Promise oracle off promise answers

Given: the objects and hypotheses in the statement above.

For promise problems $(Y,N)$ and $(Y^{\mathrm{\prime }},N^{\mathrm{\prime }})$ in the stated convention, a polynomial-time many-one promise reduction is a total polynomial-time function $f$ satisfying $f(Y)\subseteq Y^{\mathrm{\prime }}$ and $f(N)\subseteq N^{\mathrm{\prime }}$. There is no condition on $f$ outside $Y\cup N$. A polynomial-time oracle promise reduction is a polynomially clocked deterministic oracle machine $M$ which solves $(Y,N)$ for every total language $B$ satisfying $Y^{\mathrm{\prime }}\subseteq B$ and $B\cap N^{\mathrm{\prime }}=\mathrm{\varnothing }$. This uses binary, consistent membership completions: an off-promise word may have either bit, but repeated queries to that word receive the same bit. The clock is uniform over all completions. (Promise preserving reduction).

A total membership oracle in the stated convention fixes the answer on every query word. A promise target in the stated convention describes a collection of total completions. Correctness of a promise reduction must hold for each completion, including its arbitrary answers outside the target promise. A promised input to the caller does not by itself guarantee that the caller's queries satisfy the target promise. (Oracle and promise conventions are distinct).