an:01303200
Zbl 0941.03044
Cenzer, D.; Remmel, J. B.
\(\Pi_1^0\) classes in mathematics
EN
Ershov, Yu. L. (ed.) et al., Handbook of recursive mathematics. Vol. 2: Recursive algebra, analysis and combinatorics. Amsterdam: Elsevier. Stud. Logic Found. Math. 139, 623-821 (1998).
1998
a
03D45 03-02 03D15 03D30 03D35
\(\Pi^0_1\) classes; recursive algebra; recursive combinatorics; recursive graphs; survey; nonmonotonic logic
A survey (including some new results) of the occurrences and uses of \(\Pi^0_1\) classes in areas such as logic, nonmonotonic logic, algebra, analysis, and combinatorics. More specifically, a typical result here represents \(\Pi^0_1\) classes by solutions to problems in one of these areas of application. E.g., given a recursive graph, its set of \(k\)-colorings forms a recursively bounded \(\Pi^0_1\) class. The representation result, loosely stated, is that every recursively bounded \(\Pi^0_1\) class arises in this manner. Such a representation then allows the general machinery of \(\Pi^0_1\) classes to be brought to bear on, in this case, recursive graph theory. Polynomial-time versions of these ideas are also treated.
For the entire collection see [Zbl 0905.03002].
Leon Harkleroad (Poughkeepsie)