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Structure of Decidable Locally Finite Varieties (Hardcover, 1989 ed.): Ralph McKenzie, Matthew Valeriote Structure of Decidable Locally Finite Varieties (Hardcover, 1989 ed.)
Ralph McKenzie, Matthew Valeriote
R1,584 Discovery Miles 15 840 Ships in 12 - 17 working days

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.

Structure of Decidable Locally Finite Varieties (Paperback, Softcover reprint of the original 1st ed. 1989): Ralph McKenzie,... Structure of Decidable Locally Finite Varieties (Paperback, Softcover reprint of the original 1st ed. 1989)
Ralph McKenzie, Matthew Valeriote
R1,447 Discovery Miles 14 470 Ships in 10 - 15 working days

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.

The Cape Itself (Hardcover, 1st ed): Robert Finch, Ralph MacKenzie The Cape Itself (Hardcover, 1st ed)
Robert Finch, Ralph MacKenzie
R959 R725 Discovery Miles 7 250 Save R234 (24%) Out of stock

In a unique collaboration between words and images, The Cape Itself illuminates the polymorphic and often contradictory nature of the Cape: a place which epitomizes both the 'human scale' of a gentle, accessible landscape, and the chaotic, overwhelming presence of the sea; a place increasingly swamped and obscured by crowds and development, yet still affording surprising opportunities for wonder and discovery. It is precisely the interaction between these two creative modes of expression, reflecting and commenting on each other, which results in a personal, yet broadly evocative portrait of one of the most fascinating landscapes and communities in this country.

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