  Surrogate rewards for linear temporal logic (LTL) objectives are commonly
utilized in planning problems for LTL objectives. In a widely-adopted surrogate
reward approach, two discount factors are used to ensure that the expected
return approximates the satisfaction probability of the LTL objective. The
expected return then can be estimated by methods using the Bellman updates such
as reinforcement learning. However, the uniqueness of the solution to the
Bellman equation with two discount factors has not been explicitly discussed.
We demonstrate with an example that when one of the discount factors is set to
one, as allowed in many previous works, the Bellman equation may have multiple
solutions, leading to inaccurate evaluation of the expected return. We then
propose a condition for the Bellman equation to have the expected return as the
unique solution, requiring the solutions for states inside a rejecting bottom
strongly connected component (BSCC) to be 0. We prove this condition is
sufficient by showing that the solutions for the states with discounting can be
separated from those for the states without discounting under this condition
