Let $$ be the input alphabet. Let M be a non$-$deterministic Turing machine that

halts on all its inputs. Then L$($M$)=$ exactly when:

For every string in $,$ every branch of its computation tree has an accepting

configuration.

For every string in $,$ some branch of its computation tree has an accepting

configuration.

For every string in $,$ every branch of its computation tree has a rejecting

configuration.

For every string in $,$ some branch of its computation tree has a rejecting

configuration.