Given a real number uniquely defined by a finite system of equations and inequalities with rational coefficients involving the standard elementary functions only. Is it decidable whether this number is rational?
Edit: The system of equations and inequalities may have $n$ real-valued variables, and the real number to be defined is $\xi$. By decomposition using intermediate variables, we may suppose that each inequality is of the form $x_i\,\omega\, c$ where $\omega\in\{<,>,\le,\ge\}$ and $c$ is an integer, and each equation is of the form $x_i=\xi$, or $x_i=c$ where $c$ is an integer, or $x_i=x_j\circ x_k$ with $\circ\in\{+,-,*,/\}$, or $x_i=\phi(x_k)$ where $\phi\in E$ with a specified set $E$ of elementary functions, such as $E=\{\exp,\log\}$. The answer to the question might depend on the choice of $E$.
I assume that $\xi$ is uniquely determined (hence defined) by this system of equations and inequalities. One may even assume that one has a proof for this, should this help.
The algorithm may be defined according to any of various traditional models, specifying a sensible set of primitive operations (including the rational operations [e.g., on bits or on reals] and branching on an expression). I am more interested in the applicable techniques than in an answer for a particular model.
Note that the Wikipedia article on the constant problem
claims (by reference to
D.H. Bailey, Mathematics of Computation 50 (1988), 275–281)
a result from which decidability would follow, but this claim is spurious.
Best Answer
This is a long comment rather than an answer.
Your problem is what is known as a "promise problem." The input comes equipped with a promise, namely that there exists a unique real solution. Presumably, if the promise is violated, then the algorithm is "off the hook" and can do anything it likes, including failing to halt.
I'm not aware of any research on the decidability of promise problems. (There is some work on the computational complexity of decidable promise problems.) The essential difficulty of your problem seems to reside in its nature as a promise problem, rather than on anything about rational numbers. To see this, let us consider the related question: Is being an integer decidable? This looks just as difficult as your question.
In the computational complexity realm, it's often possible to pass between promise problems and existence-of-a-solution problems. That is, the questions "there exists a unique solution; what is it?" and "does there exist a solution?" are closely related, and one can often reduce one to the other. Indeed, for your problem, if we had an algorithm to answer "does there exist a rational solution?" then we could solve your problem. However, I don't see how to create a reduction in the opposite direction, because I don't see how to create suitable promise problems starting from an arbitrary system. If we could, then perhaps the fact that Hilbert's tenth problem over $\mathbb Q$ is still open could be used to show that your problem is still open.