tarski discovered these results around 1930). The de termination of precise lines of division between the classes of decidable and undecidable theories became an important goal of research in this area. algebra we mean simply any structure (A, h(i E I)} consisting of By an a nonvoid set A and a system of finitary operations Ii over A.
Author: Bradd T. Hart Publisher: Springer Science & Business Media ISBN: 9401589232 Category : Mathematics Languages : en Pages : 285
Book Description
Recent major advances in model theory include connections between model theory and Diophantine and real analytic geometry, permutation groups, and finite algebras. The present book contains lectures on recent results in algebraic model theory, covering topics from the following areas: geometric model theory, the model theory of analytic structures, permutation groups in model theory, the spectra of countable theories, and the structure of finite algebras. Audience: Graduate students in logic and others wishing to keep abreast of current trends in model theory. The lectures contain sufficient introductory material to be able to grasp the recent results presented.
Author: Leonid A. Bokut' Publisher: American Mathematical Soc. ISBN: 0821851381 Category : Mathematics Languages : en Pages : 666
Book Description
In August 1989, more than 700 Soviet algebraists and more than 200 foreign mathematicians convened in Novosibirsk in what was then the Soviet Union for the International Conference on Algebra. Dedicated to the memory of A.I. Mal'cev, the Russian algebraist and logician, the conference marked the first time since the International Congress of Mathematicians was held in Moscow in 1966 that Soviet algebraists could meet with a large number of their foreign colleagues. This volume contains the proceedings from this historic conference. Some of the Soviet contributors to this volume are not easily available from other sources. Some of the major figures in the field, including P.M. Cohn, P. Gabriel, N. Jacobson, E.R. Kolchin, and V. Platonov, contributed to this volume. The papers span a broad range of areas including groups, Lie algebras, associative and nonassociative rings, fields and skew fields, differential algebra, universal algebra, categories, combinatorics, logic, algebraic geometry, topology, and mathematical physics.
Author: Joel Berman Publisher: American Mathematical Soc. ISBN: 0821837079 Category : Mathematics Languages : en Pages : 159
Book Description
The G-spectrum or generative complexity of a class $\mathcal{C}$ of algebraic structures is the function $\mathrm{G}_\mathcal{C}(k)$ that counts the number of non-isomorphic models in $\mathcal{C}$ that are generated by at most $k$ elements. We consider the behavior of $\mathrm{G}_\mathcal{C}(k)$ when $\mathcal{C}$ is a locally finite equational class (variety) of algebras and $k$ is finite. We are interested in ways that algebraic properties of $\mathcal{C}$ lead to upper or lower bounds on generative complexity.Some of our results give sharp upper and lower bounds so as to place a particular variety or class of varieties at a precise level in an exponential hierarchy. We say $\mathcal{C}$ has many models if there exists $c>0$ such that $\mathrm{G}_\mathcal{C}(k) \ge 2^{2^{ck}}$ for all but finitely many $k$, $\mathcal{C}$ has few models if there is a polynomial $p(k)$ with $\mathrm{G}_\mathcal{C}(k) \le 2^{p(k)}$, and $\mathcal{C}$ has very few models if $\mathrm{G}_\mathcal{C}(k)$ is bounded above by a polynomial in $k$.Much of our work is motivated by a desire to know which locally finite varieties have few or very few models, and to discover conditions that force a variety to have many models. We present characterization theorems for a very broad class of varieties including most known and well-studied types of algebras, such as groups, rings, modules, lattices. Two main results of our work are: a full characterization of locally finite varieties omitting the tame congruence theory type 1 with very few models as the affine varieties over a ring of finite representation type, and a full characterization of finitely generated varieties omitting type 1 with few models. In particular, we show that a finitely generated variety of groups has few models if and only if it is nilpotent and has very few models if and only if it is Abelian.
Author: Viktor A. Gorbunov Publisher: Springer Science & Business Media ISBN: 0306110636 Category : Mathematics Languages : en Pages : 314
Book Description
The theory of quasivarieties constitutes an independent direction in algebra and mathematical logic and specializes in a fragment of first-order logic-the so-called universal Horn logic. This treatise uniformly presents the principal directions of the theory from an effective algebraic approach developed by the author himself. A revolutionary exposition, this influential text contains a number of results never before published in book form, featuring in-depth commentary for applications of quasivarieties to graphs, convex geometries, and formal languages. Key features include coverage of the Birkhoff-Mal'tsev problem on the structure of lattices of quasivarieties, helpful exercises, and an extensive list of references.
Author: Ralph McKenzie
Author: Ralph McKenzie
Author: Ralph MacKenzie
Author: Valery B. Kudryavtsev
Author: 
Author: Bradd T. Hart
Author: Leonid A. Bokut'
Author: Alessandro Andretta
Author: L.G. Kovacs
Author: Joel Berman
Author: Viktor A. Gorbunov