promise problem

A promise problem is a generalization of a decision problem. It is defined by two decision problems $L_1$ and $L_2$ with $L_1\cap L_2=\mathrm{\varnothing }$. A Turing machine decides a promise problem if, for any $x\in L_1\cup L_2$, it accepts when $x\in L_1$ and rejects if $x\in L_2$. Behavior is undefined when $x\notin L_1\cup L_2$ (this is the promise: that $x$ is in one of the two sets).

If $L_2={\mathrm{\Gamma }}^{+}\setminus L_1$ then this is just the decision problem for $L_1$.