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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.
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.
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