Continuity of capping in C

Paul Brodhead, Angsheng Li, Weilin Li

Abstract: A set A⊆ω<math><mi is="true">A</mi><mo is="true">⊆</mo><mi is="true">ω</mi></math> is called computably enumerable (c.e., for short), if there is an algorithm to enumerate the elements of it. For sets A,B⊆ω<math><mi is="true">A</mi><mo is="true">,</mo><mi is="true">B</mi><mo is="true">⊆</mo><mi is="true">ω</mi></math>, we say that A<math><mi is="true">A</mi></math> is bounded Turing reducible to (or alternatively, weakly truth table (wtt, for short) reducible to) B<math><mi is="true">B</mi></math> if there is a Turing functional, Φ<math><mi is="true">Φ</mi></math> say, with a computable bound of oracle query bits such that A<math><mi is="true">A</mi></math> is computed by Φ<math><mi is="true">Φ</mi></math> equipped with an oracle B<math><mi is="true">B</mi></math>, written A≤bTB<math><mi is="true">A</mi><msub is="true"><mrow is="true"><mo is="true">≤</mo></mrow><mrow is="true"><mstyle mathvariant="normal" is="true"><mi is="true">bT</mi></mstyle></mrow></msub><mi is="true">B</mi></math>. Let CbT<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">C</mi></mrow><mrow is="true"><mstyle mathvariant="normal" is="true"><mi is="true">bT</mi></mstyle></mrow></msub></math> be the structure of the c.e. bT-degrees, the c.e. degrees under the bounded Turing reductions. In this paper we study the continuity properties in CbT<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">C</mi></mrow><mrow is="true"><mstyle mathvariant="normal" is="true"><mi is="true">bT</mi></mstyle></mrow></msub></math>. We show that for any c.e. bT-degree b≠0,0′<math><mstyle mathvariant="bold" is="true"><mi is="true">b</mi></mstyle><mo is="true">≠</mo><mstyle mathvariant="bold" is="true"><mi is="true">0</mi></mstyle><mo is="true">,</mo><msup is="true"><mrow is="true"><mstyle mathvariant="bold" is="true"><mi is="true">0</mi></mstyle></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup></math>, there is a c.e. bT-degree a>b<math><mstyle mathvariant="bold" is="true"><mi is="true">a</mi></mstyle><mo is="true">&gt;</mo><mstyle mathvariant="bold" is="true"><mi is="true">b</mi></mstyle></math> such that for any c.e. bT-degree x<math><mstyle mathvariant="bold" is="true"><mi is="true">x</mi></mstyle></math>, b∧x=0<math><mstyle mathvariant="bold" is="true"><mi is="true">b</mi></mstyle><mo is="true">∧</mo><mstyle mathvariant="bold" is="true"><mi is="true">x</mi></mstyle><mo is="true">=</mo><mstyle mathvariant="bold" is="true"><mi is="true">0</mi></mstyle></math> if and only if a∧x=0<math><mstyle mathvariant="bold" is="true"><mi is="true">a</mi></mstyle><mo is="true">∧</mo><mstyle mathvariant="bold" is="true"><mi is="true">x</mi></mstyle><mo is="true">=</mo><mstyle mathvariant="bold" is="true"><mi is="true">0</mi></mstyle></math>. We prove that the analog of the Seetapun local noncappability theorem from the c.e. Turing degrees also holds in CbT<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">C</mi></mrow><mrow is="true"><mstyle mathvariant="normal" is="true"><mi is="true">bT</mi></mstyle></mrow></msub></math>. This theorem demonstrates that every b≠0,0′<math><mstyle mathvariant="bold" is="true"><mi is="true">b</mi></mstyle><mo is="true">≠</mo><mstyle mathvariant="bold" is="true"><mi is="true">0</mi></mstyle><mo is="true">,</mo><msup is="true"><mrow is="true"><mstyle mathvariant="bold" is="true"><mi is="true">0</mi></mstyle></mrow><mrow is="true"><mo is="true">′</mo></mrow></msup></math> is noncappable with any nontrivial degree below some a>b<math><mstyle mathvariant="bold" is="true"><mi is="true">a</mi></mstyle><mo is="true">&gt;</mo><mstyle mathvariant="bold" is="true"><mi is="true">b</mi></mstyle></math> (i.e. if x<a<math><mstyle mathvariant="bold" is="true"><mi is="true">x</mi></mstyle><mo is="true">&lt;</mo><mstyle mathvariant="bold" is="true"><mi is="true">a</mi></mstyle></math> and x∧b=0<math><mstyle mathvariant="bold" is="true"><mi is="true">x</mi></mstyle><mo is="true">∧</mo><mstyle mathvariant="bold" is="true"><mi is="true">b</mi></mstyle><mo is="true">=</mo><mstyle mathvariant="bold" is="true"><mi is="true">0</mi></mstyle></math> then x=0<math><mstyle mathvariant="bold" is="true"><mi is="true">x</mi></mstyle><mo is="true">=</mo><mstyle mathvariant="bold" is="true"><mi is="true">0</mi></mstyle></math>).